Article

Free access

Automated theorem proving: mapping logic into AI

Author:

D W LovelandAuthors Info & Claims

ISMIS '86: Proceedings of the ACM SIGART international symposium on Methodologies for intelligent systems

Pages 214 - 229

https://doi.org/10.1145/12808.12833

Published: 01 December 1986 Publication History

Abstract

Logic can be defined as the formal study of reasoning; if we replace “formal” by “mechanical” we can place almost the entire set of methodologies used in the field of automated theorem proving (ATP) within the scope of logic. Because of the goals of ATP, if not always the methodologies, ATP has been considered to be within the domain of AI. We explore the methodologies of ATP, including the logics that underlie the theorem provers, and discuss some of the mechanisms that utilize these logics. These include term rewriting systems, mathematical induction, inductionless induction and even mixed integer programming. The ATP field, via resolution, has even provided the foundation for an exciting AI and database programming language, PROLOG. We conclude with a new method for extending the PROLOG system to work with non-Horn clause sets within a positive logic format, particularly simple for “slightly non-Horn” clause sets.

References

[1]

Loveland, D.W. Automated theorem proving' a quarter century review. In {2}, 1-45.

[2]

Bledsoe, W.W. and D.W. Loveland (Eds.). Automating Theorem Proving: After 25 Years. Contemporary Mathematics Series, Vol. 29, Amer. Math. Soc., 1984.

[3]

Boyer, R.S. and J.S. Moore. A Computational Logic. Academic Press, New York, 1979.

[4]

Mendelson, E. Introduction to Mathematical Logic. van Nostrand, 2nd edition, 1979.

Digital Library

[5]

Suppes, P. Introduction to Logic, van Nostrand, 1957.

[6]

Chang, C. and R.C. Lee. Symbolic Logic and Mechanical Theorem Proving. Academic Press, New York, 1973.

Digital Library

[7]

Loveland, D.W. Automated Theorem Proving: A Logical Basis. North-Holland, Amsterdam, 1978.

Digital Library

[8]

Wos, L., R. Overbeek, E. Lusk and T. Boyle. Automated Reasoning: Introduction and Applications. Prentice-Hall, Englewood Cliffs, NJ, 1984.

Digital Library

[9]

Bibel, W. Automated Theorem Proving. Vieweg Verlag. Braunschweig, 1982.

[10]

Cohen, P.J. Set Theory and the Continuum Hypothesis. W.A. Benjamin, New York, 1966.

[11]

Newell, A., J.C. Shaw and H.A. Simon. Empirical explorations of the logic theory machine' a case study in heuristics. Proc. Western Joint Computer Conf., 1956, 218-239. Also in Computers and Thought (Feigenbaum and Feldman, Eds.), McGraw-Hill, 1963, 134-152.

Digital Library

[12]

Blair, C.E., R.G. Jeroslow and J.K. Lowe. Some results and experiments in programming techniques for propositional logic. To appear in Computer and Operations Research.

Digital Library

[13]

Jeroslow, R.G. Computation-oriented reduction of predicate calculus to propositional logic. Submitted for publication.

[14]

Tseitin, G. On the complexity of derivations in propositional calculus. Studies in Constructive Mathematics and Mathematical Logic - Part II (Slisenko, Ed.), Consultants Bureau, New York, 1970, 115-125. Reprinted in Automation of Reasoning 2: Classical Papers on Computational Logic (Seikmann and Wrightson, Eds.) Springer-Verlag, Berlin, 1983, 466-483.

[15]

Wos, L. Automated reasoning: basic research problems. Report ANL/MCS- TM-67. Argonne National Lab., Argonne, IL, March 1986.

[16]

Lusk, E.L., W.W. McCune and R.H. Overbeek. Logic machine architecture: (I) kernel functions (II) inference mechanisms. Proc. Sixth Conf. on Auto. Deduction (Loveland, Ed.), Lecture Notes in Comp. Sci. 138, Springer-Verlag, Berlin, June 1982, 70-108.

Digital Library

[17]

Kowalski, R. A proof procedure based on connection graphs. Jour. Assoc. Cornput. Mach., 1975.

Digital Library

[18]

Shostak, R.E. Refutation graphs. Artif. Intell., 1976.

