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Rewriting Logic as a Logical and Semantic Framework
, 1993
"... Rewriting logic [72] is proposed as a logical framework in which other logics can be represented, and as a semantic framework for the specification of languages and systems. Using concepts from the theory of general logics [70], representations of an object logic L in a framework logic F are und ..."
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Cited by 147 (52 self)
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Rewriting logic [72] is proposed as a logical framework in which other logics can be represented, and as a semantic framework for the specification of languages and systems. Using concepts from the theory of general logics [70], representations of an object logic L in a framework logic F are understood as mappings L ! F that translate one logic into the other in a conservative way. The ease with which such maps can be defined for a number of quite different logics of interest, including equational logic, Horn logic with equality, linear logic, logics with quantifiers, and any sequent calculus presentation of a logic for a very general notion of "sequent," is discussed in detail. Using the fact that rewriting logic is reflective, it is often possible to reify inside rewriting logic itself a representation map L ! RWLogic for the finitely presentable theories of L. Such a reification takes the form of a map between the abstract data types representing the finitary theories of...
Theorem Proving with Ordering and Equality Constrained Clauses
 Journal of Symbolic Computation
, 1995
"... constraint strategies and saturation Given a signature F , below we denote by S the set of all clauses built over F , and similarly by C the set of all constraints, and by EC the set of all equality constraints (which is a subset of C). Definition 3.1. An inference rule IR is a mapping of ntuples ..."
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Cited by 74 (19 self)
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constraint strategies and saturation Given a signature F , below we denote by S the set of all clauses built over F , and similarly by C the set of all constraints, and by EC the set of all equality constraints (which is a subset of C). Definition 3.1. An inference rule IR is a mapping of ntuples of clauses to sets of triples containing a clause, a constraint and an equality constraint: IR : S n \Gamma! P(hS; C; ECi) An inference system is a set of inference rules. Definition 3.2. A constraint inheritance strategy is a function mapping a clause, two constraints and an equality constraint to a clause and a constraint: H : S \Theta C \Theta C \Theta EC \Gamma! S \Theta C Inference systems and constraint inheritance strategies are combined to produce inferences in the usual sense: given constrained clauses C 1 [[T 1 ]]; : : : ; Cn [[T n ]], we obtain a conclusion C [[T ]] as follows. Applying an inference rule to C 1 ; : : : ; Cn we obtain a triple hD; OT;ET i. Then the constraint...
Basic Paramodulation
 Information and Computation
, 1995
"... We introduce a class of restrictions for the ordered paramodulation and superposition calculi (inspired by the basic strategy for narrowing), in which paramodulation inferences are forbidden at terms introduced by substitutions from previous inference steps. In addition we introduce restrictions bas ..."
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Cited by 68 (11 self)
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We introduce a class of restrictions for the ordered paramodulation and superposition calculi (inspired by the basic strategy for narrowing), in which paramodulation inferences are forbidden at terms introduced by substitutions from previous inference steps. In addition we introduce restrictions based on term selection rules and redex orderings, which are general criteria for delimiting the terms which are available for inferences. These refinements are compatible with standard ordering restrictions and are complete without paramodulation into variables or using functional reflexivity axioms. We prove refutational completeness in the context of deletion rules, such as simplification by rewriting (demodulation) and subsumption, and of techniques for eliminating redundant inferences.
Simple LPO constraint solving methods
 Information Processing Letters
, 1993
"... We present simple techniques for deciding the satisfiability of lexicographic path ordering constraints under two different semantics: solutions built over the given signature and solutions in extended signatures. For both cases we give the first NP algorithms, which is optimal as we prove the probl ..."
