Reduced Solutions of Boolean Equations

Brown, Frank Markham

doi:10.1109/t-c.1970.222806

Cited by 16 publications

(8 citation statements)

References 6 publications

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“…Also let β be the closest assignment to β that satisfies π J (w J ). Evaluating the RHS of (5) at β yields: k j=1 c[j]GCF(π j (f j ), π j (w j ), ) (β) = GCF(π J (f J ), π J (w J ), )(β) = π J (f J )(β ) (6) Using similar reasoning, we find that γ * (β) is such that for each v i ∈ V :…”

Section: Discussionmentioning

confidence: 99%

“…The idea of using parametric representations of boolean functions (what we term BFVs) goes back to the early 1970s [6], [9]. Such a representation, generated by the generalized co-factor operation (GCF) (a.k.a.…”

Section: Related Workmentioning

confidence: 99%

“…Logically 6 , this instruction takes two arrays s1 and s2, each having 8 entries, and each entry being a 16-bit word, and returns ind ∈ {0, . .…”

Section: Sse4 String Instructionmentioning

confidence: 99%

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Universal boolean functional vectors

Bingham

2015

2015 Formal Methods in Computer-Aided Design (FMCAD)

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Abstract-In the simplest setting, one represents a boolean function using expressions over variables, where each variable corresponds to a function input. So-called parametric representations, used to represent a function in some restricted subspace of its domain, break this correspondence by allowing inputs to be associated with functions. This can lead to more succinct representations, for example when using binary decision diagrams (BDDs). Here we introduce Universal Boolean Functional Vectors (UBFVs), which also break the correspondence, but done so such that all input vectors are accounted for. Intelligent choice of a UBFV can have a dramatic impact on BDD size; for instance we show how the hidden weighted bit function can be efficiently represented using UBFVs, whereas without UBFVs BDDs are known to be exponential for any variable order. We show several industrial examples where the UBFV approach has a huge impact on proof performance, the "killer app" being floating point addition, wherein the wide case-split used in the state-of-the-art approach is entirely done away with, resulting in 70-fold reduction in proof runtime. We give other theoretical and experimental results, and also provide two approaches to verifying the crucial "universality" aspect of a proposed UBFV. Finally, we suggest several interesting avenues of future research stemming from this program.

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Section: Discussionmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

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Universal boolean functional vectors

Bingham

2015

2015 Formal Methods in Computer-Aided Design (FMCAD)

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“…When each appearance of an atom is tagged by a particular member of its orthonormal set of tags, an auxiliary function ( , )(0 ≤ ≤ ( − 1), ≠ 0) results. The parametric solution is now given by [1,[18][19][20][21][22][23].…”

Section: ( ) ≡ ∨ ∈{ } ( )mentioning

confidence: 99%

A Novel Method for Compact Listing of All Particular Solutions of a System of Boolean Equations

Rushdi¹,

Ahmad²

2017

BJMCS

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“…These methods could be cast in pure algebraic form [24][25][26][27] , but become much easier to visualize and comprehend when presented via the natural map of a big Boolean algebra, which (for historical reasons) is called the variable-entered Karnaugh map (VEKM) [23,[28][29][30][31][32][33][34][35][36][37] . In the classical method, the number of parameters used is minimized and compact solutions are obtained [22,23,30,[38][39][40][41] . However, the parameters belong to the underlying big Boolean algebra.…”

Section: Introductionmentioning

confidence: 99%

Satisfiability in Big Boolean Algebras via Boolean-Equation Solving

Rushdi¹

2017

JKAU Earth Sci

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This paper studies Satisfiability (SAT) in finite atomic Boolean algebras larger than the two-valued one B2, which are named big Boolean algebras. Unlike the formula ݃(ࢄ (in the SAT problem over B2, which is either satisfiable or unsatisfiable, this formula for the SAT problem over a big Boolean algebra could be unconditionally satisfiable, conditionally satisfiable, or unsatisfiable depending on the nature of the consistency condition of the Boolean equation {݃(ࢄ = (1}, since this condition could be an identity, a genuine equation, or a contradiction. The paper handles this latter SAT problem by using a conventional method and a novel one for deriving parametric general solutions, and subsequently utilizing expansion trees for generating all particular solutions of the aforementioned Boolean equation. Each of these two methods could be cast in pure algebraic form, but becomes much easier to visualize and comprehend when presented via the natural map of a big Boolean algebra, which (for historical reasons) is called the variable-entered Karnaugh map (VEKM). In the classical method, the number of parameters used is minimized and compact solutions are obtained. However, the parameters belong to the underlying big Boolean algebra. By contrast, the novel method does not attempt to minimize the number of parameters used, as it uses independent parameters belonging to the two-valued Boolean algebra B2 for each asserted atom in the Boole-Shannon expansion of the formula ݃(ࢄ .(Though the method produces non-compact expressions, it is much quicker in generating particular solutions. The two methods are demonstrated via two detailed examples.

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Reduced Solutions of Boolean Equations

Cited by 16 publications

References 6 publications

Universal boolean functional vectors

Universal boolean functional vectors

A Novel Method for Compact Listing of All Particular Solutions of a System of Boolean Equations

Satisfiability in Big Boolean Algebras via Boolean-Equation Solving

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