Digital Library

[19]

Andrews, P. Refutations by matings. IEEE Trans. on Computers, C-25, 1976, 801-807.

Digital Library

[20]

Bl'~ius, K., N. Eininger, J. Siekmann, G. Smolke, A. Herold and C. Walthur. The Markgraf Karl refutation procedure (Fall 1981). Proc. Seventh Intern. Joint Conf. on Artif. Intell., Aug. 1981, 511-518.

[21]

Robinson, J.A. A machine-oriented logic based on the resolution principle. Jour. Assoc. for Comput. Mach., 1965, 23-41.

Digital Library

[22]

Haken, A. Resolution takes more than polynomial time. Ph.D. thesis, Univ. of illinois, Urbana-Champaign, IL, 1983.

[23]

Miller, D. Expansion tree proofs and their conversion to natural deduction proofs. Proc. Seventh Conf. on Auto. Deduction (Shosta.k, Ed.), Lecture Notes in Com- ' purer Science, Springer-Verlag, Berlin, May, 1984, 375-393.

Digital Library

[24]

Bledsoe, W.W. Some automatic proofs in analysis. In {2}, 89-118.

[25]

Suppes, P. University-level computer-assisted instruction at Stanford' 1968-1980. Inst. for Math. Studies in the Social Sci., Stanford Univ., Stanford, CA., 1981.

[26]

McDonald, J. and P. Suppes. Student use of an interactive theorem prover. In {2}, 015-360.

[27]

Knuth, D.E. and P.B. Bendix. Simple word problems in universal algebra. Combinatorial Problem in Abstract Algebras (Leech, Ed.), Pergamon, New York, 1970, 263-270.

[28]

Stickel, M.E. A case study of theorem proving by the Knuth-Bendix method discovering that x a~x implies ring commutativity. Proc. Seventh Conf. on Auto. Deduction (Shostak, Ed.), Lecture Notes in Computer Science, Springer-Verlag. Berlin, May, 1984, 248-258.

Digital Library

[29]

Bledsoe, W.W. Non-resolution theorem proving. Artif. Intell., 1977, 1-35. Also in Readings in Artif. Intell. (Webber and Nilsson, Eds.), Tioga, Palo Alto, 1981, 91-108.

[30]

Nelson, G. and D. Oppen. Fast decision procedures based on congruence closure. Jour. Assoc. for Comput. Math., 1980, 356-364.

Digital Library

[31]

Nelson, G. and D. Oppen. Simplification by cooperating decision procedures. ACM Trans. on Progr. Lang. and Systems, 1979.

Digital Library

[32]

Shostak, R.E. Deciding combinations of theories. Proc. Sixth Conf. on Auto Deduction (Loveland, Ed.), Lecture Notes in Comp. Sci. 138, Springer-Verlag, Berlin, June 1982, 209-223.

Digital Library

[33]

Boyer, R.S. and J.S. Moore. Proof checking, theorem proving, and program verification. In {2}, 133-167.

[34]

Musser, D. On proving inductive properties of abstract data types. Proc. Seventh ACM Conf. on Principles of Progr. Lang., Las Vegas, Jan. 1980, 154-162.

Digital Library

[35]

Goguen, J.A. How to prove algebra inductive hypotheses without induction with applications to the correctness of data type implementation. Proc. Fifth Conf. on Auto. Deduction (Bibel and Kowalski, Eds.), Lecture Notes in Comp. Sci. 87, Springer-Verlag, Berlin, July, 1980, 356-373.

Digital Library

[36]

Andrews, P.B. An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof. Academic Press, New York, 1986.

Digital Library

[37]

Andrews, P.B., D.A. Miller, E.L. Cohen and F. Pfenning. Automating higherorder logic. In {2}, 169-192.

[38]

O'Donnell, M.J. Equational Logic as a Programming Language. MIT Press, Cambridge, 1985.

Digital Library

[39]

Constable, R.L., S.F. Allen, H.M. Bromley, W.R. Cleaveland, J.F. Cremer, R.W. Harper, D.J. Howe, T.B. Knoblock, N.P. Mendler, P. Panangaden, J.T. Sasaki, S.F. Smith. Implementing Mathematics with the Nuprl Proof Development System. Prentice-Hall, Englewood Cliffs, NJ, 1986.

