Hyperresolution for Propositional Product Logic with Truth Constants

Abstract

In the paper, we generalise the well-known hyperresolution principle to the propositional product logic with explicit partial truth. We propose a hyperresolution calculus suitable for automated deduction in a useful expansion of the propositional product logic by intermediate truth constants and the equality, \({\pmb {\eqcirc }}\), strict order, \({\pmb {\prec }}\), projection, \({{\varvec{\Delta }}}\), operators. We expand the propositional product logic by a countable set of intermediate truth constants of the form \(\bar{c}\), \(c\in (0,1)\). We propose translation of a formula to an equivalent satisfiable finite order clausal theory, which consists of order clauses - finite sets of order literals of the augmented form: \(\varepsilon _1\diamond \varepsilon _2\) where \(\varepsilon _i\) is either a truth constant \(\bar{0}\) or \(\bar{1}\), or a conjunction of powers of propositional atoms or intermediate truth constants, and \(\diamond \) is a connective \(\eqcirc \) or \(\prec \). \(\eqcirc \) and \(\prec \) are interpreted by the standard equality and strict order on [0, 1], respectively. We shall investigate the canonical standard completeness, where the semantics of the propositional product logic is given by the standard \(\varvec{\Pi }\)-algebra, and truth constants are interpreted by ‘themselves’. The hyperresolution calculus over order clausal theories is refutation sound and complete for the finite case. We solve the deduction problem \(T\,\models \,\phi \) of a formula \(\phi \) from a finite theory T in this expansion.