Fuzzy Propositions Research Articles

Is the success of fuzzy logic really paradoxical?: Toward the actual logic behind expert systems

The formal concept of logical equivalence in fuzzy logic, while theoretically sound, seems impractical. The misinterpretation of this concept has led to some pessimistic conclusions. Motivated by practical interpretation of truth values for fuzzy propositions, we take the class (lattice) of all subintervals of the unit interval [0, 1] as the truth value space for fuzzy logic, subsuming the traditional class of numerical truth values from [0, 1]. The associated concept of logical equivalence is stronger than the traditional one. Technically, we are dealing with much smaller set of pairs of equivalent formulas, so that we are able to check equivalence algorithmically. The checking is done by showing that our strong equivalence notion coincides with the equivalence in logic programming.

FUZZY PETRI NETS FOR MODELING RULE-BASED REASONING

In this paper, a fuzzy Petri nets for modeling fuzzy rule-based reasoning is proposed to bring together the possibilistic entailment and the fuzzy reasoning to handle uncertain and imprecise information. The three key components in our fuzzy rule-based reasoning: fuzzy propositions, truth-qualified fuzzy rules, and truth-qualified fuzzy facts, can be formulated as fuzzy places, uncertain transitions, and uncertain fuzzy tokens, respectively. Four types of uncertain transitions, inference, aggregation, duplication and aggregation-duplication transitions, are introduced to meet the mechanism of fuzzy rule-based reasoning. A reasoning algorithm based on fuzzy Petri nets is also presented to improve the efficiency of fuzzy rule-based reasoning. The reasoning algorithm is consistent with not only the rule-based reasoning but also the execution of Petri nets.

Some considerations about a least squares model for fuzzy rules of inference

In this paper, the method of least squares is applied to the fuzzy inference rules. We begin studying the conditions in which from a fuzzy set we can build another through the method of least squares. Then we apply this technique in order to evaluate the conclusions of the generalized modus ponens and the generalized modus tollens. We present different theorems and examples that demonstrate the fundamental advantages of the method studied. We conclude presenting a method for obtaining the conclusion when working with fuzzy conditional propositions that contain two fuzzy propositions joined by the connective “and”.

Possibility theory is not fully compositional! A comment on a short note by H.J. Greenberg

A recent short note [H.J. Greenberg, Fuzzy Sets and Systems 86 (1997) 249–250] has assumed that a possibility measure is fully compositional with respect to negation, conjunction and disjunction. Up to a trivial exception, such a possibility measure does not exist. This short note recalls that the calculus of possibility measures on Boolean propositions is not compositional (except for the disjunction) and has little to do with the fully compositional max-min calculus of degrees of truth for fuzzy propositions.

Two-sided (intuitionistic) fuzzy reasoning

Fuzzy propositional logic structure is summarized; different forms of fuzzy propositional expressions and their relations are discussed. Two-sided fuzzy proposition is defined in propositional logic algebra as the counterpart of intuitionistic fuzzy set. The concept of two-sided fuzzy reasoning is discussed and its mathematical structure is developed. Using two-sided fuzzy propositions, human decision-making can be closely simulated by considering his perception of both (somewhat opposite) sides of the subject matter simultaneously. Multiuniverse two-sided fuzzy propositions are presented and multiuniverse operations are defined. Two-sided fuzzy if-then rules are investigated under different interpretations of fuzzy implications, Approximate of these different implications and the inferences in the output universe are investigated, and associated error terms are identified.

Knowledge representation using fuzzy Petri nets-revisited

In the paper by S. Chen et al. (see ibid., vol.2, no.3, p.311-19, 1990), the authors proposed an algorithm which determines whether there exists an antecedent-consequence relationship from a fuzzy proposition d/sub s/ to proposition d/sub j/ and if the degree of truth of proposition d/sub s/ is given, then the degree of truth of proposition d/sub j/ can be evaluated. The fuzzy reasoning algorithm proposed by S. Chen et al. (1990) was found not to be working with all types of data. We propose: (1) a modified form of the algorithm, and (2) a concept of hierarchical fuzzy Petri nets for data abstraction.

