Normal Forms Of Boolean Functions Research Articles

Technology independent optimization when implementing sparse systems of disjunctive normal forms of Boolean functions in ASIC

Objectives. The problem of choosing the best methods and programs for circuit implementation as part of digital ASIC (Application-Specific Integrated Circuit) sparse systems of disjunctive normal forms (DNF) of completely defined Boolean functions is considered. For matrix forms of sparse DNF systems, the ternary matrix specifying elementary conjunctions contains a large proportion of undefined values corresponding to missing literals of Boolean input variables, and the Boolean matrix specifying the occurrences of conjunctions in DNF functions contains a large proportion of zero values.Methods. It is proposed to investigate various methods of technologically independent logical optimization performed at the first stage of logical synthesis: joint minimization of systems of functions in the DNF class, separate and joint minimization in classes of multilevel representations in the form of Boolean networks and BDD representations using mutually inverse cofactors, as well as the division of a system of functions into subsystems with a limited number of input variables and the method of block cover of DNF systems, focused on minimizing the total area of the blocks forming the cover.Results. When implementing sparse DNF systems of Boolean functions in ASIC, along with traditional methods of joint minimization of systems of functions in the DNF class, methods for optimizing multilevel representations of Boolean function systems based on Shannon expansions can be used for technologically independent optimization, while separate minimization and joint minimization of the entire system as a whole turn out to be less effective compared with block partitions and coatings of the DNF system and subsequent minimization of multilevel representations. Schemes obtained as a result of synthesis using minimized representations of Boolean networks often have a smaller area than schemes obtained using minimized BDD representations.Conclusion. For the design of digital ASIC, the effectiveness of combined approach is shown, when initially the block coverage programs of the DNF system is used, followed by the use of programs to minimize multilevel block representations in the form of Boolean networks minimized based on Shannon expansion.

Developing the minimization of a polynomial normal form of boolean functions by the method of figurative transformations

This paper reports a study that has established the possibility of improving the effectiveness of the method of figurative transformations in order to minimize Boolean functions on the Reed-Muller basis. Such potential prospects in the analytical method have been identified as a sequence in the procedure of inserting the same conjuncterms of polynomial functions followed by the operation of super-gluing the variables. The extension of the method of figurative transformations to the process of simplifying the functions of the polynomial basis involved the developed algebra in terms of the rules for simplifying functions in the Reed-Muller basis. It was established that the simplification of Boolean functions of the polynomial basis by a figurative transformation method is based on a flowchart with repetition, which is actually the truth table of the predefined function. This is a sufficient resource to minimize functions that makes it possible not to refer to such auxiliary objects as Karnaugh maps, Weich charts, cubes, etc. A perfect normal form of the polynomial basis functions can be represented by binary sets or a matrix that would represent the terms of the functions and the addition operation by module two for them. The experimental study has confirmed that the method of figurative transformations that employs the systems of 2-(n, b)-design, and 2-(n, x/b)-design in the first matrix improves the efficiency of minimizing Boolean functions. That also simplifies the procedure for finding a minimum function on the Reed-Muller basis. Compared to analogs, this makes it possible to enhance the performance of minimizing Boolean functions by 100‒200 %. There is reason to assert the possibility of improving the efficiency of minimizing Boolean functions in the Reed-Muller basis by a method of figurative transformations. This is ensured by using more complex algorithms to simplify logical expressions involving a procedure of inserting the same function terms in the Reed-Muller basis, followed by the operation of super-gluing the variables.

Devising a method of figurative transformations for minimizing boolean functions in the implicative basis

