DFKI Research Report-93-06



Language: English

by Hans-Jürgen Bürckert, Bernhard Hollunder, Armin Laux

On Skolemization in Constrained Logics

40 Pages


First-order logics allows one to quantify over all elements of the universe. However, it is often more natural to quantify only over those elements which satisfy a certain condition. Constrained logics provide this possibility by introducing restricted quantifiers For_all X:R F and Exists X:R F where X is a set of variables, and which can be read as "F holds for all elements satisfying the restriction R" and "F holds if there exist elements which satisfy R", respectively.

In order to test unsatisfiability of a set of such formulas using an extended resolution principle, one needs a procedure which transforms them into a set of constrained clauses. Such a procedure causes more problems than the classical transformation of first-order formulas into a set of clauses. This is due to the fact that quantification over the empty set may occur. Especially, a modified Skolemization procedure has to be used in order to remove restricted existential quantifiers.

In this paper we will give a procedure that transforms formulas with restricted quantifiers into a set of clauses with constraints preserving unsatisfiability. Since restrictions may be given by sorts this procedure can, e.g., be applied to sorted logics where empty sorts may occur. The obtained clauses are of the form C double-vertical-line R where C is an ordinary clause and R is a restriction, and which can be read as "C holds if R holds". They can be tested on unsatisfiability via constrained resolution. Finally, we introduce so-called constraint unification which can be used for optimization of constrained resolution if certain conditions are satisfied.

This document is available as Postscript.

The next abstract is here, and the previous abstract is here.

DFKI-Bibliothek (bib@dfki.uni-kl.de)

Note: This page was written to look best with CSS stylesheet support Level 1 or higher. Since you can see this, your browser obviously doesn't support CSS, or you have turned it off. We highly recommend you use a browser that supports and uses CSS, and review this page once you do. However, don't fear, we've tried to write this page to still work and be readable without CSS.