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Solving Symbolic Ordering Constraints
, 1990
"... We show how to solve boolean combinations of inequations s ? t in the Herbrand Universe, assuming that is interpreted as a lexicographic path ordering extending a total precedence. In other words, we prove that the existential fragment of the theory of a lexicographic path ordering which extends a ..."
Abstract

Cited by 49 (10 self)
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We show how to solve boolean combinations of inequations s ? t in the Herbrand Universe, assuming that is interpreted as a lexicographic path ordering extending a total precedence. In other words, we prove that the existential fragment of the theory of a lexicographic path ordering which extends a total precedence is decidable. Keywords: simplification orderings, ordered strategies, term algebras, constraint solving. 1. Introduction The first order theory of term algebras over a language (or alphabet) with no relational symbol (other than equality) has been shown to be decidable 1;2 . See also Refs 3 and 4. Introducing into the language a binary relational symbol interpreted as the subterm ordering makes the theory undecidable 5 . Venkataraman also shows in the latter paper that the purely existential fragment of the theory, i.e. the subset of sentences whose prenex form does not contain 8, is decidable. Venkataraman was concerned with some applications in functional programm...
Syntactic Unification Problems under Constrained Substitutions
, 1996
"... ... This paper is a collection of results on the decidability and the computational complexity of a syntactic unification problem under constrained substitutions. A number of decidable, undecidable, tractable and intractable results of the problem are presented. Since a unification problem under con ..."
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Cited by 8 (0 self)
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... This paper is a collection of results on the decidability and the computational complexity of a syntactic unification problem under constrained substitutions. A number of decidable, undecidable, tractable and intractable results of the problem are presented. Since a unification problem under constrained substitutions can be regarded as an ordersorted unification problem with term declarations such that the number of sorts is only one, the results presented in this paper also indicate how the intractability of ordersorted unification problems is reduced by restricting the number of sorts to one.
Defining Soft Sortedness by Abstract Interpretation
, 1994
"... Abstract When a language refers to a universe of discourse which contains more than one sort of object, it is often useful for the language to include notations for describing sort restrictions on functions and predicates so as to achieve more flexible representations and more efficient reasoning. I ..."
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Abstract When a language refers to a universe of discourse which contains more than one sort of object, it is often useful for the language to include notations for describing sort restrictions on functions and predicates so as to achieve more flexible representations and more efficient reasoning. Increased efficiency is typically based on sort restrictions, which may be checked either statically or dynamically. Languages using syntactic sort constraints (&quot;signatures&quot;) on function and relation symbols support more static sort checking but are less expressive than languages using semantic sort constraints (&quot;sort predicates&quot;). In this paper, we describe an approach, called soft sorting, in which both static and dynamic sort checking can be performed within an unified framework. In this framework, we aim to do as much static sort checking as possible, relying on dynamic sort checking only when necessary. We describe the basic concepts and results of this approach in the context of firstorder languages. 1 Introduction Symbols in a firstorder language denote objects of an intended universe of discourse and functions and relations on them. When a universe of discourse contains more than one sort of object, it is often useful to include in the language, notations for describing sort constraints on those functions and predicates, so as to achieve more flexible representations and more efficient reasoning. Hereafter we will use the word &quot;expression &quot; to refer generally to terms and formulas. In this paper, sorting means the use of sort constraints on the symbols of a language and consequently on expressions of the language.