This paper reports a study that has established the possibility of reducing computational complexity while improving the productivity of simplification of Boolean functions in the class of perfect implied normal forms (PINF-1 and PINF-2) using a method of figurative transformations.The method of figurative transformations has been expanded to cover the process of simplifying the functions of the implicative basis by using the developed algebra of the implicative basis in the form of rules that simplify the PINF-1 and PINF-2 functions of the implicative basis. A special feature in simplifying the functions of the implicative basis on the binary structures of 2-(n, b)-designs) is the use of analogs of perfect disjunctive normal forms (PDNF) and perfect conjunctive normal forms (PCNF) of Boolean functions. The specified forms of the functions define transformation rules for the functions of the implicative basis on binary structures.It is shown that the perfect implicative normal form of n-place function of the implicative basis can be represented by the binary sets or a matrix. Logical operations over the structure of the matrix ensure the result from simplifying the functions of the implicative basis. This makes it possible to focus the minimization principle within the truth table of the assigned function and avoid auxiliary objects such as Carnot map, Weich charts, etc.The method under consideration makes it possible:– to reduce the algorithmic complexity of PINF-1 and PINF-2 simplification;– to improve the performance of simplifying the functions of the implied basis by 100‒200 %;– to visualize the process of PINF-1 or PINF-2 minimization;There is reason to argue that minimizing the functions of the implicative basis using a method of figurative transformations brings the task of PINF-1 and PINF-2 minimization to the level of well-researched problems within the class of disjunctive-conjunctive normal forms of Boolean functions

Completeness and normal form of multi-valued logical functions

Theory of completeness is essential for multi-valued logical functions. Using semi-tensor product (STP) of matrices, the algebraic form of k-valued logical functions is presented. Using algebraic form, a method is proposed to construct an adequate set of connectives (ASC), consisting of unary operators with conjunction/disjunction for k-valued logical functions, which can be used to express any k-valued logical functions. Based on it, two normal forms of k-valued logical functions are presented, which are extensions of the disjunctive normal form and conjunctive normal form of Boolean functions respectively. The ASC is then simplified to a condensed set. Finally, the normal forms are further extended to mix-valued logical functions.

Minimization of conjunctive normal forms of boolean functions by combinatorial method

The object of research is the combinatorial method of minimizing conjunctive normal forms (CNF) of Boolean functions in order to reduce its algorithmic complexity. One of the most places to minimize CNF of Boolean functions is the complexity of the minimization algorithm and the guarantee of obtaining the minimum function. In the course of the study, the method of equivalent figurative transformations based on the laws and axioms of the algebra of logic, protocols for minimizing CNF of Boolean functions is used. The reduction of the computational complexity of the process of minimization of the CNF of the Boolean functions by the combinatorial method according to the new established criteria has been obtained, thanks to the use of a number of features of the algorithm for finding minimal disjunctive normal forms (DNF) and CNF of logical functions, in particular the use of the mathematical apparatus of transforming flowcharts with repetition allows to increase the information component of the figurative transformation with respect to the orthogonality, adjacency, uniqueness of truth table blocks; equivalent figurative transformations allow with the effect to replace verbal procedures of algebraic transformations due to the greater information capacity of matrix images; result of minimization is estimated on the basis of the minimal function; minimal DNF or CNF of the functions are obtained regardless of the normal form of the given logical function; minimization protocols for CNF of Boolean functions make up a library of protocols for the process of minimization of CNF of Boolean functions as standard procedures. Due to the above, it is possible to optimally reduce the number of variables of a given function without losing its functionality. The effectiveness of the use of figurative transformations is demonstrated by examples of minimizing functions borrowed from other methods for the purpose of comparison. Compared with similar known methods of minimizing Boolean functions, the proposed method allows reduce the algorithmic complexity of minimizing CNF of Boolean functions; increase the visibility of the minimization process of DNF or CNF of Boolean functions; ensure the self-sufficiency of the combinatorial method of minimizing Boolean functions by introducing features of the minimal function and minimization on the full table of DNF and CNF.

On one class of decision diagrams

A class of decision diagrams for representation of the normal forms of Boolean functions was introduced. Consideration was given, in particular, to the disjunctive diagrams representing the disjunctive normal forms (DNF). In distinction to the binary decision diagrams (BDD) and reduced ordered binary decision diagram (ROBDD), the disjunctive diagram representing an arbitrary DNF is constructed in a time which is polynomial of the size of the DNF binary code. Corresponding algorithms were described, and the results were presented of the computer-aided experiments where the proposed diagrams were used to reduce the information content accumulated in the course of deciding hard variants of Boolean satisfiability problem (SAT).

