# A view of automated proof checking and proving

Section I Contributions From Artificial Intelligence

## Abstract

Different techniques of automated formal reasoning are described and their performance and requirements on the human user are evaluated. The main trade-off is between autonomy and flexibility in conducting proofs. Examples of the use of techniques and existing systems are given, but not attempt of an exhaustive overview is made. The goal is to provide the reader with an idea of what to look for when selecting an approach for his/her application.

### Keywords

Automated theorem proving proof checking## Preview

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### References

- [1]J.L. Bates and R.L. Constable, Proofs as programs, ACM TOPLAS 7, 1 (1985) 113–136.Google Scholar
- [2]R.S. Boyer and JS. Moore,
*A Computational Logic*, ACM Monograph Series (Academic Press, 1979).Google Scholar - [3]R.S. Boyer and JS. Moore, A verification condition generator for FORTRAN, in:
*The Correctness Problem in Computer Science*, eds. R.S. Boyer and JS. Moore, International Lecture Series in Computer Science (Academic Press, 1981) pp. 9–102.Google Scholar - [4]R.S. Boyer and JS. Moore, Overview of a theorem prover for a computational logic, in:
*8th Conf. on Automated Deduction*, ed. J.H. Siekmann, Lecture Notes in Computer Science 230 (Springer Verlag, 1986) pp. 675–678.Google Scholar - [5]A. Bundy,
*The Computer Modelling of Mathematical Reasoning*(Academic Press, 1983).Google Scholar - [6]S.-C. Chou, Proving elementary geometry theorems using Wu's algorithm, in:
*Automated Theorem Proving: After 25 Years*, eds. W.W. Bledsoe and D.W. Loveland, Contemporary Mathematics, Vol. 29 (American Mathematical Society, 1984) pp. 243–286.Google Scholar - [7]E.M. Clarke, E.A. Emerson and A.P. Sistla, Automatic verification of finite-state concurrent systems using temporal logic specifications: a practical approach,
*Proc. 10th Ann. Symp. on Principles of Programming Languages*, 1983, pp. 117–126.Google Scholar - [8]W.F. Clocksin and C.S. Mellish,
*Programming in Prolog*, 2nd ed. (Springer-Verlag, 1984).Google Scholar - [9]R.L. Constable, T.B. Knoblock and J.L. Bates, Writing programs that construct proofs, Journal of Automated Reasoning 1, 3 (1985) 285–326.Google Scholar
- [10]R.L. Constable et al.,
*Implementing Mathematics with the Nuprl Proof Development System*(Prentice-Hall, 1986).Google Scholar - [11]D.I. Good, The proof of a distributed system in Gypsy, Tech. Rep. #30, Institute for Computing Science, The University of Texas at Austin, Sept. 1982.Google Scholar
- [12]D.I. Good, Mechanical proofs about computer programs, in:
*Mathematical Logic and Programming Languages*, eds. C.A.R. Hoare and J.S. Shepherdson, Series in Computer Science (Prentice-Hall Int., 1985) pp. 55–75.Google Scholar - [13]M.J.C. Gordon, A.J. Milner and C.P. Wadsworth,
*Edinburgh LCF*, Lecture Notes in Computer Science 78 (Springer Verlag, 1979).Google Scholar - [14]D. Harel,
*First-order Dynamic Logic*, Lecture Notes in Computer Science 68 (Springer-Verlag, 1979).Google Scholar - [15]D. Hilbert,
*Foundations of Geometry*(Open Court Publ. Co., 1971); Revised by P. Bernays, 2nd ed.Google Scholar - [16]D.E. Knuth and P. Bendix, Simple world problems in universal algebras, in:
*Computational Problems in Abstract Algebra*, ed. J. Leech (Pergamon Press, 1970) pp. 263–297.Google Scholar - [17]C. Lengauer, On the role of automated theorem proving in the compile-time derivation of concurrency, Journal of Automated Reasoning 1, 1 (1985) 75–101.Google Scholar
- [18]Mathlab Group, MACSYMA Reference Manual, Computer Science Laboratory, Massachusetts Institute of Technology, 1977.Google Scholar
- [19]J. McCarthy, Applications of Circumscription to formalizing common-sense knowledge, Artificial Intelligence 28, 1 (1986) 89–116.Google Scholar
- [20]N.J. Nilsson,
*Principles of Artificial Intelligence*(Tioga Publ. Co., 1980).Google Scholar - [21]L.C. Paulson, Lessons learned from LCF: a survey of natural deduction proofs, Comp. J. 28, 5 (1985) 474–479.Google Scholar
- [22]W. Polak,
*Compiler Specification and Verification*, Lecture Notes in Computer Science 124 (Springer Verlag, 1981).Google Scholar - [23]PRL staff, PRL: Proof Refinement Logic Programmer's Manual, Department of Computer Science, Cornell University, 1984.Google Scholar
- [24]R. Reiter, A logic for default reasoning, Artificial Intelligence 13, 1–2 (1980) 81-132.Google Scholar
- [25]J.A. Robinson, A machine oriented logic based on the resolution principle, J. ACM 12, 1 (1965) 23–41.Google Scholar
- [26]A. Tarski,
*A Decision Method for Elementary Algebra and Geometry*, 2nd ed. (University of California Press, 1951).Google Scholar - [27]D.S. Touretzky,
*LISP — A Gentle Introduction to Symbolic Computation*(Harper & Row, 1983).Google Scholar - [28]L. Wos, S. Winker and E. Lusk, An automated reasoning system,
*AFIPS Conf. Proc. 50*, National Computer Conference, 1981, pp. 697–702.Google Scholar - [29]L. Wos, Solving open questions with an automated theorem-proving program, in:
*6th Conf. on Automated Deduction*, ed. D.W. Loveland, Lecture Notes in Computer Science 138 (Springer-Verlag, 1982) pp. 1–31.Google Scholar - [30]L. Wos, R. Overbeek, E. Lusk and J. Boyle,
*Automated Reasoning: Introduction and Applications*(Prentice-Hall, 1984).Google Scholar - [31]L. Wos and S. Winker, Open Questions Solved with the Assistance of AURA. In:
*Automated Theorem Proving: After 25 Years*, eds. W.W. Bledsoe and D.W. Loveland, Contemporary Mathematics, Vol. 29, American Mathematical Society, 1984, pp. 73–88.Google Scholar - [32]W.-T. Wu, On the decision problem and the mechanization of theorem-proving in elementary geometry, in:
*Automated Theorem Proving: After 25 Years*, eds. W.W. Bledsoe and D.W. Loveland, Contemporary Mathematics, Vol. 29, American Mathematical Society, 1984, pp. 213–234.Google Scholar - [33]L.A. Zadeh, A theory of approximate reasoning, in:
*Machine Intelligence 9*, eds. J. Hayes, D. Michie and L. Mikulich (Elsevier, 1979) pp. 149–194.Google Scholar

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© J.C. Baltzer AG, Scientific Publishing Company 1988