Digital Library

[40]

Lindamood, G.E. The structure of the Japanese fifth generation computer project - then and now. Future Generations Comp. Systems, July 1984, 51-55.

[41]

Kowalski, R.A. Logic for Problem Solving. North-Holland, Amsterdam, 1980.

Digital Library

[42]

Shapiro, E.Y. and A. Takeuchi. Object oriented programming in Concurrent Prolog. New Generation Computing, 1, 1, 1983.

[43]

Miller, D. and G. Nadathur. Higher-order logic programming. Report MS-CIS- 86-17, Dept. of Computer and Infor. Sci., Moore School, Univ. of Penn., Phil. PA., March, 1986.

[44]

Pl~isted, D. The occur-check problem in Prolog. New Generation Computing, 1984, 309-322.

[45]

Stickel, M.E. A Prolog technology theorem prover. New Generation Computing, 1984, 371-383.

[46]

Kowalski, R. and D. Kuehner. Linear resolution with collection function. Artif. Intell., 1971, 227-260.

[47]

Plaisted, D.A. A simplified problem reduction format. Artif. Intell., 1982, 227- 261.

[48]

Loveland, D.W. A Prolog for non-Horn programs. Forthcoming report.

Cited By

Marakakis EKondylakis HPapadakis N(2014)A knowledge-based interactive verifier for logic programsInternational Journal of Knowledge-based and Intelligent Engineering Systems10.3233/KES-14029418:3(143-156)Online publication date: 1-Jul-2014
https://dl.acm.org/doi/10.3233/KES-140294
Xu CCheung SChan W(2007)Goal-Directed Context Validation for Adaptive Ubiquitous SystemsProceedings of the 2007 International Workshop on Software Engineering for Adaptive and Self-Managing Systems10.1109/SEAMS.2007.8Online publication date: 20-May-2007
https://dl.acm.org/doi/10.1109/SEAMS.2007.8
Stickel M(1988)A Prolog technology theorem prover: implementation by an extended Prolog computerJournal of Automated Reasoning10.1007/BF002972454:4(353-380)Online publication date: 1-Dec-1988
https://dl.acm.org/doi/10.1007/BF00297245

Index Terms

Recommendations

Automated Theorem Proving: Mapping Logic Into AI
Automated Theorem Proving in Higher-Order Logic
Theorem Proving Modulo
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 general interest because it permits one to separate ...

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

ISMIS '86: Proceedings of the ACM SIGART international symposium on Methodologies for intelligent systems

December 1986

450 pages

ISBN:0897912063

DOI:10.1145/12808

Editors:
Zbigniew W. Ras
Univ. of North Caroling, Charlotte
,
Maria Zemankova

Copyright © 1986 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Sponsors

SIGAI: ACM Special Interest Group on Artificial Intelligence

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 December 1986

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Qualifiers

Article

Conference

ISMIS86

Sponsor:

SIGAI

ISMIS86: ACM SIGART International Symposium on Methodologies for Intelligent Systems

October 22 - 24, 1986

Tennessee, Knoxville, USA

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

3
Total Citations
View Citations
970
Total Downloads

Downloads (Last 12 months)154
Downloads (Last 6 weeks)29

Reflects downloads up to 05 Feb 2025

Other Metrics

View Author Metrics

Citations

Cited By

Marakakis EKondylakis HPapadakis N(2014)A knowledge-based interactive verifier for logic programsInternational Journal of Knowledge-based and Intelligent Engineering Systems10.3233/KES-14029418:3(143-156)Online publication date: 1-Jul-2014
https://dl.acm.org/doi/10.3233/KES-140294
Xu CCheung SChan W(2007)Goal-Directed Context Validation for Adaptive Ubiquitous SystemsProceedings of the 2007 International Workshop on Software Engineering for Adaptive and Self-Managing Systems10.1109/SEAMS.2007.8Online publication date: 20-May-2007
https://dl.acm.org/doi/10.1109/SEAMS.2007.8
Stickel M(1988)A Prolog technology theorem prover: implementation by an extended Prolog computerJournal of Automated Reasoning10.1007/BF002972454:4(353-380)Online publication date: 1-Dec-1988
https://dl.acm.org/doi/10.1007/BF00297245

View Options

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

Figures

Tables

Media

View Table of Conten