NON-CLAUSAL MULTI-ARY α-GENERALIZED RESOLUTION PRINCIPLE FOR A LATTICE-VALUED PROPOSITIONAL LOGIC

Abstract

As a continuation and extension of the established work on binary resolution at certain truth-value level (called α-resolution), this paper introduces nonclausal multi-ary α-generalized resolution principle and deduction for latticevalued propositional logic LP(X) based on lattice implication algebra, which is essentially a non-clausal generalized resolution avoiding the reduction to normal clausal form. Non-clausal multi-ary α-generalized resolution deduction in LP(X) is then proved to be sound and complete.

Non-Clausal Multi-ary α-Generalized Resolution Calculus for a Finite Lattice-Valued Logic

“…Inspired from all the above ideas and motivation of generalized resolution in Boolean logic and multi-ary α-resolution in lattice-valued logic, the present paper aims to propose the general generalized α-resolution principle (the reason why it is called "general" and "generalized" will be clarified further in Section 3) in order to deal with complex formulas in finitely lattice-valued logic L(X). This paper is a continuation and extension of the work in [25][26][27][28][29][30][31][32][33] , the binary α-resolution principle introduced in [25,26] for L(X) is extended to multi-ary α-generalized resolution principle in different ways as follows: (1) the resolution is based on general generalized clauses which is constructed by the generalized literals and logical connectives ″∨, ∧, ′, →, ↔″, instead of the generalized clause containing only ″ ′, →″ in [25,26]. This, in essential, is a non-clausal resolution; (2) the set of the generalized clauses, which is a complex logical formula, are not necessary to be transformed into the GCNF; (3) the above extended binary α-generalized resolution is further extended into multi-ary α-generalized resolution, i.e., extends the α-generalized resolution pair composed of two generalized literals to the α-generalized resolution group composed of multiple generalized literals based on the work in [24].…”

“…In addition, LIAs form a proper class, and include no-chain algebra and no-Boolean algebra as well. The relationship between LIA with other logical algebraic structure is discussed in [24]. It shows that all the results obtained based on LIA or related logic can be applied into Boolean logic or Łukasiewicz logic at least, as well as other logical algebras.…”

“…In this section, the outlined work in [29] will be extended and systematized, and further extended into the one for first-order logic LF(X) in Section 4.…”

“…We cannot get a binary α-resolution refutation of S 1 , but can get an alternative multi-ary α-resolution refutation of S 1 using the multi-ary α-resolution principle introduced in [24], however which apparently needs more steps or is relatively more complex, i.e., …”

“…In order to handle more than two generalized clauses simultaneously, Xu et al (2013) [24] extended the α-resolution principle in [25,26] to multi-ary α-resolution principle in L(X), which can enhance resolution automated reasoning compared with the ones in classical logic and those proposed in some many-valued logics in terms of soundness and completeness, applicability, reasoning capability and reasoning efficiency. It has specially demonstrated clearly in [24] that binary resolution has limited reasoning capability and also reasoning efficiency especially in many-valued logic. The above mentioned methods, however, can only deal with the generalized conjunctive normal form (GCNF) in L(X), but cannot handle other general forms of logical formulas.…”