A Generalisation of the Hyperresolution Principle to First Order Gödel Logic

Abstract

In the paper, we generalise the well-known hyperresolution principle to the standard first-order Gödel logic. Our approach is based on the translation of a formula of Gödel logic to an equivalent satisfiable finite order clausal theory, consisting of order clauses. We introduce a notion of quantified atom: a formula a is a quantified atom iff a = Q x p(t ₀,…,t _τ) where Q is a quantifier (∀, ∃); p(t ₀,…,t _τ) is an atom; x is a variable occurring in p(t ₀,…,t _τ); for all i ≤ τ, either t _i = x or x does not occur in t _i. Then an order clause is a finite set of order literals of the form ε ₁ ⋄ ε ₂ where ε _i is either an atom or a quantified atom, and ⋄ is a connective either $\eqcirc$ or ≺. $\eqcirc$ and ≺ are interpreted by the equality and strict linear order on [0,1], respectively. For an input theory of Gödel logic, the proposed translation produces a so-called admissible order clausal theory. On the basis of the hyperresolution principle, a calculus operating over admissible order clausal theories, is devised. The calculus is proved to be refutation sound and complete for the countable case.

This work is partially supported by VEGA Grant 1/0979/12 and Slovak Literary Fund.