Hyperresolution for Propositional Product Logic

Abstract

We provide the foundations of automated deduction in the propositional product logic. Particularly, we generalise the hyperresolution principle to the propositional product logic. 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: ε 1 ε 2 where ε i is either the truth constant 0 or 1 or a conjunction of powers of propositional atoms, and is the connective or ≺. and ≺ are interpreted by the standard equality and strict order on [0, 1], respectively. We devise a hyperresolution calculus over order clausal theories, which is refutation sound and complete for the finite case. By means of the translation and calculus, we solve the deduction problem T |= φ for a finite theory T and a formula φ. Both the rules/calculi are refutation complete and sound: a clausal theory is unsatisfiable if and only 30

Automated Deduction in Propositional Product Logic

“…The following lemma solves the satisfiability problem for the basic case where the order clausal theory in question is intermediate and unit. It introduces a criterion for satisfiability: if there does not exist an application of Rule (20) to the order clausal theory, then it is satisfiable, and we are able to construct a model of it. Lemma 9.…”

“…Lemma 9. Let S ⊆ F OrdPropCl be intermediate and unit, and there not exist an application of Rule (20) to S. S is satisfiable.…”

“…This idea is materialised in a unit contradiction rule (cf. Rule (20), Section 4). We prove that the proposed DPLL procedure is refutation sound and finitely complete.…”

“…In addition, some admissible rules will be introduced for practical DPLL inference. Preliminary results have been presented at conferences [20][21][22][23]. Other approaches to the solutions to the SAT, VAL, and DED problems in fuzzy logics with continuous t-norms rely on mixed integer programming (MIP) [24][25][26] or on satisfiability modulo theories (SMT) [27][28][29].…”

Hyperresolution for Gödel logic with truth constants

“…In the propositional case [13], we have proposed some translation of a formula to an equivalent conjunctive normal form (CNF) containing literals of the form a or a → b or (a → b) → b where a is an atom, and b is an atom or the truth constant 0. An output equivalent CNF may be of exponential size with respect to the size of the input formula; we had laid no restrictions on use of the distributivity law (4) during translation to conjunctive normal form.…”