Sample records for boolean algebra

  1. Boolean algebra essentials

    Solomon, Alan D


    REA's Essentials provide quick and easy access to critical information in a variety of different fields, ranging from the most basic to the most advanced. As its name implies, these concise, comprehensive study guides summarize the essentials of the field covered. Essentials are helpful when preparing for exams, doing homework and will remain a lasting reference source for students, teachers, and professionals. Boolean Algebra includes set theory, sentential calculus, fundamental ideas of Boolean algebras, lattices, rings and Boolean algebras, the structure of a Boolean algebra, and Boolean

  2. Cardinal invariants on Boolean algebras

    Monk, J Donald


    This book is concerned with cardinal number valued functions defined for any Boolean algebra. Examples of such functions are independence, which assigns to each Boolean algebra the supremum of the cardinalities of its free subalgebras, and cellularity, which gives the supremum of cardinalities of sets of pairwise disjoint elements. Twenty-one such functions are studied in detail, and many more in passing. The questions considered are the behaviour of these functions under algebraic operations such as products, free products, ultraproducts, and their relationships to one another. Assuming familiarity with only the basics of Boolean algebras and set theory, through simple infinite combinatorics and forcing, the book reviews current knowledge about these functions, giving complete proofs for most facts. A special feature of the book is the attention given to open problems, of which 185 are formulated. Based on Cardinal Functions on Boolean Algebras (1990) and Cardinal Invariants on Boolean Algebras (1996) by the...

  3. Cardinal invariants on Boolean algebras

    Monk, J Donald


    Deals with cardinal number valued functions defined for any Boolean algebra. This title considers the behavior of these functions under algebraic operations such as products, free products, ultraproducts, and their relationships to one another. It covers topics such as ultraproducts and Fedorchukis theorem

  4. Generalized join-hemimorphisms on Boolean algebras

    Sergio Celani


    We introduce the notions of generalized join-hemimorphism and generalized Boolean relation as an extension of the notions of join-hemimorphism and Boolean relation, respectively. We prove a duality between these two notions. We will also define a generalization of the notion of Boolean algebra with operators by considering a finite family of Boolean algebras endowed with a generalized join-hemimorphism. Finally, we define suitable notions of subalgebra, congruences, Boole...

  5. Duality theories for Boolean algebras with operators

    Givant, Steven


    In this new text, Steven Givant—the author of several acclaimed books, including works co-authored with Paul Halmos and Alfred Tarski—develops three theories of duality for Boolean algebras with operators. Givant addresses the two most recognized dualities (one algebraic and the other topological) and introduces a third duality, best understood as a hybrid of the first two. This text will be of interest to graduate students and researchers in the fields of mathematics, computer science, logic, and philosophy who are interested in exploring special or general classes of Boolean algebras with operators. Readers should be familiar with the basic arithmetic and theory of Boolean algebras, as well as the fundamentals of point-set topology.

  6. Fast Vertical Mining Using Boolean Algebra

    Hosny M. Ibrahim


    Full Text Available The vertical association rules mining algorithm is an efficient mining method, which makes use of support sets of frequent itemsets to calculate the support of candidate itemsets. It overcomes the disadvantage of scanning database many times like Apriori algorithm. In vertical mining, frequent itemsets can be represented as a set of bit vectors in memory, which enables for fast computation. The sizes of bit vectors for itemsets are the main space expense of the algorithm that restricts its expansibility. Therefore, in this paper, a proposed algorithm that compresses the bit vectors of frequent itemsets will be presented. The new bit vector schema presented here depends on Boolean algebra rules to compute the intersection of two compressed bit vectors without making any costly decompression operation. The experimental results show that the proposed algorithm, Vertical Boolean Mining (VBM algorithm is better than both Apriori algorithm and the classical vertical association rule mining algorithm in the mining time and the memory usage.

  7. On the Prime Whales of a Boolean Algebra

    Holland, Jason


    In this note, we introduce objects called prime whales and use them to represent a Boolean algebra as an algebra of sets in a way that is analogous to Stone's Representation Theorem. We also characterize the existence of prime whales in terms of the existence of prime ideals.

  8. On atomicity of free algebras in Boolean algebras with operators, and a new result on Pinter's free algebras

    Ahmed, Tarek Sayed


    We give some general theorems on free algebras of varieties of Boolean algebras with operators; a hitherto new result is obtained for Pinter's substitution algebras. For n\\geq 3, and m>1, there is a generating set of the free algebra freely generated by m elements, which is not a free set of generators.

  9. A complexity theory based on Boolean algebra

    Skyum, Sven; Valiant, Leslie


    A projection of a Boolean function is a function obtained by substituting for each of its variables a variable, the negation of a variable, or a constant. Reducibilities among computational problems under this relation of projection are considered. It is shown that much of what is of everyday rel...

  10. Reduction of Database Independence to Dividing in Atomless Boolean Algebras

    Hyttinen, Tapani; Paolini, Gianluca


    We prove that the form of conditional independence at play in database theory and independence logic is reducible to the first-order dividing calculus in the theory of atomless Boolean algebras. This establishes interesting connections between independence in database theory and stochastic independence. As indeed, in light of the aforementioned reduction and recent work of Ben-Yaacov [4], the former case of independence can be seen as the discrete version of the latter.

  11. C*-algebras associated to Boolean dynamical systems

    Pardo Espino, Enrique


    The goal of this talk is to present the C*-algebra $C^*(\\mathcal{B}, \\mathcal{L}, \\theta)$ of a Boolean dynamical system $(\\mathcal{B}, \\mathcal{L}, \\theta)$, that generalizes the $C^*$-algebra associated to a labelled graph introduced by Bates and Pask, and to determine its simplicity, its gauge invariant ideals, as well as compute its K-Theory. This is a joint work with Toke Meier Carlsen (Department of Science and Technology, University of the Faroe Islands) and Eduard Ortega (Departme...

  12. Disjunctive normal forms for any class of Boolean algebras with operators

    Khaled, Mohamed


    Disjunctive normal forms can provide elegant and constructive proofs of many standard results such as completeness, decidability and so on. They were also used to show non atomicity of some free algebras of specific Boolean algebras with operators. Here, we generalize the normal forms for any class of Boolean algebras with operators.

  13. Boolean Algebra Application in Analysis of Flight Accidents

    Casandra Venera BALAN


    Full Text Available Fault tree analysis is a deductive approach for resolving an undesired event into its causes, identifying the causes of a failure and providing a framework for a qualitative and quantitative evaluation of the top event. An alternative approach to fault tree analysis methods calculus goes to logical expressions and it is based on a graphical representation of the data structure for a logic - based binary decision diagram representation. In this analysis, such sites will be reduced to a minimal size and arranged in the sense that the variables appear in the same order in each path. An event can be defined as a statement that can be true or false. Therefore, Boolean algebra rules allow restructuring of a Fault Tree into one equivalent to it, but simpler.

  14. Boolean Functions, Quantum Gates, Hamilton Operators, Spin Systems and Computer Algebra

    Hardy, Yorick; Steeb, Willi-Hans


    We describe the construction of quantum gates (unitary operators) from boolean functions and give a number of applications. Both non-reversible and reversible boolean functions are considered. The construction of the Hamilton operator for a quantum gate is also described with the Hamilton operator expressed as spin system. Computer algebra implementations are provided.

  15. Fuzzy Boolean Algebras Based on Implication Operator%基于蕴涵算子上的模糊布尔代数



    文中给出R-模糊布尔代数的定义,讨论了其与模糊布尔代数的关系,证明在一定的条件下,有限个R-模糊布尔代数的交(并)还是R-模糊布尔代数,R-模糊布尔代数的同态像(原像)仍是R-模糊布尔代数.%In this paper ,we introduce the definition of fuzzy Boolean algebra. Based on that, the differences and connection between R-fuzzy Boolean algebra and fuzzy Boolean algebra are discussed. Furhtermore, it is proved that the finite intersection (union) of R-fuzzy Boolean algebra is still R-fuzzy Boolean algebra , and the homomorphic image (preimage) of R-fuzzy Boolean algebra is still R-fuzzy Boolean algebra.

  16. A Boolean algebra approach to the construction of snarks

    Fiol Mora, Miquel Àngel


    This work deals with the construction of snarks, that is, cubic graphs that cannot be 3-edge-colored. A natural generalization of the concept of "color", that describes in a simple way the coloring ("0" or "1") of any set of (semi)edges, is introduced. This approach allows us to apply the Boolean logic theory to find an ample family of snarks, which includes many of the previous known constructions and also some interesting ones. Peer Reviewed

  17. Ω-fuzzy subalgebra of Boolean algebra%布尔代数的Ω-模糊子代数



    Ω-fuzzy subalgebra of Boolean algebra and its properties were discussed in this paper .Ω-fuzzy subalgebra of Boolean algebra was defined and two judging theorems of Ω-fuzzy subalgebra of Boolean algebra were given , and it was stated that intersection-images and inverse-images under Boolean algebra homomorphism of Ω-fuzzy subalgebra of Boolean algebra were respectively Ω-fuzzy subalgebra of Boolean algebra . Then , by defining three operations ⊕ ,ⓧ , over RΩ where RΩexpressed all mappings from set Ωto Boolean algebra R ,a Boolean algebra was obtained ,and fuzzy subalgebra and Ω-fuzzy subalgebra about RΩwere researched .%  研究布尔代数的Ω-模糊子代数及其性质。定义布尔代数的Ω-模糊子代数,给出布尔代数的Ω-模糊子代数的两个简化判定定理,并证明布尔代数的Ω-模糊子代数的交、同态像和同态逆像等也是布尔代数的Ω-模糊子代数。然后,令RΩ表示集合Ω到布尔代数R的所有映射的集合,通过在 RΩ上定义3种运算磑,磗,,得到布尔代数枙RΩ,磑,磗,, I0,I1枛,并研究与其相关的模糊子代数和Ω-模糊子代数。

  18. Multiple Sequence Alignment using Boolean Algebra and Fuzzy Logic:A Comparative Study

    Nivit Gill


    Full Text Available Multiple sequence alignment is the most fundamental and essential task of computational biology, and forms the base for other tasks of bioinformatics. In this paper, two different approaches to sequence alignment have been discussed and compared. The first method employs Boolean algebra which is a two-valued logic whereas the second is based on Fuzzy logic which is a multi-valued logic. Both the methods perform sequence matching by direct comparison method using the operations of Boolean algebra and fuzzy logic respectively. To ensure the optimal alignment, dynamic programming is employed to align multiple sequences progressively. Both the methods are implemented and then tested on various sets of real genome sequences taken from NCBI bank. The processing time for both the methods on these data sets have been computed and compared.

  19. Boolean Differential Operators

    Catumba, Jorge; Diaz, Rafael


    We consider four combinatorial interpretations for the algebra of Boolean differential operators. We show that each interpretation yields an explicit matrix representation for Boolean differential operators.

  20. Statistical and algebraic analysis of a family of random Boolean equations

    The statistical and algebraic properties of a family of random Boolean equations are studied in this paper. Based on studying the mechanism of influence propagation from some variable fixed to 1, the giant strongly connected component and magnetization of solutions are investigated, which exhibit the splitting phenomenon of solution space and a ferromagnetic transition. Furthermore, by analyzing the semi-group property of the solution space and the influence of propagation from some variable fixed to 0, the scale of generating elements is calculated, which undergoes linear, polynomial and exponential phases. Compared with the analysis by statistical mechanics, it is suggested that these different phases of algebraic complexity correspond to the structural complexity of the solution space as replica symmetry, one-step replica symmetry breaking and further-step replica symmetry breaking phases. It is supposed that the structural complexity of solution space can be interpreted from the viewpoint of algebra

  1. Security analysis of boolean algebra based on Zhang-Wang digital signature scheme

    In 2005, Zhang and Wang proposed an improvement signature scheme without using one-way hash function and message redundancy. In this paper, we show that this scheme exits potential safety concerns through the analysis of boolean algebra, such as bitwise exclusive-or, and point out that mapping is not one to one between assembly instructions and machine code actually by means of the analysis of the result of the assembly program segment, and which possibly causes safety problems unknown to the software

  2. Ones and zeros understanding Boolean algebra digital circuits and the logic of sets

    Gregg, John


    "Ones and Zeros explains, in lay terms, Boolean algebra, the suprisingly simple system of mathematical logic used in digital computer circuitry. Ones and Zeros follows the development of this logic system from its origins in Victorian England to its rediscovery in this century as the foundation of all modern computing machinery. Readers will learn about the interesting history of the development of symbolic logic in particular, and the often misunderstood process of mathematical invention and scientific discovery, in general. Ones and Zeros also features practical exercises with answers, real-world examples of digital circuit design, and a reading list." "Ones and Zeros will be of particular interest to software engineers who want to gain a comprehensive understanding of computer hardware." "Outstanding features include: a history of mathematical logic, an explanation of the logic of digital circuits, and hands-on exercises and examples."--Jacket.

  3. Boolean reasoning the logic of boolean equations

    Brown, Frank Markham


    A systematic treatment of Boolean reasoning, this concise, newly revised edition combines the works of early logicians with recent investigations, including previously unpublished research results. Brown begins with an overview of elementary mathematical concepts and outlines the theory of Boolean algebras. Two concluding chapters deal with applications. 1990 edition.

  4. Boolean filters of distributive lattices

    M. Sambasiva Rao


    Full Text Available In this paper we introduce the notion of Boolean filters in a pseudo-complemented distributive lattice and characterize the class of all Boolean filters. Further a set of equivalent conditions are derived for a proper filter to become a prime Boolean filter. Also a set of equivalent conditions is derived for a pseudo-complemented distributive lattice to become a Boolean algebra. Finally, a Boolean filter is characterized in terms of congruences.

  5. Boole代数上的几个新度量结构%Several New Metric Structures on Boolean Algebra

    左卫兵; 娄妍


    设B是Boole代数,Ω是B到Boole代数{0,1}的全体格同态,μ是Ω上的概率测度,基于B中元素的尺寸的概念提出了元素之间的几个伪度量,建立了B上的度量结构,研究了其上运算的连续性及相互关系.%Let B be a Boolean algebra and ft the set of all homomorphisms from B into {0,1}, and μ be a probability measure on Ω. Based on the concept of sizes of elements of B several new metrics on the pairs of elements are introduced, and the metric structures on B are established, finally, the propositions of continuity of the operations on B are studied.

  6. Construction of 1-Resilient Boolean Functions with Optimal Algebraic Immunity and Good Nonlinearity

    Sen-Shan Pan; Xiao-Tong Fu; Wei-Guo Zhangx


    This paper presents a construction for a class of 1-resilient functions with optimal algebraic immunity on an even number of variables. The construction is based on the concatenation of two balanced functions in associative classes. For some n, a part of 1-resilient functions with maximum algebraic immunity constructed in the paper can achieve almost optimal nonlinearity. Apart from their high nonlinearity, the functions reach Siegenthaler's upper bound of algebraic degree. Also a class of 1-resilient functions on any number n > 2 of variables with at least sub-optimal algebraic immunity is provided.

  7. Algebraic Degree of a Class Boolean Function Annihilators%一类布尔函数零化子的代数次数

    祁传达; 俞迎达


    The effectiveness of algebraic attacks of stream ciphers depends on the algebraic degrees of annihilators of nonlinear Boolean functions.But it remains a difficult problem to construct annihilators with low degree for a given Boolean function. In this paper, we give a new proof of a result on the existence of the n-k degree annihilators formulated by Zhang Wenying,et al,and correct an error in their original proof.%序列密码代数攻击的成效取决于所使用的非线性布尔函数零化子的代数次数,但如何构造一个给定函数的低次数零化子仍是一个难题.本文对张文英等人提出的关于一类布尔函数存在n-k次零化子的结论给出了新的证明,弥补了原文证明不严密的缺陷.

  8. 基于8种常用蕴涵算子上的模糊布尔代数%Fuzzy Boolean Algebras Based on Eight Familiar Kinds of Implication Operator



    Based on the work of "Fuzzy Boolean Algebras Based on Implication Operator", present work gives eight kinds of equivalent forms of fuzzy Boolean algebras based on familiar eight kinds of implication operator by using inequalities characterizatian method. This paper promotes the results of the corresponding fuzzy algebra, and enriches riches theoretical results of fuzzy algebra.%在文献[1]的基础上,利用不等式的刻画方法,给出8种常用的R-蕴涵算子下的R-模糊布尔代数的8种等价形式,推广了现有相应模糊代数的结果,丰富了模糊代数的理论成果.

  9. 偶数变元代数免疫最优布尔函数的构造方法%Class of constructions of even variables boolean function with optimum algebraic immunity

    陈银冬; 陆佩忠


    A second order recursive construction of even variables Boolean function with optimum algebraic immunity was proposed. It could be observed that the constructed Boolean functions have well cryptographic properties, such as good balance, high algebraic degree and high nonlinearity. Further more, it was generalized to a class of constructions for Boolean functions with optimum algebraic immunity.%提出了构造偶数变元代数免疫最优的布尔函数的方法,这是一个二阶的递归构造方法.分析表明,利用该方法构造而得到的布尔函数具有优良的密码学特性,比如具有较好的平衡性,较高的代数次数和非线性度等.最后,还对该构造方法进行了推广,进一步导出了递归构造偶数变元代数免疫最优布尔函数的一类方法.

  10. Nonmonotonic logics and algebras

    CHAKRABORTY Mihir Kr; GHOSH Sujata


    Several nonmonotonie logic systems together with their algebraic semantics are discussed. NM-algebra is defined.An elegant construction of an NM-algebra starting from a Boolean algebra is described which gives rise to a few interesting algebraic issues.

  11. Ockham Algebras Arising from Monoids

    T.S. Blyth; H.J. Silva; J.C. Varlet


    An Ockham algebra (L; f) is of boolean shape if its lattice reduct L is boolean and f is not the complementation. We investigate a natural construction of Ockham algebras of boolean shape from any given monoid. Of particular interest is the question of when such algebras are subdirectly irreducible. In settling this, we obtain what is probably the first example of a subdirectly irreducible Ockham algebra that does not belong to the generalized variety Kω. We also prove that every semigroup can be embedded in the monoid of endomorphisms of an Ockham algebra of boolean shape.

  12. Boolean-Valued Belief Functions

    Kramosil, Ivan


    Roč. 31, č. 2 (2002), s. 153-181. ISSN 0308-1079 R&D Projects: GA AV ČR IAA1030803 Institutional research plan: AV0Z1030915 Keywords : Dempster-Schafer theory * Boolean algebra Subject RIV: BA - General Mathematics Impact factor: 0.241, year: 2002

  13. Boolean Logic Optimization in Majority-Inverter Graphs

    Amarù, Luca; Gaillardon, Pierre-Emmanuel; De Micheli, Giovanni


    We present a Boolean logic optimization framework based on Majority-Inverter Graph (MIG). An MIG is a directed acyclic graph consisting of three-input majority nodes and regular/complemented edges. Current MIG optimization is supported by a consistent algebraic framework. However, when algebraic methods cannot improve a result quality, stronger Boolean methods are needed to attain further optimization. For this purpose, we propose MIG Boolean methods exploiting the error masking property of m...

