逻辑系统NMG 的满足性和紧致性

紧致性是模糊逻辑的一个重要性质.现已经证明?ukasiewicz 命题逻辑、G?del 命题逻辑、乘积命题逻辑和形式系统L^*都是紧的.通过刻画逻辑系统NMG 中的极大相容理论和证明NMG 的满足性,进而证明了NMG也是紧的.     

On the strict logic foundation of fuzzy reasoning

This paper focuses on the logic foundation of fuzzy reasoning. At first, a new complete first-order fuzzy predicate calculus system K^* corresponding to the formal system L^* is built. Based on the many-sort system K_ms^* corresponding to K^*, the triple I methods of FMP and FMT for fuzzy reasoning and their consistency are formalized, thus fuzzy reasoning is put completely and rigorously into the logic framework of fuzzy logic.The author is indebted to anonymous referee for his useful comments which have helped to improve the paper.     

Undecidability of PDL with L = {a2i |; i ⩾ 0}

It is shown that the validity problem for propositional dynamic logic (PDL), which is decidable and actually DEXPTIME-complete for the usual class of regular programs, becomes highly undecidable, viz. Π₁¹-complete, when the single nonregular one-letter program L = {a²ⁱ |; i ? 0} is added. This answers a question of Harel, Pnueli, and Stavi.     

基于支持度理论的广义Modus Ponens问题的最优解

为了将模糊推理纳入逻辑的框架并从语构和语义两个方面为模糊推理奠定严格的逻辑基础,通过将模糊推理形式化的方法移植到经典命题逻辑系统中,把FMP(fuzzy modus ponens)问题转化为GMP(generalized modus ponens)问题,并基于公式的真度概念提出了公式之间的支持度,进一步利用支持度的思想引入了GMP问题以及CGMP(collective generalized modus ponens)问题的一种新型最优求解机制.证明了最优解的存在性,同时指出,在经典命题逻辑系统中存在着与模糊逻辑完全相似的推理机制.该方法是一种程度化的方法,这就使得求解过程从算法上实现成为可能,并对知识的程度化推理有所启示.     

Theory of truth degrees of propositions in the logic system L n *

By means of infinite product of uniformly distributed probability spaces of cardinal n the concept of truth degrees of propositions in the n-valued generalized Lukasiewicz propositional logic system L _n ^* is introduced in the present paper. It is proved that the set consisting of truth degrees of all formulas is dense in [0, 1], and a general expression of truth degrees of formulas as well as a deduction rule of truth degrees is then obtained. Moreover, similarity degrees among formulas are proposed and a pseudo-metric is defined therefrom on the set of formulas, and hence a possible framework suitable for developing approximate reasoning theory in n-valued generalized Lukasiewicz propositional logic is established.     

Some constructions of the join of fuzzy subgroups and certain lattices of fuzzy subgroups with sup property

In this paper, some new lattices of fuzzy substructures are constructed. For a given fuzzy set μ in a group G, a fuzzy subgroup S(μ) generated by μ is defined which helps to establish that the set L_s of all fuzzy subgroups with sup property constitutes a lattice. Consequently, many other sublattices of the lattice L of all fuzzy subgroups of G like , etc. are also obtained. The notion of infimum is used to construct a fuzzy subgroup i(μ) generated by a given fuzzy set μ, in contrast to the usual practice of using supremum. In the process a new fuzzy subgroup i(μ)^∗ is defined which we shall call a shadow fuzzy subgroup of μ. It is established that if μ has inf property, then i(μ)^∗ also has this property.     

Theory of truth degrees of propositions in the logic system Ln*

By means of infinite product of uniformly distributed probability spaces of cardinal n the concept of truth degrees of propositions in the n-valued generalized Lukasiewicz propositional logic system L _n ^* is introduced in the present paper. It is proved that the set consisting of truth degrees of all formulas is dense in [0, 1], and a general expression of truth degrees of formulas as well as a deduction rule of truth degrees is then obtained. Moreover, similarity degrees among formulas are proposed and a pseudo-metric is defined therefrom on the set of formulas, and hence a possible framework suitable for developing approximate reasoning theory in n-valued generalized Lukasiewicz propositional logic is established.     

Results on the propositional μ-calculus

In this paper we define and study a propositional μ-calculus Lμ, which consists essentially of propositional modal logic with a least fixpoint operator. Lμ is syntactically simpler yet strictly more expressive than Propositional Dynamic Logic (PDL). For a restricted version we give an exponential-time decision procedure, small model property, and complete deductive system, theory subsuming the corresponding results for PDL.     

