The Tableau–Based Theorem Prover

3T AP is an acronym for 3–valued tableau–based theorem prover. It is based on the method of analytic tableaux. 3T AP has been developed at the University of Karlsruhe in cooperation with the Institute for Knowledge Based Systems of IBM Germany in Heidelberg. Despite its name 3T AP is able to deal with “classical” — i.e. two– valued — first–order predicate logic as well as with any finite–valued first–order logic, provided the semantics is specified by truth–tables. Currently implemented versions are working for two–valued and for a certain three–valued first–order predicate logic, which is a variant of the strong Kleene logic, see [3]. The multiple–valued version implements the concept of generalized signs. These may be seen as sets of ordinary tableau signs or prefixes, see [6] and [7] for details. Without generalized signs one has to build a separate tableau for each non–designated sign to refute a formula. 3T AP needs to close only one tableau using generalized signs. The system has been implemented in Quintus Prolog and is running on SUN and IBM PS/2. The use of Prolog and the modular design makes it easy to extend or modify the prover. 3T AP ’s input is given by a set of axioms and theorems contained in a database file, which can be precompiled. The user may specify a theorem to be proved or (s)he proves the consistency of the database. The formulae need not to be in normal form, hence the database remains readable. The use of a sort hierarchy in the database to distinguish terms of different nature is supported. For inspection of proofs some output formatting utilities allow the indentation of a proof or the conversion to LTEX–syntax. During the compilation of a database static links are being built. These links are used to decide whether a formula can possibly lead to a branch closure and needs to be used in the tableau or not. The free variable tableau calculus used by 3T AP is basically the one described in [10] or [4] with a few modifications concerning the handling of free variables and the extension to the multiple–valued case. Through branch closures free variables become instantiated and they are used up. Since a formula is often needed with a different substitution of free variables, a mechanism has been employed that allows to mark formulae as universal with respect to a variable occurring freely in it. Basically, this is the case when the formula can