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Specification and Proof in Membership Equational Logic
 Theoretical Computer Science
, 1996
"... : This paper is part of a longterm effort to increase expressiveness of algebraic specification languages while at the same time having a simple semantic basis on which efficient execution by rewriting and powerful theoremproving tools can be based. In particular, our rewriting techniques provide ..."
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Cited by 131 (54 self)
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: This paper is part of a longterm effort to increase expressiveness of algebraic specification languages while at the same time having a simple semantic basis on which efficient execution by rewriting and powerful theoremproving tools can be based. In particular, our rewriting techniques provide semantic foundations for Maude's functional sublanguage, where they have been efficiently implemented. Membership equational logic is quite simple, and yet quite powerful. Its atomic formulae are equations and sort membership assertions, and its sentences are Horn clauses. It extends in a conservative way both ordersorted equational logic and partial algebra approaches, while Horn logic can be very easily encoded. After introducing the basic concepts of the logic, we give conditions and proof rules with which efficient equational deduction by rewriting can be achieved. We also give completion techniques to transform a specification into one meeting these conditions. We address the important ...
On the theory of structural subtyping
, 2003
"... We show that the firstorder theory of structural subtyping of nonrecursive types is decidable. Let Σ be a language consisting of function symbols (representing type constructors) and C a decidable structure in the relational language L containing a binary relation ≤. C represents primitive types; ..."
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Cited by 18 (8 self)
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We show that the firstorder theory of structural subtyping of nonrecursive types is decidable. Let Σ be a language consisting of function symbols (representing type constructors) and C a decidable structure in the relational language L containing a binary relation ≤. C represents primitive types; ≤ represents a subtype ordering. We introduce the notion of Σtermpower of C, which generalizes the structure arising in structural subtyping. The domain of the Σtermpower of C is the set of Σterms over the set of elements of C. We show that the decidability of the firstorder theory of C implies the decidability of the firstorder theory of the Σtermpower of C. This result implies the decidability of the firstorder theory of structural subtyping of nonrecursive types.
Tree automata with global constraints
 In 12th Int. Conf. in Developments in Lang. Theory (DLT), vol. 5257 of LNCS
, 2008
"... Abstract. A tree automaton with global tree equality and disequality constraints, TAGED for short, is an automaton on trees which allows to test (dis)equalities between subtrees which may be arbitrarily faraway. In particular, it is equipped with an (dis)equality relation on states, so that whenever ..."
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Cited by 16 (2 self)
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Abstract. A tree automaton with global tree equality and disequality constraints, TAGED for short, is an automaton on trees which allows to test (dis)equalities between subtrees which may be arbitrarily faraway. In particular, it is equipped with an (dis)equality relation on states, so that whenever two subtrees t and t ′ evaluate (in an accepting run) to two states which are in the (dis)equality relation, they must be (dis)equal. We study several properties of TAGEDs, and prove decidability of emptiness of several classes. We give two applications of TAGEDs: decidability of an extension of Monadic Second Order Logic with tree isomorphism tests and of unification with membership constraints. These results significantly improve the results of [10]. 1
Term algebras with length function and bounded quantifier alternation
 In Theorem Proving in HigherOrder Logics, volume 3223 of LNCS
, 2004
"... .)L: TA! Z. Formulae are formed from term literals and integerliterals using logical connectives and quantifications. Term literals are exactly ..."
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Cited by 12 (5 self)
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.)L: TA! Z. Formulae are formed from term literals and integerliterals using logical connectives and quantifications. Term literals are exactly
How to Win a Game with Features
 1ST INTERNATIONAL CONFERENCE ON CONSTRAINTS IN COMPUTATIONAL LOGICS, LECTURE NOTES IN COMPUTER SCIENCE
, 1994
"... We employ the modeltheoretic method of EhrenfeuchtFraisse Games to prove the completeness of the theory CFT, which has been introduced in [22] for describing rational trees in a language of selector functions. The comparison to other techniques used in this field shows that EhrenfeuchtFraisse ..."
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Cited by 8 (2 self)
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We employ the modeltheoretic method of EhrenfeuchtFraisse Games to prove the completeness of the theory CFT, which has been introduced in [22] for describing rational trees in a language of selector functions. The comparison to other techniques used in this field shows that EhrenfeuchtFraisse Games lead to simpler proofs.
