Abstract
Abstract In the paper, we generalise the well-known hyperresolution principle to the general first-order Godel logic with explicit partial truth. We propose a hyperresolution calculus suitable for automated deduction in a useful expansion of Godel 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 Godel 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 e 1 ⋄ e 2 where e i 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 Godel 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 Godel logic with truth constants and the equality, ≖, strict order, ≺, projection, Δ, operators.
Published Version
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