  14. Algebra

    Tabak, John


    Looking closely at algebra, its historical development, and its many useful applications, Algebra examines in detail the question of why this type of math is so important that it arose in different cultures at different times. The book also discusses the relationship between algebra and geometry, shows the progress of thought throughout the centuries, and offers biographical data on the key figures. Concise and comprehensive text accompanied by many illustrations presents the ideas and historical development of algebra, showcasing the relevance and evolution of this branch of mathematics.

  15. Boolean Logic with Fault Tolerant Coding

    Alagoz, B. Baykant


    Error detectable and error correctable coding in Hamming space was researched to discover possible fault tolerant coding constellations, which can implement Boolean logic with fault tolerant property. Basic logic operators of the Boolean algebra were developed to apply fault tolerant coding in the logic circuits. It was shown that application of three-bit fault tolerant codes have provided the digital system skill of auto-recovery without need for designing additional-fault tolerance mechanisms.

  16. Algebra


    Through most of Greek history, mathematicians concentrated on geometry, although Euclid considered the theory of numbers. The Greek mathematician Diophantus (3rd century),however, presented problems that had to be solved by what we would today call algebra. His book is thus the first algebra text.

  17. Algebra

    Flanders, Harley


    Algebra presents the essentials of algebra with some applications. The emphasis is on practical skills, problem solving, and computational techniques. Topics covered range from equations and inequalities to functions and graphs, polynomial and rational functions, and exponentials and logarithms. Trigonometric functions and complex numbers are also considered, together with exponentials and logarithms.Comprised of eight chapters, this book begins with a discussion on the fundamentals of algebra, each topic explained, illustrated, and accompanied by an ample set of exercises. The proper use of a

  18. Some Aspects of Boolean Valued Analysis

    Kusraev, A. G.; Kutateladze, S. S.


    This is a survey of some recent applications of Boolean valued analysis to operator theory and harmonic analysis. Under consideration are pseudoembedding operators, the noncommutative Wickstead problem, the Radon-Nikodym Theorem for JB-algebras, and the Bochner Theorem for lattice-valued positive definite mappings on locally compact groups.

  19. Stratification and enumeration of Boolean functions by canalizing depth

    He, Qijun; Macauley, Matthew


    Boolean network models have gained popularity in computational systems biology over the last dozen years. Many of these networks use canalizing Boolean functions, which has led to increased interest in the study of these functions. The canalizing depth of a function describes how many canalizing variables can be recursively "picked off", until a non-canalizing function remains. In this paper, we show how every Boolean function has a unique algebraic form involving extended monomial layers and a well-defined core polynomial. This generalizes recent work on the algebraic structure of nested canalizing functions, and it yields a stratification of all Boolean functions by their canalizing depth. As a result, we obtain closed formulas for the number of n-variable Boolean functions with depth k, which simultaneously generalizes enumeration formulas for canalizing, and nested canalizing functions.

  20. Stratification and enumeration of Boolean functions by canalizing depth

    He, Qijun


    Boolean network models have gained popularity in computational systems biology over the last dozen years. Many of these networks use canalizing Boolean functions, which has led to increased interest in the study of these functions. The canalizing depth of a function describes how many canalizing variables can be recursively picked off, until a non-canalizing function remains. In this paper, we show how every Boolean function has a unique algebraic form involving extended monomial layers and a well-defined core polynomial. This generalizes recent work on the algebraic structure of nested canalizing functions, and it yields a stratification of all Boolean functions by their canalizing depth. As a result, we obtain closed formulas for the number of n-variable Boolean functions with depth k, which simultaneously generalizes enumeration formulas for canalizing, and nested canalizing functions.

  1. Quantum boolean functions

    Montanaro, Ashley; Osborne, Tobias J.


    In this paper we introduce the study of quantum boolean functions, which are unitary operators f whose square is the identity: f^2 = I. We describe several generalisations of well-known results in the theory of boolean functions, including quantum property testing; a quantum version of the Goldreich-Levin algorithm for finding the large Fourier coefficients of boolean functions; and two quantum versions of a theorem of Friedgut, Kalai and Naor on the Fourier spectra of boolean functions. In o...

  2. Boolean Expression Diagrams

    Andersen, Henrik Reif; Hulgaard, Henrik


    This paper presents a new data structure called boolean expression diagrams (BEDs) for representing and manipulating Boolean functions. BEDs are a generalization of binary decision diagrams (BDDs) which can represent any Boolean circuit in linear space. Two algorithms are described for transforming...

  3. Monotone Boolean functions

    Monotone Boolean functions are an important object in discrete mathematics and mathematical cybernetics. Topics related to these functions have been actively studied for several decades. Many results have been obtained, and many papers published. However, until now there has been no sufficiently complete monograph or survey of results of investigations concerning monotone Boolean functions. The object of this survey is to present the main results on monotone Boolean functions obtained during the last 50 years

  4. Boolean integral calculus

    Tucker, Jerry H.; Tapia, Moiez A.; Bennett, A. Wayne


    The concept of Boolean integration is developed, and different Boolean integral operators are introduced. Given the changes in a desired function in terms of the changes in its arguments, the ways of 'integrating' (i.e. realizing) such a function, if it exists, are presented. The necessary and sufficient conditions for integrating, in different senses, the expression specifying the changes are obtained. Boolean calculus has applications in the design of logic circuits and in fault analysis.

  5. Boolean Expression Diagrams

    Andersen, Henrik Reif; Hulgaard, Henrik

    This paper presents a new data structure called Boolean Expression Diagrams (BEDs) for representing and manipulating Boolean functions. BEDs are a generalization of Binary Decision Diagrams (BDDs) which can represent any Boolean circuit in linear space and still maintain many of the desirable...... properties of BDDs. Two algorithms are described for transforming a BED into a reduced ordered BDD. One closely mimics the BDD apply-operator while the other can exploit the structural information of the Boolean expression. The efficacy of the BED representation is demonstrated by verifying that the...

  6. Integrating Boolean and Mathematical Solving: Foundations, Basic Algorithms and Requirements

    Audemard, Gilles; Bertoli, Piergiorgio; Cimatti, Alessandro; Kornilowicz, Artur; Sebastiani, Roberto


    In the last years we have witnessed an impressive advance in the efficiency of boolean solving techniques, which has brought large previously intractable problems at the reach of state-of-the-art solvers. Unfortunately, simple boolean expressions are not expressive enough for representing many real-world problems, which require handling also integer or real values and operators. On the other hand, mathematical solvers, like computer-algebra systems or constraint solvers, cannot handle efficie...

  7. Boolean representations of simplicial complexes and matroids

    Rhodes, John


    This self-contained monograph explores a new theory centered around boolean representations of simplicial complexes leading to a new class of complexes featuring matroids as central to the theory. The book illustrates these new tools to study the classical theory of matroids as well as their important geometric connections. Moreover, many geometric and topological features of the theory of matroids find their counterparts in this extended context.   Graduate students and researchers working in the areas of combinatorics, geometry, topology, algebra and lattice theory will find this monograph appealing due to the wide range of new problems raised by the theory. Combinatorialists will find this extension of the theory of matroids useful as it opens new lines of research within and beyond matroids. The geometric features and geometric/topological applications will appeal to geometers. Topologists who desire to perform algebraic topology computations will appreciate the algorithmic potential of boolean represent...

  8. Semigroups and computer algebra in algebraic structures

    Bijev, G.


    Some concepts in semigroup theory can be interpreted in several algebraic structures. A generalization fA,B,fA,B(X) = A(X')B of the complement operator (') on Boolean matrices is made, where A and B denote any rectangular Boolean matrices. While (') is an isomorphism between Boolean semilattices, the generalized complement operator is homomorphism in the general case. The map fA,B and its general inverse (fA,B)+ have quite similar properties to those in the linear algebra and are useful for solving linear equations in Boolean matrix algebras. For binary relations on a finite set, necessary and sufficient conditions for the equation αξβ = γ to have a solution ξ are proved. A generalization of Green's equivalence relations in semigroups for rectangular matrices is proposed. Relationships between them and the Moore-Penrose inverses are investigated. It is shown how any generalized Green's H-class could be constructed by given its corresponding linear subspaces and converted into a group isomorphic to a linear group. Some information about using computer algebra methods concerning this paper is given.

  9. Proposition Algebra with Projective Limits

    Bergstra, J A


    Sequential logic deviates from propositional logic by taking into account that atomic propositions yield different Boolean values at different times during the sequential evaluation of a single proposition. Reactive valuations capture this dynamics of a proposition's environment. This logic is phrased as an equationally specified algebra rather than in the form of proof rules. It is strictly more general than Boolean algebra to the extent that the classical connectives fail to be expressively complete in the sequential case. The proposition algebra PRA is developed in a fashion similar to the process algebra ACP and the program algebra PGA via an algebraic specification which has a meaningful initial algebra for which a range of courser congruences are considered important as well. In addition infinite objects (that is propositions, processes and programs respectively) are preferably dealt with by means of an inverse limit construction which allows the transfer of knowledge concerning finite objects to facts ...

  10. Reflexive Operator Algebras on Banach Spaces

    Merlevède, Florence; Peligrad, Costel; Peligrad, Magda


    In this paper we study the reflexivity of a unital strongly closed algebra of operators with complemented invariant subspace lattice on a Banach space. We prove that if such an algebra contains a complete Boolean algebra of projections of finite uniform multiplicity and with the direct sum property, then it is reflexive, i.e. it contains every operator that leaves invariant every closed subspace in the invariant subspace lattice of the algebra. In particular, such algebras coincide with their...

  11. Fast Vertical Mining Using Boolean Algebra

    Hosny M. Ibrahim; Marghny, M. H.; Noha M. A. Abdelaziz


    The vertical association rules mining algorithm is an efficient mining method, which makes use of support sets of frequent itemsets to calculate the support of candidate itemsets. It overcomes the disadvantage of scanning database many times like Apriori algorithm. In vertical mining, frequent itemsets can be represented as a set of bit vectors in memory, which enables for fast computation. The sizes of bit vectors for itemsets are the main space expense of the algorithm that restricts its ex...

  12. Towards boolean operations with thermal photons

    Ben-Abdallah, Philippe


    The Boolean algebra is the natural theoretical framework for a classical information treatment. The basic logical operations are usually performed using logic gates. In this Letter we demonstrate that NOT, OR and AND gates can be realized exploiting the near-field radiative interaction in N-body systems with phase change materials. With the recent development of a photon thermal transistor and thermal memory, this result paves the way for a full information treatment and smart solutions for active thermal management at nanoscale with photons.

  13. Random Boolean Networks

    Drossel, Barbara


    This review explains in a self-contained way the properties of random Boolean networks and their attractors, with a special focus on critical networks. Using small example networks, analytical calculations, phenomenological arguments, and problems to solve, the basic concepts are introduced and important results concerning phase diagrams, numbers of relevant nodes and attractor properties are derived.


    Ali Muhammad Ali Rushdi


    Full Text Available A total description of a synchronous Boolean network is typically achieved by a matrix recurrence relation. A simpler alternative is to use a scalar equation which is a possibly nonlinear equation that involves two or more instances of a single scalar variable and some Boolean operator(s. Further simplification is possible in terms of a linear reduced scalar equation which is the simplest two-term scalar equation that includes no Boolean operators and equates the value of a scalar variable at a latter instance t2 to its value at an earlier instance t1. This equation remains valid when the times t1 and t2 are both augmented by any integral multiple of the underlying time period. In other words, there are infinitely many versions of a reduced scalar equation, any of which is useful for deducing information about the cyclic behavior of the network. However, to obtain correct information about the transient behavior of the network, one must find the true reduced scalar equation for which instances t1 and t2 are minimal. This study investigates the nature, derivation and utilization of reduced scalar equations. It relies on Boolean-algebraic manipulations for the derivation of such equations and suggests that this derivation can be facilitated by seeking certain orthogonality relations among certain successive (albeit not necessarily consecutive instances of the same scalar variable. We demonstrate, contrary to previously published assumptions or assertions, that there is typically no common reduced scalar equation for all the scalar variables. Each variable usually satisfies its own distinct reduced scalar equation. We also demonstrate that the derivation of a reduced scalar equation is achieved not only by proving it but also by disproving an immediately preceding version of it when such a version might exist. We also demonstrate that, despite the useful insight supplied by the reduced scalar equations, they do not provide a total solution like the

  15. Fault Tolerant Boolean Satisfiability

    Roy, A


    A delta-model is a satisfying assignment of a Boolean formula for which any small alteration, such as a single bit flip, can be repaired by flips to some small number of other bits, yielding a new satisfying assignment. These satisfying assignments represent robust solutions to optimization problems (e.g., scheduling) where it is possible to recover from unforeseen events (e.g., a resource becoming unavailable). The concept of delta-models was introduced by Ginsberg, Parkes and Roy (AAAI 1998), where it was proved that finding delta-models for general Boolean formulas is NP-complete. In this paper, we extend that result by studying the complexity of finding delta-models for classes of Boolean formulas which are known to have polynomial time satisfiability solvers. In particular, we examine 2-SAT, Horn-SAT, Affine-SAT, dual-Horn-SAT, 0-valid and 1-valid SAT. We see a wide variation in the complexity of finding delta-models, e.g., while 2-SAT and Affine-SAT have polynomial time tests for delta-models, testing w...

  16. Computational complexity of Boolean functions

    Boolean functions are among the fundamental objects of discrete mathematics, especially in those of its subdisciplines which fall under mathematical logic and mathematical cybernetics. The language of Boolean functions is convenient for describing the operation of many discrete systems such as contact networks, Boolean circuits, branching programs, and some others. An important parameter of discrete systems of this kind is their complexity. This characteristic has been actively investigated starting from Shannon's works. There is a large body of scientific literature presenting many fundamental results. The purpose of this survey is to give an account of the main results over the last sixty years related to the complexity of computation (realization) of Boolean functions by contact networks, Boolean circuits, and Boolean circuits without branching. Bibliography: 165 titles.

  17. Cylindric-like algebras and algebraic logic

    Ferenczi, Miklós; Németi, István


    Algebraic logic is a subject in the interface between logic, algebra and geometry, it has strong connections with category theory and combinatorics. Tarski’s quest for finding structure in logic leads to cylindric-like algebras as studied in this book, they are among the main players in Tarskian algebraic logic. Cylindric algebra theory can be viewed in many ways:  as an algebraic form of definability theory, as a study of higher-dimensional relations, as an enrichment of Boolean Algebra theory, or, as logic in geometric form (“cylindric” in the name refers to geometric aspects). Cylindric-like algebras have a wide range of applications, in, e.g., natural language theory, data-base theory, stochastics, and even in relativity theory. The present volume, consisting of 18 survey papers, intends to give an overview of the main achievements and new research directions in the past 30 years, since the publication of the Henkin-Monk-Tarski monographs. It is dedicated to the memory of Leon Henkin.

  18. Geometric Operators on Boolean Functions

    Frisvad, Jeppe Revall; Falster, Peter


    In truth-functional propositional logic, any propositional formula represents a Boolean function (according to some valuation of the formula). We describe operators based on Decartes' concept of constructing coordinate systems, for translation of a propositional formula to the image of a Boolean function. With this image of a Boolean function corresponding to a propositional formula, we prove that the orthogonal projection operator leads to a theorem describing all rules of inference in propo...

  19. Algorithms for Weighted Boolean Optimization

    Manquinho, Vasco; Marques-Silva, Joao; Planes Cid, Jordi


    The Pseudo-Boolean Optimization (PBO) and Maximum Satisfiability (MaxSAT) problems are natural optimization extensions of Boolean Satisfiability (SAT). In the recent past, different algorithms have been proposed for PBO and for MaxSAT, despite the existence of straightforward mappings from PBO to MaxSAT and viceversa. This papers proposes Weighted Boolean Optimization (WBO), a new uni- fied framework that aggregates and extends PBO and MaxSAT. In addition, the paper proposes...

  20. Congruence Kernels of Orthoimplication Algebras

    I. Chajda


    Full Text Available Abstracting from certain properties of the implication operation in Boolean algebras leads to so-called orthoimplication algebras. These are in a natural one-to-one correspondence with families of compatible orthomodular lattices. It is proved that congruence kernels of orthoimplication algebras are in a natural one-to-one correspondence with families of compatible p-filters on the corresponding orthomodular lattices. Finally, it is proved that the lattice of all congruence kernels of an orthoimplication algebra is relatively pseudocomplemented and a simple description of the relative pseudocomplement is given.

  1. Reconstructing an atomic orthomodular lattice from the poset of its Boolean sublattices

    Constantin, Carmen; Doering, Andreas


    We show that an atomic orthomodular lattice L can be reconstructed up to isomorphism from the poset B(L) of Boolean subalgebras of L. A motivation comes from quantum theory and the so-called topos approach, where one considers the poset of Boolean sublattices of L=P(H), the projection lattice of the algebra B(H) of bounded operators on Hilbert space.

  2. Modern Algebra, Mathematics: 5293.36.

    Edwards, Raymond J.

    This guidebook covers Boolean algebra, matrices, linear transformations of the plane, characteristic values, vectors, and algebraic structures. Overall course goals and performance objectives for each unit are specified; sequencing of units and various time schedules are suggested. A sample pretest and posttest are given, and an annotated list of…

  3. Symmetry in Boolean Satisfiability

    Fadi A. Aloul


    Full Text Available This paper reviews recent approaches on how to accelerate Boolean Satisfiability (SAT search by exploiting symmetries in the problem space. SAT search algorithms traverse an exponentially large search space looking for an assignment that satisfies a set of constraints. The presence of symmetries in the search space induces equivalence classes on the set of truth assignments. The goal is to use symmetries to avoid traversing all assignments by constraining the search to visit a few representative assignments in each equivalence class. This can lead to a significant reduction in search runtime without affecting the completeness of the search.

  4. Boolean Orthogonalizing Combination Methods

    Yavuz Can


    Full Text Available In this paper a new logical operation method called “ presented. It is used to calculate the difference, but also the complement of a function as well as the EXOR and EXNOR of two minterms respectively two ternary respectively two ternary-vector logical operation method called “orthogonal OR advantages of both methods are their results, which are already available form that has an essential advantage for continuing calculations. Since it applies, an orthogonal disjunctive normal form is equal to orthogonal antivalence normal form, subsequent Boolean differential calculus will be simplified.

  5. Investigating Boolean Matrix Factorization

    Snášel, V.; Platoš, J.; Krömer, P.; Húsek, Dušan; Neruda, Roman; Frolov, A. A.

