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26
Syntacticness, CycleSyntacticness and Shallow Theories
 INFORMATION AND COMPUTATION
, 1994
"... Solving equations in the free algebra T (F; X) (i.e. unification) uses the two rules: f(~s) = f( ~ t) ! ~s = ~ t (decomposition) and s[x] = x !? (occurcheck). These two rules are not correct in quotients of T (F; X) by a finitely generated congruence =E . Following C. Kirchner, we first define cl ..."
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Solving equations in the free algebra T (F; X) (i.e. unification) uses the two rules: f(~s) = f( ~ t) ! ~s = ~ t (decomposition) and s[x] = x !? (occurcheck). These two rules are not correct in quotients of T (F; X) by a finitely generated congruence =E . Following C. Kirchner, we first define classes of equational theories (called syntactic and cycle syntactic respectively) for which it is possible to derive some rules replacing the two above ones. Then, we show that these abstract classes are relevant: all shallow theories, i.e. theories which can be generated by equations in which variables occur at depth at most one, are both syntactic and cycle syntactic. Moreover, the new set of unification rules is terminating, which proves that unification is decidable and finitary in shallow theories. We give still further extensions. If the set of equivalence classes is infinite, a problem which turns out to be decidable in shallow theories, then shallow theories fulfill Colmerauer's indep...
Complete Axiomatizations of some Quotient Term Algebras
 In Proc. 18th Int. Coll. on Automata, Languages and Programming, Madrid, LNCS 510
, 1993
"... We show that T (F )= =E can be completely axiomatized when =E is a quasifree theory. Quasifree theories are a wider class of theories than permutative theories of [Mal71] for which Mal'cev gave decision results. As an example of application, we show that the first order theory of T (F )= =E is de ..."
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Cited by 10 (3 self)
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We show that T (F )= =E can be completely axiomatized when =E is a quasifree theory. Quasifree theories are a wider class of theories than permutative theories of [Mal71] for which Mal'cev gave decision results. As an example of application, we show that the first order theory of T (F )= =E is decidable when E is a set of ground equations. Besides, we prove that the \Sigma 1 fragment of the theory of T (F )= =E is decidable when E is a compact set of axioms. In particular, the existential fragment of the theory of associativecommutative function symbols is decidable. Introduction Mal'cev studied in the early sixties classes of locally free algebras that can be completely axiomatized [Mal71]. He proved in particular that what is today known as Clark's equality theory is decidable. He also studied some classes of permutative algebras in which, roughly, the axiom f(s 1 ; : : : ; s n ) = f(t 1 ; : : : ; t n ) ) s 1 = t 1 : : : s n = t n is replaced with f(s 1 ; : : : ; s n ) = f(t ...
Feature Trees over Arbitrary Structures
 Specifying Syntactic Structures, chapter 7
, 1997
"... This paper presents a family of first order feature tree theories, indexed by the theory of the feature labels used to build the trees. A given feature label theory, which is required to carry an appropriate notion of sets, is conservatively extended to a theory of feature trees with the predicat ..."
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Cited by 9 (2 self)
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This paper presents a family of first order feature tree theories, indexed by the theory of the feature labels used to build the trees. A given feature label theory, which is required to carry an appropriate notion of sets, is conservatively extended to a theory of feature trees with the predicates x[t]y (feature t leads from the root of tree x to the tree y), where we have to require t to be a ground term, and xt# (feature t is defined at the root of tree x). In the latter case, t might be a variable. Together with the notion of sets provided by the feature label theory, this yields a firstclass status of arities.
Constraint deduction in an intervalbased temporal logic
 Executable Modal and Temporal Logics, (Proc. of the IJCAI'93 Workshop), volume 897 of LNAI
, 1995
"... Abstract. We describe reasoning methods for the intervalbased modal temporal logic LLP which employs the modal operators sometimes, always, next, and chop. We propose a constraint deduction approach and compare it with a sequent calculus, developed as the basic machinery for the deductive planning ..."
