Implication Algebra 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.

ON QUASI-LATTICE IMPLICATION ALGEBRAS

The notion of quasi-lattice implication algebras is a generalization of lattice implication algebras. In this paper, we give an optimized definition of quasi-lattice implication algebra and show that this algebra is a distributive lattice and that this algebra is a lattice implication algebra. Also, we define a congruence relation Φ_F induced by a filter F and show that every congruence relation on a quasi-lattice implication algebra is a congruence relation Φ_F induced by a filter F.

Some kinds of falling fuzzy filters of lattice implication algebras

In this paper, the concepts of falling fuzzy (implicative, associative) filters of lattice implication algebras based on the theory of falling shadows and fuzzy sets are presented at first. And then the relations between fuzzy (implicative, associative) filters and falling fuzzy (implicative, associative) filters are provided. In particular, we put forward an open question on a kind of falling fuzzy filters of lattice implication algebras. Finally, we apply falling fuzzy inference relations to lattice implication algebras and obtain some related results.

Fuzzy 1-type and 2-type positive implicative filters of pseudo-BCK algebras1

The notions of fuzzy 1-type (2-type) positive implicative filter and fuzzy normal filter of pseudo-BCK algebras are introduced, some properties and equivalent conditions are given. By using the new notions, the characterizations of 1-type positive implicative pseudo-BCK algebra and 2-type positive implicative pseudo-BCK algebra (positive implicative BCK-algebra) are displayed. Moreover, some conclusions related with positive implicative filters in previous literature are revised. Finally, the concepts of fuzzy commutative filter and fuzzy implicative filter are proposed, and the relationships among fuzzy 1-type (2-type) positive implicative filters, fuzzy commutative filters and fuzzy implicative filters of pseudo-BCK algebras are investigated.

Vague Implicative LI – Ideals of Lattice Implication Algebras

We introduce the concept of vague implicative LI – ideals of lattice implication algebra and discuss some of their properties. We study the relationship between v-implicative filters, vague ILI - ideals and ILI – ideals. Extension property of a vague implicative LI – ideal is built.

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.

Simple Logics for Basic Algebras

An MV-algebra is an algebra (A, ⊕, ¬, 0), where (A, ⊕, 0) is a commutative monoid and ¬ is an idempotent operation on A satisfying also some additional axioms. Basic algebras are similar algebras that can roughly be characterised as nonassociative (hence, also non-commutative) generalizations of MV-algebras. Basic algebras and commutative basic algebras provide an equivalent algebraic semantics in the sense of Blok and Pigozzi for two recent logical systems. Both are Hilbert-style systems, with implication and negation as the primitive connectives. We present a considerably simpler logic, Lʙ, for basic algebras, where implication and falsum are taken as primitives. We also consider some subvarieties of basic algebras known in the literature, discuss classes of implicational algebras term-equivalent to each of these varieties, and construct axiomatic extensions of Lʙ for which these classes serve as equivalent algebraic semantics.

Ideals and fuzzy ideals on residuated lattices

This paper mainly focus on building the ideals theory of non regular residuated lattices. Firstly, the notions of ideals and fuzzy ideals of a residuated lattice are introduced, their properties and equivalent characterizations are obtained; at the meantime, the relation between filter and ideal is discussed. Secondly, two types prime ideals of a residuated lattice are introduced, the relations between the two types ideals are studied, in some special residuated lattices (such as MTL-algebras, lattice implication algebras, BL-algebras), prime ideal and prime ideal of the second kind are coincide. At the meantime, the notions of fuzzy prime ideal and fuzzy prime ideal of the second kind on a residuated lattice are introduced, aiming at the relation between prime ideal and prime ideal of the second kind, we mainly investigate the fuzzy prime ideal of the second kind. Finally, we investigated the fuzzy congruence relations induced by fuzzy ideal, we construct a new residuated lattice induced by fuzzy congruences, the homomorphism theorem is given.

Notes on “A survey of fuzzy implication algebras and their axiomatization”

In this note, we present a correction of Theorem 2 in the paper [1], and show that all the conditions (I1)–(I5) in the definition of fuzzy implication algebras are independent of each other. In addition, we prove that the class of all commutative fuzzy implication algebras forms an equational algebra class.

Linguistic-Valued Approximate Reasoning With Lattice Ordered Linguistic-Valued Credibility

AbstractLinguistic terms are often used to represent the truth degree or credibility degree to manage the uncertainty or imprecision as one of popular ways of knowledge representation in the perception-based decision making problem. This present work represents the credibility of uncertain knowledge using linguistic values. The linguistic-valued credibility is then modeled based on a lattice ordered logical algebra - lattice implication algebra (LIA) and then used in the linguistic-valued based on approximated reasoning. This approach can better express both comparable and incomparable linguistic valued credibility information in knowledge base representation. And it can also fit to approximate reasoning under a fuzzy environment with both comparable and incomparable linguistic credibility. Examples illustrate that the proposed approach can retain the original information and can simulate people's way of thinking through reasoning with words.