Characterization of the ergodicity of 1-Lipschitz functions on [formula omitted] using the q-Mahler basis

The ergodicity of 1-Lipschitz functions on Z2 represented by the Mahler basis was characterized by V.S. Anashin (1994) in [1]. His results are mainly based on the so-called folklore criterion for ergodicity, depending on the algebraic normal form of Boolean functions associated with coordinate functions. In this paper, we employ the q-Mahler basis to provide q-analogues of Anashin's results whose proof does not rely on this criterion.

High order differential attacks on stream ciphers

Differential cryptanalysis is probably the most popular tool for chosen plaintext attacks on block ciphers. It also applies to chosen IV attacks on stream ciphers, but here, high order differential attacks have been surprisingly successful, namely on NLFSR-based constructions. Most approaches have been developed in terms of the algebraic normal form of Boolean functions. Prominent examples are the d-monomial test, cube attacks, and cube testers. We review the various techniques and translate them into the terminology of high order derivatives introduced by Lai. The unified view points out similarities between seemingly different approaches and naturally suggests generalizations and refinements such as conditional differential cryptanalysis.

Simple disjunctive normal forms of Boolean functions with a restricted number of zeros

This paper studies the complexity of disjunctive normal forms (DNFs) of Boolean functions defined by their matrices of zeros. The best available bounds are obtained for the minimum number of conjunc� tions making up a DNF of a Boolean function with a restricted number of zeros. The results can be used to implement characteristic functions of classes in pat� tern recognition with binary data [1]. 1. The standard method for constructing DNFs of characteristic classes of functions can be described as follows. Let , , …, and , , …, be given binary descriptions of reference objects of classes K1 and K2, respectively, where , i ∈ {0, 1}, j ∈ {1, 2, …, ki}, is a binary vector of dimension n.

Composition of Post classes and normal forms of Boolean functions

The class composition C ∘ K of Boolean clones, being the set of composite functions f ( g 1 , … , g n ) with f ∈ C , g 1 , … , g n ∈ K , is investigated. This composition C ∘ K is either the join C ∨ K in the Post Lattice or it is not a clone, and all pairs of clones C , K are classified accordingly. Factorizations of the clone Ω of all Boolean functions as a composition of minimal clones are described and seen to correspond to normal form representations of Boolean functions. The median normal form, arising from the factorization of Ω with the clone SM of self-dual monotone functions as the leftmost composition factor, is compared in terms of complexity with the well-known DNF, CNF, and Zhegalkin (Reed–Muller) polynomial representations, and it is shown to provide a more efficient normal form representation.

An upper bound for complexity of polynomial normal forms of Boolean functions

We consider the problem of minimisation of Boolean functions in the class of polynomial normal forms. We suggest an algorithm for constructing polynomial normal forms for arbitrary Boolean functions such that the length of the obtained formula depends on the number of variables of the function only. As the input of the algorithm, along with the function, we take the solution of a covering problem. The number of elementary conjunctions in the obtained formula is equal to the cardinality of the covering. For the introduced covering problem we find an approximate solution. We succeed to prove that the complexity of Boolean functions in the class of polynomial normal forms is less than 2 n +1 (log 2 n + 1)/ n , which allows us to conclude that for almost any Boolean function the complexity of its representation in the polynomial normal form is less than its representation in the disjunctive normal form.

An upper bound for complexity of polynomial normal forms of Boolean functions

We consider the problem of minimisation of Boolean functions in the class of polynomial normal forms. We suggest an algorithm for constructing polynomial normal forms for arbitrary Boolean functions such that the length of the obtained formula depends on the number of variables of the function only. As the input of the algorithm, along with the function, we take the solution of a covering problem. The number of elementary conjunctions in the obtained formula is equal to the cardinality of the covering. For the introduced covering problem we find an approximate solution. We succeed to prove that the complexity of Boolean functions in the class of polynomial normal forms is less than 2 n +1 (log 2 n + 1)/ n , which allows us to conclude that for almost any Boolean function the complexity of its representation in the polynomial normal form is less than its representation in the disjunctive normal form.

