Theory Of Approximate Reasoning Research Articles

Failure Mode and Effects Analysis on the Air System of an Aero Turbofan Engine Using the Gaussian Model and Evidence Theory.

Failure mode and effects analysis (FMEA) is a proactive risk management approach. Risk management under uncertainty with the FMEA method has attracted a lot of attention. The Dempster-Shafer (D-S) evidence theory is a popular approximate reasoning theory for addressing uncertain information and it can be adopted in FMEA for uncertain information processing because of its flexibility and superiority in coping with uncertain and subjective assessments. The assessments coming from FMEA experts may include highly conflicting evidence for information fusion in the framework of D-S evidence theory. Therefore, in this paper, we propose an improved FMEA method based on the Gaussian model and D-S evidence theory to handle the subjective assessments of FMEA experts and apply it to deal with FMEA in the air system of an aero turbofan engine. First, we define three kinds of generalized scaling by Gaussian distribution characteristics to deal with potential highly conflicting evidence in the assessments. Then, we fuse expert assessments with the Dempster combination rule. Finally, we obtain the risk priority number to rank the risk level of the FMEA items. The experimental results show that the method is effective and reasonable in dealing with risk analysis in the air system of an aero turbofan engine.

A Theory of Approximate Reasoning with Type-2 Fuzzy Set

In this paper, an attempt is made to study approximate reasoning based on a Type-2 fuzzy set theory. In the process, we have examined the underlying fuzzy logic structure on which the reasoning is formulated. We have seen that the partial/incomplete/imprecise truth-values of elements of a type-2 fuzzy set under consideration forms a lattice. We propose two new lattice operations which ultimately help us to define a residual and thereby making the structure of truth- values a residuated lattice. We have focused upon two typical rules of inference used mostly in ordinary approximate reasoning methodology based on Type-1 fuzzy set theory. Our proposal is illustrated with typical artificial examples.

Varying functional representations of a crisp discrete function under fuzzy environment to coin new siso algorithms with application to time series modeling

One of the main categories of fuzzy reasoning mechanisms for a single-input single-output (SISO) fuzzy system based on linguistically described model is the category of composition based algorithms. The main ingredients of a composition based SISO algorithm consist of two parts: a system relation matrix (representing the structure information of a fuzzy system) and a compositional rule of inference (to obtain the corresponding output for the crisp or fuzzy value of an input variable). In literature, there are various composition based algorithms based on a fuzzy-logic-based approach via different versions of the theory of approximate reasoning (AR) since it is initiated by Zadeh in 1979. In this article, we provide an alternate approach to construct composition based SISO algorithms by viewing the collection of the (antecedent part, consequent part) of IF-THEN rules of a fuzzy system as a class of weighted information and using weighted t-norm and weighted t-conorm operations to aggregate such weighted information to build its system relation matrix instead of the usual AR approach. The novelty of our approach is that not only the construction of the system relation matrix but also the process of composition can be executed in a similar way. To justify our approach, we examine various functional representations of a crisp discrete function under fuzzy environment and using them as patterns to coin four classes of composition based SISO algorithms, in which, two classes are of new type while the other two classes rediscover the well-known Mandani-type and Zadeh's logic-type algorithms. Then, as an application, in the latter part of this article, we provide a generalization of the time series modeling proposed by Štĕpnička et al. in which, they suggested using the fuzzy transform method to estimate the trend-cyclic component of a time series. To accomplish this, we will show that, under the contexts of this article, we may view the (discrete) fuzzy transform (and the corresponding inverse fuzzy transform) of a time series as a composition of specific SISO algorithms and hence, the time series modeling proposed by Štĕpnička et al. may be extended via suitable techniques of SISO algorithms.

Truth Degrees Theory and Approximate Reasoning in 3-Valued Propositional Pre-Rough Logic

By means of the function induced by a logical formulaA, the concept of truth degree of the logical formulaAis introduced in the 3-valued pre-rough logic in this paper. Moreover, similarity degrees among formulas are proposed and a pseudometric is defined on the set of formulas, and hence a possible framework suitable for developing approximate reasoning theory in 3-value logic pre-rough logic is established.

A Quantitative Method of n-valued Gödel Propositional Logic

Abstract By means of infinite product of uniformly distributed probability spaces of cardinal n the concept of pure truth degrees of propositions in the n-valued Godel logic system is introduced in the present paper. Inference rules of pure truth degrees are 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 Godel propositional logic is established.

