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SETHEO: A high-performance theorem prover

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Abstract

A sound and complete theorem prover for first-order logic is presented, which is based on the connection method. The inference machine is implemented using PROLOG technology, an approach taken also with other systems, most prominently with Stickel's PTTP. But SETHEO differs from those in essential characteristics, among which are the following ones. It incorporates a powerful preprocessing module for a reduction of the input formula. The main proof procedure is realized as a variant of Warren's abstract machine. For search pruning we perform subsumption and regular proofs. Factorization, lemma generation, and the application of proof schemata are offered as options. The entire system is implemented in C and is running on several machines. The most remarkable feature of SETHEO is its performance of up to 70 Klips on a SUN SPARC station 1 with 12 Mips. The paper comprises the theoretical background, the system architecture as well as details of the implementation.

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References

  1. R. Letz and J. Schumann, ‘Global variables in logic programming’, Technical report FKI-96-b-88, Technische Universität München (1988).

  2. W. Bibel, Automated Theorem Proving, Vieweg Verlag, Braunschweig, second edition (1987).

    Google Scholar 

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

  4. M. E. Stickel, ‘A Prolog technology theorem prover: implementation by an extended Prolog compiler’, Journal of Automated Reasoning, 4, 353–380 (1988).

    Google Scholar 

  5. D. H. D. Warren, ‘An abstract PROLOG instruction set’, Technical report, SRI, Menlo Park, Ca, USA (1983).

    Google Scholar 

  6. E. Eder, ‘An implementation of a theorem prover based on the connection method’. In W. Bibel and B. Petkoff (editors), AIMSA: Artificial Intelligence Methodology Systems Applications, 121–128. North-Holland (1985).

  7. C.-L. Chang and R. C.-T. Lee, Symbolic Logic and Mechanical Theorem Proving, Academic Press, Inc., (dy1973).

  8. J. A. Robinson, ‘A machine-oriented logic based on the resolution principle”, Journal of the ACM, 12, 23–41 (1965).

    Google Scholar 

  9. W. Bibel, R. Letz, and J. Schumann, ‘Bottom-up enhancements of deductive systems’. In I. Plander (editor), Artificial Intelligence and Information-Control Systems of Robots 87, pp. 1–9. North-Holland (1987).

  10. M. E. Stickel, ‘Schubert's steamroller problem: formulations and solutions’, Journal of Automated Reasoning, 2, 89–101 (1986).

    Google Scholar 

  11. W. McCune, ‘OTTER users' guide’, Technical report, Mathematics and Computer Sci. Division, Argonne National Laboratory, Argonne, Ill., USA, May 1988.

    Google Scholar 

  12. E. Eder, ‘Properties of substitutions and unifications’, Journal of Symbolic computation, 1, 31–46 (1985).

    Google Scholar 

  13. W. Bibel, ‘Automated inferencing’, Journal of Symbolic Computation, 1, 245–260 (1985).

    Google Scholar 

  14. K. Bläsius, N. Eisinger, J. Siekmann, G. Smolka, A. Herold, and C. Walther, ‘The Markgraf Karl refutation proof procedure’. In Proceedings of the Seventh International Joint Conference on Artificial Intelligence, pp. 511–518, Vancouver, 1981.

  15. C. B. Suttner, ‘Learning heuristics for automated theorem proving’. Diploma Thesis, Technische Universität München (1989).

  16. R. M. Smullyan, First Order Logic. Springer (1968).

  17. R. E. Korf, ‘Depth-first iterative deepening: an optimal admissible tree search’, Artificial Intelligence, 27, 97–109 (1985).

    Google Scholar 

  18. S. Fleisig, D. Loveland, A. K. Smiley III, and D. L. Yarmush, ‘An implementation of the model elimination proof procedure’, Journal of the ACM, pp. 124–139 (1974).

