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Hilbert's #Terms in Automated Theorem Proving

Summary: Hilbert's #­Terms in
Automated Theorem Proving
Martin Giese and Wolfgang Ahrendt
Institut f˜ur Logik, Komplexit˜at und Deduktionssysteme,
Universt˜at Karlsruhe, Germany
Abstract. #­terms, introduced by David Hilbert [8], have the form #x.#,
where x is a variable and # is a formula. Their syntactical structure is
thus similar to that of a quantified formulae, but they are terms, denoting
`an element for which # holds, if there is any'.
The topic of this paper is an investigation into the possibilities and lim­
its of using #­terms for automated theorem proving. We discuss the re­
lationship between #­terms and Skolem terms (which both can be used
alternatively for the purpose of #­quantifier elimination), in particular
with respect to e#ciency and intuition. We also discuss the consequences
of allowing #­terms in theorems (and cuts). This leads to a distinction
between (essentially two) semantics and corresponding calculi, one en­
abling e#cient automated proof search, and the other one requiring hu­
man guidance but enabling a very intuitive (i.e. semantic) treatment of
#­terms. We give a theoretical foundation of the usage of both variants in


Source: Ahrendt, Wolfgang - Department of Computer Science and Engineering, Chalmers University of Technology


Collections: Computer Technologies and Information Sciences