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Some applications of Gentzen's proof theory in automated deduction

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Extensions of Logic Programming (ELP 1989)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 475))

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Abstract

Reasoning and problem-solving programs must eventually allow the full use of quantifiers and sets, and have strong enough control methods to use them without combinatorial explosion.

J. McCarthy

We show that Prolog is intimately connected with Gentzen's cut-free sequent calculus G, analyzing Prolog computations as the construction of certain cut-free derivations. We introduce a theorem-proving program GENTZEN based on Gentzen's sequent calculus, which incorporates some features of Prolog's computation procedure. We show that GENTZEN has the following properties: (1) It is (non-deterministically) sound and complete for first-order intuitionistic predicate calculus; (2) Its successful computations coincide with those of Prolog on the Horn clause fragment (both deterministically and non-deterministically). The proofs of (1) and (2) contain a new proof of the completeness of (non-deterministic) Prolog for Horn clause logic, based on our analysis of Prolog in terms of Gentzen sequents instead of on resolution. GENTZEN has been implemented and tested on examples including some proofs by induction in number theory, an example constructed by J. McCarthy to show the limitations of Prolog, and “Schubert's Steamroller.” An extension of GENTZEN also provides a decision procedure for intuitionistic propositional calculus (but at some cost in efficiency).

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Authors and Affiliations

  1. Mathematics and Computer Science, San Jose State University, 95192, San Jose, California, USA

    Michael Beeson

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  1. Michael Beeson
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Peter Schroeder-Heister

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© 1991 Springer-Verlag Berlin Heidelberg

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Beeson, M. (1991). Some applications of Gentzen's proof theory in automated deduction. In: Schroeder-Heister, P. (eds) Extensions of Logic Programming. ELP 1989. Lecture Notes in Computer Science, vol 475. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0038693

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  • DOI: https://doi.org/10.1007/BFb0038693

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  • Print ISBN: 978-3-540-53590-4

  • Online ISBN: 978-3-540-46879-0

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