A Methodological View of Constraint Solving
, 1996
"... Constraints have become very popular during the last decade. Constraints allow to define sets of data by means of logical formulae. Our goal here is to survey the notion of constraint system and to give examples of constraint systems operating on various domains, such as natural, rational or real nu ..."
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Cited by 6 (2 self)
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Constraints have become very popular during the last decade. Constraints allow to define sets of data by means of logical formulae. Our goal here is to survey the notion of constraint system and to give examples of constraint systems operating on various domains, such as natural, rational or real numbers, finite domains, and term domains. We classify the different methods used for solving constraints, syntactic methods based on transformations, semantic methods based on adequate representations of constraints, hybrid methods combining transformations and enumerations. Examples are used throughout the paper to illustrate the concepts and methods. We also discuss applications of constraints to various fields, such as programming, operations research, and theorem proving. y CNRS and LRI, Bat. 490, Universit'e de Paris Sud, 91405 ORSAY Cedex, France fcomon, jouannaudg@lri.lri.fr z COSYTEC, Parc Club Orsay Universit'e, 4 Rue Jean Rostand, 91893 Orsay Cedex, France dincbas@cosytec.fr x ...
The Decidability of the Firstorder Theory of KnuthBendix Order
"... Abstract. Two kinds of orderings are widely used in term rewriting and theorem proving, namely recursive path ordering (RPO) and KnuthBendix ordering (KBO). They provide powerful tools to prove the termination of rewriting systems. They are also applied in ordered resolution to prune the search spa ..."
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Cited by 5 (0 self)
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Abstract. Two kinds of orderings are widely used in term rewriting and theorem proving, namely recursive path ordering (RPO) and KnuthBendix ordering (KBO). They provide powerful tools to prove the termination of rewriting systems. They are also applied in ordered resolution to prune the search space without compromising refutational completeness. Solving ordering constraints is therefore essential to the successful application of ordered rewriting and ordered resolution. Besides the needs for decision procedures for quantifierfree theories, situations arise in constrained deduction where the truth value of quantified formulas must be decided. Unfortunately, the full firstorder theory of recursive path orderings is undecidable. This leaves an open question whether the firstorder theory of KBO is decidable. In this paper, we give a positive answer to this question using quantifier elimination. In fact, we shall show the decidability of a theory that is more expressive than the theory of KBO. 1
Verifying Balanced Trees
, 2007
"... Balanced search trees provide guaranteed worstcase time performance and hence they form a very important class of data structures. However,... ..."
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Cited by 5 (1 self)
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Balanced search trees provide guaranteed worstcase time performance and hence they form a very important class of data structures. However,...
Feature Automata and Sets of Feature Trees
, 1993
"... Feature trees generalize firstorder trees (which are called ground terms in the general framework of universal algebra). Namely, argument positions become keywords ("features") from an infinite symbol set F. A constructor symbol becomes a node symbol that can occur with arbitrary and arbi ..."
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Feature trees generalize firstorder trees (which are called ground terms in the general framework of universal algebra). Namely, argument positions become keywords ("features") from an infinite symbol set F. A constructor symbol becomes a node symbol that can occur with arbitrary and arbitrarily many argument positions. Feature trees are used to model flexible records; the assumption that F is infinite accounts for dynamic record field additions. We develop a universal algebra framework for feature trees. We extend the classical setdefining notions: automata, regular expressions and equational systems, and show that they coincide. This extension of the regular theory of trees requires new notions and proofs. Roughly, a feature automaton reads a feature tree in two directions: along its branches and along the list of the direct descendants of each node. The second direction corresponds to an automaton on a commutative monoid (over an infinite alphabet). One motivation for this work st...
FIRSTORDER CONSTRAINT SYSTEMS WITH MULTIPLE CONGRUENCE RELATIONS
"... Abstract. We investigate the problem of deciding firstorder theories of finite trees with several distinguished congruence relations, each of them given by some equational axioms. We give an automatabased solution for the case where the different equational axiom systems are linear and variabledi ..."
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Abstract. We investigate the problem of deciding firstorder theories of finite trees with several distinguished congruence relations, each of them given by some equational axioms. We give an automatabased solution for the case where the different equational axiom systems are linear and variabledisjoint (this includes the case where all axioms are ground), and where the logic does not permit to express tree relations x=f(y,z). We show that the problem is undecidable when these restrictions are relaxed. As motivation and application, we show how to translate the model checking problem of ApiL, a spatial equational logic for the applied picalculus, to the validity of firstorder formulas in term algebras with multiple congruence relations. 1.