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InductiveDataType Systems
, 2002
"... In a previous work ("Abstract Data Type Systems", TCS 173(2), 1997), the leI two authors presented a combined lmbined made of a (strongl normal3zG9 alrmal rewrite system and a typed #calA#Ik enriched by patternmatching definitions folnitio a certain format,calat the "General Schema", whichgenera ..."
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Cited by 755 (22 self)
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In a previous work ("Abstract Data Type Systems", TCS 173(2), 1997), the leI two authors presented a combined lmbined made of a (strongl normal3zG9 alrmal rewrite system and a typed #calA#Ik enriched by patternmatching definitions folnitio a certain format,calat the "General Schema", whichgeneral39I theusual recursor definitions fornatural numbers and simil9 "basic inductive types". This combined lmbined was shown to bestrongl normalIk39f The purpose of this paper is toreformul33 and extend theGeneral Schema in order to make it easil extensibl3 to capture a more general cler of inductive types, cals, "strictly positive", and to ease the strong normalgAg9Ik proof of theresulGGg system. Thisresul provides a computation model for the combination of anal"DAfGI specification language based on abstract data types and of astrongl typed functional language with strictly positive inductive types.
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...
Completion of Rewrite Systems with Membership Constraints Part II: Constraint Solving
 J. Symbolic Computation
, 1992
"... this paper is to show how to solve the constraints that are involved in the deduction mechanism of the first part. This may be interesting in its own since this provides with a unification algorithm for an ordersorted logic with context variables and can be read independently of the first part. Thi ..."
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Cited by 66 (2 self)
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this paper is to show how to solve the constraints that are involved in the deduction mechanism of the first part. This may be interesting in its own since this provides with a unification algorithm for an ordersorted logic with context variables and can be read independently of the first part. This can also be compared with unification of term schemes of various kind (Chen & Hsiang, 1991; Salzer, 1992; Comon, 1995; R. Galbav'y and M. Hermann, 1992). Indeed,
Disunification: a Survey
 Computational Logic: Essays in Honor of Alan
, 1991
"... Solving an equation in an algebra of terms is known as unification. Solving more complex formulas combining equations and involving in particular negation is called disunification. With such a broad definition, many works fall into the scope of disunification. The goal of this paper is to survey the ..."
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Cited by 57 (9 self)
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Solving an equation in an algebra of terms is known as unification. Solving more complex formulas combining equations and involving in particular negation is called disunification. With such a broad definition, many works fall into the scope of disunification. The goal of this paper is to survey these works and bring them together in a same framework. R'esum'e On appelle habituellement (algorithme d') unification un algorithme de r'esolution d'une 'equation dans une alg`ebre de termes. La r'esolution de formules plus complexes, comportant en particulier des n'egations, est appel'ee ici disunification. Avec une d'efinition aussi 'etendue, de nombreux travaux peuvent etre consid'er'es comme portant sur la disunification. L'objet de cet article de synth`ese est de rassembler tous ces travaux dans un meme formalisme. Laboratoire de Recherche en Informatique, Bat. 490, Universit'e de ParisSud, 91405 ORSAY cedex, France. Email: comon@lri.lri.fr i Contents 1 Syntax 5 1.1 Basic Defini...
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...
A New Method for Undecidability Proofs of First Order Theories
 Journal of Symbolic Computation
, 1992
"... this paper is to define a framework for such reduction proofs. The method proposed is illustrated by proving the undecidability of the theory of a term algebra modulo the axioms of associativity and commutativity and of the theory of a partial lexicographic path ordering. 1. Introduction ..."
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Cited by 30 (7 self)
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this paper is to define a framework for such reduction proofs. The method proposed is illustrated by proving the undecidability of the theory of a term algebra modulo the axioms of associativity and commutativity and of the theory of a partial lexicographic path ordering. 1. Introduction
Equational Inference, Canonical Proofs, And Proof Orderings
 Journal of the ACM
, 1992
"... We describe the application of proof orderingsa technique for reasoning about inference systemsto various rewritebased theoremproving methods, including re#nements of the standard KnuthBendix completion procedure based on critical pair criteria; Huet's procedure for rewriting modulo a congr ..."
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Cited by 30 (11 self)
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We describe the application of proof orderingsa technique for reasoning about inference systemsto various rewritebased theoremproving methods, including re#nements of the standard KnuthBendix completion procedure based on critical pair criteria; Huet's procedure for rewriting modulo a congruence; ordered completion #a refutationally complete extension of standard completion#; and a proof by consistency procedure for proving inductive theorems. # This is a substantially revised version of the paper, #Orderings for equational proofs," coauthored with J. Hsiang and presented at the Symp. on Logic in Computer Science #Boston, Massachusetts, June 1986#. It includes material from the paper #Proof by consistency in equational theories," by the #rst author, presented at the ThirdAnnual Symp. on Logic in Computer Science #Edinburgh, Scotland, July 1988#. This researchwas supported in part by the National Science Foundation under grants CCR8901322, CCR9007195, and CCR9024271. 1 ...
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...
A total ACcompatible ordering based on RPO
 Theoretical Computer Science
, 1995
"... We define a simplification ordering on terms which is ACcompatible and total on nonAC equivalent ground terms, without any restrictions on the signature like the number of ACsymbols or free symbols. Unlike previous work by Narendran and Rusinowitch [NR91], our ACRPO ordering is not based on poly ..."
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Cited by 12 (7 self)
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We define a simplification ordering on terms which is ACcompatible and total on nonAC equivalent ground terms, without any restrictions on the signature like the number of ACsymbols or free symbols. Unlike previous work by Narendran and Rusinowitch [NR91], our ACRPO ordering is not based on polynomial interpretations, but on a simple extension of the wellknown RPO ordering (with a total (arbitrary) precedence on the function symbols). This solves an open question posed e.g. by Bachmair [Bac92]. A second difference is that this ordering is also defined on terms with variables, which makes it applicable in practice for complete theorem proving strategies with builtin ACunification and for orienting nonground rewrite systems. The ordering is defined in a simple way by means of rewrite rules, and can be easily implemented, since its main component is RPO. 1 Introduction Automated termination proofs are wellknown to be crucial for using rewritinglike methods in theorem proving an...