Linear-time algorithms for testing the satisfiability of propositional horn formulae

Dowling, William F.; Gallier, Jean

doi:10.1016/0743-1066(84)90014-1

Cited by 779 publications

(355 citation statements)

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“…By the theorem in Lankford and Ballantyne, 1983 for uniqueness of canonical associative-commutative rewriting systems, it results in a unique set of equivalences, determined by the ordering . The resultant system can be used to decide satis ability in the given propositional Horn theory, though not as fast as in Dowling and Gallier, 1984 . The equivalences can optionally be converted back to Horn form.…”

Section: Horn Theoriesmentioning

confidence: 99%

Canonical sets of horn clauses

Dershowitz

1991

Automata, Languages and Programming

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Rewrite rules are oriented equations used to replace equals-by-equals in the speci ed direction. Input terms are repeatedly rewritten according to the rules. When and if no rule applies, the resultant normal form is considered the value of the initial term. If no in nite sequences of rewrites is possible, a rewrite system is said to have t h e termination property. Con uence of a rewrite system is a property that ensures that no term has more than one normal form. A convergent rewrite system is one with both the con uence and termination properties.Let T be a set of rst-order terms, with variables taken from a set X , and G be its subset of ground variable-free terms. If t is a term in T , b y tj we signify the subterm of t rooted at position and by t s or simply t s we denote t with its subterm tj replaced by a term s. W e use the following notations for equational deduction: s ' t stands for the usual sense of equality in logical systems; s $ e t or just s $ t denotes one step of replacement of equals for equals using equation e ; s ! R t or just s ! t stands for one replacement according to the orientation of a rewrite rule in R ; s ! t, f o r a n y n umber including zero of rewrites; s $ t also stands for one rewrite step in either direction. Two terms s and t are said to be joinable if there is a term v such that s ! v t, o r s t for short. For convergent R, an identity s ' t holds in the theory de ned by R each rule viewed as an equation if and only if the normal forms of s and t are identical. Thus, validity of equations is decidable for nite convergent R, since the joinability relation is decidable. A ground convergent rewrite system is one that terminates and de nes unique normal forms for all ground terms. Ground convergent systems can be used to decide validity b y s k olemizing s and t and reducing to normal form. For a survey of the theory of rewriting, see Dershowitz and Jouannaud, 1990 .A conditional equation is a universally-quanti ed de nite Horn clause in which the only predicate symbol is equality. W e write such a clause in the form e 1^ên s ' t n 0 , meaning that equality s ' t holds whenever all the antecedent equations e i , hold. The term s will be called the left-hand side; t is the right-hand side; and the e i are the conditions. I f n = 0 , then the positive unit clause will be called an unconditional equation. Conditional equations are important for specifying abstract data types and expressing logic programs with equations. A conditional rewrite rule is an equational implication in which the equation in the consequent s ' t is oriented. To give operational semantics to such a system, the conditions under which a rewrite may be performed need to made precise. The most popular convention see Dershowitz and Okada, 1990 for conditional rewriting is that the terms in each condition be

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Section: Horn Theoriesmentioning

confidence: 99%

Canonical sets of horn clauses

Dershowitz

1991

Automata, Languages and Programming

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“…However, the notion of causality is generally lost when translating from rules to Horn sets. Horn sets can be solved in linear time using unit resolution [33,58,91] (see algorithm UR, Figure 3). An important property of Horn sets is that a model, if one exists, which is minimal in the number of variables set to 1 is a unique minimum model with respect to 1 (this is referred to as the minimal model below).…”

Section: Definition 7 a Cnf Expression Is Horn If Every Clause It Conmentioning

confidence: 99%

“…Those results have been used, for example, to show how limited most succinctly defined polynomial-time classes of SAT are. Notable examples of such classes are Horn [33,58], renameable Horn [73], q-Horn [16,15], extended Horn [18], SLUR [89], balanced [25], and matched [46], to name a few. These classes have been studied partly in the belief that they will yield some distinction between hard and easy problems.…”

Section: Introductionmentioning

confidence: 99%

Probabilistic Analysis of Satisfiability Algorithms

Franco¹,

Crama²,

Hammer³

2010

Boolean Models and Methods in Mathematics, Computer Science, and Engineering

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“…In fact, Binomial-BRSAT defines the same Boolean functions as does Horn-SAT, which has linear complexity [13].…”

Section: Theorem 2 Binomial-brsat Is Linear-time Solvablementioning

confidence: 99%

Boolean Rings for Intersection-Based Satisfiability

Dershowitz

Hsiang

Huang

et al. 2006

Logic for Programming, Artificial Intelligence, and Reasoning

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There is not a person in this courtroomwho has never told a lie, who has never done an immoral thing, and there is no man living who has never looked upon a woman without desire. 1 -Harper Lee: To Kill a MockingbirdAbstract. A potential advantage of using a Boolean-ring formalism for propositional formulae is the large measure of simplification it facilitates. We propose a combined linear and binomial representation for Booleanring polynomials with which one can easily apply Gaussian elimination and Horn-clause methods to advantage. We demonstrate that this framework, with its enhanced simplification, is especially amenable to intersection-based learning, as in recursive learning and the method of Stålmarck. Experiments support the idea that problem variables can be eliminated and search trees can be shrunk by incorporating learning in the form of Boolean-ring saturation.

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Linear-time algorithms for testing the satisfiability of propositional horn formulae

Cited by 779 publications

References 1 publication

Canonical sets of horn clauses

Canonical sets of horn clauses

Probabilistic Analysis of Satisfiability Algorithms

Boolean Rings for Intersection-Based Satisfiability

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