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Superposition Theorem Proving for Abelian Groups Represented as Integer Modules
 Theoretical Computer Science
, 1996
"... We define a superposition calculus specialized for abelian groups represented as integer modules, and show its refutational completeness. This allows to substantially reduce the number of inferences compared to a standard superposition prover which applies the axioms directly. Specifically, equation ..."
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Cited by 13 (4 self)
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We define a superposition calculus specialized for abelian groups represented as integer modules, and show its refutational completeness. This allows to substantially reduce the number of inferences compared to a standard superposition prover which applies the axioms directly. Specifically, equational literals are simplified, so that only the maximal term of the sums is on the lefthand side. Only certain minimal superpositions need to be considered; other superpositions which a standard prover would consider become redundant. This not only reduces the number of inferences, but also reduces the size of the ACunification problems which are generated. That is, ACunification is not necessary at the top of a term, only below some nonACsymbol. Further, we consider situations where the axioms give rise to variable overlaps and develop techniques to avoid these explosive cases where possible. 1 Introduction Historically, starting from plain resolution, more and more problematic axioms ha...
Cancellative Abelian Monoids in Refutational Theorem Proving. Dissertation, Universitat des Saarlandes
 PhD thesis, Institut für Informatik, Universität des Saarlandes
, 1997
"... der Technischen Fakult"at der Universit"at des Saarlandes Saarbr"ucken ..."
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Cited by 3 (1 self)
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der Technischen Fakult"at der Universit"at des Saarlandes Saarbr"ucken
Equational Reasoning using AC Constraints
 In Proceedings of the Fifteenth International Joint Conference on Artificial Intelligence (IJCAI97
, 1997
"... Unfailing completion is a commonly used technique for equational reasoning. For equational problems with associative and commutative functions, unfailing completion often generates a large number of rewrite rules. By comparing it with a ground completion procedure, we show that many of the rewrite r ..."
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Cited by 1 (1 self)
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Unfailing completion is a commonly used technique for equational reasoning. For equational problems with associative and commutative functions, unfailing completion often generates a large number of rewrite rules. By comparing it with a ground completion procedure, we show that many of the rewrite rules generated are redundant. A set of consistency constraints is formulated to detect redundant rewrite rules. We propose a new completion algorithm, consistent unfailing completion, in which only consistent rewrite rules are used for critical pair generation and rewriting. Our approach does not need to use flattened terms. Thus it avoids the double exponential worst case complexity of AC unification. It also allows the use of more flexible termination orderings. We present some sufficient conditions for detecting inconsistent rewrite rules. The proposed algorithm is implemented in PROLOG. 1 Introduction KnuthBendix completion [ Knuth and Bendix, 1970 ] and its extensions [ Bachmair et al...
Efficient FirstOrder Semantic Deduction Techniques
, 1998
"... Mathematical logic formalizes the process of mathematical reasoning. For centuries, it has been a dream of mathematicians to do mathematical reasoning mechanically. In the TPTP library, one finds thousands of problems from various domains of mathematics such as group theory, number theory, set theor ..."
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Cited by 1 (0 self)
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Mathematical logic formalizes the process of mathematical reasoning. For centuries, it has been a dream of mathematicians to do mathematical reasoning mechanically. In the TPTP library, one finds thousands of problems from various domains of mathematics such as group theory, number theory, set theory, etc. Many of these problems can now be solved with state of the art automated theorem provers. Theorem proving also has applications in artificial intelligence and formal verification. As a formal method, theorem proving has been used to verify the correctness of various hardware and software designs. In this thesis, we propose a novel firstorder theorem proving strategy  ordered semantic hyper linking (OSHL). OSHL is an instancebased theorem proving strategy. It proves firstorder unsatisfiability by generating instances of firstorder clauses and proving the set of instances to be propositionally unsatisfiable. OSHL can use semantics, i.e. domain information, to guide its search. OS...