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Integrating decision procedures into heuristic theorem provers: A case study of linear arithmetic
 Machine Intelligence
, 1988
"... We discuss the problem of incorporating into a heuristic theorem prover a decision procedure for a fragment of the logic. An obvious goal when incorporating such a procedure is to reduce the search space explored by the heuristic component of the system, as would be achieved by eliminating from the ..."
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Cited by 118 (10 self)
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We discuss the problem of incorporating into a heuristic theorem prover a decision procedure for a fragment of the logic. An obvious goal when incorporating such a procedure is to reduce the search space explored by the heuristic component of the system, as would be achieved by eliminating from the systemâ€™s data base some explicitly stated axioms. For example, if a decision procedure for linear inequalities is added, one would hope to eliminate the explicit consideration of the transitivity axioms. However, the decision procedure must then be used in all the ways the eliminated axioms might have been. The difficulty of achieving this degree of integration is more dependent upon the complexity of the heuristic component than upon that of the decision procedure. The view of the decision procedure as a &quot;black box &quot; is frequently destroyed by the need pass large amounts of search strategic information back and forth between the two components. Finally, the efficiency of the decision procedure may be virtually irrelevant; the efficiency of the final system may depend most heavily on how easy it is to communicate between the two components. This paper is a case study of how we integrated a linear arithmetic procedure into a heuristic theorem prover. By linear arithmetic here we mean the decidable subset of number theory dealing with universally quantified formulas composed of the logical connectives, the identity relation, the Peano &quot;less than &quot; relation, the Peano addition and subtraction functions, Peano constants,
Proof checking the RSA public key encryption algorithm
 American Mathematical Monthly
, 1984
"... The authors describe the use of a mechanical theoremprover to check the published proof of the invertibility of the public key encryption algorithm of Rivest, Shamir and Adleman: (M mod n) mod N=M, provided n is the product of two distinct primes p and q, M<n, and e and d are multiplicative inve ..."
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Cited by 21 (9 self)
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The authors describe the use of a mechanical theoremprover to check the published proof of the invertibility of the public key encryption algorithm of Rivest, Shamir and Adleman: (M mod n) mod N=M, provided n is the product of two distinct primes p and q, M<n, and e and d are multiplicative inverses in the ring of integers modulo (p1)*(q1). Among the lemmas proved mechanically and used in the main proof are many familiar theorems of number theory, including Fermatâ€™s theorem: M mod p=1, when p M. The axioms underlying the proofs are those of Peano arithmetic and ordered pairs. The development of mathematics toward greater precision has led, as is well known, to the formalization of large tracts of it, so that one can prove any theorem using nothing but a few mechanical rules. Godel [11] But formalized mathematics cannot in practice be written down in full, and therefore we must have confidence in what might be called the common sense of the mathematician... We shall therefore very quickly abandon formalized mathematics... Bourbaki [1] 1.