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

Abstract

Due to the need of the logical foundation for uncertain information processing, development of efficient automated reasoning system based on non-classical logics is always an active research area. The present paper focuses on the resolution-based automated reasoning theory in a many-valued logic with truth-values defined in a lattice-ordered many-valued algebraic structure -lattice implication algebras (LIA). Specifically, as a continuation and extension of the established work on binary resolution at a certain truth-value level α (called α-resolution), a non-clausal multi-ary α-generalized resolution calculus is introduced for a lattice-valued propositional logic LP(X) based on LIA, which is essentially a non-clausal generalized resolution avoiding reduction to normal clausal form. The new resolution calculus in LP(X) is then proved to be sound and complete. The concepts and theoretical results are further extended and established in the corresponding lattice-valued first-order logic LF(X) based on LIA.

A logical reasoning based decision making method for handling qualitative knowledge

“…LIA, by combining lattice and implication algebra, serves as not only a class of efficient algebraic structure for representing uncertain information, but also an appropriate link with the non-classical logic systems. In order to make the general LIA based lattice-valued logic more specific for decision making problems under qualitative and uncertain environment, a series of linguistic truth-valued LIAs (L-LIAs) [28,45,48] were constructed for representing qualitative information, and the corresponding linguistic truth-valued logic system [29,46], approximate reasoning approaches [10,48,50] and related decision making methods [30,53,54] were then proposed. Liu et al [28] summarized some ideas on lattice ordered linguistic decision making, and presented a systematic framework for lattice ordered linguistic decision making problems from the viewpoint of lattice structure representation and logical reasoning.…”

The Possibilistic Horn Non-Clausal Knowledge Bases

“…Developing methods for NC reasoning is an actual concern in the principlal fields of classical logic, namely satisfiability solving [54,43], logic programming [17,14], theorem proving [31,53] and quantified boolean formulas [30,13], and in many other fields (see [40] and the references thereof). And within non-classical logics, NC formulas with different functionalities have been studied in a profusion of languages: signed many-valued logic [47,8,58], Lukasiewicz logic [42], Levesque's three-valued logic [15], Belnap's four-valued logic [15], M3 logic [1], fuzzy logic [35], fuzzy description logic [34], intuitionistic logic [55], modal logic [55], lattice-valued logic [60] and regular many-valued logic [39]. We highlight the proposal in [49,50] as is the only existing approach, to our knowledge, to deal with possibilistic non-clausal formulas, concretely within the answer set programming field.…”

A First Polynomial Non-Clausal Class in Many-Valued Logic

“…Indeed, within classical logic, non-clausal formulas are found in numerous scenarios and reasoning problems such as quantified boolean formulas [45], DPLL [97], nested logic programming [90], knowledge compilation [35], description logics [69], numeric planning [92] and many other fields that are mentioned in [66]. In the particular case of firstorder logic, one can find approaches on non-clausal theorem proving in the former steps of automated reasoning e.g., [25,6] but such area is still the object of current research activity as the regularly reported novel results show e.g., [48,99,88].…”

“…And within non-classical logics, non-clausal formulas having different roles and functionalities have been studied in a profusion of languages: signed many-valued logic [86,18,96], Lukasiewicz logic [72], Levesque's three-valued logic [30], Belnap's four-valued logic [30], M3 logic [2], fuzzy logic [55], fuzzy description logic [54], intuitionistic logic [89], modal logic [89], lattice-valued logic [99] and more.…”