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Automatic Recognition of Tractability in Inference Relations
 Journal of the ACM
, 1990
"... This report describes research done at the Artificial Intelligence Laboratory of the Massachusetts Institute of Technology. Support for the laboratory's artificial intelligence research is provided in part the National Science Foundation contract IRI8819624 and in part by the Advanced Research ..."
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Cited by 70 (14 self)
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This report describes research done at the Artificial Intelligence Laboratory of the Massachusetts Institute of Technology. Support for the laboratory's artificial intelligence research is provided in part the National Science Foundation contract IRI8819624 and in part by the Advanced Research Projects Agency of the Department of Defense under Office of Naval Research contract N0001486K0124
Efficient Ematching for SMT solvers
, 2007
"... Satisfiability Modulo Theories (SMT) solvers have proven highly scalable, efficient and suitable for integrating theory reasoning. However, for numerous applications from program analysis and verification, the ground fragment is insufficient, as proof obligations often include quantifiers. A well ..."
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Cited by 59 (10 self)
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Satisfiability Modulo Theories (SMT) solvers have proven highly scalable, efficient and suitable for integrating theory reasoning. However, for numerous applications from program analysis and verification, the ground fragment is insufficient, as proof obligations often include quantifiers. A well known approach for quantifier reasoning uses a matching algorithm that works against an Egraph to instantiate quantified variables. This paper introduces algorithms that identify matches on Egraphs incrementally and efficiently. In particular, we introduce an index that works on Egraphs, called Ematching code trees that combine features of substitution and code trees, used in saturation based theorem provers. Ematching code trees allow performing matching against several patterns simultaneously. The code trees are combined with an additional index, called the inverted path index, which filters Egraph terms that may potentially match patterns when the Egraph is updated. Experimental results show substantial performance improvements over existing stateoftheart SMT solvers.
Taxonomic Syntax for First Order Inference
 Journal of the ACM
, 1989
"... Most knowledge representation languages are based on classes and taxonomic relationships between classes. Taxonomic hierarchies without defaults or exceptions are semantically equivalent to a collection of formulas in first or der predicate calculus. Although designers of knowledge representation l ..."
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Cited by 39 (15 self)
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Most knowledge representation languages are based on classes and taxonomic relationships between classes. Taxonomic hierarchies without defaults or exceptions are semantically equivalent to a collection of formulas in first or der predicate calculus. Although designers of knowledge representation lan guages often express an intuitive feeling that there must be some advantage to representing facts as taxonomic relationships rather than first order for mulas, there are few,, if any, technical results supporting this intuition. We attempt to remedy this situation by presenting a taxonomic syntax for first order predicate calculus and a series of theorems that support the claim that taxonomic syntax is superior to classical syntax.
Theorem proving using equational matings and rigid E–unification
 Journal of the ACM
"... In this paper, it is shown that the method of matings due to Andrews and Bibel can be extended to (firstorder) languages with equality. A decidable version of Eunification called rigid Eunification is introduced, and it is shown that the method of equational matings remains complete when used in ..."
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Cited by 38 (1 self)
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In this paper, it is shown that the method of matings due to Andrews and Bibel can be extended to (firstorder) languages with equality. A decidable version of Eunification called rigid Eunification is introduced, and it is shown that the method of equational matings remains complete when used in conjunction with rigid Eunification. Checking that a family of mated sets is an equational mating is equivalent to the following restricted kind of Eunification. Problem: Given →/E = {Ei  1 ≤ i ≤ n} a family of n finite sets of equations and S = {〈ui, vi 〉  1 ≤ i ≤ n} a set of n pairs of terms, is there a substitution θ such that, treating each set θ(Ei) as a set of ground equations (i.e. holding the variables in θ(Ei) "rigid"), θ(ui) and θ(vi) are provably equal from θ(Ei) for i = 1,...,n? Equivalently, is there a substitution θ such that θ(ui) and θ(vi) can be shown congruent from θ(Ei) by the congruence closure method for i 1,..., n? A substitution θ solving the above problem is called a rigid →/Eunifier of S, and a pair (→/E, S) such that S has some rigid →/Eunifier is called an equational premating. It is shown that deciding whether a pair 〈→/E, S 〉 is an
Shostak's Congruence Closure as Completion
 Proceedings of the 8th International Conference on Rewriting Techniques and Applications, volume 1232 of Lecture Notes in Computer Science
, 1997
"... . Shostak's congruence closure algorithm is demystified, using the framework of ground completion on (possibly nonterminating, nonreduced) rewrite rules. In particular, the canonical rewriting relation induced by the algorithm on ground terms by a given set of ground equations is precisely ..."
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Cited by 35 (3 self)
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. Shostak's congruence closure algorithm is demystified, using the framework of ground completion on (possibly nonterminating, nonreduced) rewrite rules. In particular, the canonical rewriting relation induced by the algorithm on ground terms by a given set of ground equations is precisely constructed. The main idea is to extend the signature of the original input to include new constant symbols for nonconstant subterms appearing in the input. A byproduct of this approach is (i) an algorithm for associating a confluent rewriting system with possibly nonterminating ground rewrite rules, and (ii) a new quadratic algorithm for computing a canonical rewriting system from ground equations. 1 Introduction Equality reasoning has been found critical in many applications including compiler optimization, functional languages, and reasoning about data bases, most importantly, reasoning about different aspects of software and hardware  circuits, programs and specifications. Signific...
