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Halting and Equivalence of Schemes over Recursive Theories
"... Let Σ be a fixed firstorder signature. In this note we consider the following decision problems. (i) Given a recursive ground theory T over Σ, a program scheme p over Σ, and input values specified by ground terms t1,...,tn, doesp halt on input t1,...,tn in all models of T? (ii) Given a recursive gr ..."
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Let Σ be a fixed firstorder signature. In this note we consider the following decision problems. (i) Given a recursive ground theory T over Σ, a program scheme p over Σ, and input values specified by ground terms t1,...,tn, doesp halt on input t1,...,tn in all models of T? (ii) Given a recursive ground theory T over Σ and two program schemes p and q over Σ, are p and q equivalent in all models of T? When T is empty, these two problems are the classical halting and equivalence problems for program schemes, respectively. We show that problem (i) is r.e.complete and problem (ii) is Π0 2complete. Both these problems remain hard for their respective complexity classes even if T is empty and Σ is restricted to contain only a single constant, a single unary function symbol, and a single monadic predicate. It follows from (ii) that there can exist no relatively complete deductive system for scheme equivalence. Key words: model theory, Kleene algebra, dynamic logic
Guarded Quantification in Least Fixed Point Logic
, 2002
"... We develop a variant of Least Fixed Point logic based on First Order logic with a relaxed version of guarded quantification. We develop a Game Theoretic Semantics of this logic, and find that under reasonable conditions, guarding quantification does not reduce the expressibility of Least Fixed Point ..."
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We develop a variant of Least Fixed Point logic based on First Order logic with a relaxed version of guarded quantification. We develop a Game Theoretic Semantics of this logic, and find that under reasonable conditions, guarding quantification does not reduce the expressibility of Least Fixed Point logic. But guarding quantification increases worstcase time complexity.
Game Representations of Complexity Classes
 Proc. Eur. Summer School on Logic, Language and Information (European Assoc. Logic, Language and Information
, 2001
"... Many descriptive and computational complexity classes have gametheoretic representations. These can be used to study the relation between different logics and complexity classes in finite model theory. ..."
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Many descriptive and computational complexity classes have gametheoretic representations. These can be used to study the relation between different logics and complexity classes in finite model theory.
J. LOGIC PROGRAMMING 1985:1:115 1 HORN CLAUSE QUERIES AND GENERALIZATIONS* ASHOK K. CHANDRA* * AND DAVID HAREL+
"... A logic program consists of a set of Horn clauses, and can be used to express a query on relational data bases. It is shown that logic programs express precisely the queries in YE+ (the set of queries representable by a fixpoint applied to a positive existential query). Queries expressible by logic ..."
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A logic program consists of a set of Horn clauses, and can be used to express a query on relational data bases. It is shown that logic programs express precisely the queries in YE+ (the set of queries representable by a fixpoint applied to a positive existential query). Queries expressible by logic programs are thus not firstorder queries in general, nor are all the firstorder queries expressible as logic programs. Several ways of adding negation to logic programs are examined. The most general case is where arbitrary firstorder formulas (with “nonterminal ” relation symbols) are allowed. The resulting class has the expressive power of universally quantified secondorder logic. a 1. INTRODUCIION Kowalski [12] has introduced a programming language based on predicate calculus. A computation in this language is analogous to a resolutiondriven attempt at proving a theorem of the form “atomic formula C follows from sentences C,,..., Cm”. In [12], as well as in subsequent papers on the topic, e.g., [l, 6,7], the sentences C, are taken to be (closures of) Horn clauses in the predicate calculus; i.e., each C, is of the form (VF)(A V,B, v.* * V,B”) for some atomic formulas A, B,, where F consists of all variables appearing in A and in the Bj. Such a sentence is usually written simply as