    - : ACM, 2008 - (Ding, C.; Li, T.; Zhu, S.), s. 18-25 ISBN 978-1-60558-307-5. [DMMT'08. Workshop in Conjunction with SIGKDD 2008 /14./. Las Vegas (US), 24.08.2008-24.08.2008] Institutional research plan: CEZ:AV0Z10300504 Keywords : Boolean factor analysis * nonnegative matrix factorization * neural networks * information retrieval * data mining * binary data Subject RIV: BB - Applied Statistics, Operational Research

  6. Fault Tolerant Boolean Satisfiability

    Roy, A


    A delta-model is a satisfying assignment of a Boolean formula for which any small alteration, such as a single bit flip, can be repaired by flips to some small number of other bits, yielding a new satisfying assignment. These satisfying assignments represent robust solutions to optimization problems (e.g., scheduling) where it is possible to recover from unforeseen events (e.g., a resource becoming unavailable). The concept of delta-models was introduced by Ginsberg, Parkes and Roy (AAAI 1998...

  7. Approximate Reasoning with Fuzzy Booleans

    Broek, van den P.M.; Noppen, J.A.R.


    This paper introduces, in analogy to the concept of fuzzy numbers, the concept of fuzzy booleans, and examines approximate reasoning with the compositional rule of inference using fuzzy booleans. It is shown that each set of fuzzy rules is equivalent to a set of fuzzy rules with singleton crisp ante

  8. Geometric Operators on Boolean Functions

    Frisvad, Jeppe Revall; Falster, Peter

    of a few geometric operators working on the images of Boolean functions. The operators we describe, arise from the niche area of array-based logic and have previously been tightly bound to an array-based representation of Boolean functions. We redefine the operators in an abstract form to make them...

  9. Cryptographic Boolean functions and applications

    Cusick, Thomas W


    Boolean functions are the building blocks of symmetric cryptographic systems. Symmetrical cryptographic algorithms are fundamental tools in the design of all types of digital security systems (i.e. communications, financial and e-commerce).Cryptographic Boolean Functions and Applications is a concise reference that shows how Boolean functions are used in cryptography. Currently, practitioners who need to apply Boolean functions in the design of cryptographic algorithms and protocols need to patch together needed information from a variety of resources (books, journal articles and other sources). This book compiles the key essential information in one easy to use, step-by-step reference. Beginning with the basics of the necessary theory the book goes on to examine more technical topics, some of which are at the frontier of current research.-Serves as a complete resource for the successful design or implementation of cryptographic algorithms or protocols using Boolean functions -Provides engineers and scient...

  10. Polynomial threshold functions and Boolean threshold circuits

    Hansen, Kristoffer Arnsfelt; Podolskii, Vladimir V.


    secondary interest. We show that PTFs on general Boolean domains are tightly connected to depth two threshold circuits. Our main results in regard to this connection are: PTFs of polynomial length and polynomial degree compute exactly the functions computed by THRMAJ circuits. An exponential length lower...... bound for PTFs that holds regardless of degree, thereby extending known lower bounds for THRMAJ circuits. We generalize two-party unbounded error communication complexity to the multi-party number-on-the-forehead setting, and show that communication lower bounds for 3-player protocols would yield size...... lower bounds for THRTHR circuits. We obtain several other results about PTFs. These include relationships between weight and degree of PTFs, and a degree lower bound for PTFs of constant length. We also consider a variant of PTFs over the max-plus algebra. We show that they are connected to PTFs over...

  11. Consistent stabilizability of switched Boolean networks.

    Li, Haitao; Wang, Yuzhen


    This paper investigates the consistent stabilizability of switched Boolean networks (SBNs) by using the semi-tensor product method, and presents a number of new results. First, an algebraic expression of SBNs is obtained by the semi-tensor product, based on which the consistent stabilizability is then studied for SBNs and some necessary and sufficient conditions are presented for the design of free-form and state-feedback switching signals, respectively. Finally, the consistent stabilizability of SBNs with state constraints is considered and some necessary and sufficient conditions are proposed. The study of illustrative examples shows that the new results obtained in this paper are very effective in designing switching signals for the consistent stabilizability of SBNs. PMID:23787170

  12. Quantum computation using geometric algebra

    Matzke, Douglas James

    This dissertation reports that arbitrary Boolean logic equations and operators can be represented in geometric algebra as linear equations composed entirely of orthonormal vectors using only addition and multiplication Geometric algebra is a topologically based algebraic system that naturally incorporates the inner and anticommutative outer products into a real valued geometric product, yet does not rely on complex numbers or matrices. A series of custom tools was designed and built to simplify geometric algebra expressions into a standard sum of products form, and automate the anticommutative geometric product and operations. Using this infrastructure, quantum bits (qubits), quantum registers and EPR-bits (ebits) are expressed symmetrically as geometric algebra expressions. Many known quantum computing gates, measurement operators, and especially the Bell/magic operators are also expressed as geometric products. These results demonstrate that geometric algebra can naturally and faithfully represent the central concepts, objects, and operators necessary for quantum computing, and can facilitate the design and construction of quantum computing tools.

  13. Techniques for solving Boolean equation systems

    Keinänen, Misa


    Boolean equation systems are ordered sequences of Boolean equations decorated with least and greatest fixpoint operators. Boolean equation systems provide a useful framework for formal verification because various specification and verification problems, for instance, μ-calculus model checking can be represented as the problem of solving Boolean equation systems. The general problem of solving a Boolean equation system is a computationally hard task, and no polynomial time solution technique ...

  14. A Note on the Inversion Complexity of Boolean Functions in Boolean Formulas

    Morizumi, Hiroki


    In this note, we consider the minimum number of NOT operators in a Boolean formula representing a Boolean function. In circuit complexity theory, the minimum number of NOT gates in a Boolean circuit computing a Boolean function $f$ is called the inversion complexity of $f$. In 1958, Markov determined the inversion complexity of every Boolean function and particularly proved that $\\lceil \\log_2(n+1) \\rceil$ NOT gates are sufficient to compute any Boolean function on $n$ variables. As far as we...

  15. Strengthening Effect Algebras in a Logical Perspective: Heyting-Wajsberg Algebras

    Konig, Martinvaldo


    Heyting effect algebras are lattice-ordered pseudoboolean effect algebras endowed with a pseudocomplementation that maps on the center (i.e. Boolean elements). They are the algebraic counterpart of an extension of both Łukasiewicz many-valued logic and intuitionistic logic. We show that Heyting effect algebras are termwise equivalent to Heyting-Wajsberg algebras where the two different logical implications are defined as primitive operators. We prove this logic to be decidable, to be strongly complete and to have the deduction-detachment theorem.

  16. Boolean Operations on Conic Polygons

    Yong-Xi Gong; Yu Liu; Lun Wu; Yu-Bo Xie


    An algorithm for Boolean operations on conic polygons is proposed. Conic polygons are polygons consisting of conic segments or bounded conics with directions. Preliminaries of Boolean operations on general polygons are presented. In our algorithm, the intersection points and the topological relationships between two conic polygons are computed. Boundaries are obtained by tracking path and selecting uncrossed boundaries following rule tables to build resulting conic polygons.We define a set of rules for the intersection, union, and subtraction operations on conic polygons. The algorithm considers degeneration cases such as homology, complement, interior, and exterior. The algorithm is also evaluated and implemented.

  17. Evolutionary Design of Boolean Functions

    WANG Zhang-yi; ZHANG Huan-guo; QIN Zhong-ping; MENG Qing-shu


    We use evolutionary computing to synthesize Boolean functions randomly. By using specific crossover and mutation operator in evolving process and modifying search space and fitness function, we get some high non-linearity functions which have other good cryptography characteristics such as autocorrelation etc. Comparing to other heuristic search techniques, evolutionary computing approach is more effective because of global search strategy and implicit parallelism.

  18. Constant communication complexity protocols for multiparty accumulative boolean functions

    pal, S P; Kumar, S; Das, Sima; Kumar, Somesh; pal, Sudebkumar Prasant


    Generalizing a boolean function from Cleve and Buhrman \\cite{cb:sqec}, we consider the class of {\\it accumulative boolean functions} of the form $f_B(X_1,X_2,..., X_m)=\\bigoplus_{i=1}^n t_B(x_i^1x_i^2... x_i^m)$, where $X_j=(x^j_1,x^j_2,..., x^j_n), 1\\leq j\\leq m$ and $t_B(x_i^1x_i^2... x_i^m)=1$ for input $m$-tuples $x_i^1x_i^2...x_i^m\\in B\\subseteq A\\subseteq \\{0,1\\}^n$, and 0, otherwise. Here the set $A$ is the promise set for function $f_B$. The input vectors $X_j, 1\\leq j\\leq m$ are given to the $m\\geq 2$ parties respectively, who communicate classical bits in a distributed environment so that one of them (say Alice) comes up with the value of the function. We algebraically characterize entanglement assisted LOCC protocols requiring only $m-1$ cbits of communication, for certain classes of such multiparty boolean functions for $m\\geq 2$ parties under appropriate uniform parity promise restrictions on input $m$-tuples $x_i^1x_i^2...x_i^m, 1\\leq i\\leq n$. In contrast, for certain $m$-party accumulative boo...

  19. Binary higher order neural networks for realizing Boolean functions.

    Zhang, Chao; Yang, Jie; Wu, Wei


    In order to more efficiently realize Boolean functions by using neural networks, we propose a binary product-unit neural network (BPUNN) and a binary π-ς neural network (BPSNN). The network weights can be determined by one-step training. It is shown that the addition " σ," the multiplication " π," and two kinds of special weighting operations in BPUNN and BPSNN can implement the logical operators " ∨," " ∧," and " ¬" on Boolean algebra 〈Z(2),∨,∧,¬,0,1〉 (Z(2)={0,1}), respectively. The proposed two neural networks enjoy the following advantages over the existing networks: 1) for a complete truth table of N variables with both truth and false assignments, the corresponding Boolean function can be realized by accordingly choosing a BPUNN or a BPSNN such that at most 2(N-1) hidden nodes are needed, while O(2(N)), precisely 2(N) or at most 2(N), hidden nodes are needed by existing networks; 2) a new network BPUPS based on a collaboration of BPUNN and BPSNN can be defined to deal with incomplete truth tables, while the existing networks can only deal with complete truth tables; and 3) the values of the weights are all simply -1 or 1, while the weights of all the existing networks are real numbers. Supporting numerical experiments are provided as well. Finally, we present the risk bounds of BPUNN, BPSNN, and BPUPS, and then analyze their probably approximately correct learnability. PMID:21427020

  20. Left Artinian Algebraic Algebras

    S. Akbari; M. Arian-Nejad


    Let R be a left artinian central F-algebra, T(R) = J(R) + [R, R],and U(R) the group of units of R. As one of our results, we show that, if R is algebraic and char F = 0, then the number of simple components of -R = R/J(R)is greater than or equal to dimF R/T(R). We show that, when char F = 0 or F is uncountable, R is algebraic over F if and only if [R, R] is algebraic over F. As another approach, we prove that R is algebraic over F if and only if the derived subgroup of U(R) is algebraic over F. Also, we present an elementary proof for a special case of an old question due to Jacobson.

  1. Bounded Algebra and Current-Mode Digital Circuits

    WU Xunwei; Massoud Pedram


    This paper proposes two boundedarithmetic operations, which are easily realized with current signals.Based on these two operations, a bounded algebra system suitable fordescribing current-mode digital circuits is developed and itsrelationship with the Boolean algebra, which is suitable for representingvoltage-mode digital circuits, is investigated. Design procedure forcurrent-mode circuits using the proposed algebra system is demonstratedon a number of common circuit elements which are used to realizearithmetic operations, such as adders and multipliers.

  2. Quantum algorithms for testing Boolean functions

    Erika Andersson


    Full Text Available We discuss quantum algorithms, based on the Bernstein-Vazirani algorithm, for finding which variables a Boolean function depends on. There are 2^n possible linear Boolean functions of n variables; given a linear Boolean function, the Bernstein-Vazirani quantum algorithm can deterministically identify which one of these Boolean functions we are given using just one single function query. The same quantum algorithm can also be used to learn which input variables other types of Boolean functions depend on, with a success probability that depends on the form of the Boolean function that is tested, but does not depend on the total number of input variables. We also outline a procedure to futher amplify the success probability, based on another quantum algorithm, the Grover search.

  3. Boolean Reasoning with Graphs of Partitions

    Goossens, Daniel


    version longue du papier court "A Dynamic Boolean Knowledge Base" accepté à ICTAI 2010. This paper presents an implemented architecture for easy learning, reorganizing and navigation into a Boolean knowledge base. As the base grows with new definitions and constraints, it is normalized by the closure of a completion operator. This normalization allows arbitrary formats for Boolean expressions. It ensures basic reasoning abilities and spontaneously organizes intermingled taxonomies of conce...

  4. Model Checking of Boolean Process Models

    Schneider, Christoph; Wehler, Joachim


    In the field of Business Process Management formal models for the control flow of business processes have been designed since more than 15 years. Which methods are best suited to verify the bulk of these models? The first step is to select a formal language which fixes the semantics of the models. We adopt the language of Boolean systems as reference language for Boolean process models. Boolean systems form a simple subclass of coloured Petri nets. Their characteristics are low tokens to mode...

  5. Using computer algebra and SMT solvers in algebraic biology

    Pineda Osorio, Mateo


    Biologic processes are represented as Boolean networks, in a discrete time. The dynamics within these networks are approached with the help of SMT Solvers and the use of computer algebra. Software such as Maple and Z3 was used in this case. The number of stationary states for each network was calculated. The network studied here corresponds to the immune system under the effects of drastic mood changes. Mood is considered as a Boolean variable that affects the entire dynamics of the immune system, changing the Boolean satisfiability and the number of stationary states of the immune network. Results obtained show Z3's great potential as a SMT Solver. Some of these results were verified in Maple, even though it showed not to be as suitable for the problem approach. The solving code was constructed using Z3-Python and Z3-SMT-LiB. Results obtained are important in biology systems and are expected to help in the design of immune therapies. As a future line of research, more complex Boolean network representations of the immune system as well as the whole psychological apparatus are suggested.

  6. Progress in Applications of Boolean Functions

    Sasao, Tsutomu


    This book brings together five topics on the application of Boolean functions. They are 1. Equivalence classes of Boolean functions: The number of n-variable functions is large, even for values as small as n = 6, and there has been much research on classifying functions. There are many classifications, each with their own distinct merit. 2. Boolean functions for cryptography: The process of encrypting/decrypting plain text messages often depends on Boolean functions with specific properties. For example, highly nonlinear functions are valued because they are less susceptible to linear attacks.

  7. Maiorana-McFarland class: Degree optimization and algebraic properties

    Pasalic, Enes


    In this paper, we consider a subclass of the Maiorana-McFarland class used in the design of resilient nonlinear Boolean functions. We show that these functions allow a simple modification so that resilient Boolean functions of maximum algebraic degree may be generated instead of suboptimized degree...... in the original class. Preserving a high-nonlinearity value immanent to the original construction method, together with the degree optimization gives in many cases functions with cryptographic properties superior to all previously known construction methods. This approach is then used to increase the...... algebraic degree of functions in the extended Maiorana-McFarland (MM) class (nonlinear resilient functions F : GF (2)(n) -> GF (2)(m) derived from linear codes). We also show that in the Boolean case, the same subclass seems not to have an optimized algebraic immunity, hence not providing a maximum...

  8. Algebraic geometry

    Lefschetz, Solomon


    An introduction to algebraic geometry and a bridge between its analytical-topological and algebraical aspects, this text for advanced undergraduate students is particularly relevant to those more familiar with analysis than algebra. 1953 edition.

  9. Boolean networks as modelling framework

    Florian eGreil


    Full Text Available In a network, the components of a given system are represented as nodes, the interactions are abstracted as links between the nodes. Boolean networks refer to a class of dynamics on networks, in fact it is the simplest possible dynamics where each node has a value 0 or 1. This allows to investigate extensively the dynamics both analytically and by numerical experiments. The present article focuses on the theoretical concept of relevant components and the immediate application in plant biology, references for more in-depths treatment of the mathematical details are also given.

  10. Boolean Search: Current State and Perspectives.

    Frants, Valery I.; Shapiro, Jacob; Taksa, Isak; Voiskunskii, Vladimir G.


    Discusses the use of Boolean logic in information-retrieval systems and analyzes existing criticisms of operational systems. Considers users' ability to use and understand Boolean operators, ranking, the quality of query formulations, and negative effects of criticism; and concludes that criticism is directed at the methodology employed in…

  11. Version Spaces and Generalized Monotone Boolean Functions

    J.C. Bioch (Cor); T. Ibaraki


    textabstractWe consider generalized monotone functions f: X --> {0,1} defined for an arbitrary binary relation <= on X by the property x <= y implies f(x) <= f(y). These include the standard monotone (or positive) Boolean functions, regular Boolean functions and other interesting functions as speci

  12. Boolean gates on actin filaments

    Siccardi, Stefano; Tuszynski, Jack A.; Adamatzky, Andrew


    Actin is a globular protein which forms long polar filaments in the eukaryotic cytoskeleton. Actin networks play a key role in cell mechanics and cell motility. They have also been implicated in information transmission and processing, memory and learning in neuronal cells. The actin filaments have been shown to support propagation of voltage pulses. Here we apply a coupled nonlinear transmission line model of actin filaments to study interactions between voltage pulses. To represent digital information we assign a logical TRUTH value to the presence of a voltage pulse in a given location of the actin filament, and FALSE to the pulse's absence, so that information flows along the filament with pulse transmission. When two pulses, representing Boolean values of input variables, interact, then they can facilitate or inhibit further propagation of each other. We explore this phenomenon to construct Boolean logical gates and a one-bit half-adder with interacting voltage pulses. We discuss implications of these findings on cellular process and technological applications.

  13. Efficient Instantiation of Parameterised Boolean Equation Systems to Parity Games

    Gijs Kant


    Full Text Available Parameterised Boolean Equation Systems (PBESs are sequences of Boolean fixed point equations with data variables, used for, e.g., verification of modal mu-calculus formulae for process algebraic specifications with data. Solving a PBES is usually done by instantiation to a Parity Game and then solving the game. Practical game solvers exist, but the instantiation step is the bottleneck. We enhance the instantiation in two steps. First, we transform the PBES to a Parameterised Parity Game (PPG, a PBES with each equation either conjunctive or disjunctive. Then we use LTSmin, that offers transition caching, efficient storage of states and both distributed and symbolic state space generation, for generating the game graph. To that end we define a language module for LTSmin, consisting of an encoding of variables with parameters into state vectors, a grouped transition relation and a dependency matrix to indicate the dependencies between parts of the state vector and transition groups. Benchmarks on some large case studies, show that the method speeds up the instantiation significantly and decreases memory usage drastically.

