Solution of systems of Boolean equations via the integer domain

Abdel-Gawad, Ahmed H.; Atiya, Amir F.; Darwish, Nevin M.

doi:10.1016/j.ins.2009.09.010

Cited by 16 publications

(9 citation statements)

References 19 publications

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“…Few packages for minimizing representations of Boolean functions use a Karnaugh-map like strategy [27], but most modern packages use algebraic techniques based on divide-and-conquer strategies like the unate recursive paradigm [5]. Useful discussions on computational techniques to solve Boolean and switching equations are available in [1,10].…”

Section: Discussionmentioning

confidence: 99%

Using variable-entered Karnaugh maps to produce compact parametric general solutions of Boolean equations

Rushdi

Amashah

2011

International Journal of Computer Mathematics

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The variable-entered Karnaugh map (VEKM) is shown to be the natural map for representing and manipulating general 'big' Boolean functions that are not restricted to the switching or two-valued case. The VEKM is utilized herein in producing a compact general solution of a system of Boolean equations. It serves as a powerful manual tool for function inversion, implementation of the solution procedure, handling don't-care conditions and minimization of the final expressions. The rules of using the VEKM are semi-algebraic and collective in nature, and hence are much easier to state, remember and implement than are the tabular and per-cell rules of classical maps. As a result, the maps used are significantly smaller than those required by classical methods. As an offshoot, the paper contributes some pictorial insight into the representation of 'big' Boolean algebras and functions. It also predicts the correct number of particular solutions of a Boolean equation, and produces an exhaustive list of particular solutions. Details of the method are carefully explained and further demonstrated via an illustrative example.

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Section: Discussionmentioning

confidence: 99%

Using variable-entered Karnaugh maps to produce compact parametric general solutions of Boolean equations

Rushdi

Amashah

2011

International Journal of Computer Mathematics

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“…Solving a system of logical equations has many applications, such as synthesis, output data coding, the state assignment of finite automata, the modeling and testing of digital networks, automatic test pattern generation and the determination of the initial state in circuits, timing analysis, and the generation of delayed failure tests for combinational circuits. The solution of a system of logical equations in the field of cryptography is used to analyze and break block ciphers, since they can be reduced to the problem of solving large-scale systems of logical equations [1][2][3][4][5][6]. This is because, for a specific cipher, algebraic cryptanalysis consists of two stages: the transformation of the cipher into a system of polynomial equations (usually over Boolean ring), and the solution of the resulting system of polynomial equations [7][8][9].…”

Section: Introductionmentioning

confidence: 99%

“…The real continuous domain is a richer domain to work with, as it involves many, well developed algorithms. Already in a real continuous domain, the transformed system can be reduced to a numerical optimization problem, making it possible to apply, analyze and combine such methods as the steepest descent algorithm, Newton's method and the coordinate descent algorithm [1][2][3][4][5][6]14].…”

Section: Introductionmentioning

confidence: 99%

Polylinear Transformation Method for Solving Systems of Logical Equations

Barotov

Баротов

2022

Mathematics

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In connection with applications, the solution of a system of logical equations plays an important role in computational mathematics and in many other areas. As a result, many new directions and algorithms for solving systems of logical equations are being developed. One of these directions is transformation into the real continuous domain. The real continuous domain is a richer domain to work with because it features many algorithms, which are well designed. In this study, firstly, we transformed any system of logical equations in the unit -dimensional cube into a system of polylinear–polynomial equations in a mathematically constructive way. Secondly, we proved that if we slightly modify the system of logical equations, namely, add no more than one special equation to the system, then the resulting system of logical equations and the corresponding system of polylinear–polynomial equations in is equivalent. The paper proposes an algorithm and proves its correctness. Based on these results, further research plans are developed to adapt the proposed method.

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“…A recent paper [1] in this journal by Abdel-Gawad, Atiya, and Darwish (whom we shall call AAD) presents a method of solving Boolean equations using an integer polynomial algebra. The authors write, ''Several approaches have been proposed in the literature for the problem of solving simultaneous Boolean equations.…”

Section: Introductionmentioning

confidence: 99%

“…Boole's solution-procedure, and most contemporary algorithms, begin by reducing a system of equations (Boolean or proto-Boolean) to an equivalent single equation. E-mail address: FMBrown@1953.USNA.com 1 If m (the number of equations) is greater than n (the number of variables), AAD propose to ''select any n equations and apply the proposed methodology" after which the generated solutions are checked against the remaining m À n equations. If m < n, ''we apply the proposed methodology to eliminate as many variables as possible".…”

Section: Introductionmentioning

confidence: 99%

Comments on a numerical method for solving Boolean equations

Brown

2011

Information Sciences

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Solution of systems of Boolean equations via the integer domain

Cited by 16 publications

References 19 publications

Using variable-entered Karnaugh maps to produce compact parametric general solutions of Boolean equations

Using variable-entered Karnaugh maps to produce compact parametric general solutions of Boolean equations

Polylinear Transformation Method for Solving Systems of Logical Equations

Comments on a numerical method for solving Boolean equations

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