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92
Theorem Proving Modulo
 Journal of Automated Reasoning
"... Abstract. Deduction modulo is a way to remove computational arguments from proofs by reasoning modulo a congruence on propositions. Such a technique, issued from automated theorem proving, is of much wider interest because it permits to separate computations and deductions in a clean way. The first ..."
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Cited by 75 (14 self)
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Abstract. Deduction modulo is a way to remove computational arguments from proofs by reasoning modulo a congruence on propositions. Such a technique, issued from automated theorem proving, is of much wider interest because it permits to separate computations and deductions in a clean way. The first contribution of this paper is to define a sequent calculus modulo that gives a proof theoretic account of the combination of computations and deductions. The congruence on propositions is handled via rewrite rules and equational axioms. Rewrite rules apply to terms and also directly to atomic propositions. The second contribution is to give a complete proof search method, called Extended Narrowing and Resolution (ENAR), for theorem proving modulo such congruences. The completeness of this method is proved with respect to provability in sequent calculus modulo. An important application is that higherorder logic can be presented as a theory modulo. Applying the Extended Narrowing and Resolution method to this presentation of higherorder logic subsumes full higherorder resolution.
Problem Structure in the Presence of Perturbations
 In Proceedings of the 14th National Conference on AI
, 1997
"... Recent progress on search and reasoning procedures has been driven by experimentation on computationally hard problem instances. Hard random problem distributions are an important source of such instances. Challenge problems from the area of finite algebra have also stimulated research on searc ..."
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Cited by 72 (17 self)
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Recent progress on search and reasoning procedures has been driven by experimentation on computationally hard problem instances. Hard random problem distributions are an important source of such instances. Challenge problems from the area of finite algebra have also stimulated research on search and reasoning procedures. Nevertheless, the relation of such problems to practical applications is somewhat unclear. Realistic problem instances clearly have more structure than the random problem instances, but, on the other hand, they are not as regular as the structured mathematical problems. We propose a new benchmark domain that bridges the gap between the purely random instances and the highly structured problems, by introducing perturbations into a structured domain. We will show how to obtain interesting search problems in this manner, and how such problems can be used to study the robustness of search control mechanisms. Our experiments demonstrate that the performan...
Short Single Axioms for Boolean Algebra
 J. Automated Reasoning
, 2002
"... We present short single equational axioms for Boolean algebra in terms of disjunction and negation and in terms of the Sheffer stroke. Previously known single axioms for these theories are much longer than the ones we present. We show that there is no shorter axiom in terms of the Sheffer stroke tha ..."
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Cited by 21 (11 self)
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We present short single equational axioms for Boolean algebra in terms of disjunction and negation and in terms of the Sheffer stroke. Previously known single axioms for these theories are much longer than the ones we present. We show that there is no shorter axiom in terms of the Sheffer stroke than the ones we present. Automated deduction techniques were used for several different aspects of the work. Keywords: Boolean algebra, Sheffer stroke, single axiom 1. Background and
Learning Search Control Knowledge for Equational Theorem Proving
 Fakultat fur Informatik, Technische Universitat Munchen
, 2001
"... One of the major problems in clausal theorem proving is the control of the proof search. In the presence of equality, this problem is particularly hard, since nearly all stateoftheart systems perform the proof search by saturating a mostly unstructured set of clauses. We describe an approach that ..."
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Cited by 19 (5 self)
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One of the major problems in clausal theorem proving is the control of the proof search. In the presence of equality, this problem is particularly hard, since nearly all stateoftheart systems perform the proof search by saturating a mostly unstructured set of clauses. We describe an approach that enables a superpositionbased prover to pick good clauses for generating inferences based on experiences from previous successful proof searches for other problems. Information about good and bad search decisions (useful and superfluous clauses) is automatically collected from search protocols and represented in the form of annotated clause patterns. At run time, new clauses are compared with stored patterns and evaluated according to the associated information found. We describe our implementation of the system. Experimental results demonstrate that a learned heuristic significantly outperforms the conventional base strategy, especially in domains where enough training examples are available.
A Taxonomy of Parallel Strategies for Deduction
 Annals of Mathematics and Artificial Intelligence
, 1999
"... This paper presents a taxonomy of parallel theoremproving methods based on the control of search (e.g., masterslaves versus peer processes), the granularity of parallelism (e.g., fine, medium and coarse grain) and the nature of the method (e.g., orderingbased versus subgoalreduction) . We anal ..."
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Cited by 14 (1 self)
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This paper presents a taxonomy of parallel theoremproving methods based on the control of search (e.g., masterslaves versus peer processes), the granularity of parallelism (e.g., fine, medium and coarse grain) and the nature of the method (e.g., orderingbased versus subgoalreduction) . We analyze how the di#erent approaches to parallelization a#ect the control of search: while fine and mediumgrain methods, as well as masterslaves methods, generally do not modify the sequential search plan, parallelsearch methods may combine sequential search plans (multisearch) or extend the search plan with the capability of subdividing the search space (distributed search). Precisely because the search plan is modified, the latter methods may produce radically di#erent searches than their sequential base, as exemplified by the first distributed proof of the Robbins theorem generated by the Modified ClauseDi#usion prover Peersmcd. An overview of the state of the field and directions...
Formal proof—theory and practice
 Notices AMS
, 2008
"... Aformal proof is a proof written in a precise artificial language that admits only a fixed repertoire of stylized steps. This formal language is usually designed so that there is a purely mechanical process by which the correctness of a proof in the language can be verified. Nowadays, there are nume ..."
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Cited by 12 (1 self)
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Aformal proof is a proof written in a precise artificial language that admits only a fixed repertoire of stylized steps. This formal language is usually designed so that there is a purely mechanical process by which the correctness of a proof in the language can be verified. Nowadays, there are numerous computer programs known as proof assistants that can check, or even partially construct, formal proofs written in their preferred proof language. These can be considered as practical, computerbased realizations of the traditional systems of formal symbolic logic and set theory proposed as foundations for mathematics. Why should we wish to create formal proofs?
Knowledge Representation and Classical Logic
"... Mathematical logicians had developed the art of formalizing declarative knowledge long before the advent of the computer age. But they were interested primarily in formalizing mathematics. Because of the important role of nonmathematical knowledge in AI, their emphasis was too narrow from the perspe ..."
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Cited by 10 (4 self)
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Mathematical logicians had developed the art of formalizing declarative knowledge long before the advent of the computer age. But they were interested primarily in formalizing mathematics. Because of the important role of nonmathematical knowledge in AI, their emphasis was too narrow from the perspective of knowledge representation, their formal languages were not sufficiently expressive. On the other hand, most logicians were not concerned about the possibility of automated reasoning; from the perspective of knowledge representation, they were often too generous in the choice of syntactic constructs. In spite of these differences, classical mathematical logic has exerted significant influence on knowledge representation research, and it is appropriate to begin this handbook with a discussion of the relationship between these fields. The language of classical logic that is most widely used in the theory of knowledge representation is the language of firstorder (predicate) formulas. These are the formulas that John McCarthy proposed to use for representing declarative knowledge in his advice taker paper [176], and Alan Robinson proposed to prove automatically using resolution [236]. Propositional logic is, of course, the most important subset of firstorder logic; recent