  14. Mining TCGA data using Boolean implications.

    Subarna Sinha

    Full Text Available Boolean implications (if-then rules provide a conceptually simple, uniform and highly scalable way to find associations between pairs of random variables. In this paper, we propose to use Boolean implications to find relationships between variables of different data types (mutation, copy number alteration, DNA methylation and gene expression from the glioblastoma (GBM and ovarian serous cystadenoma (OV data sets from The Cancer Genome Atlas (TCGA. We find hundreds of thousands of Boolean implications from these data sets. A direct comparison of the relationships found by Boolean implications and those found by commonly used methods for mining associations show that existing methods would miss relationships found by Boolean implications. Furthermore, many relationships exposed by Boolean implications reflect important aspects of cancer biology. Examples of our findings include cis relationships between copy number alteration, DNA methylation and expression of genes, a new hierarchy of mutations and recurrent copy number alterations, loss-of-heterozygosity of well-known tumor suppressors, and the hypermethylation phenotype associated with IDH1 mutations in GBM. The Boolean implication results used in the paper can be accessed at

  15. Monomial algebras

    Villarreal, Rafael


    The book stresses the interplay between several areas of pure and applied mathematics, emphasizing the central role of monomial algebras. It unifies the classical results of commutative algebra with central results and notions from graph theory, combinatorics, linear algebra, integer programming, and combinatorial optimization. The book introduces various methods to study monomial algebras and their presentation ideals, including Stanley-Reisner rings, subrings and blowup algebra-emphasizing square free quadratics, hypergraph clutters, and effective computational methods.

  16. Generalizing Boolean Satisfiability II: Theory

    Dixon, H E; Luks, E M; Parkes, A J; 10.1613/jair.1555


    This is the second of three planned papers describing ZAP, a satisfiability engine that substantially generalizes existing tools while retaining the performance characteristics of modern high performance solvers. The fundamental idea underlying ZAP is that many problems passed to such engines contain rich internal structure that is obscured by the Boolean representation used; our goal is to define a representation in which this structure is apparent and can easily be exploited to improve computational performance. This paper presents the theoretical basis for the ideas underlying ZAP, arguing that existing ideas in this area exploit a single, recurring structure in that multiple database axioms can be obtained by operating on a single axiom using a subgroup of the group of permutations on the literals in the problem. We argue that the group structure precisely captures the general structure at which earlier approaches hinted, and give numerous examples of its use. We go on to extend the Davis-Putnam-Logemann-...

  17. Boolean networks with reliable dynamics

    Peixoto, Tiago P


    We investigated the properties of Boolean networks that follow a given reliable trajectory in state space. A reliable trajectory is defined as a sequence of states which is independent of the order in which the nodes are updated. We explored numerically the topology, the update functions, and the state space structure of these networks, which we constructed using a minimum number of links and the simplest update functions. We found that the clustering coefficient is larger than in random networks, and that the probability distribution of three-node motifs is similar to that found in gene regulation networks. Among the update functions, only a subset of all possible functions occur, and they can be classified according to their probability. More homogeneous functions occur more often, leading to a dominance of canalyzing functions. Finally, we studied the entire state space of the networks. We observed that with increasing systems size, fixed points become more dominant, moving the networks close to the frozen...

  18. Local Correction of Boolean Functions

    Alon, Noga


    A Boolean function f over n variables is said to be q-locally correctable if, given a black-box access to a function g which is "close" to an isomorphism f_sigma of f, we can compute f_sigma(x) for any x in Z_2^n with good probability using q queries to g. We observe that any k-junta, that is, any function which depends only on k of its input variables, is O(2^k)-locally correctable. Moreover, we show that there are examples where this is essentially best possible, and locally correcting some k-juntas requires a number of queries which is exponential in k. These examples, however, are far from being typical, and indeed we prove that for almost every k-junta, O(k log k) queries suffice.

  19. Densities of mixed volumes for Boolean models

    Weil, Wolfgang


    In generalization of the well-known formulae for quermass densities of stationary and isotropic Boolean models, we prove corresponding results for densities of mixed volumes in the stationary situation and show how they can be used to determine the intensity of non-isotropic Boolean models Z in d-dimensional space for d = 2, 3, 4. We then consider non-stationary Boolean models and extend results of Fallert on quermass densities to densities of mixed volumes. In particular, we present explicit...

  20. Translating Pseudo-Boolean Constraints into CNF

    Aavani, Amir


    A Pseudo-Boolean constraint is a linear constraint over Boolean variables. This kind of constraints has been widely used in expressing NP-complete problems. This paper introduces a new algorithm for translating Pseudo-Boolean constraints into CNF clauses. The CNF produced by the proposed encoding has small size, and we also characterize the constraints for which one can expect the SAT solvers to perform well on the produced CNF. We show that there are many constraints for which the proposed encoding has a good performance.

  1. On a Boolean-valued Model of the Strict Implication System(Continuous)

    LI Na; LIU Hua-ke


    The reference [4] proved the consistency of S 1 and S 2 among Lewis'five strict implicationsystems in the modal logic by using the method of the Boolean-valued model. But, in this method, the consistency of S 3 , S 4 and S 5 in Lewis'five strictimplication systems is not decided. This paper makes use of the properties : (1) the equivalence of the modal systems S 3 andP 3 , S 4 and P 4 ; (2) the modal systems P 3 and P 4 all contained the modal axiom T(□p) ; (3) the modal axiom T is correspondence to the reflexiveproperty in VB . Hence, the paper proves: (a) |A S 31|=1 ; (b) |A S 41|=1 ;(c) |A S 51|=1 in the model (where B is a complete Boolean algebra, R is reflexive property in VB ). Therefore, the paper finallyproves that the Boolean-valued model VB of the ZFC axiomsystem in set theory is also a Boolean-valued model of Lewis'the strict implication system S 3 , S 4 and S 5 .

  2. Combinatorics of Boolean automata circuits dynamics

    Demongeot, Jacques; Noual, Mathilde; Sené, Sylvain


    International audience In line with fields of theoretical computer science and biology that study Boolean automata networks to model regulation networks, we present some results concerning the dynamics of networks whose underlying structures are oriented cycles, that is, Boolean automata circuits. In the context of biological regulation, former studies have highlighted the importance of circuits on the asymptotic dynamical behaviour of the biological networks that contain them. Our work fo...

  3. Sensitivity versus block sensitivity of Boolean functions

    Virza, Madars


    Determining relationship between sensitivity and block sensitivity of Boolean functions is of interest for computational complexity theory. We construct a sequence of Boolean functions with bs(f) = 1/2 (s(f))^2+ 1/2 s(f). The best known separation previously was bs(f) = 1/2 (s(f))^2 due to Rubinstein (1995). We also report results of computer search for functions with at most 12 variables.

  4. An Algebra of Synchronous Scheduling Interfaces

    Michael Mendler


    Full Text Available In this paper we propose an algebra of synchronous scheduling interfaces which combines the expressiveness of Boolean algebra for logical and functional behaviour with the min-max-plus arithmetic for quantifying the non-functional aspects of synchronous interfaces. The interface theory arises from a realisability interpretation of intuitionistic modal logic (also known as Curry-Howard-Isomorphism or propositions-as-types principle. The resulting algebra of interface types aims to provide a general setting for specifying type-directed and compositional analyses of worst-case scheduling bounds. It covers synchronous control flow under concurrent, multi-processing or multi-threading execution and permits precise statements about exactness and coverage of the analyses supporting a variety of abstractions. The paper illustrates the expressiveness of the algebra by way of some examples taken from network flow problems, shortest-path, task scheduling and worst-case reaction times in synchronous programming.

  5. An Algebra of Synchronous Scheduling Interfaces

    Mendler, Michael


    In this paper we propose an algebra of synchronous scheduling interfaces which combines the expressiveness of Boolean algebra for logical and functional behaviour with the min-max-plus arithmetic for quantifying the non-functional aspects of synchronous interfaces. The interface theory arises from a realisability interpretation of intuitionistic modal logic (also known as Curry-Howard-Isomorphism or propositions-as-types principle). The resulting algebra of interface types aims to provide a general setting for specifying type-directed and compositional analyses of worst-case scheduling bounds. It covers synchronous control flow under concurrent, multi-processing or multi-threading execution and permits precise statements about exactness and coverage of the analyses supporting a variety of abstractions. The paper illustrates the expressiveness of the algebra by way of some examples taken from network flow problems, shortest-path, task scheduling and worst-case reaction times in synchronous programming.

  6. Supertropical algebra

    Izhakian, Zur; Rowen, Louis


    We develop the algebraic polynomial theory for "supertropical algebra," as initiated earlier over the real numbers by the first author. The main innovation there was the introduction of "ghost elements," which also play the key role in our structure theory. Here, we work somewhat more generally over an ordered monoid, and develop a theory which contains the analogs of several basic theorems of classical commutative algebra. This structure enables one to develop a Zariski-type algebraic geomet...

  7. A Multiple—Valued Algebra for Modeling MOS VLSI Circuits at Switch—Level



    A multiple-valued algebra for modeling MOS VLSI circuits at switch-level is proposed in this paper,Its structure and properties are studied.This algebra can be used to transform a MOS digital circuit to a swith-level algebraic expression so as to generate the truth table for the circuit and to derive a Boolean expression for it.In the paper,methods to construct a switch-level algebraic expression for a circuit and methods to simplify expressions are given.This algebra provides a new tool for MOS VLSI circuit design and analysis.

  8. 高阶布尔网络的结构%Structure of higher order Boolean networks*

    李志强; 赵寅; 程代展


    The higher order Boolean (control) network is introduced and its topological structure is studied.Using semi-tensor product of matrices,its dynamics is converted into two algebraic forms,which are standard discrete-time dynamic systems.The one-to-one correspondence of the network dynamics and its first algebraic form is proved,and certain topological structures,including fixed points,cycles,and transient time,of higher order Boolean (control) networks are revealed.The relationship between the original system and its second algebraic form is also studied.%介绍高阶布尔(控制)网络,并研究了其拓扑结构.以矩阵的半张量积作为工具,把高阶布尔网络的动态过程转化为2种标准离散事件动态系统的代数形式.证明了高阶布尔网络和第1代数形式的一一对应关系,并由此得到其拓扑结构(不动点、极限圈以及暂态期等).还研究了高阶布尔网络系统与它第2代数形式的关系.

  9. Forced synchronization of autonomous dynamical Boolean networks

    We present the design of an autonomous time-delay Boolean network realized with readily available electronic components. Through simulations and experiments that account for the detailed nonlinear response of each circuit element, we demonstrate that a network with five Boolean nodes displays complex behavior. Furthermore, we show that the dynamics of two identical networks display near-instantaneous synchronization to a periodic state when forced by a common periodic Boolean signal. A theoretical analysis of the network reveals the conditions under which complex behavior is expected in an individual network and the occurrence of synchronization in the forced networks. This research will enable future experiments on autonomous time-delay networks using readily available electronic components with dynamics on a slow enough time-scale so that inexpensive data collection systems can faithfully record the dynamics

  10. Boolean nested canalizing functions: a comprehensive analysis

    Li, Yuan; Murrugarra, David; Aguilar, Boris; Laubenbacher, Reinhard


    Boolean network models of molecular regulatory networks have been used successfully in computational systems biology. The Boolean functions that appear in published models tend to have special properties, in particular the property of being nested canalizing, a property inspired by the concept of canalization in evolutionary biology. It has been shown that networks comprised of nested canalizing functions have dynamic properties that make them suitable for modeling molecular regulatory networks, namely a small number of (large) attractors, as well as relatively short limit cycles. This paper contains a detailed analysis of this class of functions, based on a novel normal form as polynomial functions over the Boolean field. The concept of layer is introduced that stratifies variables into different classes depending on their level of dominance. Using this layer concept a closed form formula is derived for the number of nested canalizing functions with a given number of variables. Additional metrics analyzed in...

  11. Forced synchronization of autonomous dynamical Boolean networks

    Rivera-Durón, R. R., E-mail:; Campos-Cantón, E., E-mail: [División de Matemáticas Aplicadas, Instituto Potosino de Investigación Científica y Tecnológica A. C., Camino a la Presa San José 2055, Col. Lomas 4 Sección, C.P. 78216, San Luis Potosí, S.L.P. (Mexico); Campos-Cantón, I. [Facultad de Ciencias, Universidad Autónoma de San Luis Potosí, Álvaro Obregón 64, C.P. 78000, San Luis Potosí, S.L.P. (Mexico); Gauthier, Daniel J. [Department of Physics and Center for Nonlinear and Complex Systems, Duke University, Box 90305, Durham, North Carolina 27708 (United States)


    We present the design of an autonomous time-delay Boolean network realized with readily available electronic components. Through simulations and experiments that account for the detailed nonlinear response of each circuit element, we demonstrate that a network with five Boolean nodes displays complex behavior. Furthermore, we show that the dynamics of two identical networks display near-instantaneous synchronization to a periodic state when forced by a common periodic Boolean signal. A theoretical analysis of the network reveals the conditions under which complex behavior is expected in an individual network and the occurrence of synchronization in the forced networks. This research will enable future experiments on autonomous time-delay networks using readily available electronic components with dynamics on a slow enough time-scale so that inexpensive data collection systems can faithfully record the dynamics.

  12. New Measure of Boolean Factor Analysis Quality

    Frolov, A. A.; Húsek, Dušan; Polyakov, P.Y.

    Vol. 1. Heidelberg: Springer, 2011 - (Dobnikar, A.; Lotrič, U.; Šter, B.), s. 100-109. (Lecture Notes in Computer Science. 6593). ISBN 978-3-642-20281-0. ISSN 0302-9743. [ICANNGA'2011. International Conference /10./. Ljubljana (SI), 14.04.2011-16.04.2011] R&D Projects: GA ČR GAP202/10/0262; GA ČR GA205/09/1079 Institutional research plan: CEZ:AV0Z10300504 Keywords : Boolean factor analysis * information gain * expectation-maximization * associative memory * neural network application * Boolean matrix factorization * bars problem * Hopfield neural network Subject RIV: IN - Informatics, Computer Science

  13. Algorithms for Boolean Function Query Properties

    Aaronson, Scott


    We present new algorithms to compute fundamental properties of a Boolean function given in truth-table form. Specifically, we give an O(N^2.322 log N) algorithm for block sensitivity, an O(N^1.585 log N) algorithm for `tree decomposition,' and an O(N) algorithm for `quasisymmetry.' These algorithms are based on new insights into the structure of Boolean functions that may be of independent interest. We also give a subexponential-time algorithm for the space-bounded quantum query complexity of...

  14. Fast Gr\\"obner Basis Computation for Boolean Polynomials

    Hinkelmann, Franziska


    We introduce the Macaulay2 package BooleanGB, which computes a Gr\\"obner basis for Boolean polynomials using a binary representation rather than symbolic. We compare the runtime of several Boolean models from systems in biology and give an application to Sudoku.

  15. A finite alternation result for reversible boolean circuits

    Selinger, Peter


    We say that a reversible boolean function on n bits has alternation depth d if it can be written as the sequential composition of d reversible boolean functions, each of which acts only on the top n-1 bits or on the bottom n-1 bits. We show that every reversible boolean function of n >= 4 bits has alternation depth 9.

  16. Algebraic Groups


    The workshop continued a series of Oberwolfach meetings on algebraic groups, started in 1971 by Tonny Springer and Jacques Tits who both attended the present conference. This time, the organizers were Michel Brion, Jens Carsten Jantzen, and Raphaël Rouquier. During the last years, the subject of...... algebraic groups (in a broad sense) has seen important developments in several directions, also related to representation theory and algebraic geometry. The workshop aimed at presenting some of these developments in order to make them accessible to a "general audience" of algebraic group-theorists, and to...

  17. Demonstrating Boolean Logic Using Simple Electrical Circuits

    McElhaney, Kevin W.


    While exploring the subject of geometric proofs, boolean logic operators AND and OR can be used to allow students to visualize their true-or-false patterns. An activity in the form of constructing electrical circuits is illustrated to explain the concept.

  18. Boolean Queries Optimization by Genetic Algorithms

    Húsek, Dušan; Owais, S.S.J.; Krömer, P.; Snášel, Václav


    Roč. 15, - (2005), s. 395-409. ISSN 1210-0552 R&D Projects: GA AV ČR 1ET100300414 Institutional research plan: CEZ:AV0Z10300504 Keywords : evolutionary algorithms * genetic algorithms * genetic programming * information retrieval * Boolean query Subject RIV: BB - Applied Statistics, Operational Research

  19. Evolutionary Algorithms for Boolean Queries Optimization

    Húsek, Dušan; Snášel, Václav; Neruda, Roman; Owais, S.S.J.; Krömer, P.


    Roč. 3, č. 1 (2006), s. 15-20. ISSN 1790-0832 R&D Projects: GA AV ČR 1ET100300414 Institutional research plan: CEZ:AV0Z10300504 Keywords : evolutionary algorithms * genetic algorithms * information retrieval * Boolean query Subject RIV: BA - General Mathematics

  20. A Boolean Map Theory of Visual Attention

    Huang, Liqiang; Pashler, Harold


    A theory is presented that attempts to answer two questions. What visual contents can an observer consciously access at one moment? Answer: only one feature value (e.g., green) per dimension, but those feature values can be associated (as a group) with multiple spatially precise locations (comprising a single labeled Boolean map). How can an…

  1. Linear Algebra and Smarandache Linear Algebra

    Vasantha, Kandasamy


    The present book, on Smarandache linear algebra, not only studies the Smarandache analogues of linear algebra and its applications, it also aims to bridge the need for new research topics pertaining to linear algebra, purely in the algebraic sense. We have introduced Smarandache semilinear algebra, Smarandache bilinear algebra and Smarandache anti-linear algebra and their fuzzy equivalents. Moreover, in this book, we have brought out the study of linear algebra and ve...

  2. Segal algebras in commutative Banach algebras

    INOUE, Jyunji; TAKAHASI, Sin-Ei


    The notion of Reiter's Segal algebra in commutative group algebras is generalized to a notion of Segal algebra in more general classes of commutative Banach algebras. Then we introduce a family of Segal algebras in commutative Banach algebras under considerations and study some properties of them.

  3. Algebra-Geometry of Piecewise Algebraic Varieties

    Chun Gang ZHU; Ren Hong WANG


    Algebraic variety is the most important subject in classical algebraic geometry.As the zero set of multivariate splines,the piecewise algebraic variety is a kind generalization of the classical algebraic variety.This paper studies the correspondence between spline ideals and piecewise algebraic varieties based on the knowledge of algebraic geometry and multivariate splines.

  4. The algebraic structure of the Onsager algebra

    DATE, ETSURO; Roan, Shi-shyr


    We study the Lie algebra structure of the Onsager algebra from the ideal theoretic point of view. A structure theorem of ideals in the Onsager algebra is obtained with the connection to the finite-dimensional representations. We also discuss the solvable algebra aspect of the Onsager algebra through the formal Lie algebra theory.

