 
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
{gieseahrendt}@ira.uka.de
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 efficiency 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 efficient 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
