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Natural termination
 Theoretical Computer Science
"... Abstract. We generalize the various path orderings and the conditions under which they work, and describe an implementation of this general ordering. We look at methods for proving termination of orthogonal systems and give a new solution to a problem of Zantema's. 1 ..."
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Cited by 84 (11 self)
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Abstract. We generalize the various path orderings and the conditions under which they work, and describe an implementation of this general ordering. We look at methods for proving termination of orthogonal systems and give a new solution to a problem of Zantema's. 1
Paramodulation with NonMonotonic Orderings
 In 14th IEEE Symposium on Logic in Computer Science (LICS
, 1999
"... All current completeness results for ordered paramodulation require the term ordering Ø to be wellfounded, monotonic and total(izable) on ground terms. Here we introduce a new proof technique where the only properties required for Ø are wellfoundedness and the subterm property 1 . The technique ..."
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Cited by 9 (7 self)
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All current completeness results for ordered paramodulation require the term ordering Ø to be wellfounded, monotonic and total(izable) on ground terms. Here we introduce a new proof technique where the only properties required for Ø are wellfoundedness and the subterm property 1 . The technique is a relatively simple and elegant application of some fundamental results on the termination and confluence of ground term rewrite systems (TRS). By a careful further analysis of our technique, we obtain the first KnuthBendix completion procedure that finds a convergent TRS for a given set of equations E and a (possibly nontotalizable) reduction ordering Ø whenever it exists 2 . Note that being a reduction ordering is the minimal possible requirement on Ø, since a TRS terminates if, and only if, it is contained in a reduction ordering. Keywords: term rewriting, automated deduction. 1 Introduction Deduction with equality is fundamental in mathematics, logics and many applications of ...
Canonical Sets of Horn Clauses
, 1990
"... Rewrite rules are oriented equations used to replace equalsbyequals in the specified direction. Input terms are repeatedly rewritten according to the rules. When and if no rule applies... ..."
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Cited by 6 (2 self)
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Rewrite rules are oriented equations used to replace equalsbyequals in the specified direction. Input terms are repeatedly rewritten according to the rules. When and if no rule applies...
On the Completeness of Arbitrary Selection Strategies for Paramodulation
 In Proceedings of ICALP 2001
, 2001
"... A crucial way for reducing the search space in automated deduction are the socalled selection strategies: in each clause, the subset of selected literals are the only ones involved in inferences. For firstorder Horn clauses without equality, resolution is complete with an arbitrary selection o ..."
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Cited by 5 (1 self)
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A crucial way for reducing the search space in automated deduction are the socalled selection strategies: in each clause, the subset of selected literals are the only ones involved in inferences. For firstorder Horn clauses without equality, resolution is complete with an arbitrary selection of one single literal in each clause [dN96]. For Horn clauses with builtin equality, i.e., paramodulationbased inference systems, the situation is far more complex. Here we show that if a paramodulationbased inference system is complete with eager selection of negative equations and, moreover, it is compatible with equality constraint inheritance, then it is complete with arbitrary selection strategies. A first important application of this result is the one for paramodulation wrt. nonmonotonic orderings, which was left open in [BGNR99]. 1
WellFoundedness is Sufficient for Completeness of Ordered Paramodulation
, 2002
"... For many years all known completeness results for KnuthBendix completion and ordered paramodulation required the term ordering to be wellfounded, monotonic and total(izable) on ground terms. ..."
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Cited by 2 (0 self)
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For many years all known completeness results for KnuthBendix completion and ordered paramodulation required the term ordering to be wellfounded, monotonic and total(izable) on ground terms.
Canonical Inference for Implicational Systems ⋆
"... Abstract. Completion is a general paradigm for applying inferences to generate a canonical presentation of a logical theory, or to semidecide the validity of theorems, or to answer queries. We investigate what canonicity means for implicational systems that are axiomatizations of Moore families – o ..."
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Cited by 2 (2 self)
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Abstract. Completion is a general paradigm for applying inferences to generate a canonical presentation of a logical theory, or to semidecide the validity of theorems, or to answer queries. We investigate what canonicity means for implicational systems that are axiomatizations of Moore families – or, equivalently, of propositional Horn theories. We build a correspondence between implicational systems and associativecommutative rewrite systems, give deduction mechanisms for both, and show how their respective inferences correspond. Thus, we exhibit completion procedures designed to generate canonical systems that are “optimal ” for forward chaining, to compute minimal models, and to generate canonical systems that are rewriteoptimal. Rewriteoptimality is a new notion of “optimality ” for implicational systems, one that takes contraction by simplification into account. 1
A Survey of Some Recent Trends in RewriteBased and ParamodulationBased Deduction
"... Introduction Deduction with equality is fundamental in mathematics, logics and many applications of formal methods in computer science. During the last two decades this field has importantly progressed through new KnuthBendixlike completion techniques and their extensions to ordered paramodulatio ..."
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Cited by 2 (0 self)
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Introduction Deduction with equality is fundamental in mathematics, logics and many applications of formal methods in computer science. During the last two decades this field has importantly progressed through new KnuthBendixlike completion techniques and their extensions to ordered paramodulation for firstorder clauses. These techniques have lead to important results on deduction in firstorder logic with equality, like [HR91,BDH86,BD94,BG94,BG98,NR01], results that have been applied to stateoftheart theorem provers like Spass [Wei97] and Vampire [RV01]. These techniques have also led to results on logicbased complexity and decidability analysis [BG01,Nie98], on deduction with constrained clauses [KKR90,NR95], on inductive theorem proving [CN00], and on many other applications like symbolic constraint solving, or equationallogic programming. In the handbook chapter [NR01] the fundamental techniques in this area are reviewed and presented in a uniform fr
unknown title
, 2003
"... They are not capable to ground a canonicity of universal consistency. —Alexandra Deligiorgi (ΠAI∆EIA, 1998) We explore how different proof orderings induce different notions of saturation. We relate completion, paramodulation, saturation, redundancy elimination, and rewrite system reduction to proof ..."
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They are not capable to ground a canonicity of universal consistency. —Alexandra Deligiorgi (ΠAI∆EIA, 1998) We explore how different proof orderings induce different notions of saturation. We relate completion, paramodulation, saturation, redundancy elimination, and rewrite system reduction to proof orderings. 1