  5. A Representation of Quantum Measurement in Nonassociative Algebras

    Niestegge, Gerd


    Starting from an abstract setting for the Lueders - von Neumann quantum measurement process and its interpretation as a probability conditionalization rule in a non-Boolean event structure, the author derived a certain generalization of operator algebras in a preceding paper. This is an order-unit space with some specific properties. It becomes a Jordan operator algebra under a certain set of additional conditions, but does not own a multiplication operation in the most general case. A major objective of the present paper is the search for such examples of the structure mentioned above that do not stem from Jordan operator algebras; first natural candidates are matrix algebras over the octonions and other nonassociative rings. Therefore, the case when a nonassociative commutative multiplication exists is studied without assuming that it satisfies the Jordan condition. The characteristics of the resulting algebra are analyzed. This includes the uniqueness of the spectral resolution as well as a criterion for i...

  6. Design of Logic Controllers Thanks to Symbolic Computation of Simultaneously Asserted Boolean Equations

    Jean-Marc Roussel


    Full Text Available Formal methods can strongly contribute to improve dependability of controllers during design, by providing means to avoid flaws due to designers' omissions or specifications misinterpretations. This paper presents a synthesis method dedicated to logic controllers. Its goal is to obtain the control laws from specifications given in natural language by symbolic computation. The formal framework that underlies this method is the Boolean algebra of n-variable switching functions. In this algebra, thanks to relations and theorems presented in this paper, it is possible to formally express logical controllers specifications, to automatically detect inconsistencies in specifications, and to obtain automatically the set of solutions or to choose an optimal solution according to given optimization criteria. The application of this synthesis method to an example allows illustrating its main advantages.

  7. College algebra

    Kolman, Bernard


    College Algebra, Second Edition is a comprehensive presentation of the fundamental concepts and techniques of algebra. The book incorporates some improvements from the previous edition to provide a better learning experience. It provides sufficient materials for use in the study of college algebra. It contains chapters that are devoted to various mathematical concepts, such as the real number system, the theory of polynomial equations, exponential and logarithmic functions, and the geometric definition of each conic section. Progress checks, warnings, and features are inserted. Every chapter c

  8. Zonotopal algebra

    Holtz, Olga; Ron, Amos


    A wealth of geometric and combinatorial properties of a given linear endomorphism $X$ of $\\R^N$ is captured in the study of its associated zonotope $Z(X)$, and, by duality, its associated hyperplane arrangement ${\\cal H}(X)$. This well-known line of study is particularly interesting in case $n\\eqbd\\rank X \\ll N$. We enhance this study to an algebraic level, and associate $X$ with three algebraic structures, referred herein as {\\it external, central, and internal.} Each algebraic structure is ...

  9. Hom-Akivis algebras

    Issa, A. Nourou


    Hom-Akivis algebras are introduced. The commutator-Hom-associator algebra of a non-Hom-associative algebra (i.e. a Hom-nonassociative algebra) is a Hom-Akivis algebra. It is shown that non-Hom-associative algebras can be obtained from nonassociative algebras by twisting along algebra automorphisms while Hom-Akivis algebras can be obtained from Akivis algebras by twisting along algebra endomorphisms. It is pointed out that a Hom-Akivis algebra associated to a Hom-alternative algebra is a Hom-M...

  10. Algebraic models of deviant modal operators based on de Morgan and Kleene lattices

    Cattaneo, G.; Ciucci, DE; Dubois, D.


    An algebraic model of a kind of modal extension of de Morgan logic is described under the name MDS5 algebra. The main properties of this algebra can be summarized as follows: (1) it is based on a de Morgan lattice, rather than a Boolean algebra; (2) a modal necessity operator that satisfies the axioms N, K, T, and 5 (and as a consequence also B and 4) of modal logic is introduced; it allows one to introduce a modal possibility by the usual combination of necessity operation and...

  11. Enveloping algebras

    Since the works of Gelfand, Harish-Chandra, Kostant and Duflo, a new theory has earned its place in the field of mathematics, due to the abundance of its results and the coherence of its methods: the theory of enveloping algebras. This study is the first to present the whole subject in textbook form. The most recent results are included, as well as complete proofs, starting from the elementary theory of Lie algebras. (Auth.)

  12. Boolean network robotics: a proof of concept

    Roli, Andrea; Pinciroli, Carlo; Birattari, Mauro


    Dynamical systems theory and complexity science provide powerful tools for analysing artificial agents and robots. Furthermore, they have been recently proposed also as a source of design principles and guidelines. Boolean networks are a prominent example of complex dynamical systems and they have been shown to effectively capture important phenomena in gene regulation. From an engineering perspective, these models are very compelling, because they can exhibit rich and complex behaviours, in spite of the compactness of their description. In this paper, we propose the use of Boolean networks for controlling robots' behaviour. The network is designed by means of an automatic procedure based on stochastic local search techniques. We show that this approach makes it possible to design a network which enables the robot to accomplish a task that requires the capability of navigating the space using a light stimulus, as well as the formation and use of an internal memory.

  13. 布尔函数与形态算子关系的研究%On the Relationship between Boolean Function and Morphology Operator

    段汕; 罗敬; 徐文; 贺兴


      以布尔代数理论和欧式空间中二值形态变换理论为基础,通过布尔函数引入一个结构化映射,对二值形态变换的基本运算(腐蚀、膨胀)进行了描述,探讨了布尔函数与形态变换的关系,以期为二值形态变换的扩展提供新的途径。%This paper presents binary morphological transformation ( corrosion and expansion) on the basis of the Boolean algebra theory and the theory of binary morphological transformation in Euclidean space, and introduces a structural mapping resting on Boolean function, then researches the relationship between Boolean function and morphological transformation, which will provide a new way for extending morphological transformation.

  14. Efficient Analog Circuits for Boolean Satisfiability

    Yin, Xunzhao; Sedighi, Behnam; Varga, Melinda; Ercsey-Ravasz, Maria; Toroczkai, Zoltan; Hu, Xiaobo Sharon


    Efficient solutions to NP-complete problems would significantly benefit both science and industry. However, such problems are intractable on digital computers based on the von Neumann architecture, thus creating the need for alternative solutions to tackle such problems. Recently, a deterministic, continuous-time dynamical system (CTDS) was proposed (Nature Physics, 7(12), 966 (2011)) to solve a representative NP-complete problem, Boolean Satisfiability (SAT). This solver shows polynomial ana...

  15. Using Genetic Algorithms for Boolean Queries Optimization

    Húsek, Dušan; Snášel, Václav; Owais, S.S.J.; Krömer, P.

    Calgary: ACTA Press, 2005 - (Hamza, M.), s. 178-184 ISBN 0-88986-510-8. [IASTED International Conference on Internet and Multimedia Systems and Applications /9./. Honolulu (US), 15.08.2005-17.08.2005] R&D Projects: GA AV ČR 1ET100300419 Institutional research plan: CEZ:AV0Z10300504 Keywords : genetic algorithms * information retrieval * Boolean query * genetic programming Subject RIV: BA - General Mathematics

  16. On the Implementation of Boolean Matrix Factorization

    Snášel, V.; Krömer, P.; Platoš, J.; Húsek, Dušan

    Los Alamitos: IEEE, 2008, s. 554-558. ISBN 978-0-7695-3299-8. [ETID '08. International Workshop on Evolutionary Techniques /2./, in collocation with DEXA 2008 International Conference /19./. Turin (IT), 01.09.2008-05.09.2008] Institutional research plan: CEZ:AV0Z10300504 Keywords : data mining * genetic algorithms * Boolean factorization * binary data * machine learning * feature extraction Subject RIV: IN - Informatics, Computer Science

  17. Effect of memory in non-Markovian Boolean networks

    Ebadi, Haleh; Ausloos, Marcel; Jafari, GholamReza


    One successful model of interacting biological systems is the Boolean network. The dynamics of a Boolean network, controlled with Boolean functions, is usually considered to be a Markovian (memory-less) process. However, both self organizing features of biological phenomena and their intelligent nature should raise some doubt about ignoring the history of their time evolution. Here, we extend the Boolean network Markovian approach: we involve the effect of memory on the dynamics. This can be explored by modifying Boolean functions into non-Markovian functions, for example, by investigating the usual non-Markovian threshold function, - one of the most applied Boolean functions. By applying the non-Markovian threshold function on the dynamical process of a cell cycle network, we discover a power law memory with a more robust dynamics than the Markovian dynamics.

  18. Which multiplier algebras are $W^*$-algebras?

    Akemann, Charles A.; Amini, Massoud; Asadi, Mohammad B.


    We consider the question of when the multiplier algebra $M(\\mathcal{A})$ of a $C^*$-algebra $\\mathcal{A}$ is a $ W^*$-algebra, and show that it holds for a stable $C^*$-algebra exactly when it is a $C^*$-algebra of compact operators. This implies that if for every Hilbert $C^*$-module $E$ over a $C^*$-algebra $\\mathcal{A}$, the algebra $B(E)$ of adjointable operators on $E$ is a $ W^*$-algebra, then $\\mathcal{A}$ is a $C^*$-algebra of compact operators. Also we show that a unital $C^*$-algebr...

  19. Algebraic entropy for algebraic maps

    We propose an extension of the concept of algebraic entropy, as introduced by Bellon and Viallet for rational maps, to algebraic maps (or correspondences) of a certain kind. The corresponding entropy is an index of the complexity of the map. The definition inherits the basic properties from the definition of entropy for rational maps. We give an example with positive entropy, as well as two examples taken from the theory of Bäcklund transformations. (letter)

  20. Homotopy DG algebras induce homotopy BV algebras

    Terilla, John; Tradler, Thomas; Wilson, Scott O.


    Let TA denote the space underlying the tensor algebra of a vector space A. In this short note, we show that if A is a differential graded algebra, then TA is a differential Batalin-Vilkovisky algebra. Moreover, if A is an A-infinity algebra, then TA is a commutative BV-infinity algebra.


    Ali Muhammad Ali Rushdi; Adnan Ahmad Alsogati


    A total description of a synchronous Boolean network is typically achieved by a matrix recurrence relation. A simpler alternative is to use a scalar equation which is a possibly nonlinear equation that involves two or more instances of a single scalar variable and some Boolean operator(s). Further simplification is possible in terms of a linear reduced scalar equation which is the simplest two-term scalar equation that includes no Boolean operators and equates the value of a scalar variable a...

  2. Using Boolean Constraint Propagation for Sub-clause Deduction

    Darras, Sylvain; Dequen, Gilles; Devendeville, Laure; Mazure, Bertrand; Ostrowski, Richard; Sais, Lahkdar


    Boolean Constraint Propagation (BCP) is recognized as one of the most use- ful technique for efficient satisfiability checking. In this paper a new extension of the scope of boolean constraint propagation is proposed. It makes an original use of BCP to achieve further reduction of boolean formulas. Considering the impli- cation graph generated by the constraint propagation process as a resolution tree, sub-clauses from the original formula can be deduced. Then, we show how such extension can ...

  3. Combinational Logic-Level Verification using Boolean Expression Diagrams

    Hulgaard, Henrik; Williams, Poul Frederick; Andersen, Henrik Reif


    Boolean Expression Diagrams (BEDs) is a new data structure for representing and manipulating Boolean functions. BEDs are a generalization of Binary Decision Diagrams (BDDs) that are capable of representing any Boolean circuit in linear space and still maintain many of the desirable properties of...... BDDs. This paper demonstrates that BEDs are well suited for solving the combinational logic-level verification problem which is, given two combinational circuits, to determine whether they implement the same Boolean functions. Based on all combinational circuits in the ISCAS 85 and LGSynth 91...

  4. Linear algebra and probability for computer science applications

    Davis, Ernest


    MATLABDesk calculator operations Booleans Nonstandard numbers Loops and conditionals Script file Functions Variable scope and parameter passingI: Linear Algebra Vectors Definition of vectors Applications of vectorsBasic operations on vectorsDot productVectors in MATLAB: Basic operationsPlotting vectors in MATLABVectors in other programming languagesMatrices Definition of matrices Applications of matrices Simple operations on matrices Multiplying a matrix times a vector Linear transformation Systems of linear equations Matrix multiplication Vectors as matrices Algebraic properties of matrix mul

  5. Piecewise algebraic varieties

    WANG Renhong; ZHU Chungang


    The piecewise algebraic variety is a generalization of the classical algebraic variety. This paper discusses some properties of piecewise algebraic varieties and their coordinate rings based on the knowledge of algebraic geometry.

  6. Linear algebra

    Edwards, Harold M


    In his new undergraduate textbook, Harold M Edwards proposes a radically new and thoroughly algorithmic approach to linear algebra Originally inspired by the constructive philosophy of mathematics championed in the 19th century by Leopold Kronecker, the approach is well suited to students in the computer-dominated late 20th century Each proof is an algorithm described in English that can be translated into the computer language the class is using and put to work solving problems and generating new examples, making the study of linear algebra a truly interactive experience Designed for a one-semester course, this text adopts an algorithmic approach to linear algebra giving the student many examples to work through and copious exercises to test their skills and extend their knowledge of the subject Students at all levels will find much interactive instruction in this text while teachers will find stimulating examples and methods of approach to the subject

  7. Linear algebra

    Liesen, Jörg


    This self-contained textbook takes a matrix-oriented approach to linear algebra and presents a complete theory, including all details and proofs, culminating in the Jordan canonical form and its proof. Throughout the development, the applicability of the results is highlighted. Additionally, the book presents special topics from applied linear algebra including matrix functions, the singular value decomposition, the Kronecker product and linear matrix equations. The matrix-oriented approach to linear algebra leads to a better intuition and a deeper understanding of the abstract concepts, and therefore simplifies their use in real world applications. Some of these applications are presented in detailed examples. In several ‘MATLAB-Minutes’ students can comprehend the concepts and results using computational experiments. Necessary basics for the use of MATLAB are presented in a short introduction. Students can also actively work with the material and practice their mathematical skills in more than 300 exerc...

  8. Clifford algebra, geometric algebra, and applications

    Lundholm, Douglas; Svensson, Lars


    These are lecture notes for a course on the theory of Clifford algebras, with special emphasis on their wide range of applications in mathematics and physics. Clifford algebra is introduced both through a conventional tensor algebra construction (then called geometric algebra) with geometric applications in mind, as well as in an algebraically more general form which is well suited for combinatorics, and for defining and understanding the numerous products and operations of the algebra. The v...

  9. Cofree Hopf algebras on Hopf bimodule algebras

    Fang, Xin; Jian, Run-Qiang


    We investigate a Hopf algebra structure on the cotensor coalgebra associated to a Hopf bimodule algebra which contains universal version of Clifford algebras and quantum groups as examples. It is shown to be the bosonization of the quantum quasi-shuffle algebra built on the space of its right coinvariants. The universal property and a Rota-Baxter algebra structure are established on this new algebra.

  10. Why Do the Quantum Observables Form a Jordan Operator Algebra?

    Niestegge, Gerd


    The Jordan algebra structure of the bounded real quantum observables was recognized already in the early days of quantum mechanics. While there are plausible reasons for most parts of this structure, the existence of the distributive nonassociative multiplication operation is hard to justify from a physical or statistical point of view. Considering the non-Boolean extension of classical probabilities, presented in a recent paper, it is shown in this paper that such a multiplication operation can be derived from certain properties of the conditional probabilities and the observables, i.e., from postulates with a clear statistical interpretation. The well-known close relation between Jordan operator algebras and C*-algebras then provides the connection to the quantum-mechanical Hilbert space formalism, thus resulting in a novel axiomatic approach to general quantum mechanics that includes the types II and III von Neumann algebras.

  11. On uniform topological algebras

    Azhari, M. El


    The uniform norm on a uniform normed Q-algebra is the only uniform Q-algebra norm on it. The uniform norm on a regular uniform normed Q-algebra with unit is the only uniform norm on it. Let A be a uniform topological algebra whose spectrum M (A) is equicontinuous, then A is a uniform normed algebra. Let A be a regular semisimple commutative Banach algebra, then every algebra norm on A is a Q-algebra norm on A.

  12. Generalized exterior algebras

    Marchuk, Nikolay


    Exterior algebras and differential forms are widely used in many fields of modern mathematics and theoretical physics. In this paper we define a notion of $N$-metric exterior algebra, which depends on $N$ matrices of structure constants. The usual exterior algebra (Grassmann algebra) can be considered as 0-metric exterior algebra. Clifford algebra can be considered as 1-metric exterior algebra. $N$-metric exterior algebras for $N\\geq2$ can be considered as generalizations of the Grassmann alg...

  13. Word Hopf algebras

    Hazewinkel, Michiel


    Two important generalizations of the Hopf algebra of symmetric functions are the Hopf algebra of noncommutative symmetric functions and its graded dual the Hopf algebra of quasisymmetric functions. A common generalization of the latter is the selfdual Hopf algebra of permutations (MPR Hopf algebra). This latter Hopf algebra can be seen as a Hopf algebra of endomorphisms of a Hopf algebra. That turns out to be a fruitful way of looking at things and gives rise to wide ranging further generaliz...

  14. Linear algebra

    Allenby, Reg


    As the basis of equations (and therefore problem-solving), linear algebra is the most widely taught sub-division of pure mathematics. Dr Allenby has used his experience of teaching linear algebra to write a lively book on the subject that includes historical information about the founders of the subject as well as giving a basic introduction to the mathematics undergraduate. The whole text has been written in a connected way with ideas introduced as they occur naturally. As with the other books in the series, there are many worked examples.Solutions to the exercises are available onlin

  15. Lie algebras

    Jacobson, Nathan


    Lie group theory, developed by M. Sophus Lie in the 19th century, ranks among the more important developments in modern mathematics. Lie algebras comprise a significant part of Lie group theory and are being actively studied today. This book, by Professor Nathan Jacobson of Yale, is the definitive treatment of the subject and can be used as a textbook for graduate courses.Chapter I introduces basic concepts that are necessary for an understanding of structure theory, while the following three chapters present the theory itself: solvable and nilpotent Lie algebras, Carlan's criterion and its

  16. Linear algebra

    Stoll, R R


    Linear Algebra is intended to be used as a text for a one-semester course in linear algebra at the undergraduate level. The treatment of the subject will be both useful to students of mathematics and those interested primarily in applications of the theory. The major prerequisite for mastering the material is the readiness of the student to reason abstractly. Specifically, this calls for an understanding of the fact that axioms are assumptions and that theorems are logical consequences of one or more axioms. Familiarity with calculus and linear differential equations is required for understand

  17. Abstract algebra

    Deskins, W E


    This excellent textbook provides undergraduates with an accessible introduction to the basic concepts of abstract algebra and to the analysis of abstract algebraic systems. These systems, which consist of sets of elements, operations, and relations among the elements, and prescriptive axioms, are abstractions and generalizations of various models which evolved from efforts to explain or discuss physical phenomena.In Chapter 1, the author discusses the essential ingredients of a mathematical system, and in the next four chapters covers the basic number systems, decompositions of integers, diop


    Labourie, François


    We define a Poisson Algebra called the {\\em swapping algebra} using the intersection of curves in the disk. We interpret a subalgebra of the fraction algebra of the swapping algebra -- called the {\\em algebra of multifractions} -- as an algebra of functions on the space of cross ratios and thus as an algebra of functions on the Hitchin component as well as on the space of $\\mathsf{SL}_n(\\mathbb R)$-opers with trivial holonomy. We relate this Poisson algebra to the Atiyah--Bott--Goldman symple...