Endomorphisms of Implication Algebras

Abstract In this note we prove that if two implication algebras have isomorphic monoids of endomorphisms then they are isomorphic.

TRANSITIVE AND ABSORBENT FILTERS OF LATTICE IMPLICATION ALGEBRAS

The notion of transitive filters is introduced in lattice implication algebras. A necessary and sufficient condition is derived for every filter to become a transitive filter. Some sufficient conditions are also derived for a filter to become a transitive filter. The concept of absorbent filters is introduced and their properties are studied. A set of equivalent conditions is obtained for a filter to become an absorbent filter.

Monadic MV-algebras II: Monadic implicational subreducts

In this paper, we study the class of all monadic implicational subreducts, that is, the \({\{\rightarrow, \forall,1\}}\)-subreducts of the class of monadic MV-algebras. We prove that this class is an equational class, which we denote by \({\mathcal{ML}}\), and we give an equational basis for this variety. An algebra in \({\mathcal{ML}}\) is called a monadic Łukasiewicz implication algebra. We characterize the subdirectly irreducible members of \({\mathcal{ML}}\) and the congruences of every monadic Łukasiewicz implication algebra by monadic filters. We prove that \({\mathcal{ML}}\) is generated by its finite members. Finally, we completely describe the lattice of subvarieties, and we give an equational basis for each proper subvariety.

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.

On power idealization filter topologies of lattice implication algebras.

The aim of this paper is to introduce power idealization filter topologies with respect to filter topologies and power ideals of lattice implication algebras, and to investigate some properties of power idealization filter topological spaces and their quotient spaces.

Vague congruences and quotient lattice implication algebras.

The aim of this paper is to further develop the congruence theory on lattice implication algebras. Firstly, we introduce the notions of vague similarity relations based on vague relations and vague congruence relations. Secondly, the equivalent characterizations of vague congruence relations are investigated. Thirdly, the relation between the set of vague filters and the set of vague congruences is studied. Finally, we construct a new lattice implication algebra induced by a vague congruence, and the homomorphism theorem is given.

Attribute Extended Algorithm of Lattice-Valued Concept Lattice Based on Congener Formal Context

This paper is the continuation of our research work about lattice-valued concept lattice based on lattice implication algebra. For a better application of lattice-valued concept lattice into data distributed storage and parallel processing, it is necessary to research attribute extended algorithm based on congener formal context. The definitions of attribute extended formal context and congener formal context are proposed. On condition that the extent set stays invariable when the new attribute is increased, the necessary and sufficient conditions of forming attribute values are researched. Based on these conditions, the algorithms of generating lattice-valued congener formal context and establishing concept lattice are given, by which we can provide a useful basis for union algorithm and constructing algorithm of lattice-valued concept lattices in distributed and parallel system.

A unified algorithm for finding $$k$$ k -IESFs in linguistic truth-valued lattice-valued propositional logic

As a symbolic approach for computing with words, linguistic truth-valued lattice-valued propositional logic $$\fancyscript{L}_{V(n\times 2)}P(X)$$ L V ( n × 2 ) P ( X ) can represent and handle both imprecise and incomparable linguistic value-based information. Indecomposable extremely simple form (IESF) is a basic concept of $$\alpha $$ ? -resolution automated reasoning in lattice-valued logic based in lattice implication algebra (LIA). In this paper we establish a unified method for finding the structure of $$k$$ k -IESF in $$\fancyscript{L}_{V(n\times 2)}P(X)$$ L V ( n × 2 ) P ( X ) . Firstly, some operational properties of logical formulae in $$L_6P(X)$$ L 6 P ( X ) are studied, and some rules are obtained for judging whether a given logical formula is a $$k$$ k -IESF, which are used to contrive an algorithm for finding $$k$$ k -IESF in $$L_6P(X)$$ L 6 P ( X ) . Then, all the results are extended into $$\fancyscript{L}_{V(n\times 2)}P(X)$$ L V ( n × 2 ) P ( X ) . Finally, a unified algorithm for finding all $$k$$ k -IESFs in $$\fancyscript{L}_{V(n\times 2)}P(X)$$ L V ( n × 2 ) P ( X ) is proposed. This work provides theoretical foundations and algorithms for $$\alpha $$ ? -resolution automated reasoning in linguistic truth-valued lattice-valued logic based in linguistic truth-valued LIAs and formal tools for symbolic natural language processing.

Research of Unary Lattice Implication Algebra Equations on Interval Sets

The concepts are re-defined on the interval sets which are filter, prime filter, LI-ideal, dual atom and convex sub-lattice in the lattice implication algebra. Three basic unary lattice implication algebra equations on the interval sets are researched. The necessary and sufficient conditions for existence of solutions for the equations are presented. And some properties of equation sets also are given.