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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 107 (9 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 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 "black box " 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 "less than " relation, the Peano addition and subtraction functions, Peano constants,
The BoyerMoore Theorem Prover and Its Interactive Enhancement
, 1995
"... . The socalled "BoyerMoore Theorem Prover" (otherwise known as "Nqthm") has been used to perform a variety of verification tasks for two decades. We give an overview of both this system and an interactive enhancement of it, "PcNqthm," from a number of perspectives. First we introduce the logic in ..."
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. The socalled "BoyerMoore Theorem Prover" (otherwise known as "Nqthm") has been used to perform a variety of verification tasks for two decades. We give an overview of both this system and an interactive enhancement of it, "PcNqthm," from a number of perspectives. First we introduce the logic in which theorems are proved. Then we briefly describe the two mechanized theorem proving systems. Next, we present a simple but illustrative example in some detail in order to give an impression of how these systems may be used successfully. Finally, we give extremely short descriptions of a large number of applications of these systems, in order to give an idea of the breadth of their uses. This paper is intended as an informal introduction to systems that have been described in detail and similarly summarized in many other books and papers; no new results are reported here. Our intention here is merely to present Nqthm to a new audience. This research was supported in part by ONR Contract N...
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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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.
A mechanical proof of the unsolvability of the halting problem
 JACM
, 1984
"... We describe a proof by a computer program of the unsolvability of the halting problem. The halting problem is posed in a constructive, formal language. The computational paradigm formalized is Pure LISP, not Turing machines. The machine was led to the proof by the authors, who suggested certain func ..."
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Cited by 15 (8 self)
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We describe a proof by a computer program of the unsolvability of the halting problem. The halting problem is posed in a constructive, formal language. The computational paradigm formalized is Pure LISP, not Turing machines. The machine was led to the proof by the authors, who suggested certain function definitions and stated certain intermediate lemmas. The machine checked that every suggested definition was admissible and the machine proved the main theorem and every lemma. We believe this is the first instance of a machine checking that a given problem is not solvable by machine.
Program verification
 Journal of Automated Reasoning
, 1985
"... Computer programs may be regarded as formal mathematical objects whose properties are subject to mathematical proof. Program verification is the use of formal, mathematical techniques to debug software and software specifications. 1. Code Verification How are the properties of computer programs prov ..."
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Cited by 14 (4 self)
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Computer programs may be regarded as formal mathematical objects whose properties are subject to mathematical proof. Program verification is the use of formal, mathematical techniques to debug software and software specifications. 1. Code Verification How are the properties of computer programs proved? We discuss three approaches in this article: inductive invariants, functional semantics, and explicit semantics. Because the first approach has received by far the most attention, it has produced the most impressive results to date. However, the field is now moving away from the inductive invariant approach. 1.1. Inductive Assertions The socalled FloydHoare inductive assertion method of program verification [25, 33] has its roots in the classic Goldstine and von Neumann reports [53] and handles the usual kind of programming language, of which FORTRAN is perhaps the best example. In this style of verification, the specifier "annotates " certain points in the program with mathematical assertions that are supposed to describe relations that hold between the program variables and the initial input values each time "control " reaches the annotated point. Among these assertions are some that characterize acceptable input and the desired output. By exploring all possible paths from one assertion to the next and analyzing the effects of intervening program statements it is possible to reduce the correctness of the program to the problem of proving certain derived formulas called verification conditions. Below we illustrate the idea with a simple program for computing the factorial of its integer input N flowchart assertion start with input(N) input N A: = 1 N = 0 yes stop with? answer A
Verification of the MillerRabin Probabilistic Primality Test
, 2003
"... Using the HOL theorem prover, we apply our formalization of probability theory to specify and verify the MillerRabin probabilistic primality test. The version of the test commonly found in algorithm textbooks implicitly accepts probabilistic termination, but our own verified implementation satisfie ..."
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Cited by 14 (3 self)
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Using the HOL theorem prover, we apply our formalization of probability theory to specify and verify the MillerRabin probabilistic primality test. The version of the test commonly found in algorithm textbooks implicitly accepts probabilistic termination, but our own verified implementation satisfies the stronger property of guaranteed termination. Completing the proof of correctness requires a significant body of group theory and computational number theory to be formalized in the theorem prover. Once verified, the primality test can either be executed in the logic (using rewriting) and used to prove the compositeness of numbers, or manually extracted to Standard ML and used to find highly probable primes.
Functional correctness proofs of encryption algorithms
 In Proceedings of 12th Conference on Logic for Programming Artificial Intelligence and Reasoning (LPAR 2005), number 3835 in LNAI
, 2005
"... Abstract. We discuss a collection of mechanized formal proofs of symmetric key block encryption algorithms (AES, MARS, Twofish, RC6, Serpent, IDEA, and TEA), performed in an implementation of higher order logic. For each algorithm, functional correctness, namely that decryption inverts encryption, i ..."
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Cited by 10 (3 self)
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Abstract. We discuss a collection of mechanized formal proofs of symmetric key block encryption algorithms (AES, MARS, Twofish, RC6, Serpent, IDEA, and TEA), performed in an implementation of higher order logic. For each algorithm, functional correctness, namely that decryption inverts encryption, is formally proved by a simple but effective proof methodology involving application of invertibility lemmas in the course of symbolic evaluation. Block ciphers are then lifted to the encryption of arbitrary datatypes by using modes of operation to encrypt lists of bits produced by a polytypic encoding method. 1
Proofchecking, theoremproving and program verification
, 1983
"... This article consists of three parts: a tutorial introduction to a computer program that proves theorems by induction; a brief description of recent applications of that theoremprover; and a discussion of several nontechnical aspects of the problem of building automatic theoremprovers. The theorem ..."
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Cited by 8 (0 self)
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This article consists of three parts: a tutorial introduction to a computer program that proves theorems by induction; a brief description of recent applications of that theoremprover; and a discussion of several nontechnical aspects of the problem of building automatic theoremprovers. The theoremprover described has proved theorems such as the uniqueness of prime factorizations, Fermat’s theorem, and the recursive unsolvability of the halting problem. The article is addressed to those who know nothing about automatic theoremproving but would like a glimpse of one such system. This article definitely does not provide a balanced view of all automatic theoremproving, the literature of which is already rather large and technical. 1Good places to start on the technical literature are [Loveland 78] and [Bledsoe 77]. Nor do we describe the details of our theoremproving system, but they can be found in the books, articles, and technical reports that we reference. In our opinion, progress in automatic theoremproving is largely a function of the mathematical ability of those attempting to build such systems. We encourage good mathematicians to work in the field.
A mechanized program verifier
 In IFIP Working Conference on the Program Verifier Challenge
, 2005
"... Abstract. In my view, the “verification problem ” is the theorem proving problem, restricted to a computational logic. My approach is: adopt a functional programming language, build a general purpose formal reasoning engine around it, integrate it into a program and proof development environment, an ..."
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Abstract. In my view, the “verification problem ” is the theorem proving problem, restricted to a computational logic. My approach is: adopt a functional programming language, build a general purpose formal reasoning engine around it, integrate it into a program and proof development environment, and apply it to model and verify a wide variety of computing artifacts, usually modeled operationally within the functional programming language. Everything done in this approach is software verification since the models are runnable programs in a subset of an ANSI standard programming language (Common Lisp). But this approach is of interest to proponents of other approaches (e.g., verification of procedural programs or synthesis) because of the nature of the mathematics of computing. I summarize the progress so far using this approach, sketch the key research challenges ahead and describe my vision of the role and shape of a useful verification system. 1