  19. Smarandache Jordan Algebras - abstract

    Vasantha Kandasamy, W. B.; Christopher, S.; A. Victor Devadoss


    We prove a S-commutative Jordan Algebra is a S-weakly commutative Jordan algebra. We define a S-Jordan algebra to be S-simple Jordan algebras if the S-Jordan algebra has no S-Jordan ideals. We obtain several other interesting notions and results on S-Jordan algebras.

  20. Learning Boolean functions with concentrated spectra

    Mixon, Dustin G.; Peterson, Jesse


    This paper discusses the theory and application of learning Boolean functions that are concentrated in the Fourier domain. We first estimate the VC dimension of this function class in order to establish a small sample complexity of learning in this case. Next, we propose a computationally efficient method of empirical risk minimization, and we apply this method to the MNIST database of handwritten digits. These results demonstrate the effectiveness of our model for modern classification tasks. We conclude with a list of open problems for future investigation.

  1. Boolean Factor Analysis by Attractor Neural Network

    Frolov, A. A.; Húsek, Dušan; Muraviev, I. P.; Polyakov, P.Y.


    Roč. 18, č. 3 (2007), s. 698-707. ISSN 1045-9227 R&D Projects: GA AV ČR 1ET100300419; GA ČR GA201/05/0079 Institutional research plan: CEZ:AV0Z10300504 Keywords : recurrent neural network * Hopfield-like neural network * associative memory * unsupervised learning * neural network architecture * neural network application * statistics * Boolean factor analysis * dimensionality reduction * features clustering * concepts search * information retrieval Subject RIV: BB - Applied Statistics, Operational Research Impact factor: 2.769, year: 2007

  2. Neural Network Boolean Factor Analysis and Applications

    Húsek, Dušan; Frolov, A.; Polyakov, P.Y.; Snášel, V.

    -: WSEAS Press, 2007 - (Katehakis, M.; And ina, D.; Mastorakis, M.), s. 30-35. (Electrical and Computer Engineering Series). ISBN 978-960-6766-21-3. [CIMMACS'07. WSEAS International Conference on Computational Intelligence, Man-Machine Systems and Cybernetics. Tenerife (ES), 14.12.2007-16.12.2007] R&D Projects: GA MŠk 1M0567; GA AV ČR 1ET100300414; GA ČR GA201/05/0079 Institutional research plan: CEZ:AV0Z10300504 Keywords : Hopfield neural network * boolean factor analysis * unsupervised learning * dimension reduction * data mining Subject RIV: BB - Applied Statistics, Operational Research

  3. Solving the Satisfiability Problem Through Boolean Networks

    Roli, Andrea; Milano, Michela


    In this paper we present a new approach to solve the satisfiability problem (SAT), based on boolean networks (BN). We define a mapping between a SAT instance and a BN, and we solve SAT problem by simulating the BN dynamics. We prove that BN fixed points correspond to the SAT solutions. The mapping presented allows to develop a new class of algorithms to solve SAT. Moreover, this new approach suggests new ways to combine symbolic and connectionist computation and provides a general framework f...

  4. Solving the Satisfiability Problem Through Boolean Networks

    Roli, Andrea


    In this paper we present a new approach to solve the satisfiability problem (SAT), based on boolean networks (BN). We define a mapping between a SAT instance and a BN, and we solve SAT problem by simulating the BN dynamics. We prove that BN fixed points correspond to the SAT solutions. The mapping presented allows to develop a new class of algorithms to solve SAT. Moreover, this new approach suggests new ways to combine symbolic and connectionist computation and provides a general framework for local search algorithms.

  5. Representations of Boolean Functions by Perceptron Networks

    Kůrková, Věra

    Prague : Institute of Computer Science AS CR, 2014 - (Kůrková, V.; Bajer, L.; Peška, L.; Vojtáš, R.; Holeňa, M.; Nehéz, M.), s. 68-70 ISBN 978-80-87136-19-5. [ITAT 2014. European Conference on Information Technologies - Applications and Theory /14./. Demänovská dolina (SK), 25.09.2014-29.09.2014] R&D Projects: GA MŠk(CZ) LD13002 Institutional support: RVO:67985807 Keywords : perceptron networks * model complexity * Boolean functions Subject RIV: IN - Informatics, Computer Science

  6. Algebraic Stacks

    Tomás L Gómez


    This is an expository article on the theory of algebraic stacks. After introducing the general theory, we concentrate in the example of the moduli stack of vector bundles, giving a detailed comparison with the moduli scheme obtained via geometric invariant theory.

  7. Algebraic Topology

    Oliver, Bob; Pawałowski, Krzystof


    As part of the scientific activity in connection with the 70th birthday of the Adam Mickiewicz University in Poznan, an international conference on algebraic topology was held. In the resulting proceedings volume, the emphasis is on substantial survey papers, some presented at the conference, some written subsequently.

  8. Constructions of vector output Boolean functions with high generalized nonlinearity

    KE Pin-hui; ZHANG Sheng-yuan


    Carlet et al. recently introduced generalized nonlinearity to measure the ability to resist the improved correlation attack of a vector output Boolean function. This article presents a construction of vector output Boolean functions with high generalized nonlinearity using the sample space. The relation between the resilient order and generalized nonlinearity is also discussed.

  9. E-Referencer: Transforming Boolean OPACs to Web Search Engines.

    Khoo, Christopher S. G.; Poo, Danny C. C.; Toh, Teck-Kang; Hong, Glenn

    E-Referencer is an expert intermediary system for searching library online public access catalogs (OPACs) on the World Wide Web. It is implemented as a proxy server that mediates the interaction between the user and Boolean OPACs. It transforms a Boolean OPAC into a retrieval system with many of the search capabilities of Web search engines.…

  10. Terse Integer Linear Programs for Boolean Optimization

    Christoph Buchheim


    Full Text Available We present a new polyhedral approach to nonlinear Boolean optimization. Compared to other methods, it produces much smaller integer programming models, making it more efficient from a practical point of view. We mainly obtain this by two different ideas: first, we do not require the objective function to be in any normal form. The transformation into a normal form usually leads to the introduction of many additional variables or constraints. Second, we reduce the problem to the degree-two case in a very efficient way, by slightly extending the dimension of the original variable space. The resulting model turns out to be closely related to the maximum cut problem; we show that the corresponding polytope is a face of a suitable cut polytope in most cases. In particular, our separation problem reduces to the one for the maximum cut problem. In practice, the approach appears to be very competitive for unconstrained Boolean optimization problems. First experimental results, which have been obtained for some particularly hard instances of the Max-SAT Evaluation 2007, show that our very general implementation can outperform even special-purpose Max-SAT solvers. The software is accessible online under “”.

  11. Wavelets and quantum algebras

    A non-linear associative algebra is realized in terms of translation and dilation operators, and a wavelet structure generating algebra is obtained. We show that this algebra is a q-deformation of the Fourier series generating algebra, and reduces to this for certain value of the deformation parameter. This algebra is also homeomorphic with the q-deformed suq(2) algebra and some of its extensions. Through this algebraic approach new methods for obtaining the wavelets are introduced. (author). 20 refs

  12. Central simple Poisson algebras

    SU; Yucai; XU; Xiaoping


    Poisson algebras are fundamental algebraic structures in physics and symplectic geometry. However, the structure theory of Poisson algebras has not been well developed. In this paper, we determine the structure of the central simple Poisson algebras related to locally finite derivations, over an algebraically closed field of characteristic zero.The Lie algebra structures of these Poisson algebras are in general not finitely-graded.

  13. The Onsager Algebra

    El-Chaar, Caroline


    In this thesis, four realizations of the Onsager algebra are explored. We begin with its original definition as introduced by Lars Onsager. We then examine how the Onsager algebra can be presented as a Lie algebra with two generators and two relations. The third realization of the Onsager algebra consists of viewing it as an equivariant map algebra which then gives us the tools to classify its closed ideals. Finally, we examine the Onsager algebra as a subalgebra of the tetrahedron algebra. U...

  14. Ordered Boolean List (OBL): reducing the footprint for evaluating Boolean expressions.

    Rossignac, Jaroslaw Jarek


    An Expanded Boolean Expression (EBE) does not contain any XOR or EQUAL operators. The occurrence of each variable is a different literal. We provide a linear time algorithm that converts an EBE of n literals into a logically equivalent Ordered Boolean List (OBL) and show how to use the OBL to evaluate the EBE in n steps and O(log log n) space, if the values of the literals are each read once in the order prescribed by the OBL. (An evaluation workspace of 5 bits suffices for all EBEs of up to six billion literals.) The primary application is the SIMD architecture, where the same EBE is evaluated in parallel for different input vectors when rendering solid models on the GPU directly from their Constructive Solid Geometry (CSG) representation. We compare OBL to the Reduced Ordered Binary Decision Diagram (ROBDD) and suggest possible applications of OBL to logic verification and to circuit design. PMID:21737862

  15. Exotic Elliptic Algebras

    Chirvasitu, Alex; Smith, S. Paul


    This paper examines a general method for producing twists of a comodule algebra by tensoring it with a torsor then taking co-invariants. We examine the properties that pass from the original algebra to the twisted algebra and vice versa. We then examine the special case where the algebra is a 4-dimensional Sklyanin algebra viewed as a comodule algebra over the Hopf algebra of functions on the non-cyclic group of order 4 with the torsor being the 2x2 matrix algebra. The twisted algebra is an "...

  16. Fibered F-Algebra

    Kleyn, Aleks


    The concept of F-algebra and its representation can be extended to an arbitrary bundle. We define operations of fibered F-algebra in fiber. The paper presents the representation theory of of fibered F-algebra as well as a comparison of representation of F-algebra and of representation of fibered F-algebra.

  17. Representing Boolean Functions by Decision Trees

    Chikalov, Igor


    A Boolean or discrete function can be represented by a decision tree. A compact form of decision tree named binary decision diagram or branching program is widely known in logic design [2, 40]. This representation is equivalent to other forms, and in some cases it is more compact than values table or even the formula [44]. Representing a function in the form of decision tree allows applying graph algorithms for various transformations [10]. Decision trees and branching programs are used for effective hardware [15] and software [5] implementation of functions. For the implementation to be effective, the function representation should have minimal time and space complexity. The average depth of decision tree characterizes the expected computing time, and the number of nodes in branching program characterizes the number of functional elements required for implementation. Often these two criteria are incompatible, i.e. there is no solution that is optimal on both time and space complexity. © Springer-Verlag Berlin Heidelberg 2011.

  18. Robust Boolean Operation for Sculptured Models


    To enhance the ability of current modeling system, an uniformed representation is designed to represent wire-frame, solid, surface models. We present an algorithm for Boolean operation between the models under this representation. Accuracy, efficiency and robustness are the main consideration. The geometric information is represented with trimmed parametric patches and trimmed parametric splines. The topological information is represented with an extended half-edge data structure. In the process of intersection calculation, hierarchy intersection method is applied for unified classification. Tracing the intersection curve to overcome degenerate cases that occur frequently in practice. The algorithm has been implemented as the modeling kernel of a feature based modeling system named GS-CAD98, which was developed on Windows/NT platform.


    A. V. Sokolov


    Full Text Available The rapid development of methods of error-correcting coding, cryptography, and signal synthesis theory based on the principles of many-valued logic determines the need for a more detailed study of the forms of representation of functions of many-valued logic. In particular the algebraic normal form of Boolean functions, also known as Zhegalkin polynomial, that well describe many of the cryptographic properties of Boolean functions is widely used. In this article, we formalized the notion of algebraic normal form for many-valued logic functions. We developed a fast method of synthesis of algebraic normal form of 3-functions and 5-functions that work similarly to the Reed-Muller transform for Boolean functions: on the basis of recurrently synthesized transform matrices. We propose the hypothesis, which determines the rules of the synthesis of these matrices for the transformation from the truth table to the coefficients of the algebraic normal form and the inverse transform for any given number of variables of 3-functions or 5-functions. The article also introduces the definition of algebraic degree of nonlinearity of the functions of many-valued logic and the S-box, based on the principles of many-valued logic. Thus, the methods of synthesis of algebraic normal form of 3-functions applied to the known construction of recurrent synthesis of S-boxes of length N = 3k, whereby their algebraic degrees of nonlinearity are computed. The results could be the basis for further theoretical research and practical applications such as: the development of new cryptographic primitives, error-correcting codes, algorithms of data compression, signal structures, and algorithms of block and stream encryption, all based on the perspective principles of many-valued logic. In addition, the fast method of synthesis of algebraic normal form of many-valued logic functions is the basis for their software and hardware implementation.

  20. Real Algebraic Geometry

    Mahé, Louis; Roy, Marie-Françoise


    Ten years after the first Rennes international meeting on real algebraic geometry, the second one looked at the developments in the subject during the intervening decade - see the 6 survey papers listed below. Further contributions from the participants on recent research covered real algebra and geometry, topology of real algebraic varieties and 16thHilbert problem, classical algebraic geometry, techniques in real algebraic geometry, algorithms in real algebraic geometry, semialgebraic geometry, real analytic geometry. CONTENTS: Survey papers: M. Knebusch: Semialgebraic topology in the last ten years.- R. Parimala: Algebraic and topological invariants of real algebraic varieties.- Polotovskii, G.M.: On the classification of decomposing plane algebraic curves.- Scheiderer, C.: Real algebra and its applications to geometry in the last ten years: some major developments and results.- Shustin, E.L.: Topology of real plane algebraic curves.- Silhol, R.: Moduli problems in real algebraic geometry. Further contribu...

  1. Categorical Formulation of Finite-Dimensional Quantum Algebras

    Vicary, Jamie


    We describe how †-Frobenius monoids give the correct categorical description of certain kinds of finite-dimensional `quantum algebras'. We develop the concept of an involution monoid, and use it to construct a correspondence between finite-dimensional C*-algebras and certain types of †-Frobenius monoids in the category of Hilbert spaces. Using this technology, we recast the spectral theorems for commutative C*-algebras and for normal operators into an explicitly categorical language, and we examine the case that the results of measurements do not form finite sets, but rather objects in a finite Boolean topos. We describe the relevance of these results for topological quantum field theory.

  2. Rigid current Lie algebras

    Goze, Michel; Remm, Elisabeth


    A current Lie algebra is contructed from a tensor product of a Lie algebra and a commutative associative algebra of dimension greater than 2. In this work we are interested in deformations of such algebras and in the problem of rigidity. In particular we prove that a current Lie algebra is rigid if it is isomorphic to a direct product gxg...xg where g is a rigid Lie algebra.

  3. Solvable quadratic Lie algebras


    A Lie algebra endowed with a nondegenerate, symmetric, invariant bilinear form is called a quadratic Lie algebra. In this paper, the author investigates the structure of solvable quadratic Lie algebras, in particular, the solvable quadratic Lie algebras whose Cartan subalgebras consist of semi-simple elements, the author presents a procedure to construct a class of quadratic Lie algebras from the point of view of cohomology and shows that all solvable quadratic Lie algebras can be obtained in this way.

  4. Graded cluster algebras

    Grabowski, Jan


    In the cluster algebra literature, the notion of a graded cluster algebra has been implicit since the origin of the subject. In this work, we wish to bring this aspect of cluster algebra theory to the foreground and promote its study. We transfer a definition of Gekhtman, Shapiro and Vainshtein to the algebraic setting, yielding the notion of a multi-graded cluster algebra. We then study gradings for finite type cluster algebras without coefficients, giving a full classification. Translating ...

  5. Approximate Counting for Complex-Weighted Boolean Constraint Satisfaction Problems

    Yamakami, Tomoyuki


    Constraint satisfaction problems (or CSPs) have been extensively studied in AI, database theory, graph theory, etc. From an approximation viewpoint, it has been important to approximate the total number of assignments that satisfy all given Boolean constraints. There is a trichotomy theorem for such approximate counting for (non-weighted) Boolean CSPs; namely, all such counting problems are neatly classified into three categories under polynomial-time approximation-preserving reductions [Dyer, Goldberg, and Jerrum, 2010]. We extend this result to approximate counting for complex-weighted Boolean CSPs, provided that all arity-1 constraints are freely available to use. This makes a significant progress in the quest for the approximation classification of all counting Boolean CSPs in the most general form. To deal with complex weights, we employ proof techniques along the line of solving Holant problems [Valiant, 2002, 2008]. Our result also gives an approximation version of the dichotomy theorem of the complexi...

  6. Automatic Ranked Output from Boolean Searches in SIRE

    Noreault, Terry; And Others


    This study examined the effectiveness using an automatic algorithm to rank the results of Boolean searches of an inverted file design document retrieval system. Relevant documents were ranked significantly higher than nonrelevant documents on output lists. (Author/KP)

  7. Boolean Burritos: How the Faculty Ate Up Keyword Searching.

    York, Sherry


    Describes an activity that librarians can use to acquaint teachers with keyword searching and Boolean operators to more successfully use the library's online catalog. Uses food ingredients to represent various possible combinations. (LRW)

  8. Boolean functions with a vertex-transitive group of automorphisms

    Savický, Petr

    -, submitted 2015 (2016) R&D Projects: GA ČR GAP202/10/1333 Institutional support: RVO:67985807 Keywords : Boolean Functions * hypercube * isometric transformation * vertex-transitive group of automorphisms Subject RIV: BA - General Mathematics

  9. Linear Programming Formulation of the Boolean Satisfiability Problem

    Diaby, Moustapha


    In this paper, we present a new, graph-based modeling approach and a polynomial-sized linear programming (LP) formulation of the Boolean satisfiability problem (SAT). The approach is illustrated with a numerical example.

  10. Piecewise-Koszul algebras


    It is a small step toward the Koszul-type algebras. The piecewise-Koszul algebras are,in general, a new class of quadratic algebras but not the classical Koszul ones, simultaneously they agree with both the classical Koszul and higher Koszul algebras in special cases. We give a criteria theorem for a graded algebra A to be piecewise-Koszul in terms of its Yoneda-Ext algebra E(A), and show an A∞-structure on E(A). Relations between Koszul algebras and piecewise-Koszul algebras are discussed. In particular, our results are related to the third question of Green-Marcos.

