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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 14 (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.
A Modelbased Completeness Proof of Extended Narrowing and Resolution
, 2000
"... We give a proof of refutational completeness for Extended Narrowing And Resolution (ENAR), a calculus introduced by Dowek, Hardin and Kirchner in the context of Theorem Proving Modulo. ENAR integrates narrowing with respect to a set of rewrite rules on propositions into automated firstorder theorem ..."
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Cited by 8 (3 self)
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We give a proof of refutational completeness for Extended Narrowing And Resolution (ENAR), a calculus introduced by Dowek, Hardin and Kirchner in the context of Theorem Proving Modulo. ENAR integrates narrowing with respect to a set of rewrite rules on propositions into automated firstorder theorem proving by resolution. Our proof allows to impose ordering restrictions on ENAR and provides general redundancy criteria, which are crucial for finding nontrivial proofs. On the other hand, it requires conuence and termination of the rewrite system, and in addition the existence of a wellfounded ordering on propositions that is compatible with rewriting, compatible with ground inferences, total on ground clauses, and has some additional technical properties. We show that such an ordering exists for a fragment of set theory. This example falls outside the scope of a previous completeness proof for ENAR that requires cut elimination for a sequent calculus modulo the rewrite rules....
Paramodulation with BuiltIn Abelian Groups
 in `15th IEEE Symposium on Logic in Computer Science (LICS
, 2000
"... A new technique is presented for superposition with firstorder clauses with builtin abelian groups (AG). Compared with previous approaches, it is simpler, and no inferences with the AG axioms or abstraction rules are needed. Furthermore, AGunification is used instead of the computationally more ex ..."
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Cited by 6 (4 self)
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A new technique is presented for superposition with firstorder clauses with builtin abelian groups (AG). Compared with previous approaches, it is simpler, and no inferences with the AG axioms or abstraction rules are needed. Furthermore, AGunification is used instead of the computationally more expensive unification modulo associativity and commutativity. Due to the simplicity and restrictiveness of our inference system, its compatibility with redundancy notions and constraints, and the fact that standard term orderings like RPO can be used, we believe that our technique will become the method of choice for practice, as well as a basis for new theoretical developments like logicbased complexity and decidability analysis. Keywords: term rewriting, automated deduction. 1 Introduction It is crucial for the performance of a deduction system that it incorporates specialized techniques to work efficiently with standard algebraic theories, since a nave handling of some axioms (like assoc...
Cancellative Abelian Monoids in Refutational Theorem Proving
 PHD THESIS, INSTITUT FÜR INFORMATIK, UNIVERSITÄT DES SAARLANDES
, 1997
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Superposition with Completely Builtin Abelian Groups
"... A new technique is presented for superposition with firstorder clauses with builtin abelian groups (AG). Compared with previous approaches, it is simpler, and AGunification is used instead of the computationally more expensive unification modulo associativity and commutativity. Furthermore, n ..."
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Cited by 3 (0 self)
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A new technique is presented for superposition with firstorder clauses with builtin abelian groups (AG). Compared with previous approaches, it is simpler, and AGunification is used instead of the computationally more expensive unification modulo associativity and commutativity. Furthermore, no inferences with the AG axioms or abstraction rules are needed; in this sense this is the first approach where AG is completely built in. 1.
Deriving Theory Superposition Calculi from Convergent Term Rewriting Systems
, 1999
"... We show how to derive refutationally complete ground superposition calculi systematically from convergent term rewriting systems for equational theories, in order to make automated theorem proving in these theories more eective. In particular we consider abelian groups and commutative rings. Thes ..."
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Cited by 2 (1 self)
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We show how to derive refutationally complete ground superposition calculi systematically from convergent term rewriting systems for equational theories, in order to make automated theorem proving in these theories more eective. In particular we consider abelian groups and commutative rings. These are dicult for automated theorem provers, since their axioms of associativity, commutativity, distributivity and the inverse law can generate many variations of the same equation. For these theories ordering restrictions can be strengthened so that inferences apply only to maximal summands, and superpositions into the inverse law that move summands from one side of an equation to the other can be replaced by an isolation rule that isolates the maximal terms on one side. Additional inferences arise from superpositions of extended clauses, but we can show that most of these are redundant. In particular, none are needed in the case of abelian groups, and at most one for any pair of ...
Exploring the Domain of Residue Classes
, 2000
"... We report on a major case study on combining proof planning with computer algebra systems. We construct proofs for basic algebraic properties of residue classes as well as for isomorphisms between residue classes using different proving techniques, which are implemented as strategies in a multistra ..."
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We report on a major case study on combining proof planning with computer algebra systems. We construct proofs for basic algebraic properties of residue classes as well as for isomorphisms between residue classes using different proving techniques, which are implemented as strategies in a multistrategy proof planner. We show how these techniques help to successfully derive proofs in our domain and explain how the search space of the proof planner can be drastically reduced by employing computations of two computer algebra systems during the planning process. Moreover, we discuss the results of experiments we conducted which give evidence that with the help of the computer algebra systems the planner is able to solve problems for which it would fail to create a proof otherwise.
5 On FirstOrder ModelBased Reasoning
, 2015
"... All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately. ..."
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All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately.