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 IRI-8819624 and in part by the Advanced Research Projects Agency of the Department of Defense under Office of Naval Research contract N00014-86-K-0124

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.

. Shostak's congruence closure algorithm is demystified, using the framework of ground completion on (possibly nonterminating, non-reduced) 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...

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 E-graph to instantiate quantified variables. This paper introduces algorithms that identify matches on E-graphs incrementally and efficiently. In particular, we introduce an index that works on E-graphs, called E-matching code trees that combine features of substitution and code trees, used in saturation based theorem provers. E-matching 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 E-graph terms that may potentially match patterns when the E-graph is updated. Experimental results show substantial performance improvements over existing state-of-the-art SMT solvers.

: 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 n-ary 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 ...

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 KR-92. 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 ...

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 Nelson--Oppen 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.

We have argued elsewhere that first order inference can be made more efficient by using non-standard syntax for first order logic. In this paper we define a syntax for first order logic based on the structure of natural language under Montague semantics. We show that, for a certain fairly expressive fragment of this language, satisfiability is polynomial time decidable. The polynomial time decision procedure can be used as a subroutine in general purpose inference systems and seems to be more powerful than analogous procedures based on either classical or taxonomic syntax.

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 proof-theoretic 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.

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 cannot 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. Keywords: Descriptive Complexity Theory, Decision Procedures, Automated Reasoning, 1. Introduction Under what conditions does a given set of inference rules define a computationally tractable inference relation? This is a syntactic question about syntactic inference rules. There are a va...