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Decision procedures for recursive data structures with integer constraints
 In International Joint Conference on Automated Reasoning, volume 3097 of LNCS
, 2004
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The Defining Power of Stratified and Hierarchical Logic Programs
"... We investigate the defining power of stratified and hierarchical logic programs. As an example for the treatment of negative information in the context of these structured programs we also introduce a stratified and hierarchical closedworld assumption. Our analysis tries to relate the defining powe ..."
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We investigate the defining power of stratified and hierarchical logic programs. As an example for the treatment of negative information in the context of these structured programs we also introduce a stratified and hierarchical closedworld assumption. Our analysis tries to relate the defining power of stratified and hierarchical programs (with and without an appropriate closedworld assumption) very precisely to notions and hierarchies in classical definability theory. Stratified and hierarchical logic programs are two wellknown and typical candidates of what one may more generally denote as structured programs. In both cases we have to deal with normal logic programs which satisfy certain syntactic conditions with respect to the occurrence of negative literals. Recently they have gained a lot of importance in connection with the search for nice declarative semantics for logic programs and the treatment of negative information in logic programming (e.g., Lloyd [10]). Stratified programs were introduced into logic programming by Apt, Blair, and Walker [2] and van Gelder [17] not long ago. In mathematical logic, however, theories of this kind have been studied for more than 20 years under the general theme of iterated inductive definability. Indeed, stratified programs can be understood as systems for (finitely) iterated inductive definitions where the definition clauses are of very low logical complexity. The notion of hierarchical program (e.g., Clark [6], Shepherdson [15]), on the other hand, is motivated by database theory and tries to reflect the idea of iterated explicit definability by simple principles. From a conceptual point of view we are interested in the relationship between logic programming, inductive definability and equational definability. By making u...
Finite Queries do not Have Effective Syntax
, 1995
"... A relational query is called finite, or sometimes safe, iff it yields a finite answer in every database state. The set of finite queries of relational calculus is known to be unsolvable. However, in many cases it is possible to impose syntactical restrictions on the class of queries that guarantee f ..."
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Cited by 14 (3 self)
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A relational query is called finite, or sometimes safe, iff it yields a finite answer in every database state. The set of finite queries of relational calculus is known to be unsolvable. However, in many cases it is possible to impose syntactical restrictions on the class of queries that guarantee finiteness and do not reduce the expressive power of the calculus. We show that unfortunately this is not always the case, as we construct a recursive domain with decidable theory where any solvable (or enumerable, for that matter) subclass of queries either contains an infinite query, or misses a finite one. We show that although any domain can always be extended to a domain with an effective syntax for finite A preliminary version of this paper appeared in the Proc. of the 14th ACM SIGACTSIGMOD SIGART Symp. on Principles of Database Systems, San Jose, CA, May 2225, 1995. y This work has been partially supported by NSF Grant CCR 9403809. z A part of this research was carried out whil...
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 (4 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
A uniform approach to constraintsolving for lists, multisets, compact lists, and sets
 ACM TRANS. COMPUT. LOG
, 2008
"... Lists, multisets, and sets are wellknown data structures whose usefulness is widely recognized in various areas of Computer Science. They have been analyzed from an axiomatic point of view with a parametric approach in [Dovier et al. 1998] where the relevant unification algorithms have been develop ..."
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Lists, multisets, and sets are wellknown data structures whose usefulness is widely recognized in various areas of Computer Science. They have been analyzed from an axiomatic point of view with a parametric approach in [Dovier et al. 1998] where the relevant unification algorithms have been developed. In this paper we extend these results considering more general constraints, namely equality and membership constraints and their negative counterparts.
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 ..."
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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 10 (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.
A CLP View of Logic Programming
 In Proc. Conf. on Algebraic and Logic Programming
, 1992
"... . We address the problem of determining those constraint domains A for which the traditional logic programming semantics and the constraint logic programming semantics CLP (A) coincide. This reduces to a study of nonstandard models of Clark's axioms and the notion of solution compactness i ..."
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. We address the problem of determining those constraint domains A for which the traditional logic programming semantics and the constraint logic programming semantics CLP (A) coincide. This reduces to a study of nonstandard models of Clark's axioms and the notion of solution compactness introduced in the CLP scheme. The results of this study include the proof of the existence of a free product in the class of algebras defined by Clark's axioms, a characterization of when Clark's axioms form a model complete theory, and a limited characterization of those models of Clark's axioms which form solution compact constraint domains. 1 Introduction Appropriate semantics for definite logic programs are now largely agreed upon [17]. They involve a completed program, SLDresolution, a onestep consequence function, a least Herbrand model and numerous relationships between them: soundness and completeness of SLDrefutations, soundness and completeness of the negationasfailure rule, .....
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
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 4 (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,...