Lattice-valued Propositional Logic LP Research Articles

α-Quasi-Lock Semantic Resolution Method Based on Lattice-Valued Logic

AbstractBased on the general form of α-resolution principle for a lattice-valued logic with truth-values defined in a lattice-valued logical algebra structure - lattice implication algebra, the further extended α-resolution method in this lattice-valued logic is discussed in the present paper in order to increase the efficiency of the resolution method. Firstly, α-quasi-lock semantic resolution method in lattice-valued propositional logic LP(X) is established by combining the lock and semantic resolution simultaneously, and its theorems of soundness and conditional completeness are proved. Secondly, this α-quasi-lock semantic resolution method is extended into the corresponding lattice-valued first-order logic LF(X), and its soundness and conditional completeness are also established. This extended resolution method will provide a theoretical basis for automated soft theorem proving and program verification based on lattice-valued logic.

Multiary α-Resolution Principle for a Lattice-Valued Logic

This paper focuses on resolution-based automated reasoning theory in a lattice-valued logic system with truth values that are defined in a lattice-valued logical algebraic structure-lattice implication algebras (LIAs) - which essentially aims to extend the classical logic to handle automated deduction under an uncertain environment. Concretely, we investigate a generalization of the known conjunctive normal form (CNF) in the classical logic representation that we call the generalized conjunctive normal form (GCNF), which aims to characterize the constants and implication connectives that are essentially different from the ones in classical logic. We then extend the established resolution principle at a certain truth-value level α (called α-resolution) in this lattice-valued logic to a more general form, i.e., from binary α -resolution to multiary α-resolution. The extension to multiary α-resolution starts from lattice-valued propositional logic LP(X), while its theorems of both soundness and completeness are proved. Multiary α-resolution principle is then further established in the corresponding lattice-valued first-order logic LF(X), along with its soundness theorem, lifting lemma, and completeness theorem. Meanwhile, an important result that multiary α-resolution principle in LF(X) can be equivalently transformed into that in LP(X) to some extent is obtained. All these works will theoretically support the establishment of an automated reasoning algorithm and its implementation with further applications into automated deduction and decision-making problems under uncertainty.

Determination of 3-Ary -Resolution in Lattice-valued Propositional Logic LP(X)

Abstract One of key issues for α-n(t) ary resolution automated reasoning based on lattice-valued logic with truth-value in a lattice implication algebra is to investigate the α-n(t) ary resolution of some generalized literals. In this article, the determination of α-resolution of any 3-ary generalized literals which include the implication operators not more than 2 in LP(X). It not only lay the foundation for practical implementation of automated reasoning algorithm in LP(X), but also provides the strong support for α-n(t) ary resolution automated reasoning approaches.

Formula omitted] resolution method for a lattice-valued first-order logic

This paper focuses on resolution-based automated reasoning approaches in a lattice-valued first-order logic LF( X) with truth-values defined in a logical algebraic structure—lattice implication algebra (LIA), which aims at providing the logic foundation to represent and handle both imprecision and incomparability. In order to improve the efficiency of α - resolution approach proposed for LF( X), firstly the concepts of α - lock resolution principle and deduction are introduced for lattice-valued propositional logic LP( X) based on LIA, along with its soundness and weak completeness theorems. Then all the results are extended into LF( X) by using Lifting Lemma. Finally an α - lock resolution automated reasoning algorithm in LF( X) is proposed for the implementation purpose. This work provides a theoretical foundation for more efficient resolution-based automated reasoning algorithm in lattice-valued logic LF( X).

Determination of α-resolution in lattice-valued first-order logic LF(X)

Key issues for resolution-based automated reasoning in lattice-valued first-order logic LF(X) are investigated with truth-values in a lattice-valued logical algebraic structure–lattice implication algebra (LIA). The determination of resolution at a certain truth-value level (called α-resolution) in LF(X) is proved to be equivalently transformed into the determination of α-resolution in lattice-valued propositional logic LP(X) based on LIA. The determination of α-resolution of any quasi-regular generalized literals and constants under various cases in LP(X) is further analyzed, specified, and subsequently verified. Hence the determination of α-resolution in LF(X) can be accordingly solved to a very broad extent, which not only lays a foundation for the practical implementation of automated reasoning algorithms in LF(X), but also provides a key support for α-resolution-based automated reasoning approaches and algorithms in LIA based linguistic truth-valued logics.

Filter-based resolution principle for lattice-valued propositional logic LP( X)

As one of most powerful approaches in automated reasoning, resolution principle has been introduced to non-classical logics, such as many-valued logic. However, most of the existing works are limited to the chain-type truth-value fields. Lattice-valued logic is a kind of important non-classical logic, which can be applied to describe and handle incomparability by the incomparable elements in its truth-value field. In this paper, a filter-based resolution principle for the lattice-valued propositional logic LP( X) based on lattice implication algebra is presented, where filter of the truth-value field being a lattice implication algebra is taken as the criterion for measuring the satisfiability of a lattice-valued logical formula. The notions and properties of lattice implication algebra, filter of lattice implication algebra, and the lattice-valued propositional logic LP( X) are given firstly. The definitions and structures of two kinds of lattice-valued logical formulae, i.e., the simple generalized clauses and complex generalized clauses, are presented then. Finally, the filter-based resolution principle is given and after that the soundness theorem and weak completeness theorems for the presented approach are proved.

α-Resolution principle based on first-order lattice-valued logic LF(X)

Abstract In the present paper, as a continuous work about α-resolution principle based on lattice-valued propositional logic LP(X) (Information Sciences 130 (2000) 1–29) whose algebra of truth-values is a relatively general lattice – lattice implication algebra (LIA), the lattice-valued resolution principle for the corresponding first-order lattice-valued logic system LF(X) is focused. Firstly, some concepts about lattice-valued resolution principle for LF(X) are introduced and the Herbrand theorem for LF(X) is proved. Then, an α-resolution principle, which can be used to judge if a first-order lattice-valued logical formula in LF(X) is false at a truth-valued level α (i.e., α-false), is established. Finally, the completeness theorem of this α-resolution principle and the soundness theorem for the strong α-resolution are also proved. It is hoped that the current work would serve as a foundation for constructing resolution-based automated reasoning methods for lattice-valued logic capable of dealing with both comparable and incomparable uncertain information.

α-Resolution principle based on lattice-valued propositional logic LP( X)

In the present paper, resolution-based automated reasoning theory in an L-type fuzzy logic is focused. Concretely, the α-resolution principle, which is based on lattice-valued propositional logic LP( X) with truth-value in a logical algebra – lattice implication algebra, is investigated. Finally, an α-resolution principle that can be used to judge if a lattice-valued logical formula in LP( X) is always false at a truth-valued level α (i.e., α-false), is established, and the theorems of both soundness and completeness of this α-resolution principle are also proved. This will become the theoretical foundation for automated reasoning based on lattice-valued logical LP( X).