Finite Representation of Infinite Query Answers
, 1992
"... : We define here a formal notion of finite representation of infinite query answers in logic programs. We apply this notion to Datalog nS (Datalog with n successors): an extension of Datalog capable of representing infinite phenomena like flow of time or plan construction. Predicates in Datalog nS ..."
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Cited by 33 (5 self)
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: We define here a formal notion of finite representation of infinite query answers in logic programs. We apply this notion to Datalog nS (Datalog with n successors): an extension of Datalog capable of representing infinite phenomena like flow of time or plan construction. Predicates in Datalog nS can have arbitrary unary and limited nary function symbols in one fixed position. This class of logic programs is known to be decidable. However, least Herbrand models of Datalog nS programs may be infinite and consequently queries may have infinite answers. We present a method to finitely represent infinite least Herbrand models of Datalog nS programs as relational specifications. A relational specification consists of a finite set of facts and a finitely specified congruence relation. A relational specification has the following desirable properties. First, it is explicit in the sense that once it is computed, the original Datalog nS program (and its underlying computational engine) can ...
New Results on Local Inference Relations
 In Principles of Knolwedge Representation and Reasoning: Proceedings of the Third International Conference
, 1992
"... We consider the concept of a local set of inference rules. A local rule set can be automatically transformed into a rule set for which bottom up evaluation terminates in polynomial time. The local rule set transformation gives polynomial time evaluation strategies for a large variety of rule sets th ..."
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Cited by 32 (9 self)
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We consider the concept of a local set of inference rules. A local rule set can be automatically transformed into a rule set for which bottom up evaluation terminates in polynomial time. The local rule set transformation gives polynomial time evaluation strategies for a large variety of rule sets that can not be given terminating evaluation strategies by any other known automatic technique. This paper discusses three new results. First, it is shown that every polynomial time predicate can be defined by an (unstratified) local rule set. Second, a new machine recognizable subclass of the local rule sets is identified. Finally we show that locality, as a property of rule sets, is undecidable in general. This paper appeared in KR92. A postscript electronic source for this paper can be found in ftp.ai.mit.edu:/pub/dam/kr92.ps. A bibtex reference can be found in internet file ftp.ai.mit.edu:/pub/dam/dam.bib. 1 INTRODUCTION Under what conditions does a given set of inference rules define ...
Relating Semantic and ProofTheoretic Concepts for Polynomial Time Decidability of Uniform Word Problems
 In Proceedings 16th IEEE Symposium on Logic in Computer Science, LICS'2001
, 2001
"... In this paper we compare three approaches to polynomial time decidability for uniform word problems for quasivarieties. Two of the approaches, by Evans and Burris, respectively, are semantical, referring to certain embeddability and axiomatizability properties. The third approach is more prooftheor ..."
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Cited by 26 (2 self)
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In this paper we compare three approaches to polynomial time decidability for uniform word problems for quasivarieties. Two of the approaches, by Evans and Burris, respectively, are semantical, referring to certain embeddability and axiomatizability properties. The third approach is more prooftheoretic in nature, inspired by McAllester's concept of local inference. We define two closely related notions of locality for equational Horn theories and show that both the criteria by Evans and Burris lie in between these two concepts. In particular, the variant we call stable locality will be shown to subsume both Evans' and Burris' method.
Combining Shostak Theories
, 2002
"... Ground decision procedures for combinations of theories are used in many systems for automated deduction. There are two basic paradigms for combining decision procedures. The NelsonOppen method combines decision procedures for disjoint theories by exchanging equality information on the shared vari ..."
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Cited by 24 (3 self)
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Ground decision procedures for combinations of theories are used in many systems for automated deduction. There are two basic paradigms for combining decision procedures. The NelsonOppen method combines decision procedures for disjoint theories by exchanging equality information on the shared variables. In Shostak's method, the combination of the theory of pure equality with canonizable and solvable theories is decided through an extension of congruence closure that yields a canonizer for the combined theory. Shostak's original presentation, and others that followed it, contained serious errors which were corrected for the basic procedure by the present authors. Shostak also claimed that it was possible to combine canonizers and solvers for disjoint theories. This claim is easily verifiable for canonizers, but is unsubstantiated in the case of solvers. We show how our earlier procedure can be extended to combine multiple disjoint canonizable, solvable theories within the Shostak framework.
Polynomialtime Computation via Local Inference Relations
 ACM TRANS. COMPUT. LOGIC
, 2000
"... We consider the concept of a local set of inference rules. A local rule set can be automatically transformed into a rule set for which bottomup evaluation terminates in polynomial time. The localruleset transformation gives polynomialtime evaluation strategies for a large variety of rule sets th ..."
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Cited by 23 (0 self)
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We consider the concept of a local set of inference rules. A local rule set can be automatically transformed into a rule set for which bottomup evaluation terminates in polynomial time. The localruleset transformation gives polynomialtime evaluation strategies for a large variety of rule sets that cannot be given terminating evaluation strategies by any other known automatic technique. This paper discusses three new results. First, it is shown that every polynomialtime predicate can be defined by an (unstratified) local rule set. Second, a new machinerecognizable subclass of the local rule sets is identified. Finally we show that locality, as a property of rule sets, is undecidable in general.