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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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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 system’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,
Termination of Nested and Mutually Recursive Algorithms
, 1996
"... This paper deals with automated termination analysis for functional programs. Previously developed methods for automated termination proofs of functional programs often fail for algorithms with nested recursion and they cannot handle algorithms with mutual recursion. We show that termination proofs ..."
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Cited by 40 (9 self)
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This paper deals with automated termination analysis for functional programs. Previously developed methods for automated termination proofs of functional programs often fail for algorithms with nested recursion and they cannot handle algorithms with mutual recursion. We show that termination proofs for nested and mutually recursive algorithms can be performed without having to prove the correctness of the algorithms simultaneously. Using this result, nested and mutually recursive algorithms do no longer constitute a special problem and the existing methods for automated termination analysis can be extended to nested and mutual recursion in a straightforward way. We give some examples of algorithms whose termination can now be proved automatically (including wellknown challenge problems such as McCarthy's f_91 function).
A Theorem Prover for a Computational Logic
, 1990
"... We briefly review a mechanical theoremprover for a logic of recursive functions over finitely generated objects including the integers, ordered pairs, and symbols. The prover, known both as NQTHM and as the BoyerMoore prover, contains a mechanized principle of induction and implementations of line ..."
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Cited by 26 (0 self)
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We briefly review a mechanical theoremprover for a logic of recursive functions over finitely generated objects including the integers, ordered pairs, and symbols. The prover, known both as NQTHM and as the BoyerMoore prover, contains a mechanized principle of induction and implementations of linear resolution, rewriting, and arithmetic decision procedures. We describe some applications of the prover, including a proof of the correct implementation of a higher level language on a microprocessor defined at the gate level. We also describe the ongoing project of recoding the entire prover as an applicative function within its own logic.
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 22 (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 Fermat’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.
Proving termination of normalization functions for conditional expressions
 Journal of Automated Reasoning
, 1986
"... ..."
Textbook Examples of Recursion
 In Artificial Intelligence and Theory of Computation
, 1991
"... Abstract. We discuss properties of recursive schemas related to McCarthy’s “91 function” and to Takeuchi’s triple recursion. Several theorems are proposed as interesting candidates for machine verification, and some intriguing open questions are raised. ..."
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Abstract. We discuss properties of recursive schemas related to McCarthy’s “91 function” and to Takeuchi’s triple recursion. Several theorems are proposed as interesting candidates for machine verification, and some intriguing open questions are raised.
Inductive invariants for nested recursion
 Theorem Proving in Higher Order Logics (TPHOLS'03), volume 2758 of LNCS
, 2003
"... Abstract. We show that certain inputoutput relations, termed inductive invariants are of central importance for termination proofs of algorithms defined by nested recursion. Inductive invariants can be used to enhance recursive function definition packages in higherorder logic mechanizations. We d ..."
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Cited by 5 (2 self)
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Abstract. We show that certain inputoutput relations, termed inductive invariants are of central importance for termination proofs of algorithms defined by nested recursion. Inductive invariants can be used to enhance recursive function definition packages in higherorder logic mechanizations. We demonstrate the usefulness of inductive invariants on a large example of the BDD algorithm Apply. Finally, we introduce a related concept of inductive fixpoints with the property that for every functional in higherorder logic there exists a largest partial function that is such a fixpoint. 1
Checking Termination of Queries to Logic Programs
, 1996
"... Termination of programs is known to be undecidable. However in the case of logic programs, where the only possible cause for nontermination is infinite recursion, termination can actually be proved automatically for a large class of programs. This paper describes an algorithm for automatically chec ..."
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Cited by 3 (3 self)
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Termination of programs is known to be undecidable. However in the case of logic programs, where the only possible cause for nontermination is infinite recursion, termination can actually be proved automatically for a large class of programs. This paper describes an algorithm for automatically checking termination of queries to logic programs. Given a program and query the algorithm either answers that the query terminates or that there may be nontermination due to infinite recursion. The algorithm can use any norm of a wide family of norms for proving termination. It has been implemented in SICStus Prolog. The implementation can handle automatically most of the examples we encountered in the literature on termination of logic programs, and about half the programs in a collection of benchmarks, which were originally used for purposes other than termination (for 11 of the 24 programs termination can be decided automatically and for 3 more it can be decided after suitable transformatio...
Assuming Termination
"... A requirement for the admission of a recursive function definition in ACL2 is a proof that the recursion terminates. Establishing termination involves identifying a wellfounded relation and an appropriate measure function that decreases according to the wellfounded relation with each recursive cal ..."
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A requirement for the admission of a recursive function definition in ACL2 is a proof that the recursion terminates. Establishing termination involves identifying a wellfounded relation and an appropriate measure function that decreases according to the wellfounded relation with each recursive call. Depending on the domain this process may prove difficult, unnecessary or even impossible. Manolios and Moore introduced a method for admitting tailrecursive function definitions that does not require the specification of a measure. Their method, however, does not produce an induction scheme. We present an extension of their method that enables the admission of arbitrary recursive function definitions and their associated induction schemes augmented with a termination check and justified by a partial measure. We demonstrate the use of this method by defining the tarai function and proving by induction its unwinding under the assumption that it terminates.