Hyperresolution for Gödel logic with truth constants

Abstract

In the paper, we generalise the well-known hyperresolution principle to the general first-order Gödel logic with explicit partial truth. We propose a hyperresolution calculus suitable for automated deduction in a useful expansion of Gödel logic by intermediate truth constants and the equality, ≖, strict order, ≺, projection, Δ, operators. We solve the deduction problem of a formula from a countable theory in this expansion. We expand Gödel logic by a countable set of intermediate truth constants of the form c¯, c∈(0,1). Our approach is based on translation of a formula to an equivalent satisfiable finite order clausal theory, consisting of order clauses. An order clause is a finite set of order literals of the form ε1⋄ε2 where εi is an atom or a truth constant or a quantified atom, and ⋄ is a connective ≖, equality, or ≺, strict order. ≖ and ≺ are interpreted by the equality and standard strict linear order on [0,1], respectively. We shall investigate the canonical standard completeness, where the semantics of Gödel logic is given by the standard G-algebra, and truth constants are interpreted by ‘themselves’. The hyperresolution calculus is refutation sound and complete for a countable order clausal theory under a certain condition for the set of truth constants occurring in the theory. As an interesting consequence, we get an affirmative solution to the open problem of recursive enumerability of unsatisfiable formulae in Gödel logic with truth constants and the equality, ≖, strict order, ≺, projection, Δ, operators.