Global Skolemization with grouped quantifiers (Abstract)

Elimination of quantifiers from formulae of classical first-order logic is a process with many implications in automated deduction [6, 1] and in foundational issues [4, 7]. When no particular theory is considered, quantifiers are usually eliminated by adopting Skolemization or the ε-operator. Traditionally, Skolemization and the ε-symbol have different, if not complementary, employments. Skolemization is one of the most widespread techniques to eliminate existential quantifiers in automated proofs [6]. This is motivated by the fact that Skolem terms can easily be manipulated in deductions. In fact, they can be treated (both at the syntactical and at the semantical level) in the same way as the terms of the initial language. Skolem terms are generally unrelated to the formulae that generate them and to the context they are introduced in. This may become a drawback because some information which could be crucial for the automatic discovery of shorter automatic proofs might rely on relationships over Skolem terms. On the other hand, terms originated by the ε-operator, the so called ε-terms, are instead strictly related to their corresponding formulae and, for this reason, used for investigations on foundational issues [4, 5], non-classical logics [3], and linguistics [8]. However, ε-terms are not naturally suited to automation because of their complicated structure. In [2], Davis and Fechter propose a quantifier elimination technique which can be regarded as a bridge between Skolemization and the ε-operator. It consists in the definition of a free-variable calculus, proved equivalent to the standard predicate calculus, where quantifiers are implicitly defined by means of suitable Skolem terms. These terms, that can be seen as a (func-