The Application of Fuzzy Rating Modus Ponens to Construction Operations

ABSTRACTIn this paper, the concept of rating space, constructed based on the idea of angular models developed earlier, is employed as a tool for approximate reasoning and applied to fuzzy modus ponens deduction techniques. This approach infers IF-THEN rules by transforming a problem into the rating space. The problem can then be solved by computing the rating values of the conclusion when a new value of an antecedent is available. Using this technique, the rating space transformation allows a uniform treatment of fuzzy propositions using a simple calculation procedure. Examples of applications to construction operations are presented for illustration purposes.

A new structure for fuzzy systems: Fuzzy propositional logic and multi-universe representation of fuzzy decision processes

A new structure has been developed for fuzzy logical algebra. This is called as “Fuzzy propositional logic structure” in which “Proposition” is defined as a more generalized form of fuzzy proposition defined by Zadeh. With this structure, the fuzzy reasoning mathematics becomes very simple, and very easy to understand. The fuzzy sets and fuzzy propositions are represented in the extended universe and all logical operations can be performed by the known logical formulas already developed for nonfuzzy logic. Mutual exclusive and complementary properties of propositions have been defined. Notion of a fuzzy basis for the universe is developed. The decision formula is transformed into the output universe, hence all calculations become one-universe operations. In short, this new structure brings about unification and simplification in all stages of fuzzy decision process.

Truth-qualification and fuzzy relations in natural languages, application to medical diagnosis

Given two fuzzy propositions, how to truth-qualify one of them to induce the other one in a semantical equivalence is the first basic issue addressed in this paper. In the most general case, the set of solutions of this converse problem is fully characterized. Two fuzzy subsets of the unit interval ( τ 0 and τ 1, representing linguistic truth values) are introduced that provide best lower and upper approximations when no exact solution can be found: best semantic entailments of propositions are thus derived. In a new approach, the problem is reformulated in terms of fuzzy relation equations, from which results are retrieved and extended. The second part of this paper introduces a truth-possibility index defined from τ 0 and τ 1, that serves pattern-matching purposes, in addition to the usualv possibility and necessity measures. A biomedical application, in which medical knowledge is expressed in a rule form, with ANDed fuzzy propositions in the antecedent, illustrates the aggregation of these measures, for medical diagnosis assistance.

Knowledge representation using fuzzy deduction graphs

A new knowledge representation model, known as fuzzy deduction graph (FDG), is introduced in this paper. An FDG can represent a knowledge base containing the fuzzy propositions and fuzzy rules. In an FDG, a systematic method of finding the fuzzy reasoning path (FRP) is given which is based on Dijkstra's shortest path framework. The FRP gives a relationship between the antecedent (source) proposition and consequent (goal) proposition, such that the consequent proposition is reached with the greatest fuzzy value. The process of finding the FRP is illustrated with examples.

APPROXIMATE REASONING AND INTERPRETATION OF LABORATORY TESTS IN MEDICAL DIAGNOSTICS

In many real-world domains, such as medicine, human knowledge is by nature imprecise. As a consequence, the expert systems oriented to these domains must have specific tools lo deal with the uncertainty. A commonly used approach is to equip the expert system with a computational capability to analyze the transmission of uncertainty from the premises lo the conclusion. The theory of fuzzy sets provides a systematic framework for dealing with fuzzy quantifiers, such as many, few, and most, and linguistic variables like normal, increased, tall, small, and old. In this framework the results of laboratory tests (we refer to the area of medicai diagnostics) that are precise may be interpreted in the terms of fuzzy propositions by means of membership functions. This approach permits the expert system to employ the same inference engine used to deal with the imprecise information provided by physicians. In this paper we describe a method for the construction of the membership functions using the statistical data ...

Relating and extending semantical approaches to possibilistic reasoning

Two main semantical approaches to possibilistic reasoning with classical propositions have been proposed in the literature. Namely, Dubois-Prade's approach known as possibilistic logic, whose semantics is based on a preference ordering in the set of possible worlds, and Ruspini's approach that we redefine and call similarity logic, which relies on the notion of similarity or resemblance between worlds. In this article we put into relation both approaches, and it is shown that the monotonic fragment of possibilistic logic can be semantically embedded into similarity logic. Furthermore, to extend possibilistic reasoning to deal with fuzzy propositions, a semantical reasoning framework, called fuzzy truth-valued logic, is also introduced and proved to capture the semantics of both possibilistic and similarity logics.

