# Experiments in automated deduction with condensed detachment

## Abstract

This paper contains the results of experiments with several search strategies on 112 problems involving condensed detachment. The problems are taken from nine different logic calculi: three versions of the two-valued sentential calculus, the many-valued sentential calculus, the implicational calculus, the equivalential calculus, the R calculus, the left group calculus, and the right group calculus. Each problem was given to the theorem prover Otter and was run with at least three strategies: (1) a basic strategy, (2) a strategy with a more refined method for selecting clauses on which to focus, and (3) a strategy that uses the refined selection mechanism and deletes deduced formulas according to some simple rules. Two new features of Otter are also presented: the refined method for selecting the next formula on which to focus, and a method for controlling memory usage.

## Preview

Unable to display preview. Download preview PDF.

## References

- [1]A. Church.
*Introduction to Mathematical Logic*, volume I. Princeton University Press, 1956.Google Scholar - [2]J. A. Kalman. Axiomatizations of logics with values in groups.
*Journal of the London Math. Society*, 2(14):193–199, 1975.Google Scholar - [3]J. A. Kalman. Computer studies of T→-W-I.
*Relevance Logic Newsletter*, 1:181–188, 1976.Google Scholar - [4]J. A. Kalman. A shortest single axiom for the classical equivalential calculus.
*Notre Dame Journal of Formal Logic*, 19:141–144, 1978.Google Scholar - [5]J. A. Kalman. Condensed detachment as a rule of inference.
*Studia Logica*, LXII(4):443–451, 1983.Google Scholar - [6]E. J. Lemmon, C. A. Meredith, D. Meredith, A. N. Prior, and I. Thomas. Calculi of pure strict implication. Technical report, Canterbury University College, Christchurch, 1957. Reprinted in
*Philosophical Logic*, Reidel, 1970.Google Scholar - [7]J. Łukasiewicz.
*Elements of Mathematical Logic*. Pergamon Press, 1963. English translation of the second edition (1958) of*Elementy logiki matematycznej*, PWN, Warsaw.Google Scholar - [8]J. Łukasiewicz.
*Selected Works*. North-Holland, 1970. Edited by L. Borkowski.Google Scholar - [9]W. McCune. Otter 2.0 Users Guide. Tech. Report ANL-90/9, Argonne National Laboratory, Argonne, IL, March 1990.Google Scholar
- [10]W. McCune. Automated discovery of new axiomatizations of the left group and right group calculi. Preprint MCS-P220-0391, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL, 1991.Google Scholar
- [11]W. McCune. Single axioms for the left group and right group calculi. Preprint MCS-P219-0391, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL, 1991.Google Scholar
- [12]C. A. Meredith. Single axioms for the systems (C,N), (C,0), and (A,N) of the two-valued propositional calculus.
*Journal of Computing Systems*, 1:155–164, 1953.Google Scholar - [13]C. A. Meredith and A. N. Prior. Equational logic.
*Notre Dame Journal of Formal Logic*, 9:212–226, 1968.Google Scholar - [14]D. Meredith. In memoriam Carew Arthur Meredith (1904–1976).
*Notre Dame Journal of Formal Logic*, 18:513–516, 1977.Google Scholar - [15]J. G. Peterson. Shortest single axioms for the classical equivalential calculus.
*Notre Dame Journal of Formal Logic*, 17(2):267–271, 1976.Google Scholar - [16]J. G. Peterson. The possible shortest single axioms for EC-tautologies. Report 105, Dept. of Mathematics, University of Auckland, 1977.Google Scholar
- [17]F. Pfenning. Single axioms in the implicational propositional calculus. In E. Lusk and R. Overbeek, editors,
*Proceedings of the 9th International Conference on Automated Deduction*,*Lecture Notes in Computer Science, Vol. 310*, pages 710–713, New York, 1988. Springer-Verlag.Google Scholar - [18]L. Wos. Meeting the challenge of fifty years of logic.
*Journal of Automated Reasoning*, 6(2):213–232, 1990.Google Scholar - [19]L. Wos. Automated reasoning and Bledsoe's dream for the field. In R. S. Boyer, editor,
*Automated Reasoning: Essays in Honor of Woody Bledsoe*, chapter 15, pages 297–342. Kluwer Academic Publishers, 1991.Google Scholar - [20]L. Wos and W. McCune. The application of automated reasoning to proof translation and to finding proofs with specified properties: A case study in many-valued sentential calculus. Tech. Report ANL-91/19, Argonne National Laboratory, Argonne, IL, 1991. In preparation.Google Scholar
- [21]L. Wos, S. Winker, W. McCune, R. Overbeek, E. Lusk, R. Stevens, and R. Butler. Automated reasoning contributes to mathematics and logic. In M. Stickel, editor,
*Proceedings of the 10th International Conference on Automated Deduction*,*Lecture Notes in Artificial Intelligence, Vol. 449*, pages 485–499, New York, July 1990. Springer-Verlag.Google Scholar - [22]L. Wos, S. Winker, B. Smith, R. Veroff, and L. Henschen. A new use of an automated reasoning assistant: Open questions in equivalential calculus and the study of infinite domains.
*Artificial Intelligence*, 22:303–356, 1984.Google Scholar