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Cited by 36 (11 self)
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We present simple techniques for deciding the satisfiability of lexicographic path ordering constraints under two different semantics: solutions built over the given signature and solutions in extended signatures. For both cases we give the first NP algorithms, which is optimal as we prove the problems to be NPcomplete. We discuss the efficient applicability of the techniques in practice, where, as far as we know, their simply exponential bound improves upon the existing methods, and describe some optimizations. Keywords: Automatic theorem proving. 1 Terminology Let F and X be sets of function symbols and variables respectively, and let ØF be a total ordering on F (the precedence). We sometimes write pairs (F ; ØF ). The lexicographic path ordering (LPO) generated by ØF , denoted Ø F lpo , is a total simplification ordering on T (F). It is defined as follows: s = f(s 1 ; : : : ; s m ) Ø F lpo g(t 1 ; : : : ; t n ) = t if 1. s i F lpo t, for some i with 1 i m or 2. f ØF g...
ACsuperposition with constraints: No ACunifiers needed
 Proceedings 12th International Conference on Automated Deduction
, 1990
"... We prove the completeness of (basic) deduction strategies with constrained clauses modulo associativity and commutativity (AC). Here each inference generates one single conclusion with an additional equality s = AC t in its constraint (instead of one conclusion for each minimal ACunifier, i.e. expo ..."
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Cited by 29 (5 self)
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We prove the completeness of (basic) deduction strategies with constrained clauses modulo associativity and commutativity (AC). Here each inference generates one single conclusion with an additional equality s = AC t in its constraint (instead of one conclusion for each minimal ACunifier, i.e. exponentially many). Furthermore, computing ACunifiers is not needed at all. A clause C [[ T ]] is redundant if the constraint T is not ACunifiable. If C is the empty clause this has to be decided to know whether C [[ T ]] denotes an inconsistency. In all other cases any sound method to detect unsatisfiable constraints can be used. 1 Introduction Some fundamental ideas on applying symbolic constraints to theorem proving were given in [KKR90], where a constrained clause is a shorthand for its (infinite) set of ground instances satisfying the constraint. In a constrained equation f(x) ' a [[ x = g(y) ]], the equality `=' of the constraint is usually interpreted in T (F) (syntactic equality), ...
Ordering Constraints on Trees
 Colloquium on Trees in Algebra and Programming
, 1994
"... . We survey recent results about ordering constraints on trees and discuss their applications. Our main interest lies in the family of recursive path orderings which enjoy the properties of being total, wellfounded and compatible with the tree constructors. The paper includes some new results, in p ..."
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Cited by 21 (1 self)
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. We survey recent results about ordering constraints on trees and discuss their applications. Our main interest lies in the family of recursive path orderings which enjoy the properties of being total, wellfounded and compatible with the tree constructors. The paper includes some new results, in particular the undecidability of the theory of lexicographic path orderings in case of a nonunary signature. 1 Symbolic Constraints Constraints on trees are becoming popular in automated theorem proving, logic programming and in other fields thanks to their potential to represent large or even infinite sets of formulae in a nice and compact way. More precisely, a symbolic constraint system, also called a constraint system on trees, consists of a fragment of firstorder logic over a set of predicate symbols P and a set of function symbols F , together with a fixed interpretation of the predicate symbols in the algebra of finite trees T (F) (or sometimes the algebra of infinite trees I(F)) ov...
The Saturate System
, 1998
"... The Saturate system is an experimental theorem prover for firstorder logic, primarily based on saturation. Saturate uses techniques of ordered chaining for arbitrary transitive relations, including orderings, equivalence relations and congruences, and integrates CNF transformation lazily into satur ..."
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Cited by 18 (11 self)
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The Saturate system is an experimental theorem prover for firstorder logic, primarily based on saturation. Saturate uses techniques of ordered chaining for arbitrary transitive relations, including orderings, equivalence relations and congruences, and integrates CNF transformation lazily into saturation.
Saturation of FirstOrder (Constrained) Clauses With The Saturate System
 REWRITING TECHNIQUES AND APPLICATIONS, 5TH INTERNATIONAL CONFERENCE, RTA93, VOLUME 690 OF LECTURE NOTES IN COMPUTER SCIENCE
, 1993
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