Automated Deduction in Gödel Logic

This article addresses the deduction problem of a formula from a countable theory in the first-order Gödel logic. We generalise the well-known hyperresolution principle for deduction in Gödel logic. Our approach is based on translation of a formula 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 if a = Qx, p(t₀, …, t_n), where Q is a quantifier (∀, ∃), p(t₀, …, t_n) is an atom, and x is a variable occurring in p(t₀, …, t_n); for all i ≤ n, 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 ϵ₁ ⋄ ϵ ₂, where ϵ_i is either an atom, or a truth constant (0, 1), or a quantified atom, and ⋄ is either a connective ≖, equality, or ≺ strict order. ≖ and ≺ are interpreted by the equality and standard strict linear order on [0, 1], respectively. On the basis of the hyperresolution principle, a calculus operating over order clausal theories is devised. The calculus is proved to be refutation sound and complete for the countable case. As an interesting consequence, we get an affirmative solution to the open problem of recursive enumerability of unsatisfiable formulae in Gödel logic.