A model of the loop formation process on knitting machines using finite automata theory

Computer control systems, currently available for industrial knitting machines, have been developed by individual machine manufacturers. The result is that each system is applicable only to a particular machine type. This work is aimed toward the determination of a generic control system that could be applied to a wide range of knitting machine types. A model of a knitting needle operation is proposed. This takes the form of a multitape controlling automaton and an operational automaton functioning together. The conformity between the technological functions of the separate parts of knitting machines and the functions of an automaton is determined. Diagrams of states and alphabets of input, stack, and output symbols of an operational automaton are defined. The functioning of this automaton is reduced to the performance of nine operations, which are its system of instructions. This enables the processes of basic structural element formation, i.e., knitted loop, tuck loop, and float, on needles of different types and on knitting machines of various constructions to be modelled uniformly. The model proposed also describes the process of a loop transfer, the new knitted fabric commencing on empty needles and finished fabric press-off. Furthermore, consideration has been given to the operation of a multitape controlling automaton that would be integrated in the design of a knitting machine. An algorithm for output symbol determination has been developed that provides independent automaton control of the number of cam layers, the identification of cam track, and the number of needles in use. The software for minimization of the disjunctive normal form of Boolean functions, based on the methods of Quine and Gorbatov, has been developed. The possibility of modelling knitting machine malfunctions is described, enabling deviations from the correct process, such as hook or latch damage, to be studied.

The problem of constructing reduced disjunctive normal forms of Boolean functions from the point of view of lattice theory

The problem of constructing reduced disjunctive normal forms of Boolean functions from the point of view of lattice theory

Disjunctive normal forms of Boolean functions with a small number of zeros

An algorithm is proposed for constructing reduced disjunctive normal forms (DNF) of Boolean functions given with zero sets wthat is most effective for a small number of zeros. Sufficiently accurate constructive upper estimates are obtained for the length of the shortest DNFs and the complexity of the minimum DNFs of almost all Boolean functions with a small number of zeros.

On lower bounds for the complexity of disjunctive normal forms of Boolean functions with a small number of zeros

A method is given for constructing efficient lower bounds for the complexity of disjunctive normal forms of Boolean functions with a fixed number of zeros. The method reduces to solving problems of integervalued linear programming of a special kind. Examples are given.

An algorithm for constructing a disjunctive normal form equivalent to the product of the left-hand sides of Nelson Boolean equations

An algorithm is given for constructing disjunctive normal forms of Boolean functions, specified by enumeration of zero sets. The complexity of the construction is estimated. Examples are given of using the algorithm to construct the shortest disjunctive normal form for some special classes of functions.

Evaluations asymptotiques du nombre des cliques des hypergraphes uniformes

In this paper it is shown that, whenn→∞, for almost all (h+1)-hypergraphs withn nodes every clique contains at most $$\left\{ {((h + 1)!\log n)^{{1 \mathord{\left/ {\vphantom {1 h}} \right. \kern-\nulldelimiterspace} h}} + \frac{h}{2}} \right\}$$ nodes. By using the method developed by V. V. Glagolev for some asymptotical estimations of disjunctive normal forms of Boolean functions, which implies the use of Tchebychev's inequality, method applied to the study of graphs (h=1) by Phan Dinh Dieu, it is proved that whenn→∞ for almost all (h+1)-hypergraphs withn nodes (h≥2), the number of cliques (or of maximal cliques) is of degreen (h/(h+1))(h!logn)1/h. This fact implies that every enumerative algorithm for determining the maximum cardinality of a clique of a hypergraph cannot work in polynomial time.

Local algorithms simplifying the disjunctive normal forms of Boolean functions

A BRIEF exposition of the theory of local algorithms for simplifying the disjunctive normal forms of Boolean functions is presented. A proof of the essential nature of all the conditions used in the construction of one of the algorithms is presented, and its majorantness is proved.

A new normal form of Boolean functions

A new normal form of Boolean functions based on the sum (mod 2), product and negation is presented. Let n = {1, 2,…, n}, let A s be the family of s-element subsets of a set A and let π aϵφ x a = 1. Then every Boolean function ƒ( x 1, x 2,…, x n ) has a normal form ƒ(x 1,x 2,…,x n= ∑ s=0 n ⊕ Π A∈n s 1⊕d A Π a∈A x a with unique coefficients d A ϵ {0, 1}. A transformation of Galois normal form into the present normal form is also shown.