  19. L. M. Pereira and A. Porto, ‘Selective backtracking’. In K. L. Clark and S.-A. Tärnlund (editors), Logic Programming, number 16 in A. P. I. C. Studies in Data Processing, pp. 107–114. Academic Press Inc. (1982).

  20. G. S. Tseitin, ‘On the complexity of derivations in the propositional calculus’. In A. O. Silsenko (editor), Studies in Mathematics and Mathematical Logic II, pp. 115–125 (1970).

  21. A. Haken, ‘The intractability of resolution’, Theoretical Computer Science, 39, 297–308 (1985).

    Google Scholar 

  22. S. A. Cook and R. A. Reckhow, ‘On the lengths of proofs in the propositional calculus’, ACM Sigact News, 6, 15–22 (1974).

    Google Scholar 

  23. W. Bibel, ‘Short proofs of the pigeonhole formulas based on the connection method’ Journal of Automated Reasoning, 6, 287–297 (1990).

    Google Scholar 

  24. D. Hilbert and W. Ackermann, Grundzüge der theoretischen Logik, Springer (1928). Engl. translation: Mathematical Logic, Chelsea (1950).

  25. J. Vlahavas and C. Halatsis, ‘A new abstract Prolog instruction set. In 7th International Workshop on Expert Systems and Applications, pp. 1025–1050, Avignon (1987).

  26. J. Schumann, N. Trapp, and M. van der Koelen, ‘SETHEO: User's manual’, Technical report FKI-121-89, Technische Universität München, 1989.

  27. J. Corbin and M. Bidoit, ‘A rehabilitation of Robinson's unification algorithm’, In Information Processing, pp. 909–914. North-Holland (1983).

  28. D. A. Plaisted, ‘The occur-check problem in Prolog’, New Generation Computing, 2, 309–322 (1984).

    Google Scholar 

  29. L. Sterling and E. Shapiro, The Art of Prolog, MIT Press (1986).

  30. G. A. Wilson and J. Minker, ‘Resolution, refinements and search strategies: a comparative study’, IEEE Transactions on Computers, C25, 782–801 (1976).

    Google Scholar 

  31. R. Reboh, B. Raphael, R. A. Yates, R. E. Kling and C. Verlarde, ‘Study of automatic theorem-proving programs’ Technical report 75, SRI AI Center, November 1972.

  32. D. Michie, R. Ross, and G. J. Shannan, ‘G-Deduction’. In B. Meltzer and D. Michie (editors), Machine Intelligence, pp. 141–165. John Wiley and Sons (1972).

  33. L. Wos, Unpublished Notes. Argonne National Laboratory (1965).

  34. J. D. Lawrence and J. D. Starkey, ‘Experimental results of resolution based theorem-proving strategies’. Technical report, Computer Science Department, Washington State University, Pullman (1974).

    Google Scholar 

  35. J. Pelletier and P. Rudnicki, ‘Non-obviousness’, AAR Newsletter6, pp. 4–5 (1986).

    Google Scholar 

  36. L. Wos, Automated Reasoning: 33 Basic Research Problems, Prentice Hall (1988).

  37. E. Eder. Personal communication (1986).

  38. J. Schumann and R. Letz, ‘PARTHEO: a high performance parallel theorem prover’. In Cade 90, 40–56, Springer (1990).

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Authors and Affiliations

  1. Institut für Informatik, Technische Universität München, Germany

    R. Letz, J. Schumann & S. Bayerl

  2. Technische Hochschule Darmstadt, Germany

    W. Bibel

  3. University of British Columbia, Vancouver, Canada

    W. Bibel

  4. Canadian Institute for Advanced Research, Canada

    W. Bibel

Authors
  1. R. Letz
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  2. J. Schumann
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  3. S. Bayerl
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  4. W. Bibel
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Letz, R., Schumann, J., Bayerl, S. et al. SETHEO: A high-performance theorem prover. J Autom Reasoning 8, 183–212 (1992). https://doi.org/10.1007/BF00244282

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