Lattice-valued Logic Research Articles

Topological Characterizations of Rough Set Theory Based on Quantum Logic

Rough set theory is a mathematical approach for dealing with uncertain and imperfect knowledge processing. This paper investigates rough sets based on quantum logic, i.e., orthomodular lattice-valued logic. Some properties of the lower and upper approximations are proposed. Finally, some topological characterizations of the new rough set model are discussed.

A Topology Graph Algorithm Based on Lattice-Valued Logic to Solve Satisfiability Problems

Automatic reasoning is a very challenging research area in the field of artificial intelligence. Automatic inference based on lattice-valued logic is an important basic research content in the field of intelligent information processing. It is of great academic significance to deal with incomparability, linguistic valued information, and formal verification of the correctness of corresponding software and hardware systems in various applications. Based on the extensive work of α-resolution automated reasoning in linguistic truth-valued lattice-valued logic, this paper discusses the structure and its α-resolvability of the generalized literal. For further improving the efficiency, a topology graph deduction is proposed as well as its soundness, completeness, and universal algorithm. The use of distributed computing and resource sharing is the core idea of this deduction. In the decision process, as long as one satisfiability solution is found by subtask, the decision result can be given directly, shortening the decision time. The algorithm reduces the time complexity from factorial level to square.

Limited approximate bisimulations and the corresponding rough approximations

To measure the similarity of nodes in the neighboring subgraphs, Milner introduced the notion of k-limited bisimilarity. Recently, as a weaker version of k-limited bisimilarity, the notion of k-limited similarity was proposed and applied to graph pattern matching. (Bi)simulations have been widely used in comparing the behavior of fuzzy transition systems. In order to study the (bi)simulation semantics of labeled fuzzy transition systems in the residuated lattice-valued logic setting, we introduce an extension of labeled approximation spaces, called the quantitative fuzzy approximation spaces (QFASs), whose labels are equipped with a residuated lattice-valued equality relation. In a QFAS, we define a new notion of limited approximate similarity, to quantify to what extent one state is simulated by another in the neighboring subgraphs, and provide its properties. Based on the new notion, we give a definition of limited approximate simulation and discuss its properties. Then we introduce an ordered pair of relations, one on the state (vertex) set (limited approximate simulation) and one on the edge (transition) set induced by the relation on state set, called VE limited approximate simulation in this paper. We also present a new notion of limited approximate bisimilarity in a QFAS, to quantify to what extent two states are similar in the neighboring subgraphs, and give its properties. One main contribution of the paper is to give a condition for two states to be limited approximate bisimilar and investigate the degree of similarity between two states in a QFAS. Finally, we discuss the relationships between the rough approximations based on the underlying crisp relation induced by underlying labeled fuzzy relation and the rough approximations based on limited approximate bisimilarity.

Categorical study for algebras of Fitting’s lattice-valued logic and lattice-valued modal logic

The paper explores categorical interconnections between lattice-valued relational systems and algebras of Fitting’s lattice-valued modal logic. We define lattice-valued Boolean systems, and then we study adjointness and co-adjointness of functors defined on them. As a result, we get a duality for algebras of lattice-valued logic. Following this duality result, we establish a duality for algebras of lattice-valued modal logic.

$$\alpha $$-Paramodulation method for a lattice-valued logic $$L_nF(X)$$ with equality

In this paper, \(\alpha \)-paramodulation and \(\alpha \)-GH paramodulation methods are proposed for handling logical formulae with equality in a lattice-valued logic \(L_nF(X)\), which has unique ability for representing and reasoning uncertain information from a logical point of view. As an extension of the work of He et al. (in: 2015 10th international conference on intelligent systems and knowledge engineering (ISKE), pp 18–20. IEEE, 2015; Uncertainty modelling in knowledge engineering and decision making: proceedings of the 12th international FLINS conference, pp 477–482. World Scientific, 2016), a new form of \(\alpha \)-equality axioms set is proposed. The equivalence between \(\alpha \)-equality axioms set and \(E_{\alpha }\)-interpretation in \(L_nF(X)\) with an appropriate level is also established, which may provide a key foundation for equality reasoning in lattice-valued logic. Based on its equivalence, \(E_{\alpha }\)-unsatisfiability equivalent transformation is given. Furthermore, \(\alpha \)-paramodulation and its restricted method (i.e., \(\alpha \)-GH paramodulation) are given. The soundness and completeness of the proposed methods are also examined.

Skolemization and Herbrand theorems for lattice-valued logics

Skolemization and Herbrand theorems are obtained for first-order logics based on algebras with a complete lattice reduct and operations that are monotone or antitone in each argument. These lattice-valued logics, defined as consequence relations on inequations between formulas, typically lack properties underlying automated reasoning in classical first-order logic such as prenexation, deduction theorems, or reductions from consequence to satisfiability. Skolemization and Herbrand theorems for the logics therefore take various forms, applying to the left or right of consequences, and restricted classes of inequations. In particular, in the presence of certain witnessing conditions, they admit sound “parallel” Skolemization procedures where a strong quantifier is removed by introducing a finite disjunction or conjunction of formulas with new function symbols. A general expansion lemma is also established that reduces consequence in a lattice-valued logic between inequations containing only strong occurrences of quantifiers on the left and weak occurrences on the right to consequence between inequations in the corresponding propositional logic. If propositional consequence is finitary, this lemma yields a Herbrand theorem for the logic.