Reasoning with doubly uncertain soft constraints

We describe the basic ideas of the theory of approximate reasoning and indicate how it provides a framework for representing human sourced soft information. We discuss how to translate linguistic knowledge into formal representations using generalized constraints. We consider the inference process within the theory of approximate reasoning and introduce the entailment principle and describe its centrality to this inference process. Next we introduce the idea of doubly uncertain statements such as John’s friend is young. In these statements there exists uncertainty both with respect to value of the age, young, and the object associated with the age, John’s friend. We suggest a method for representing these complex statements and investigate the problem of making inferences about specific objects.

A framework for reasoning with soft information

In order to provide for the representation and manipulation of human sourced soft information we turn to the fuzzy set based theory of approximate reasoning. We describe how approximate reasoning provides a framework for representing and manipulating a wide body linguistically expressed information. We then suggest a number of extensions of the theory to enhance its representational capacity. One such extension focuses on the ability to model imprecise variables as well as imprecise values for the variable. We consider the representation of possible qualified propositions. We look at the issue of deduction in the face of conflict in our knowledge base and suggest an approach compatible with human behavior.

Comparing approximate reasoning and probabilistic reasoning using the Dempster–Shafer framework

We investigate the problem of inferring information about the value of a variable V from its relationship with another variable U and information about U. We consider two approaches, one using the fuzzy set based theory of approximate reasoning and the other using probabilistic reasoning. Both of these approaches allow the inclusion of imprecise granular type information. The inferred values from each of these methods are then represented using a Dempster–Shafer belief structure. We then compare these values and show an underling unity between these two approaches.

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 proposi-tional logic is established.

Knowledge trees and protoforms in question‐answering systems

AbstractWe point out that question‐answering systems differ from other information‐seeking applications, such as search engines, by having a deduction capability, an ability to answer questions by a synthesis of information residing in different parts of its knowledge base. This capability requires appropriate representation of various types of human knowledge, rules for locally manipulating this knowledge, and a framework for providing a global plan for appropriately mobilizing the information in the knowledge to address the question posed. In this article we suggest tools to provide these capabilities. We describe how the fuzzy set–based theory of approximate reasoning can aid in the process of representing knowledge. We discuss how protoforms can be used to aid in deduction and local manipulation of knowledge. The idea of a knowledge tree is introduced to provide a global framework for mobilizing the knowledge base in response to a query. We look at some types of commonsense and default knowledge. This requires us to address the complexity of the nonmonotonicity that these types of knowledge often display. We also briefly discuss the role that Dempster‐Shafer structures can play in representing knowledge.

A unified approximate reasoning theory suitable for both propositional calculus system L* and predicate calculus system K*

The concepts of metric R 0-algebra and Hilbert cube of type R 0 are introduced. A unified approximate reasoning theory in propositional caculus system \(\mathcal{L}^* \) and predicate calculus system \(\mathcal{K}^* \) is established semantically as well as syntactically, and a unified complete theorem is obtained.

On some new classes of implication operators and their role in approximate reasoning

The basic properties required of the multivalued implication operator are introduced. The generation of these operators using the S and R implications is discussed. We suggest two new approaches for obtaining multivalued implications based on the direct use of additive generating functions. Some new classes of implication operators using these generators methods are derived. The role of the multivalued implication within the fuzzy logic based theory of approximate reasoning is described. A discussion of the modus ponens and modus tollens inference paradigms within the framework of fuzzy logic is provided. A visualization method is then used to look at various properties of the implication operator and suggest variations of standard operators. Finally we suggest a rule based approach to defining multivalued implications which allows a user to specify an implication to met their customized performance requirements.

Querying databases containing multivalued attributes using veristic variables

Veristic variables are variables that can assume multiple values such as the languages a person speaks. Different types of statements providing information about veristic variables are described. A methodology for representing and manipulating veristic information within the fuzzy set based theory of approximate reasoning is presented. We then turn to the role of these types of variables in databases. We consider methods for representing and evaluating queries involving veristic variables. Methodologies are presented for various types of queries.

Measuring information in possibilistic logic

We provide an overview of the theory of approximate reasoning and discuss the measurement of information in this reasoning system using specificity. It is then shown how to represent the binary propositional logic in the framework of approximate reasoning. Using the measure of specificity we show how to measure the information contained in the propositions of binary logic. Our measure essentially measures the proportion of possibilities eliminated by the proposition. Next we turn to the possibilistic logic and show how to represent this within the framework of approximate reasoning. We again, using specificity, provide for a measure of information of propositions in this logic. Finally we turn to the issue of quantified statements and show how to represent general quantified statements involving predicates within these two logics. ©1999 John Wiley & Sons, Inc.

