Minimal resolution proof systems for finitely-valued Lukasiewicz logics

Abstract

Ab st rac tIn lhis paper we describe an eflective method for, constructing minimal (in the sense described in thi1, paper) non-clausal resolution proof systems for tudosiewicz logics. W e show that every minimal resolution counterpart of the n-valued tukasiewicz logic, n > 2 , has 2n verifiers, and we provide a polynomial time algorithm to generate them. "has paper extends the resinlts reported in [9]. Introduction:The results reported in [l, 2, 5, 81 suggcsst that a number of well-known automated theorcm I'roving techniques, originally developed for the clas4cal locic, can also be successfully applied to many-valiicd log i t -3 and to finitely-valued calculi in particular. In [5, r j ] it is shown that for every finitely-valiwtl logic P one can effectively construct a resolution based automated proof system Rs refutationally equivalent to ' P. The main rule of inference of Rs, the so-called resolution rule, is defined on a propositional level by the following scheme: a o ( p ) , ,am(p) a o ( p l u 0 ) V V a m ( P l f " ) ' In this rule, CQ, ...,a, are arbitrary (not necessarily different) formulas with a common variable p , and V O , ..., vm are preselected formulas called verzfiers. The conclusion of this rule is obtained by replacing every occurrence of p in a l (~) with the it h verifier o, and taking the disjunction of the results. Loosely speaking, this rule says that if a w t A' = { a o ( p ) , .,., a , ( p ) } of formulas is consistent, then so is X U {ao(p/vo) V ... v a,(p/vnz)}. (This formalization of the resolution rule assumes the existence of a disjunction connective in a given language. When such a connective is absent, the resolution rule can be written as a multiple-conclusion rule (cf . [ 7 ] ) . ) The verifiers as well as their order in the resolution rule are fixed, however, the selection of the set of verifiers is by no means unique. Since the computational complexity of the deductive process based on (lie resolution rule depends, among other factors, on tlie the nuniber of verifiers occurring in the rule, i he scvirch for the smallest set of verifiers for a. givrii logic is of a primary concern. In [9] it is shown that the smalkst number of verifiers required by a resolut,ion rule of the n-valued Lukasiewicz logic is 271, provided tha.t n -1 is prime and n > 2. In this paper we show that thc assumption that, n-1 is prime can be dropped. Formally, we show that jor every n > 2, the minimal namber of verifiers required b y resolution rule of the n-valued Lukasiewicz logic is 2n.In fact, by slight modification of the notjion of a verifier, we can prove a sharper result, that only n verifiers are required. This result, together with the fact that the two-valued Lukasiewicz logic (i.e., the classical propositional logic) requires only two verifiers (cf.[4]), closes the search for the lower bound of the nrimber of verifiers for finikly-valued tukasiewicz logics. The solution of this problem constitutes the first part of this paper. In the second part, we present a polynomial time algorithm for g...

Lattices of resolution logics

“…In [3,8,9,10], P is defined to be a resolution logic, if it has a resolution proof system, i.e., if there exists a refutationally equivalent deductive proof system for P based on the resolution rule. In [7,8] it is shown that a propositional logic P is a resolution logic, if and only if, K p contains a logical system defined by a finite logical matrix.…”

“…For more complete treatments of the material of this section, one can consult [I23 for everything pertaining to consequence operations or logical matrices, and one of [3,4,7,8,9,10,11] where the resolution proof systems for resolution logics are introduced and studied.…”

“…The reader may refer to [3,7,8,9,10,11] for the discussion concerning the conditions (rl)-(r3). THEOREM 2.1: (cf.…”