Source: Notre Dame J. Formal Logic Volume 50, Number 3
(2009), 261-273.
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.
Full-text: Access denied (no subscription
detected)
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.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription.
Read more about accessing full-text
References
[1] Anderson, A. R., and N. D. Belnap, Jr., Entailment. Volume I. The Logic of Relevance and Necessity, Princeton University Press, Princeton, 1975.
[2] Bunder, M. W., "Lambda terms definable as combinators", Theoretical Computer Science, vol. 169 (1996), pp. 3--21.
[3] Bunder, M. W., "Proof finding algorithms for implicational logics", Theoretical Computer Science, vol. 232 (2000), pp. 165--86.
[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.
[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.
[8] Hindley, J. R., Basic Simple Type Theory, vol. 42 of Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, Cambridge, 1997.
[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
[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).
[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.