Representation and use of imprecise temporal knowledge in dynamic systems

In this paper, the importance of representation and use of imprecise temporal knowledge for problem solving in industrial dynamic systems is emphasized first. A representation of imprecise temporal knowledge is then given for describing the dynamic behavior of systems. A novel problem on on-line temporal matching and numeric matching is presented, and a method for solving this problem is proposed. A method for fuzzy proposition reasoning and temporal reasoning is put forward. Finally, as two application examples, the fault diagnosis of a fluidized catalytic cracking unit is taken up to illustrate the usefulness of the proposed knowledge representation and methods.

A new approach to inference in approximate reasoning

Abstract We introduce a new approach to inference in approximate reasoning based on truth value restriction. The degree to which the actual given value A of a variable X agrees with the antecedent value B in a production “If X is B then Y is C ” is represented as a fuzzy subset of a truth space. We introduce a new form of implication based on the exponential operation. A new compatibility relation based on the input value A and the antecedent value B is defined. A fuzzy truth value true derived from the antecedent value B is also defined. It is shown that by mapping this compatibility relation into the truth space, a fuzzy truth value is generated which produces an exponential family of the fuzzy truth value true . We show that the inferred consequent can be generated from this truth value. Theoretical results showing the validity of this method to (1) chain rules consisting of several single fuzzy conditional propositions, and (2) conjunctive fuzzy conditional propositions, are also presented. Theoretical and simulation results demonstrate that our method is superior to the existing methods when intuitively correct responses are required.

PLAUSIBLE APPROXIMATE REASONING

Modifications to Zadeh's theory of approximate reasoning are proposed to enhance the theory's ability to produce plausible inference results. The modifications include the introduction of a different implication definition and a certainty qualification for fuzzy propositions. Plausible reasoning is characterized by a set of reasoning patterns of practical significance. This set is then used to validate the overall approximate reasoning method.

A fuzzy expert system shell using both exact and inexact reasoning

This paper presents a comprehensive expert system shell which can deal with both exact and inexact reasoning. A prototype of this proposed shell, code named as SYSTEM Z-IIe, has been implemented successfully. It is a rule-based system which employs fuzzy logic and numbers for its reasoning. Two basic inexact concepts, fuzziness and uncertainty, are both used and distinct from each other clearly in the system. Moreover, these two concepts have been built into two levels for inexact reasoning, i.e. the level of the rules and facts, and the level of the values of the objects of these rules and facts. Other features of Z-IIe include multiple fuzzy propositions in rules and dual fact input mechanisms. It also allows any combinations of fuzzy and normal terms and uncertainties. Fuzzy numeric comparison logic control is also available for the rules and facts. Its natural language interface which uses English with restricted syntax improves the efficiency of knowledge engineering. Z-IIe is also coupled to a Database Management System for supplying facts from existing databases if appropriate. All these features can be combined to build very powerful expert systems and are illustrated by an example.

The measure of the degree of truth and of the grade of membership

A method is presented to define the ‘meaning’ of a fuzzy proposition and of a fuzzy set through the construction of fuzzy propositions with non-ambiguous intermediate truth values.

Implication in fuzzy logic

Intermediate truth values and the order relation “as true as” are interpreted. The material implication A → B quantifies the degree by which “B is at least as true as A.” Axioms for the → operator lead to a representation of → by the pseudo-Lukasiewicz model. A canonical scale for the truth value of a fuzzy proposition is selected such that the → operator is the Lukasiewicz operator and the negation is the classical 1−. operator. The mathematical structure of some conjunction and disjunction operators related to → are derived.

Necessity Measures and the Resolution Principle

A careful distinction is made between fuzzy propositions (i.e., propositions involving vague predicates) that may have intermediary degrees of truth, and uncertain propositions (with nonvague predicates) the truth or falsity of which cannot definitely be established due to the incompleteness of the available information. Then the resolution principle is extended in the case of uncertain propositions where the uncertainty is modeled in terms of necessity measures. The alternative use of probability measures or of Shafer's belief functions is also discussed.

An application of possibility theory: The evaluation of availability of acupuncture anaesthesia

It is difficult to evaluate availability of anaesthesia in operations under acupuncture anaesthesia because, except for the linguistic method, no definite method is known. In this paper, we propose a method which is based on Zadeh's possibility theory, and give a definition of the Pain Index, a quantity which can reflect the availability of anaesthesia. This method is in agreement with the traditional one and it can be applied better in some special cases. Besides, the efficiency of this method also reflects the stability of the grade of availability which is represented by the form of language and it may be described by a suitable possibility-qualified fuzzy proposition.