Hilbert's epsilon-Terms in Automated Theorem Proving

  • Authors:

  • Martin Giese;Wolfgang Ahrendt

  • Affiliations:

  • -;-

  • Venue:
  • TABLEAUX '99 Proceedings of the International Conference on Automated Reasoning with Analytic Tableaux and Related Methods
  • Year:
  • 1999

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Abstract

Ɛ-terms, introduced by David Hilbert [8], have the form Ɛx.Φ, where x is a variable and Φ is a formula. Their syntactical structure is thus similar to that of a quantified formulae, but they are terms, denoting 'an element for which Φ holds, if there is any'. The topic of this paper is an investigation into the possibilities and limits of using Ɛ-terms for automated theorem proving. We discuss the relationship between Ɛ-terms and Skolem terms (which both can be used alternatively for the purpose of ∃-quantifier elimination), in particular with respect to efficiency and intuition. We also discuss the consequences of allowing Ɛ-terms in theorems (and cuts). This leads to a distinction between (essentially two) semantics and corresponding calculi, one enabling efficient automated proof search, and the other one requiring human guidance but enabling a very intuitive (i.e. semantic) treatment of Ɛ-terms. We give a theoretical foundation of the usage of both variants in a single framework. Finally, we argue that these two approaches to Ɛ are just the extremes of a range of Ɛ-treatments, corresponding to a range of different possible Skolemization variants.