 
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 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