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Cited by 7 (1 self)
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Abstract. We describe reasoning methods for the intervalbased modal temporal logic LLP which employs the modal operators sometimes, always, next, and chop. We propose a constraint deduction approach and compare it with a sequent calculus, developed as the basic machinery for the deductive planning system PHI which uses LLP as underlying formalism. 1
About the Theory of Tree Embedding
 Proc. Int. Joint Conf. on Theory and Practice of Software Development, Lecture Notes in Computer Science
, 1993
"... . We show that the positive existential fragment of the theory of tree embedding is decidable. 1 Introduction Symbolic Constraints, i.e. formulae interpreted in some term structure, have been revealed to be extremely useful in logic programming and theorem proving. Among such constraints, the order ..."
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. We show that the positive existential fragment of the theory of tree embedding is decidable. 1 Introduction Symbolic Constraints, i.e. formulae interpreted in some term structure, have been revealed to be extremely useful in logic programming and theorem proving. Among such constraints, the ordering constraints can be used in expressing ordered strategies at the formula level instead of the inference level. This allows to cut further the search space, while keeping the completeness of the strategy [7]. Solving ordering constraints also allows for a nice lifting of orderings from the ground level to the terms with variables: define s ? t by 8~x:s ? t where ~x, the variables of s; t, range over all ground terms. This provides with more powerful orderings for termination proofs in rewriting theory. Up to now, the satisfiability of ordering constraints has been studied for some orderings on terms: Venkataraman showed that the existential fragment of the theory of the subterm ordering i...
Increasing Model Building Capabilities by Constraint Solving on Terms with Integer Exponents
 Journal of Symbolic Computation
, 1997
"... this paper the decidability of first order theory of the language of Iterms. ..."
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this paper the decidability of first order theory of the language of Iterms.
How to make SQL stand for String Query Language
 IN PROCEEDINGS OF DBPL'99, SPRINGER LNCS
, 1999
"... A string database is simply a collection of tables, the columns of which contain strings over some given alphabet. We address in this paper the issue of designing a simple, user friendly query language for string databases. We focus on the language FO(ffl), which is classical first order logic exten ..."
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A string database is simply a collection of tables, the columns of which contain strings over some given alphabet. We address in this paper the issue of designing a simple, user friendly query language for string databases. We focus on the language FO(ffl), which is classical first order logic extended with a concatenation operator, and where quantifiers range over the set of all strings. We wish to capture all string queries, i.e., welltyped and computable mappings involving a notion of string genericity. Unfortunately, unrestricted quantification may allow some queries to have infinite output. This leads us to study the "safety" problem for FO(ffl), that is, how to build syntactic and/or semantic restrictions so as to obtain a language expressing only queries with finite output, hopefully all string queries. We introduce a family of such restrictions and study their expressivness and complexity. We prove that none of these languages express all string queries. We prov...
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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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
Constraints in Term Algebras (Short Survey)
 Proc. Conf. on Algebraic Methodology and Software Technology, Univ. of Twente
, 1993
"... this paper. References ..."
Negation in Combining Constraint Systems
 Communications of the ACM
, 1998
"... In a recent paper, Baader and Schulz presented a general method for the combination of constraint systems for purely positive constraints. But negation plays an important role in constraint solving. E.g., it is vital for constraint entailment. Therefore it is of interest to extend their results to t ..."
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In a recent paper, Baader and Schulz presented a general method for the combination of constraint systems for purely positive constraints. But negation plays an important role in constraint solving. E.g., it is vital for constraint entailment. Therefore it is of interest to extend their results to the combination of constraint problems containing negative constraints. We show that the combined solution domain introduced by Baader and Schulz is a domain in which one can solve positive and negative "mixed" constraints by presenting an algorithm that reduces solvability of positive and negative "mixed" constraints to solvability of pure constraints in the components. The existential theory in the combined solution domain is decidable if solvability of literals with socalled linear constant restrictions is decidable in the components. We also give a criterion for ground solvability of mixed constraints in the combined solution domain. The handling of negative constraints can be signific...