Solution of Boolean Equations

Brown, Frank Markham

doi:10.1007/978-1-4757-2078-5_6

Cited by 27 publications

(97 citation statements)

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“…A Boolean algebra is a quintuple B = (B, ∨, ∧, 0, 1) in which B is a set, called the carrier; ∨ and ∧ are binary operations on B and the zero (0) and unit (1) elements are distinct members of B, with certain postulates on commutativity, distributivity, identities and complementation being satisfied [6,7,11,15,18]. The following facts about a Boolean algebra can be deduced:…”

Section: Representations Of Boolean Algebras and Functionsmentioning

confidence: 99%

“…Each element of FB(Y) is a disjunction of a subset of the 2 k minterms built from Y 1 , Y 2 , …, Y k . In particular, the null disjunction is the 0-element of FB(Y), and the disjunction of all the minterms is its 1-element [6,7]. The free Boolean algebra FB(Y) is isomorphic to the Boolean algebra of switching functions of k variables, and has 2 2 k elements [6,7].…”

Section: Representations Of Boolean Algebras and Functionsmentioning

confidence: 99%

“…Boolean-equation solving permeates many areas of modern science such as logical design, biology, grammars, chemistry, law, medicine, spectroscopy and graph theory [6,7]. Many important problems in operations research can be reduced to the problem of solving a system of Boolean equations.…”

Section: Introductionmentioning

confidence: 99%

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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: Representations Of Boolean Algebras and Functionsmentioning

confidence: 99%

Section: Representations Of Boolean Algebras and Functionsmentioning

confidence: 99%

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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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“…In other words, an upper-case variable refers to a proposition that is evaluated along the whole run, while a lower-case one refers only to the final (terminal) marking. Under this explanations, the formal definitions of syntax and semantics of used specifications is not included, and it can be found in any study including Boolean formulae [18]. From now on, we will assume that any Boolean-based requirement ϕ is expressed into a Conjunctive Normal Form (CNF), the conversion into such a form being possible for any logical expression [18].…”

Section: B Boolean-based Specificationsmentioning

confidence: 99%

“…2 guarantees that the team can produce the sequence of outputs fromr, although some outputs are repeated in Q. This repetitions do not affect the satisfiability of Boolean-based ϕ [18].…”

Section: Suboptimal Solutionmentioning

confidence: 99%

Planning mobile robots with Boolean-based specifications

Mahulea

Kloetzer

2014

53rd IEEE Conference on Decision and Control

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Abstract-This research proposes an automated method for planning a team of mobile robots such that a Boolean-based mission is accomplished. The specification consists of logical requirements over some regions of interest for the agents' trajectories and for their final states. A Petri net with outputs models the movement capabilities of the team and the active regions of interest. The imposed specification is translated to a set of linear restrictions for some binary variables, the robot movement capabilities are formulated as linear constraints on Petri net markings, and the evaluations of the binary variables are linked with Petri net markings via linear inequalities. This allows us to solve a Mixed Integer Linear Programming problem whose solution yields robotic trajectories satisfying the task. The method is implemented as a software package and simulation results are included.

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Implication and Inhibition Boolean Logic Gates Mimicked with Enzyme Reactions

2020

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The enzyme system mimicking Implication (IMPLY) and Inhibition (INHIB) Boolean logic gates has been designed. The same enzyme system was used to operate as the IMPLY or INHIB gate simply by reformulating the input signals. The optical analysis of the logic operation confirmed the output generation as expected for the studied logic gates. The conceptual approach to the IMPLY and INHIB logic gates allows their construction with many other enzymes operating in a similar way. Molecular [1-4] and biomolecular [5] computing, as a subarea of unconventional computing, [6] have attracted high attention and rapidly progressed in the last two decades. Enzyme-based logic systems, [7] together with DNA/RNA computing systems, [8-10] are the most important areas of research in the general framework of the biomolecular information processing systems. Enzymecatalyzed reactions, including sophisticated multi-step/multienzyme reaction cascades, have been used to mimic almost all known Boolean logic gates, [7,11] such as Yes (Identity), Not (Inversion), OR, NOR, XOR, NXOR, AND, NAND, INHIB, including reversible logic gates, [12,13] such as Feynman, Toffoli, Fredkin and Peres gates. While the vast majority of Boolean logic gates mimicked with enzyme systems have been already formulated, [7,11] optimized [14-17] and reported for various applications, [18-20] some unique logic operations still require additional studies. Particularly, Implication (IMPLY) logic gate (also named "material implication") has not achieved enough attention. This gate was first described more than 100 years ago and recognized as a basic logic operation. [21] For some technical reasons this gate was ignored in digital electronics for many years, but recently interest to this gate has been returned based on its straightforward realization in memristive switches. [22] The IMPLY logic gate has been realized in many chemical systems, [23-36] including biomolecular systems based on DNA molecules, [37-49] particularly operated as cell biology processes. [50] However, the IMPLY gate was rarely demonstrated with enzyme-catalyzed reactions. [51] The present study aimed at the general approach to the IMPLY logic gate. This approach allows the use of various

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Solution of Boolean Equations

Cited by 27 publications

References 0 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

Planning mobile robots with Boolean-based specifications

Implication and Inhibition Boolean Logic Gates Mimicked with Enzyme Reactions

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