基于蕴涵算子Lp的模糊推理的FMP反向三I算法

研究了基于蕴涵算子L_p模糊推理的FMP反向三I支持算法及α-反向三I支持算法,给出了FMP模型的反向三I算法及α-反向三I算法的计算公式。     

Complexity of some problems in positive and related calculi

A problem of recognizing important properties of propositional calculi is considered, and complexity bounds for some decidable properties are found. For a given logical system L, a property P of logical calculi is called decidable over L if there is an algorithm which for any finite set Ax of new axiom schemes decides whether the calculus L+Ax has the property P or not. In Maksimova and Voronkov (Bull. Symbol. Logic 6 (2000) 118) the complexity of tabularity, pretabularity, and interpolation problems over the intuitionistic logic (Int) and over modal logic S4 was studied.In the present paper, positive and positively axiomatizable calculi are investigated. We prove NP-completeness of tabularity, DP-hardness of pretabularity and PSPACE-completeness of interpolation and projective Beth's property over the positive fragment Int⁺ of the intuitionistic logic. Some complexity bounds for properties of propositional calculi over the intuitionistic or the minimal logic are found.     

Arithmetic operators in interval-valued fuzzy set theory

We introduce the addition, subtraction, multiplication and division on L^I, where L^I is the underlying lattice of both interval-valued fuzzy set theory [R. Sambuc, Fonctions Φ-floues. Application à l’aide au diagnostic en pathologie thyroidienne, Ph.D. Thesis, Université de Marseille, France, 1975] and intuitionistic fuzzy set theory [K.T. Atanassov, Intuitionistic fuzzy sets, 1983, VII ITKR’s Session, Sofia (deposed in Central Sci. Technical Library of Bulg. Acad. of Sci., 1697/84) (in Bulgarian)]. We investigate some algebraic properties of these operators. We show that using these operators the pseudo-t-representable extensions of the ?ukasiewicz t-norm and the product t-norm on the unit interval to L^I and some related operators can be written in a similar way as their counterparts on ([0,1],?).     

Analytic fuzzy tableaux

 In this paper Beth–Smullyan's tableaux method is extended to the fuzzy propositional logic. The fuzzy tableaux method is based on the concepts of t-truth and extended graded formula. As in classical logic, it is a refutation procedure. A closed fuzzy tableau beginning with the extended graded formula [r, A] asserting that this is not t-true, is a tableau proof of the graded formula (A, r). The theorems of soundness, completeness, and decidability are proved.     

Gdel区间值逻辑系统的广义拟重言式

吴洪博博士将王国俊教授在R0逻辑系统中的广义重言式理论推广到Gdel逻辑系统中,通过定义两个同构映射,得到其逻辑系统F(S)的一个分划。将这一理论推广到区间值模糊命题逻辑系统中,定义了两个新的区间同构映射,最终得到区间值逻辑系统F(S)的一个分划。     

On the ideal convergence of sequences of fuzzy numbers

In this paper we introduce and study the concepts of I-convergence, I^∗-convergence and I-Cauchy sequence for sequences of fuzzy numbers where I denotes the ideal of subsets of N, the set of positive integers.     

Semigroup homomorphisms and fuzzy automata

A generalized Ω-fuzzy automaton over a complete residuated lattice Ω and a monoid (M,*) and with a set S of states is introduced as a monoid homomorphism F:(M,*)→(?,°), where (?,°) is a monoid of Ω-fuzzy sets in a set S×S. An extension principle depending of proper filters Φ in Ω is introduced which then enables to introduce morphisms between generalized Ω-fuzzy automata and to introduce the category ?_Φ of these automata. It is proved that categories of classical fuzzy automata, non-deterministic automata and some other systems are equivalent to subcategories of ?_Φ for a suitable filter Φ.     

Is there a need for fuzzy logic?

