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A Mechanization of Strong Kleene Logic for Partial Functions
 PROCEEDINGS OF THE 12TH CADE
, 1994
"... Even though it is not very often admitted, partial functions do play a significant role in many practical applications of deduction systems. Kleene has already given a semantic account of partial functions using threevalued logic decades ago, but there has not been a satisfactory mechanization. ..."
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Even though it is not very often admitted, partial functions do play a significant role in many practical applications of deduction systems. Kleene has already given a semantic account of partial functions using threevalued logic decades ago, but there has not been a satisfactory mechanization. Recent years have seen a thorough investigation of the framework of manyvalued truthfunctional logics. However, strong Kleene logic, where quantification is restricted and therefore not truthfunctional, does not fit the framework directly. We solve this problem by applying recent methods from sorted logics. This paper presents a resolution calculus that combines the proper treatment of partial functions with the efficiency of sorted calculi.
Superposition with Equivalence Reasoning and Delayed Clause Normal Form Transformation
 Inf. Comput
, 2003
"... This paper describes a superposition calculus where quantifiers are eliminated lazily. Superposition and simplification inferences may employ equivalences that have arbitrary formulas at their smaller side. A closely related calculus is implemented in the Saturate system and has shown useful on many ..."
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Cited by 14 (0 self)
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This paper describes a superposition calculus where quantifiers are eliminated lazily. Superposition and simplification inferences may employ equivalences that have arbitrary formulas at their smaller side. A closely related calculus is implemented in the Saturate system and has shown useful on many examples, in particular in set theory. The paper presents a completeness proof and reports on practical experience obtained with the Saturate system.
On the reconstruction of proofs in distributed theorem proving: a modified ClauseDiffusion method
 J. of Symbolic Computation
, 1996
"... Proof reconstruction is the operation of extracting the computed proof from the trace of a theoremproving run. We study the problem of proof reconstruction in distributed theorem proving: because of the distributed nature of the derivation and especially because of deletions of clauses by contracti ..."
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Cited by 8 (4 self)
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Proof reconstruction is the operation of extracting the computed proof from the trace of a theoremproving run. We study the problem of proof reconstruction in distributed theorem proving: because of the distributed nature of the derivation and especially because of deletions of clauses by contraction, it may happen that a deductive process generates the empty clause, but does not have all the necessary information to reconstruct the proof. We analyze this problem and we present a method for distributed theorem proving, called Modified ClauseDiffusion, which guarantees that the deductive process that generates the empty clause will be able to reconstruct the distributed proof. This result is obtained without imposing a centralized control on the deductive processes or resorting to a round of postprocessing with ad hoc communication. We prove that Modified ClauseDiffusion is fair (hence complete) and guarantees proof reconstruction. First we define a set of conditions, next we prove that they are sufficient for proof reconstruction, then we show that Modified ClauseDiffusion satisfies them. Fairness is proved in the same way, which has the advantage that the sufficient conditions provide a treatment of the problem relevant for distributed theorem proving in general. 1.
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....
On the Representation of Parallel Search in Theorem Proving
 Johannes Kepler Universitat
, 1997
"... This extended abstract summarizes two contributions from ongoing work on parallel search in theorem proving. First, we give a framework of definitions for parallel theorem proving, including inference system, communication operators, parallel search plan, subdivision function, parallel strategy, ..."
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This extended abstract summarizes two contributions from ongoing work on parallel search in theorem proving. First, we give a framework of definitions for parallel theorem proving, including inference system, communication operators, parallel search plan, subdivision function, parallel strategy, parallel derivation, fairness and propagation of redundancy for parallel derivations. A notion of a parallel strategy being a parallelization of a sequential strategy, and a theorem establishing a general relation between sequential fairness and parallel fairness are also given. Second, we extend our approach to the modelling of search to parallel search, covering inferences (expansion and contraction), behaviour of the search plan, subdivision of the search space and communication among the processes. This model allows us to study the behavior of many search processes on a single marked search graph. In the full paper, we plan to extend our methodology for the measure of the complex...
A model and a first analysis of distributedsearch contractionbased strategies ∗
"... While various approaches to parallel theorem proving have been proposed, their usefulness is evaluated only empirically. This research is a contribution towards the goal of machineindependent analysis of theoremproving strategies. This paper considers clausal contractionbased strategies and their ..."
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While various approaches to parallel theorem proving have been proposed, their usefulness is evaluated only empirically. This research is a contribution towards the goal of machineindependent analysis of theoremproving strategies. This paper considers clausal contractionbased strategies and their parallelization by distributed search, with subdivision of the search space and propagation of clauses by messagepassing (e.g., à la ClauseDiffusion). A model for the representation of the parallel searches produced by such strategies is presented, and the boundedsearchspaces approach to the measurement of search complexity in infinite search spaces is extended to distributed search. This involves capturing both its advantages, e.g., the subdivision of work, and disadvantages, e.g., the cost of communication, in terms of search space. These tools are applied to compare the evolution of the search space of a contractionbased strategy with that of its parallelization in the above sense. 1.