Lattice-valued First-order Logic Research Articles

Multi-ary α-semantic resolution automated reasoning based on lattice-valued first-order logic LF (X)1

This paper focuses on multi-ary α-semantic resolution automated reasoning method based on multi-ary α-resolution principle for a lattice-valued logic with truth-value in a lattice implication algebra. Firstly, the definitions of the multi-aryα-semantic resolution and multi-ary α-semantic resolution deduction in LF (X) are given, respectively, the soundness and completeness of this ground case are gotten. Secondly, the validity of multi-ary α-semantic resolution method is analyzed. Thirdly, the algorithm for multi-ary α-semantic resolution methods based on lattice-valued first-order logicLF (X) is constructed. This work will provide a theoretical foundation for the more efficient resolution based automated reasoning in lattice-valued logic.

α-Resolution Method for Lattice-valued Horn Generalized Clauses in Lattice-valued Propositional Logic Systems

AbstractIn this paper, an α-resolution method for a set of lattice-valued Horn generalized clauses is established in lattice-valued propositional logic system L P(X ) based on lattice implication algebra. Firstly, the notions of lattice-valued Horn generalized clause, normal lattice-valued Horn generalized clause and unit lattice-valued Horn generalized clause are given in L P(X ). Then, the α-resolution of two lattice-valued Horn generalized clauses is represented in L P(X ). It indicates the reasoning rules in a resolution process, which aims at deleting α-resolution literals and obtaining a resolvent. Finally, we build an α-resolution algorithm for a set of lattice-valued Horn generalized clauses in L P(X ). It provides a foundation for automated reasoning in lattice-valued first-order logic system and an application for designing an inference system in the field of intelligent decision support.

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.

On the algebraic structure of binary lattice-valued fuzzy relations

From a general algebraic point of view, this paper aims at providing an algebraic analysis for binary lattice-valued relations based on lattice implication algebras--a kind of lattice-valued propositional logical algebra. By abstracting away from the concrete lattice-valued relations and the operations on them, such as composition and converse, the notion of lattice-valued relation algebra is introduced, LRA for short. The reduct of an LRA is a lattice implication algebra. Such an algebra generalizes Boolean relation algebras by general distributive lattices and can provide a fundamental algebraic theory for establishing lattice-valued first-order logic. Some important results are generalized from the classical case. The notion of cylindric filter is introduced and the generated cylindric filters are characterized.

Ideal resolution principle for lattice-valued first-order logic based on lattice implication algebra

As a continuate work, ideal-based resolution principle for lattice-valued first-order logic system LF(X) is proposed, which is an extension of α-resolution principle in lattice-valued logic system based on lattice implication algebra. In this principle, the resolution level is an ideal of lattice implication algebra, instead of an element in truth-value field. Moreover, the soundness theorem is given. In the light of lifting lemma, the completeness theorem is established. This can provide a new tool for automated reasoning.

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.

On the consistency of rule bases based on lattice-valued first-order logic LF(X)

The consistency of a rule base is an essential issue for rule-based intelligent information processing. Due to the uncertainty inevitably included in the rule base, it is necessary to verify the consistency of the rule base while investigating, designing, and applying a rule-based intelligent system. In the framework of the lattice-valued first-order logic system LF(X), which attempts to handle fuzziness and incomparability, this article focuses on how to verify and increase the consistency degree of the rule base in the intelligent information processing system. First, the representations of eight kinds of rule bases in LF(X) as the generalized clause set forms based on these rule bases' nonredundant generalized Skolem standard forms are presented. Then an α-automated reasoning algorithm in LF(X), also used as an automated simplification algorithm, is proposed. Furthermore, the α-consistency and the α-simplification theories of the rule base in LF(X) are formulated, and especially the coherence between these two theories is proved. Therefore, the verification of the α-consistency of the rule base, often an infinity problem that is difficult to solve, can be transformed into a finite and achievable α-simplification problem. Finally, an α-simplification stepwise search algorithm for verifying the consistency of the rule base as well as a kind of filtering algorithm for increasing the consistency level of the rule base are proposed. © 2006 Wiley Periodicals, Inc. Int J Int Syst 21: 399–424, 2006.

ON SEMANTICS OF L-VALUED FIRST-ORDER LOGIC Lvft

Many-valued logic system always plays a crucial role in artificial intelligence. Many researchers have paid considerable attention to lattice-valued logic with truth values in a lattice. In this paper, based on lattice implication algebras introduced by Xu (Journal of Southwest Jiaolong University (in Chinese), Sum. No. 89(1), 20-27, 1993, and L-valued propositional logic vft, established by Xu et al. (Information Sciences, 114, 20S-235, 1999a), the semantics of a L-type lattice-valued first-order logic Lvft, with truth values in lattice implication algebras were investigated. Some basic concepts about semantics of Lvftsuch as the language and the interpretation etc. were given and some semantic properties also were discussed. Finally, a concept of g-Skolem standard form was introduced, and it was shown that the unsatisfiability of a given lattice-valued formula was equivalent to that of its g-Skolem standard form. It will become a foundation to investigate the resolution principle based on first-order logic Lvft