### Proof-finding Algorithms for Classical and Subclassical Propositional Logics

M. W. Bunder and R. M. Rizkalla
Source: Notre Dame J. Formal Logic Volume 50, Number 3 (2009), 261-273.

#### Abstract

The formulas-as-types isomorphism tells us that every proof and theorem, in the intuitionistic implicational logic $H_\rightarrow$, corresponds to a lambda term or combinator and its type. The algorithms of Bunder very efficiently find a lambda term inhabitant, if any, of any given type of $H_\rightarrow$ and of many of its subsystems. In most cases the search procedure has a simple bound based roughly on the length of the formula involved. Computer implementations of some of these procedures were done in Dekker. In this paper we extend these methods to full classical propositional logic as well as to its various subsystems. This extension has partly been implemented by Oostdijk.

First Page:
Primary Subjects: 03B20, 03B35, 03B40, 03B47
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.

Permanent link to this document: http://projecteuclid.org/euclid.ndjfl/1257862038
Digital Object Identifier: doi:10.1215/00294527-2009-011
Zentralblatt MATH identifier: 05657219
Mathematical Reviews number (MathSciNet): MR2572974

### References

[1] Anderson, A. R., and N. D. Belnap, Jr., Entailment. Volume I. The Logic of Relevance and Necessity, Princeton University Press, Princeton, 1975.
Mathematical Reviews (MathSciNet): MR0406756
Zentralblatt MATH: 0323.02030
[2] Bunder, M. W., "Lambda terms definable as combinators", Theoretical Computer Science, vol. 169 (1996), pp. 3--21.
Mathematical Reviews (MathSciNet): MR1424925
Zentralblatt MATH: 0868.03008
Digital Object Identifier: doi:10.1016/S0304-3975(96)00111-9
[3] Bunder, M. W., "Proof finding algorithms for implicational logics", Theoretical Computer Science, vol. 232 (2000), pp. 165--86.
Mathematical Reviews (MathSciNet): MR1734553
Zentralblatt MATH: 0972.03022
Digital Object Identifier: doi:10.1016/S0304-3975(99)00174-7
[4] Bunder, M. W., and S. Hirokawa, Classical Formulas As Types of $\lambda \nu$-terms, Department of Mathematics, University of Wollongong, Preprint series 14/96, 1996.
[5] Curry, H. B., A Theory of Formal Deducibility, vol. 6 of Notre Dame Mathematical Lectures, University of Notre Dame, Notre Dame, 1950.
Mathematical Reviews (MathSciNet): MR0033779
Zentralblatt MATH: 0041.34807
[6] Dekker, A. H., "Brouwer 7.9.0 - A Proof Finding Program for Intuitionistic, BCI", BCK and Classical Logic, 1996.
[7] Gabbay, D. M., and R. J. G. B. de Queiroz, "Extending the Curry-Howard interpretation to linear, relevant and other resource logics", The Journal of Symbolic Logic, vol. 57 (1992), pp. 1319--65.
Mathematical Reviews (MathSciNet): MR1195274
Zentralblatt MATH: 0765.03005
Digital Object Identifier: doi:10.2307/2275370
[8] Hindley, J. R., Basic Simple Type Theory, vol. 42 of Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, Cambridge, 1997.
Mathematical Reviews (MathSciNet): MR1466699
Zentralblatt MATH: 0906.03012
[9] Hindley, J. R., and J. P. Seldin, Introduction to Combinators and $\lambda$-Calculus, vol. 1 of London Mathematical Society Student Texts, Cambridge University Press, Cambridge, 1986.
Mathematical Reviews (MathSciNet): MR879272
Zentralblatt MATH: 0614.03014
[10] Oostdijk, M., Lambdacal 2.''
[11] Popper, K. R., "New foundations for logic", Mind, vol. 56 (1947), pp. 193--235; errata 57, 69--70 (1948).
Mathematical Reviews (MathSciNet): MR0021924
Digital Object Identifier: doi:10.1093/mind/LVI.223.193
[12] Trigg, P., J. R. Hindley, and M. W. Bunder, "Combinatory abstraction using $\bf B$", $\bf B'$ and friends, Theoretical Computer Science, vol. 135 (1994), pp. 405--22.
Mathematical Reviews (MathSciNet): MR1311210
Zentralblatt MATH: 0838.03012
Digital Object Identifier: doi:10.1016/0304-3975(94)90114-7