L-Almost separation axioms in L-fuzzifying-bitopologies via complete residuated lattice-valued logic

In this work, an extension of almost separation axioms is introduced in lattice valued theory. Such a generalized almost separation axioms is a structure of (i, j)-regular open set of its lattice-valued satisfying axioms. It uses two lattices one complete residuated lattice, and another strictly two-sided commutative quantale. It is a logical generalization of the notions of almost separation axioms. In our definition of generalized fuzzifying separation axioms, each notions can be regarded as relations between the one another to some degree.

Multi-ary α-ordered linear minimal resolution method in lattice-valued logic system1

Multi-ary α-ordered linear minimal resolution method in lattice-valued logic system1

State hyperstructures of tree automata based on lattice-valued logic

In this paper, an association is organized between the theory of tree automata on one hand and the hyperstructures on the other hand, over complete residuated lattices. To this end, the concept of order of the states of a complete residuated lattice-valued tree automaton (simply L-valued tree automaton) is introduced along with several equivalence relations in the set of the states of an L-valued tree automaton. We obtain two main results from this study: one of the relations can lead to the creation of Kleene’s theorem for L-valued tree automata, and the other one leads to the creation of a minimal v-valued tree automaton that accepts the same language as the given one.

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

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.

Nondeterministic fuzzy automata with membership values in complete residuated lattices

Automata theory based on complete residuated lattice-valued logic has been initiated by Qiu, and further systematically established by Qiu et al. In this paper, we propose the notion of nondeterministic fuzzy automata with membership values in a complete residuated lattice L, called lattice-valued nondeterministic fuzzy automata (L-NFAs). In our setting, a state of an L-NFA may have more than one transition labeled by the same input symbol, which reflects nondeterminism. To compare the behaviors of L-NFAs, we introduce two language equivalence relations which have different discriminating power. Furthermore, we extend the two relations to the complete residuated lattice-valued setting and investigate their properties such as robustness and compositionality. The theory developed here is applicable to the quantitative modeling and verification of fuzzy systems.

Coding tree languages based on lattice-valued logic

We consider tree automata based on complete residuated lattice-valued (for simplicity we write L-valued) logic. First, we define the concepts of response function and accessible states (with threshold c) of an L-valued tree automaton. Thereafter, we consider coding of trees and investigate the relation between response function on trees and their coding. Using the provided theorems, we give a pumping lemma for recognizable coding tree languages with threshold c. Moreover, we consider closure properties of recognizable coding tree languages. In this regard, we show that the class of recognizable coding tree languages with threshold c is closed under projection, intersection and union.

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.

Researches on Six Lattice-Valued Logic

Based on the direct product of Boolean algebra and Lukasiewicz algebra, six lattice-valued logic is put forward in this paper. The algebraic structure and properties of the lattice are analyzed profoundly and the tautologies of six-valued logic system L6P(X) are discussed deeply. The researches of this paper can be used in lattice-valued logic systems and can be helpful to automated reasoning systems.

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-Minimal Resolution Principle For A Lattice-Valued Logic

α-Minimal Resolution Principle For A Lattice-Valued Logic

Lattice-valued simulations for quantitative transition systems

Quantitative (bi)simulations taking values from non-negative real numbers enjoy numerous applications in the analysis of labeled transition systems, whose transitions, states, or labels contain quantitative information. To investigate the simulation semantics of labeled transition systems in the residuated lattice-valued logic setting, we introduce an extension of labeled transition systems, called the quantitative transition systems (QTSs), whose labels are equipped with a residuated lattice-valued equality relation. We then establish a lattice-valued relation between states of a QTS, called approximate similarity, to quantify to what extent one state is simulated by another. One main contribution of this paper is to show that unlike the classic setting where similarity has both fixed point and logical characterizations, these results do not hold for approximate similarity on QTSs in general, but they hold for QTSs having truth values from finite Heyting algebras.

α-Minimal Resolution Principle For A Lattice-Valued Logic

Based on the academic ideas of resolution-based automated reasoning and the previously established research work on binary α-resolution based automated reasoning schemes in the framework of lattice...

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.

α-Generalized Semantic Resolution Method in Linguistic Truth-valued Propositional Logic P(X)

AbstractThis paper is focused on α-generalized semantic resolution automated reasoning method in linguistic truth-valued lattice-valued propositional logic. Concretely, α-generalized semantic resolution for lattice-valued propositional logic (Ln× L2)P(X) is equivalently transformed into that for lattice-valued propositional logic LnP(X)(i ∊ {1,2…,n}). A similar conclusion is obtained between the α-generalized semantic resolution for linguistic truth-valued lattice-valued propositional logic LV(n×2)P(X) and that for lattice-valued propositional logic LV(n)P(X)(i ∊ {1,2,…,n}). Secondly, the generalized semantic resolution for lattice-valued propositional logic LnP(X) based on a chain-type truth-valued field is investigated and its soundness and weak completeness are given. The Presented work provides some foundations for resolution-based automated reasoning in linguistic truth-valued lattice-valued logic based on lattice implication algebra.