EVIDENCE SETS: MODELING SUBJECTIVE CATEGORIES

Zadeh's Fuzzy Sets are extended with the Dempster-Shafer Theory of Evidence into a new mathematical structure called Evidence Sets, which can capture more efficiently all recognized forms of uncertainty in a formalism that explicitly models the subjective context dependencies of linguistic categories. A belief-based theory of Approximate Reasoning is proposed for these structures. Evidence sets are then used in the development of a relational database architecture useful for the data mining of information stored in several networked databases. This useful data mining application establishes an Artificial Intelligence model of Cognitive Categorization with a hybrid architecture that possesses both connectionist and symbolic attributes.

Open theories, consistency and related results in fuzzy logic

The paper deals with fuzzy logic in the narrow sense of Lukasiewicz style, i.e. a special kind of many-valued logic which is aimed at modeling of vagueness phenomenon. This logic is a generalization of Lukasiewicz logic with many interesting properties (for example, generalization of the deduction and completeness theorems hold in it). Our aim is to prepare the background for the resolution in fuzzy logic. We prove an analogue of the classical Herbrand theorem as well as some other related theorems which are interesting also for the theory of approximate reasoning. The possible applications and consequences for the latter are also discussed.

S-norm aggregation of infinite collections

The aggregation of membership values by means of the iterate application of an s-norm is of cardinal relevance for the development of any general theory of approximate reasoning which relies on a sup- t model for relation composition. Here we present a definition of s-norm aggregation for possibly infinite collections, which can be demonstrated to generalize previous approaches consistently. It provides a common framework in which classical sup-t and finite s- t compositions can be analyzed as well as other significant cases. General properties of the s-norm aggregation concept, its specialization for the class of continuous s-norms and its relationship with fuzzy measure theory are discussed.

What does fuzzy logic bring to AI?

The term “fuzzy logic” often refers to a particular control-engineering methodology that exploits a numerical representation of commensense control rules in order to synthesize, via interpolation, a control law. This approach has many features in common with neural networks. It is currently concerned mainly with the efficient encoding and approximation of numerical functions and has less and less relationship to knowledge representation issues. This is, however, a very narrow view of fuzzy logic that has little to do with AI. Scanning the fuzzy set literature, one realizes that fuzzy logic may also refer to two other topics: multiplevalued Iogics and approximate reasoning. Although the multiple-valued logic stream is very mathematically oriented, the notion of approximate reasoning as imagined by Zadeh is much more closely related to the program of AI research: he wrote in 1979 that “the theory of approximate reasoning is concerned with the deduction of possibly imprecise conclusions from a set of imprecise premises.” In the following, we use the term “fuzzy logic” to mean any kind of fuzzy set-based method intended to be used in reasoning systems. Fuzzy logic is 30 years old and has a long-term misunderstanding with AI. As a consequence, fuzzy logic methods have not been considered to belong to mainstream AI tools until now, although an important part of fuzzy logic research concentrates on issues in approximate reasoning and reasoning under uncertainty. Some reasons for this situation may be found in the antagonism which existed for a long time between purely symbolic methods advocated by AI and the numerically oriented approaches that were involved in fuzzy rule-based systems. Besides, fuzzy sets were a new emerging approach not yet firmly settled, but apparently challenging the monopoly of probability theory on being the unique proper framework for handling uncertainty. In spite of the fact that fuzzy sets have received more recognition recently, there is still a lack of appreciation by AI researchers of what fuzzy logic really is, as, for instance, recently exemplified by Elkan [ 1994].

Fuzzy control from the logical point of view

This paper is a discussion of the logical aspect in fuzzy control which is an application of the theory of approximate reasoning. First we analyze approximate reasoning and show that it is a concise logical theory based on many-valued logic integrated with some parts of linguistics. Then we demonstrate that fuzzy control based on equality relations can transparently be formulated as a result of the logical theory. Finally, we demonstrate that fuzzy control can be seen as a repeated derivation in a continuously modified logical theory and propose a hypothesis on the role of linguistics in the controllability problem.

Applications of fuzzy sets and approximate reasoning

This paper discusses recent applications of fuzzy sets and the theory of approximate reasoning. The primary focus is on fuzzy logic control (FLC). We begin with a brief history of the key ideas, a survey of recent applications, and a discussion of the genesis of FLC in Japan. We then turn to a study of the general principles of FLC, considering it as a combination of ideas from conventional control theory, artificial intelligence, and fuzzy sets theory. We next provide a detailed analysis of a simple application in consumer electronics, namely, a fuzzy washing machine developed by Hitachi Corporation. In concluding sections we briefly consider other types of applications, including recent work on pilotless helicopters, fuzzy expert systems, and the concept of a fuzzy computer, and we discuss the potential for future developments. It is our opinion that the subject of FLC is still very much in its infancy, and that recent events mark the beginning of an entirely new genre of intelligent control. >