  11. On vertex Leibniz algebras

    Li, Haisheng; Tan, Shaobin; Wang, Qing


    In this paper, we study a notion of what we call vertex Leibniz algebra. This notion naturally extends that of vertex algebra without vacuum, which was previously introduced by Huang and Lepowsky. We show that every vertex algebra without vacuum can be naturally extended to a vertex algebra. On the other hand, we show that a vertex Leibniz algebra can be embedded into a vertex algebra if and only if it admits a faithful module. To each vertex Leibniz algebra we associate a vertex algebra with...

  12. Universal algebra

    Grätzer, George


    Universal Algebra, heralded as ". . . the standard reference in a field notorious for the lack of standardization . . .," has become the most authoritative, consistently relied on text in a field with applications in other branches of algebra and other fields such as combinatorics, geometry, and computer science. Each chapter is followed by an extensive list of exercises and problems. The "state of the art" account also includes new appendices (with contributions from B. Jónsson, R. Quackenbush, W. Taylor, and G. Wenzel) and a well-selected additional bibliography of over 1250 papers and books which makes this a fine work for students, instructors, and researchers in the field. "This book will certainly be, in the years to come, the basic reference to the subject." --- The American Mathematical Monthly (First Edition) "In this reviewer's opinion [the author] has more than succeeded in his aim. The problems at the end of each chapter are well-chosen; there are more than 650 of them. The book is especially sui...

  13. On the number of attractors of Boolean automata circuits

    Demongeot, Jacques; Noual, Mathilde; Sené, Sylvain


    In line with fields of theoretical computer science and biology that study Boolean automata networks often seen as models of regulation networks, we present some results concerning the dynamics of networks whose underlying interaction graphs are circuits, that is Boolean automata circuits. In the context of biological regulation, former studies have highlighted the importance of circuits on the asymptotic dynamical behaviour of the biological networks that contain them. Our work focuses on th...

  14. Boolean Equi-propagation for Optimized SAT Encoding

    Metodi, Amit; Lagoon, Vitaly; Stuckey, Peter J


    We present an approach to propagation based solving, Boolean equi-propagation, where constraints are modelled as propagators of information about equalities between Boolean literals. Propagation based solving applies this information as a form of partial evaluation resulting in optimized SAT encodings. We demonstrate for a variety of benchmarks that our approach results in smaller CNF encodings and leads to speed-ups in solving times.

  15. Inadmissible Class of Boolean Functions under Stuck-at Faults

    Das, Debesh K.; Chowdhury, Debabani; Bhattacharya, Bhargab B; Sasao, Tsutomu


    Many underlying structural and functional factors that determine the fault behavior of a combinational network, are not yet fully understood. In this paper, we show that there exists a large class of Boolean functions, called root functions, which can never appear as faulty response in irredundant two-level circuits even when any arbitrary multiple stuck-at faults are injected. Conversely, we show that any other Boolean function can appear as a faulty response from an irredundant realization ...

  16. Enhancing Boolean networks with fuzzy operators and edge tuning

    Poret, Arnaud; Monteiro Sousa, Claudio; Boissel, Jean-Pierre


    Quantitative modeling in systems biology can be difficult due to the scarcity of quantitative details about biological phenomenons, especially at the subcellular scale. An alternative to escape this difficulty is qualitative modeling since it requires few to no quantitative information. Among the qualitative modeling approaches, the Boolean network formalism is one of the most popular. However, Boolean models allow variables to be valued at only true or false, which can appear too simplistic ...

  17. IMS Algorithm for Learning Representations in Boolean Neural Networks

    Biswas, Nripendra N; Murthy, TVMK; Chandrasekhar, M.


    A new algorithm for learning representations in Boolean neural networks, where the inputs and outputs are binary bits, is presented. The algorithm has become feasible because of a newly discovered theorem which states that any non-linearly separable Boolean function can be expressed as a convergent series of linearly separable functions connected by the logical OR (+) and the logical INHIBIT (-) operators. The formation of the series is carried out by many important properties exhibited by th...

  18. Yoneda algebras of almost Koszul algebras

    Zheng Lijing


    Let be an algebraically closed field, a finite dimensional connected (, )-Koszul self-injective algebra with , ≥ 2. In this paper, we prove that the Yoneda algebra of is isomorphic to a twisted polynomial algebra $A^!$ [ ; ] in one indeterminate of degree +1 in which $A^!$ is the quadratic dual of , is an automorphism of $A^!$, and = () for each $t \\in A^!$. As a corollary, we recover Theorem 5.3 of [2].

  19. Boolean network model predicts knockout mutant phenotypes of fission yeast.

    Maria I Davidich

    Full Text Available BOOLEAN NETWORKS (OR: networks of switches are extremely simple mathematical models of biochemical signaling networks. Under certain circumstances, Boolean networks, despite their simplicity, are capable of predicting dynamical activation patterns of gene regulatory networks in living cells. For example, the temporal sequence of cell cycle activation patterns in yeasts S. pombe and S. cerevisiae are faithfully reproduced by Boolean network models. An interesting question is whether this simple model class could also predict a more complex cellular phenomenology as, for example, the cell cycle dynamics under various knockout mutants instead of the wild type dynamics, only. Here we show that a Boolean network model for the cell cycle control network of yeast S. pombe correctly predicts viability of a large number of known mutants. So far this had been left to the more detailed differential equation models of the biochemical kinetics of the yeast cell cycle network and was commonly thought to be out of reach for models as simplistic as Boolean networks. The new results support our vision that Boolean networks may complement other mathematical models in systems biology to a larger extent than expected so far, and may fill a gap where simplicity of the model and a preference for an overall dynamical blueprint of cellular regulation, instead of biochemical details, are in the focus.

  20. Algebra cohomology over a commutative algebra revisited

    Pirashvili, Teimuraz


    The aim of this paper is to give a relatively easy bicomplex which computes the Shukla, or Quillen cohomology in the category of associative algebras over a commutative algebra $A$, in the case when $A$ is an algebra over a field.


    TaoChangli; LuShijie; ChenPeixin


    Algebraic reflexivity introduced by Hadwin is related to linear interpolation. In this paper, the concepts of weakly algebraic reflexivity and strongly algebraic reflexivity which are also related to linear interpolation are introduced. Some properties of them are obtained and some relations between them revealed.

  2. 基于布尔语义的Gentzen推导模型%Gentzen Deduction Model Based on Boolean Logic Semantics

    陈博; 眭跃飞


    Deduction systems are important arts of searching technology. This paper gives a new correspondence between the propositional logic and Boolean algebra, where an inequation is corresponding to a Gentzen sequent, so that the inequation is true in every Boolean algebra if and only if the Gentzen sequent is provable. In information retrieval, the information inference can effectively turn into the operation on poset. Precisely, the logical language for the propositional logic contains operators Ø'Ù'Ú;the terms instead of formulas are defined (a|Øt|t1 Ù t2|t1 Ú t2 , where a is an element) and used to represent elements in Boolean algebra. This paper defines an assignment v using Boolean algebra as its domain, and assigns the terms to be the element in Boolean algebra. The sequence ΓÞΔ is satisfied if tv £tv. Finally, this paper gives a Gentzen system to prove the soundness and completeness theorem.%布尔模型是信息检索系统的一种基础模型。给出了命题逻辑和布尔代数间的一种新的对应关系,其中布尔代数中的不等式对应Gentzen系统中的矢列式,使得当一个不等式在任意布尔代数中为真,当且仅当它所对应的矢列式是可证的。并且使得在信息检索中,针对信息的推理可以有效地转为偏序集上的运算。讨论的命题逻辑语言的运算符为Ø、Ù、Ú;并且定义了项(a|Øt|t1Ù t2|t1Ú t2'其中a是一个元素)来替代原先的公式和表示布尔代数中的元素。此外,定义了以布尔代数为论域的赋值v,将命题逻辑中的项赋值为布尔代数中的元素,并且如果tv £t v ,则矢列式ΓÞ D为真。最后给出了Gentzen系统下的可靠性和完备性定理的证明。tÎΓtÎΔ

  3. Enveloping algebras of some quantum Lie algebras

    Pourkia, Arash


    We define a family of Hopf algebra objects, $H$, in the braided category of $\\mathbb{Z}_n$-modules (known as anyonic vector spaces), for which the property $\\psi^2_{H\\otimes H}=id_{H\\otimes H}$ holds. We will show that these anyonic Hopf algebras are, in fact, the enveloping (Hopf) algebras of particular quantum Lie algebras, also with the property $\\psi^2=id$. Then we compute the braided periodic Hopf cyclic cohomology of these Hopf algebras. For that, we will show the following fact: analog...

  4. The Yoneda algebra of a K2 algebra need not be another K2 algebra

    Cassidy, T.; Phan, C.; Shelton, B.


    The Yoneda algebra of a Koszul algebra or a D-Koszul algebra is Koszul. K2 algebras are a natural generalization of Koszul algebras, and one would hope that the Yoneda algebra of a K2 algebra would be another K2 algebra. We show that this is not necessarily the case by constructing a monomial K2 algebra for which the corresponding Yoneda algebra is not K2.

  5. Efficient Analog Circuits for Boolean Satisfiability

    Yin, Xunzhao; Varga, Melinda; Ercsey-Ravasz, Maria; Toroczkai, Zoltan; Hu, Xiaobo Sharon


    Efficient solutions to NP-complete problems would significantly benefit both science and industry. However, such problems are intractable on digital computers based on the von Neumann architecture, thus creating the need for alternative solutions to tackle such problems. Recently, a deterministic, continuous-time dynamical system (CTDS) was proposed (Nature Physics, 7(12), 966 (2011)) to solve a representative NP-complete problem, Boolean Satisfiability (SAT). This solver shows polynomial analog time-complexity on even the hardest benchmark $k$-SAT ($k \\geq 3$) formulas, but at an energy cost through exponentially driven auxiliary variables. With some modifications to the CTDS equations, here we present a novel analog hardware SAT solver, AC-SAT, implementing the CTDS. AC-SAT is intended to be used as a co-processor, and with its modular design can be readily extended to different problem sizes. The circuit is designed and simulated based on a 32nm CMOS technology. SPICE simulation results show speedup factor...

  6. Measuring Mutual Information in Random Boolean Networks

    Luque, B; Luque, Bartolo; Ferrera, Antonio


    During the last few years an area of active research in the field of complex systems is that of their information storing and processing abilities. Common opinion has it that the most interesting beaviour of these systems is found ``at the edge of chaos'', which would seem to suggest that complex systems may have inherently non-trivial information proccesing abilities in the vicinity of sharp phase transitions. A comprenhensive, quantitative understanding of why this is the case is however still lacking. Indeed, even ``experimental'' (i.e., often numerical) evidence that this is so has been questioned for a number of systems. In this paper we will investigate, both numerically and analitically, the behavior of Random Boolean Networks (RBN's) as they undergo their order-disorder phase transition. We will use a simple mean field approximation to treat the problem, and without lack of generality we will concentrate on a particular value for the connectivity of the system. In spite of the simplicity of our argume...

  7. Position Automata for Kleene Algebra with Tests

    A. Silva


    Full Text Available Kleene algebra with tests (KAT is an equational system that combines Kleene and Boolean algebras. One can model basic programming constructs and assertions in KAT, which allowed for its application in compiler optimization, program transformation and dataflow analysis. To provide semantics for KAT expressions, Kozen first introduced emph{automata on guarded strings}, showing that the regular sets of guarded strings plays the same role in KAT as regular languages play in Kleene algebra. Recently, Kozen described an elegant algorithm, based on ``derivatives'', to construct a deterministic automaton that accepts the guarded strings denoted by a KAT expression. This algorithm generalizes Brzozowski's algorithm for regular expressions and inherits its inefficiency arising from the explicit computation of derivatives. In the context of classical regular expressions, many efficient algorithms to compile expressions to automata have been proposed. One of those algorithms was devised by Berry and Sethi in the 80's (we shall refer to it as Berry-Sethi construction/algorithm, but in the literature it is also referred to as position or Glushkov automata algorithm. In this paper, we show how the Berry-Sethi algorithm can be used to compile a $KAT$ expression to an automaton on guarded strings. Moreover, we propose a new automata model for KAT expressions and adapt the construction of Berry and Sethi to this new model.

  8. Workshop on Commutative Algebra

    Simis, Aron


    The central theme of this volume is commutative algebra, with emphasis on special graded algebras, which are increasingly of interest in problems of algebraic geometry, combinatorics and computer algebra. Most of the papers have partly survey character, but are research-oriented, aiming at classification and structural results.

  9. Idempotents of Clifford Algebras

    Ablamowicz, R.; Fauser, B.; Podlaski, K.; Rembielinski, J.


    A classification of idempotents in Clifford algebras C(p,q) is presented. It is shown that using isomorphisms between Clifford algebras C(p,q) and appropriate matrix rings, it is possible to classify idempotents in any Clifford algebra into continuous families. These families include primitive idempotents used to generate minimal one sided ideals in Clifford algebras. Some low dimensional examples are discussed.

  10. Historical Topics in Algebra.

    National Council of Teachers of Mathematics, Inc., Reston, VA.

    This is a reprint of the historical capsules dealing with algebra from the 31st Yearbook of NCTM,"Historical Topics for the Mathematics Classroom." Included are such themes as the change from a geometric to an algebraic solution of problems, the development of algebraic symbolism, the algebraic contributions of different countries, the origin and…

  11. Generalized Quantum Current Algebras

    ZHAO Liu


    Two general families of new quantum-deformed current algebras are proposed and identified both as infinite Hopf family of algebras, a structure which enables one to define "tensor products" of these algebras. The standard quantum affine algebras turn out to be a very special case of the two algebra families, in which case the infinite Hopf family structure degenerates into a standard Hopf algebra. The relationship between the two algebraic families as well as thefr various special examples are discussed, and the free boson representation is also considered.

  12. On fermionic Novikov algebras

    Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic type and Hamiltonian operators in formal variational calculus. They are a class of left-symmetric algebras with commutative right multiplication operators, which can be viewed as bosonic. Fermionic Novikov algebras are a class of left-symmetric algebras with anti-commutative right multiplication operators. They correspond to a certain Hamiltonian superoperator in a supervariable. In this paper, we commence a study on fermionic Novikov algebras from the algebraic point of view. We will show that any fermionic Novikov algebra in dimension ≤3 must be bosonic. Moreover, we give the classification of real fermionic Novikov algebras on four-dimensional nilpotent Lie algebras and some examples in higher dimensions. As a corollary, we obtain kinds of four-dimensional real fermionic Novikov algebras which are not bosonic. All of these examples will serve as a guide for further development including the application in physics

  13. Algebraically periodic translation surfaces

    Calta, Kariane; Smillie, John


    Algebraically periodic directions on translation surfaces were introduced by Calta in her study of genus two translation surfaces. We say that a translation surface with three or more algebraically periodic directions is an algebraically periodic surface. We show that for an algebraically periodic surface the slopes of the algebraically periodic directions are given by a number field which we call the periodic direction field. We show that translation surfaces with pseudo-Anosov automorphisms...

  14. Clifford Algebra with Mathematica

    Aragon-Camarasa, G.; Aragon-Gonzalez, G; Aragon, J. L.; Rodriguez-Andrade, M. A.


    The Clifford algebra of a n-dimensional Euclidean vector space provides a general language comprising vectors, complex numbers, quaternions, Grassman algebra, Pauli and Dirac matrices. In this work, a package for Clifford algebra calculations for the computer algebra program Mathematica is introduced through a presentation of the main ideas of Clifford algebras and illustrative examples. This package can be a useful computational tool since allows the manipulation of all these mathematical ob...

  15. Second moment method for a family of boolean CSP

    Boufkhad, Yacine


    The estimation of phase transitions in random boolean Constraint Satisfaction Problems (CSP) is based on two fundamental tools: the first and second moment methods. While the first moment method on the number of solutions permits to compute upper bounds on any boolean CSP, the second moment method used for computing lower bounds proves to be more tricky and in most cases gives only the trivial lower bound 0. In this paper, we define a subclass of boolean CSP covering the monotone versions of many known NP-Complete boolean CSPs. We give a method for computing non trivial lower bounds for any member of this subclass. This is achieved thanks to an application of the second moment method to some selected solutions called characteristic solutions that depend on the boolean CSP considered. We apply, as an example, this method to establish that the threshold r_{k} of monotone 1-in-k-SAT is \\log k/k\\leq r_{k}\\leq\\log^{2}k/k

  16. Efficient Algorithms for Membership in Boolean Hierarchies of Regular Languages

    Glasser, Christian; Selivanov, Victor


    The purpose of this paper is to provide efficient algorithms that decide membership for classes of several Boolean hierarchies for which efficiency (or even decidability) were previously not known. We develop new forbidden-chain characterizations for the single levels of these hierarchies and obtain the following results: - The classes of the Boolean hierarchy over level $\\Sigma_1$ of the dot-depth hierarchy are decidable in $NL$ (previously only the decidability was known). The same remains true if predicates mod $d$ for fixed $d$ are allowed. - If modular predicates for arbitrary $d$ are allowed, then the classes of the Boolean hierarchy over level $\\Sigma_1$ are decidable. - For the restricted case of a two-letter alphabet, the classes of the Boolean hierarchy over level $\\Sigma_2$ of the Straubing-Th\\'erien hierarchy are decidable in $NL$. This is the first decidability result for this hierarchy. - The membership problems for all mentioned Boolean-hierarchy classes are logspace many-one hard for $NL$. - T...

  17. Piecewise-Koszul algebras

    Jia-feng; Lü


    [1]Priddy S.Koszul resolutions.Trans Amer Math Soc,152:39-60 (1970)[2]Beilinson A,Ginszburg V,Soergel W.Koszul duality patterns in representation theory.J Amer Math Soc,9:473-525 (1996)[3]Aquino R M,Green E L.On modules with linear presentations over Koszul algebras.Comm Algebra,33:19-36 (2005)[4]Green E L,Martinez-Villa R.Koszul and Yoneda algebras.Representation theory of algebras (Cocoyoc,1994).In:CMS Conference Proceedings,Vol 18.Providence,RI:American Mathematical Society,1996,247-297[5]Berger R.Koszulity for nonquadratic algebras.J Algebra,239:705-734 (2001)[6]Green E L,Marcos E N,Martinez-Villa R,et al.D-Koszul algebras.J Pure Appl Algebra,193:141-162(2004)[7]He J W,Lu D M.Higher Koszul Algebras and A-infinity Algebras.J Algebra,293:335-362 (2005)[8]Green E L,Marcos E N.δ-Koszul algebras.Comm Algebra,33(6):1753-1764 (2005)[9]Keller B.Introduction to A-infinity algebras and modules.Homology Homotopy Appl,3:1-35 (2001)[10]Green E L,Martinez-Villa R,Reiten I,et al.On modules with linear presentations.J Algebra,205(2):578-604 (1998)[11]Keller B.A-infinity algebras in representation theory.Contribution to the Proceedings of ICRA Ⅸ.Beijing:Peking University Press,2000[12]Lu D M,Palmieri J H,Wu Q S,et al.A∞-algebras for ring theorists.Algebra Colloq,11:91-128 (2004)[13]Weibel C A.An Introduction to homological algebra.Cambridge Studies in Avanced Mathematics,Vol 38.Cambridge:Cambridge University Press,1995

  18. Maps from the enveloping algebra of the positive Witt algebra to regular algebras

    Sierra, Susan J.; Walton, Chelsea


    We construct homomorphisms from the universal enveloping algebra of the positive (part of the) Witt algebra to several different Artin-Schelter regular algebras, and determine their kernels and images. As a result, we produce elementary proofs that the universal enveloping algebras of the Virasoro algebra, the Witt algebra, and the positive Witt algebra are neither left nor right noetherian.

