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Rewrite Techniques for Transitive Relations
 IN PROC., 9TH IEEE SYMPOSIUM ON LOGIC IN COMPUTER SCIENCE
, 1994
"... We propose inference systems for dealing with transitive relations in the context of resolutiontype theorem proving. These inference mechanisms are based on standard techniques from term rewriting and represent a refinement of chaining methods. We establish their refutational completeness and al ..."
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Cited by 37 (5 self)
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We propose inference systems for dealing with transitive relations in the context of resolutiontype theorem proving. These inference mechanisms are based on standard techniques from term rewriting and represent a refinement of chaining methods. We establish their refutational completeness and also prove their compatibility with the usual simplification techniques used in rewritebased theorem provers. A key to the practicality of chaining techniques is the extent to which socalled variable chainings can be restricted. We demonstrate that rewrite techniques considerably restrict variable chaining, though we also show that they cannot be completely avoided for transitive relations in general. If the given relation satisfies additional properties, such as symmetry, further restrictions are possible. In particular, we discuss (partial) equivalence relations and congruence relations.
Ordered Chainings for Total Orderings
, 1995
"... We design new inference systems for total orderings by applying rewrite techniques to chaining calculi. Equality relations may either be specified axiomatically or built into the deductive calculus via paramodulation or superposition. We demonstrate that our inference systems are compatible with ..."
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Cited by 21 (5 self)
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We design new inference systems for total orderings by applying rewrite techniques to chaining calculi. Equality relations may either be specified axiomatically or built into the deductive calculus via paramodulation or superposition. We demonstrate that our inference systems are compatible with a concept of (global) redundancy for clauses and inferences that covers such widely used simplification techniques as tautology deletion, subsumption, and demodulation. A key to the practicality of chaining techniques is the extent to which socalled variable chainings can be restricted. Syntactic ordering restrictions on terms and the rewrite techniques which account for their completeness considerably restrict variable chaining. We show that variable elimination is an admissible simplification techniques within our redundancy framework, and that consequently for dense total orderings without endpoints no variable chaining is needed at all.
Linear and UnitResulting Refutations for Horn Theories
 Journal of Automated Reasoning
, 1995
"... . We present a new transformation method by which a given Horn theory is transformed in such a way that resolution derivations can be carried out which are both linear (in the sense of Prologs SLDresolution) and unitresulting (i.e. the resolvents are unit clauses). This is not trivial since altho ..."
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Cited by 2 (0 self)
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. We present a new transformation method by which a given Horn theory is transformed in such a way that resolution derivations can be carried out which are both linear (in the sense of Prologs SLDresolution) and unitresulting (i.e. the resolvents are unit clauses). This is not trivial since although both strategies alone are complete, their na ve combination is not. Completeness is recovered by our method through a completion procedure in the spirit of KnuthBendix completion, however with different ordering criteria. A powerful redundancy criterion helps to find a finite system quite often. The transformed theory can be used in combination with linear calculi such as e.g. (theory) model elimination to yield sound, complete and efficient calculi for full first order clause logic over the given Horn theory. As an example application, our method discovers a generalization of the wellknown linear paramodulation calculus for the combined theory of equality and strict orderings. The met...
Proving Ground Completeness of Resolution by Proof Planning
, 1997
"... A lot of the human ability to prove hard mathematical theorems can be ascribed to a problemspecific problem solving knowhow. Such knowledge is intrinsically incomplete. In order to prove related problems human mathematicians, however, can go beyond the acquired knowledge by adapting their knowhow ..."
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Cited by 2 (1 self)
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A lot of the human ability to prove hard mathematical theorems can be ascribed to a problemspecific problem solving knowhow. Such knowledge is intrinsically incomplete. In order to prove related problems human mathematicians, however, can go beyond the acquired knowledge by adapting their knowhow to new related problems. These two aspects, having rich experience and extending it by need, can be simulated in a proof planning framework: the problemspecific reasoning knowledge is represented in form of declarative planning operators, called methods; since these are declarative, they can be mechanically adapted to new situations by socalled metamethods. In this contribution we apply this framework to two prominent proofs in theorem proving, first, we present methods for proving the ground completeness of binary resolution, which essentially correspond to key lemmata, and then, we show how these methods can be reused for the proof of the ground completeness of lock resolution.