“Is there a need for fuzzy logic?” is an issue which is associated with a long history of spirited discussions and debate. There are many misconceptions about fuzzy logic. Fuzzy logic is not fuzzy. Basically, fuzzy logic is a precise logic of imprecision and approximate reasoning. More specifically, fuzzy logic may be viewed as an attempt at formalization/mechanization of two remarkable human capabilities. First, the capability to converse, reason and make rational decisions in an environment of imprecision, uncertainty, incompleteness of information, conflicting information, partiality of truth and partiality of possibility - in short, in an environment of imperfect information. And second, the capability to perform a wide variety of physical and mental tasks without any measurements and any computations [L.A. Zadeh, From computing with numbers to computing with words - from manipulation of measurements to manipulation of perceptions, IEEE Transactions on Circuits and Systems 45 (1999) 105-119; L.A. Zadeh, A new direction in AI - toward a computational theory of perceptions, AI Magazine 22 (1) (2001) 73-84]. In fact, one of the principal contributions of fuzzy logic - a contribution which is widely unrecognized - is its high power of precisiation.Fuzzy logic is much more than a logical system. It has many facets. The principal facets are: logical, fuzzy-set-theoretic, epistemic and relational. Most of the practical applications of fuzzy logic are associated with its relational facet.In this paper, fuzzy logic is viewed in a nonstandard perspective. In this perspective, the cornerstones of fuzzy logic - and its principal distinguishing features - are: graduation, granulation, precisiation and the concept of a generalized constraint.A concept which has a position of centrality in the nontraditional view of fuzzy logic is that of precisiation. Informally, precisiation is an operation which transforms an object, p, into an object, p^∗, which in some specified sense is defined more precisely than p. The object of precisiation and the result of precisiation are referred to as precisiend and precisiand, respectively. In fuzzy logic, a differentiation is made between two meanings of precision - precision of value, v-precision, and precision of meaning, m-precision. Furthermore, in the case of m-precisiation a differentiation is made between mh-precisiation, which is human-oriented (nonmathematical), and mm-precisiation, which is machine-oriented (mathematical). A dictionary definition is a form of mh-precisiation, with the definiens and definiendum playing the roles of precisiend and precisiand, respectively. Cointension is a qualitative measure of the proximity of meanings of the precisiend and precisiand. A precisiand is cointensive if its meaning is close to the meaning of the precisiend.A concept which plays a key role in the nontraditional view of fuzzy logic is that of a generalized constraint. If X is a variable then a generalized constraint on X, GC(X), is expressed as X isr R, where R is the constraining relation and r is an indexical variable which defines the modality of the constraint, that is, its semantics. The primary constraints are: possibilistic, (r = blank), probabilistic (r = p) and veristic (r = v). The standard constraints are: bivalent possibilistic, probabilistic and bivalent veristic. In large measure, science is based on standard constraints.Generalized constraints may be combined, qualified, projected, propagated and counterpropagated. The set of all generalized constraints, together with the rules which govern generation of generalized constraints, is referred to as the generalized constraint language, GCL. The standard constraint language, SCL, is a subset of GCL.In fuzzy logic, propositions, predicates and other semantic entities are precisiated through translation into GCL. Equivalently, a semantic entity, p, may be precisiated by representing its meaning as a generalized constraint.By construction, fuzzy logic has a much higher level of generality than bivalent logic. It is the generality of fuzzy logic that underlies much of what fuzzy logic has to offer. Among the important contributions of fuzzy logic are the following:

1.

FL-generalization. Any bivalent-logic-based theory, T, may be FL-generalized, and hence upgraded, through addition to T of concepts and techniques drawn from fuzzy logic. Examples: fuzzy control, fuzzy linear programming, fuzzy probability theory and fuzzy topology.

2.

Linguistic variables and fuzzy if-then rules. The formalism of linguistic variables and fuzzy if-then rules is, in effect, a powerful modeling language which is widely used in applications of fuzzy logic. Basically, the formalism serves as a means of summarization and information compression through the use of granulation.

3.

Cointensive precisiation. Fuzzy logic has a high power of cointensive precisiation. This power is needed for a formulation of cointensive definitions of scientific concepts and cointensive formalization of human-centric fields such as economics, linguistics, law, conflict resolution, psychology and medicine.

4.

NL-Computation (computing with words). Fuzzy logic serves as a basis for NL-Computation, that is, computation with information described in natural language. NL-Computation is of direct relevance to mechanization of natural language understanding and computation with imprecise probabilities. More generally, NL-Computation is needed for dealing with second-order uncertainty, that is, uncertainty about uncertainty, or uncertainty² for short.

In summary, progression from bivalent logic to fuzzy logic is a significant positive step in the evolution of science. In large measure, the real-world is a fuzzy world. To deal with fuzzy reality what is needed is fuzzy logic. In coming years, fuzzy logic is likely to grow in visibility, importance and acceptance.     

Decider: A fuzzy multi-criteria group decision support system

Multi-criteria group decision making (MCGDM) aims to support preference-based decision over the available alternatives that are characterized by multiple criteria in a group. To increase the level of overall satisfaction for the final decision across the group and deal with uncertainty in decision process, a fuzzy MCGDM process (FMP) model is established in this study. This FMP model can also aggregate both subjective and objective information under multi-level hierarchies of criteria and evaluators. Based on the FMP model, a fuzzy MCGDM decision support system (called Decider) is developed, which can handle information expressed in linguistic terms, boolean values, as well as numeric values to assess and rank a set of alternatives within a group of decision makers. Real applications indicate that the presented FMP model and the Decider  software are able to effectively handle fuzziness in both subjective and objective information and support group decision-making under multi-level criteria with a higher level of satisfaction by decision makers.     

A fuzzy logic system based on Schweizer-Sklar t-norm

In recent years, the basic research of fuzzy logic and fuzzy reasoning is growing ac- tively day by day, such as the basic logic system BL proposed by Hajek[1]; fuzzy logic system MTL proposed by Esteva and Godo[2]; fuzzy reasoning, implication operators …     

Reasoning with propositional knowledge based on fuzzy neural logic

In this article, a new kind of reasoning for propositional knowledge, which is based on the fuzzy neural logic initialed by Teh, is introduced. A fundamental theorem is presented showing that any fuzzy neural logic network can be represented by operations: bounded sum, complement, and scalar product. Propositional calculus of fuzzy neural logic is also investigated. Linear programming problems risen from the propositional calculus of fuzzy neural logic show a great advantage in applying fuzzy neural logic to answer imprecise questions in knowledge-based systems. An example is reconsidered here to illustrate the theory. © 1996 John Wiley & Sons, Inc.     

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.