  19. Algebraic and relational models for a system based on a poset of two elements

    Iturrioz, Luisa


    The aim of this paper is to present a very simple set of conditions, necessary for the management of knowledge of a poset $T$ of two agents, which are partially ordered by the capabilities available in the system. We build up a formal system and we elaborate suitable semantic models in order to derive information from the poset. The system is related to three-valued Heyting algebras with Boolean operators.

  20. A more robust Boolean model describing inhibitor binding

    Zhaoqian Steven XIE; Chao TANG


    From the first application of the Boolean model to the cell cycle regulation network of budding yeast, new regulative pathways have been discovered, par-ticularly in the G1/S transition circuit. This discovery called for finer modeling to study the essential biology, and the resulting outcomes are first introduced in the ar-ticle. A traditional Boolean network model set up for the new G1/S transition circuit shows that it cannot correctly simulate real biology unless the model parameters are fine tuned. The deficiency is caused by an overly coarse-grained description of the inhibitor binding process, which shall be overcome by a two-vector model proposed whose robustness is surveyed using random perturba-tions. Simulations show that the proposed two-vector model is much more robust in describing inhibitor binding processes within the Boolean framework.

  1. The Influence of Canalization on the Robustness of Boolean Networks

    Kadelka, Claus; Laubenbacher, Reinhard


    Time- and state-discrete dynamical systems are frequently used to model molecular networks. This paper provides a collection of mathematical and computational tools for the study of robustness in Boolean network models. The focus is on networks governed by $k$-canalizing functions, a recently introduced class of Boolean functions that contains the well-studied class of nested canalizing functions. The activities and sensitivity of a function quantify the impact of input changes on the function output. This paper generalizes the latter concept to $c$-sensitivity and provides formulas for the activities and $c$-sensitivity of general $k$-canalizing functions as well as canalizing functions with more precisely defined structure. A popular measure for the robustness of a network, the Derrida value, can be expressed as a weighted sum of the $c$-sensitivities of the governing canalizing functions, and can also be calculated for a stochastic extension of Boolean networks. These findings provide a computationally eff...

  2. Optimal Computation of Symmetric Boolean Functions in Collocated Networks

    Kowshik, Hemant


    We consider collocated wireless sensor networks, where each node has a Boolean measurement and the goal is to compute a given Boolean function of these measurements. We first consider the worst case setting and study optimal block computation strategies for computing symmetric Boolean functions. We study three classes of functions: threshold functions, delta functions and interval functions. We provide exactly optimal strategies for the first two classes, and a scaling law order-optimal strategy with optimal preconstant for interval functions. We also extend the results to the case of integer measurements and certain integer-valued functions. We use lower bounds from communication complexity theory, and provide an achievable scheme using information theoretic tools. Next, we consider the case where nodes measurements are random and drawn from independent Bernoulli distributions. We address the problem of optimal function computation so as to minimize the expected total number of bits that are transmitted. In ...

  3. Exploiting Surroundedness for Saliency Detection: A Boolean Map Approach.

    Zhang, Jianming; Sclaroff, Stan


    We demonstrate the usefulness of surroundedness for eye fixation prediction by proposing a Boolean Map based Saliency model (BMS). In our formulation, an image is characterized by a set of binary images, which are generated by randomly thresholding the image's feature maps in a whitened feature space. Based on a Gestalt principle of figure-ground segregation, BMS computes a saliency map by discovering surrounded regions via topological analysis of Boolean maps. Furthermore, we draw a connection between BMS and the Minimum Barrier Distance to provide insight into why and how BMS can properly captures the surroundedness cue via Boolean maps. The strength of BMS is verified by its simplicity, efficiency and superior performance compared with 10 state-of-the-art methods on seven eye tracking benchmark datasets. PMID:26336114

  4. Order-to-chaos transition in the hardness of random Boolean satisfiability problems

    Varga, Melinda; Sumi, Róbert; Toroczkai, Zoltán; Ercsey-Ravasz, Mária


    Transient chaos is a ubiquitous phenomenon characterizing the dynamics of phase-space trajectories evolving towards a steady-state attractor in physical systems as diverse as fluids, chemical reactions, and condensed matter systems. Here we show that transient chaos also appears in the dynamics of certain efficient algorithms searching for solutions of constraint satisfaction problems that include scheduling, circuit design, routing, database problems, and even Sudoku. In particular, we present a study of the emergence of hardness in Boolean satisfiability (k -SAT), a canonical class of constraint satisfaction problems, by using an analog deterministic algorithm based on a system of ordinary differential equations. Problem hardness is defined through the escape rate κ , an invariant measure of transient chaos of the dynamical system corresponding to the analog algorithm, and it expresses the rate at which the trajectory approaches a solution. We show that for a given density of constraints and fixed number of Boolean variables N , the hardness of formulas in random k -SAT ensembles has a wide variation, approximable by a lognormal distribution. We also show that when increasing the density of constraints α , hardness appears through a second-order phase transition at αχ in the random 3-SAT ensemble where dynamical trajectories become transiently chaotic. A similar behavior is found in 4-SAT as well, however, such a transition does not occur for 2-SAT. This behavior also implies a novel type of transient chaos in which the escape rate has an exponential-algebraic dependence on the critical parameter κ ˜NB |α - αχ|1-γ with 0 <γ <1 . We demonstrate that the transition is generated by the appearance of metastable basins in the solution space as the density of constraints α is increased.

  5. Brackets in representation algebras of Hopf algebras

    Massuyeau, Gwenael; Turaev, Vladimir


    For any graded bialgebras $A$ and $B$, we define a commutative graded algebra $A_B$ representing the functor of so-called $B$-representations of $A$. When $A$ is a cocommutative graded Hopf algebra and $B$ is a commutative ungraded Hopf algebra, we introduce a method deriving a Gerstenhaber bracket in $A_B$ from a Fox pairing in $A$ and a balanced biderivation in $B$. Our construction is inspired by Van den Bergh's non-commutative Poisson geometry, and may be viewed as an algebraic generaliza...

  6. Algebraic theory of numbers

    Samuel, Pierre


    Algebraic number theory introduces students not only to new algebraic notions but also to related concepts: groups, rings, fields, ideals, quotient rings and quotient fields, homomorphisms and isomorphisms, modules, and vector spaces. Author Pierre Samuel notes that students benefit from their studies of algebraic number theory by encountering many concepts fundamental to other branches of mathematics - algebraic geometry, in particular.This book assumes a knowledge of basic algebra but supplements its teachings with brief, clear explanations of integrality, algebraic extensions of fields, Gal

  7. Lukasiewicz-Moisil algebras

    Boicescu, V; Georgescu, G; Rudeanu, S


    The Lukasiewicz-Moisil algebras were created by Moisil as an algebraic counterpart for the many-valued logics of Lukasiewicz. The theory of LM-algebras has developed to a considerable extent both as an algebraic theory of intrinsic interest and in view of its applications to logic and switching theory.This book gives an overview of the theory, comprising both classical results and recent contributions, including those of the authors. N-valued and &THgr;-valued algebras are presented, as well as &THgr;-algebras with negation.Mathematicians interested in lattice theory or symbolic logic, and computer scientists, will find in this monograph stimulating material for further research.

  8. Workshop on Lie Algebras

    Osborn, J


    During the academic year 1987-1988 the University of Wisconsin in Madison hosted a Special Year of Lie Algebras. A Workshop on Lie Algebras, of which these are the proceedings, inaugurated the special year. The principal focus of the year and of the workshop was the long-standing problem of classifying the simple finite-dimensional Lie algebras over algebraically closed field of prime characteristic. However, other lectures at the workshop dealt with the related areas of algebraic groups, representation theory, and Kac-Moody Lie algebras. Fourteen papers were presented and nine of these (eight research articles and one expository article) make up this volume.

  9. Relation between dual S-algebras and BE-algebras

    Arsham Borumand Saeid


    Full Text Available In this paper, we investigate the relationship between dual (Weak Subtraction algebras, Heyting algebras and BE-algebras. In fact, the purpose of this paper is to show that BE-algebra is a generalization of Heyting algebra and dual (Weak Subtraction algebras. Also, we show that a bounded commutative self distributive BE-algebra is equivalent to the Heyting algebra.  

  10. A Boolean Approach to Airline Business Model Innovation

    Hvass, Kristian Anders

    Research in business model innovation has identified its significance in creating a sustainable competitive advantage for a firm, yet there are few empirical studies identifying which combination of business model activities lead to success and therefore deserve innovative attention. This study...... analyzes the business models of North America low-cost carriers from 2001 to 2010 using a Boolean minimization algorithm to identify which combinations of business model activities lead to operational profitability. The research aim is threefold: complement airline literature in the realm of business model...... innovation, introduce Boolean minimization methods to the field, and propose alternative business model activities to North American carriers striving for positive operating results....

  11. On Kolmogorov's superpositions and Boolean functions

    Beiu, V.


    The paper overviews results dealing with the approximation capabilities of neural networks, as well as bounds on the size of threshold gate circuits. Based on an explicit numerical (i.e., constructive) algorithm for Kolmogorov's superpositions they will show that for obtaining minimum size neutral networks for implementing any Boolean function, the activation function of the neurons is the identity function. Because classical AND-OR implementations, as well as threshold gate implementations require exponential size (in the worst case), it will follow that size-optimal solutions for implementing arbitrary Boolean functions require analog circuitry. Conclusions and several comments on the required precision are ending the paper.

  12. Experimental Comparison of Schemes for Interpreting Boolean Queries

    Lee, Whay C.; Edward A Fox


    The standard interpretation of the logical operators in a Boolean retrieval system is in general too strict. A standard Boolean query rarely comes close to retrieving all and only those documents which are relevant to the user. An AND query is often too narrow and an OR query is often too broad. The choice of the AND results in retrieving on the left end of a typical average recall-precision graph, while the choice of the OR results in retrieving on the right end, implying a tradeoff between ...

  13. Some Properties of Inclusions of Multisets and Contractive Boolean Operators

    Hyvernat, Pierre


    10 pages, including appendix Consider the following curious puzzle: call an n-tuple X=(X_1, ..., X_n) of sets smaller than another n-tuple Y if it has fewer //unordered sections//. We show that equivalence classes for this preorder are very easy to describe and characterize the preorder in terms of the simpler pointwise inclusion and the existence of a special increasing boolean operator f:B^n -> B^n. We also show that contrary to increasing boolean operators, the relevant operators are no...

  14. Constant-Overhead Secure Computation of Boolean Circuits using Preprocessing

    Damgård, Ivan Bjerre; Zakarias, Sarah Nouhad Haddad

    We present a protocol for securely computing a Boolean circuit $C$ in presence of a dishonest and malicious majority. The protocol is unconditionally secure, assuming access to a preprocessing functionality that is not given the inputs to compute on. For a large number of players the work done by...... each player is the same as the work needed to compute the circuit in the clear, up to a constant factor. Our protocol is the first to obtain these properties for Boolean circuits. On the technical side, we develop new homomorphic authentication schemes based on asymptotically good codes with an...

  15. Boolean Logic: An Aid for Searching Computer Databases in Special Education and Rehabilitation.

    Summers, Edward G.


    The article discusses using Boolean logic as a tool for searching computerized information retrieval systems in special education and rehabilitation technology. It includes discussion of the Boolean search operators AND, OR, and NOT; Venn diagrams; and disambiguating parentheses. Six suggestions are offered for development of good Boolean logic…

  16. Representations of twisted current algebras

    Lau, Michael


    We use evaluation representations to give a complete classification of the finite-dimensional simple modules of twisted current algebras. This generalizes and unifies recent work on multiloop algebras, current algebras, equivariant map algebras, and twisted forms.

  17. Hom-alternative algebras and Hom-Jordan algebras

    Makhlouf, Abdenacer


    The purpose of this paper is to introduce Hom-alternative algebras and Hom-Jordan algebras. We discuss some of their properties and provide construction procedures using ordinary alternative algebras or Jordan algebras. Also, we show that a polarization of Hom-associative algebra leads to Hom-Jordan algebra.

  18. Cellularity of diagram algebras as twisted semigroup algebras

    Wilcox, Stewart


    The Temperley-Lieb and Brauer algebras and their cyclotomic analogues, as well as the partition algebra, are all examples of twisted semigroup algebras. We prove a general theorem about the cellularity of twisted semigroup algebras of regular semigroups. This theorem, which generalises a recent result of East about semigroup algebras of inverse semigroups, allows us to easily reproduce the cellularity of these algebras.

  19. ADAM: Analysis of Discrete Models of Biological Systems Using Computer Algebra

    Hinkelmann, Franziska; Guang, Bonny; McNeill, Rustin; Blekherman, Grigoriy; Veliz-Cuba, Alan; Laubenbacher, Reinhard


    Motivation: Many biological systems are modeled qualitatively with discrete models, such as probabilistic Boolean networks, logical models, bounded Petri nets, and agent-based models. Simulation is a common practice for analyzing discrete models, but many systems are far too large to capture all the relevant dynamical features through simulation alone. Results: We convert discrete models into algebraic models and apply tools from computational algebra to analyze their dynamics. The key feature of biological systems that is exploited by our algorithms is their sparsity: while the number of nodes in a biological network may be quite large, each node is affected only by a small number of other nodes. In our experience with models arising in systems biology and random models, this structure leads to fast computations when using algebraic models, and thus efficient analysis. Availability: All algorithms and methods are available in our package Analysis of Dynamic Algebraic Models (ADAM), a user friendly web-interf...

  20. Lie Algebra of Noncommutative Inhomogeneous Hopf Algebra

    Lagraa, M.; Touhami, N.


    We construct the vector space dual to the space of right-invariant differential forms construct from a first order differential calculus on inhomogeneous quantum group. We show that this vector space is equipped with a structure of a Hopf algebra which closes on a noncommutative Lie algebra satisfying a Jacobi identity.

  1. Realizations of Galilei algebras

    All inequivalent realizations of the Galilei algebras of dimensions not greater than five are constructed using the algebraic approach proposed by Shirokov. The varieties of the deformed Galilei algebras are discussed and families of one-parametric deformations are presented in explicit form. It is also shown that a number of well-known and physically interesting equations and systems are invariant with respect to the considered Galilei algebras or their deformations. (paper)

  2. Homotopy Algebras for Operads

    Leinster, Tom


    We present a definition of homotopy algebra for an operad, and explore its consequences. The paper should be accessible to topologists, category theorists, and anyone acquainted with operads. After a review of operads and monoidal categories, the definition of homotopy algebra is given. Specifically, suppose that M is a monoidal category in which it makes sense to talk about algebras for some operad P. Then our definition says what a homotopy P-algebra in M is, provided only that some of the ...

  3. Algebraic statistics computational commutative algebra in statistics

    Pistone, Giovanni; Wynn, Henry P


    Written by pioneers in this exciting new field, Algebraic Statistics introduces the application of polynomial algebra to experimental design, discrete probability, and statistics. It begins with an introduction to Gröbner bases and a thorough description of their applications to experimental design. A special chapter covers the binary case with new application to coherent systems in reliability and two level factorial designs. The work paves the way, in the last two chapters, for the application of computer algebra to discrete probability and statistical modelling through the important concept of an algebraic statistical model.As the first book on the subject, Algebraic Statistics presents many opportunities for spin-off research and applications and should become a landmark work welcomed by both the statistical community and its relatives in mathematics and computer science.

  4. Quantum Lie algebra solitons

    We construct a special type of quantum soliton solutions for quantized affine Toda models. The elements of the principal Heisenberg subalgebra in the affinised quantum Lie algebra are found. Their eigenoperators inside the quantized universal enveloping algebra for an affine Lie algebra are constructed to generate quantum soliton solutions

  5. Deficiently Extremal Gorenstein Algebras

    Pavinder Singh


    The aim of this article is to study the homological properties of deficiently extremal Gorenstein algebras. We prove that if / is an odd deficiently extremal Gorenstein algebra with pure minimal free resolution, then the codimension of / must be odd. As an application, the structure of pure minimal free resolution of a nearly extremal Gorenstein algebra is obtained.

  6. Connecting Arithmetic to Algebra

    Darley, Joy W.; Leapard, Barbara B.


    Algebraic thinking is a top priority in mathematics classrooms today. Because elementary school teachers lay the groundwork to develop students' capacity to think algebraically, it is crucial for teachers to have a conceptual understanding of the connections between arithmetic and algebra and be confident in communicating these connections. Many…

  7. Clifford Algebras and Graphs

    Khovanova, Tanya


    I show how to associate a Clifford algebra to a graph. I describe the structure of these Clifford graph algebras and provide many examples and pictures. I describe which graphs correspond to isomorphic Clifford algebras and also discuss other related sets of graphs. This construction can be used to build models of representations of simply-laced compact Lie groups.

  8. Algebraic formulation of duality

    Two dimensional lattice spin (chiral) models over (possibly non-abelian) compact groups are formulated in terms of a generalized Pauli algebra. Such models over cyclic groups are written in terms of the generalized Clifford algebra. An automorphism of this algebra is shown to exist and to lead to the duality transformation

  9. Crisp Sets and Boolean Algebra: A Research Strategy for Student Affairs

    Banning, James; Eversole, Barbara; Most, David; Kuk, Linda


    A review of student affairs journals clearly points out that most, if not all, research strategies within the field fall within traditional approaches based on quantitative methods and, more recently, qualitative methods. The purpose of this article is not to discourage use of these time honored research strategies, but to suggest the inclusion of…

  10. Generalized Flip-Flop Input Equations Based on a Four-Valued Boolean Algebra

    Tucker, Jerry H.; Tapia, Moiez A.


    A procedure is developed for obtaining generalized flip-flop input equations, and a concise method is presented for representing these equations. The procedure is based on solving a four-valued characteristic equation of the flip-flop, and can encompass flip-flops that are too complex to approach intuitively. The technique is presented using Karnaugh maps, but could easily be implemented in software.