How to improve mamdani's approach to fuzzy control

Fuzzy control is a methodology that translates “if”-“then” rules, A_ji (x₁) &…& A_jn(x_n) → B_j(u), formulated in terms of a natural language, into an actual control strategy u(x). Implication of uncertain statements is much more difficult to understand than “and,” “or,” and “not.” So, the fuzzy control methodologies usually start with translating “if”-“then” rules into statements that contain only “and,” “not,” and “or.” the first such translation was proposed by Mamdani in his pioneer article on fuzzy control. According to this article, a fuzzy control is reasonable iff one of the rules is applicable, i.e., either the first rule is applicable (A₁₁(x₁) &…& A_1n(x_n) & B₁(u)), or the second one is applicable, etc. This approach turned out to be very successful, and it is still used in the majority of fuzzy control applications. However, as R. Yager noticed, in some cases, this approach is not ideal: Namely, if for some x, we know what u(x) should be, and add this crisp rule to our rules, then the resulting fuzzy control for this x may be different from the desired value u(x). to overcome this drawback, Yager proposed to assign priorities to the rules, so that crisp rules get the highest priority, and use these priorities while translating the rules into a control strategy u(x). In this article, we show that a natural modification of Mamdani's approach can solve this problem without adding any ad hoc priorities. © 1995 John Wiley & Sons, Inc.     

Fuzzy implication can be arbitrarily complicated: A theorem

In fuzzy logic, there are several methods of representing implication in terms of &, ∨, and ¬; in particular, explicit representations define a class of S implications, implicit representations define a class of R implications. Some reasonable implication operations have been proposed, such as Yager's a^b, that are difficult to represent as S or R implications. For such operations, a new class of representations has recently been proposed, called A implications, for which the relationship between implications and the basic operations &, ∨, and ¬ is even more complicated. A natural question is: Is this complexity really necessary? In other words, is it true that A operations cannot be described as S or R operations, or they can, but we simply have not found these representations? In this paper we show that yes, the complexity is necessary, because there are operations that cannot be represented in a simpler form. © 1998 John Wiley & Sons, Inc.     

Interpreting linguistically quantified propositions

We discuss the idea of a linguistic quantifier and fuzzy set representations of these objects. We describe two formalisms for evaluating the truth of linguistically quantified propositions such as Most winter days are cold. the first approach is based upon a probabilistic interpretation and the second is based upon a logical interpretation, and uses a generalization of the “and” and “or” operations via OWA operators. We suggest an application of these quantified statements for the representation of the quotient operator in fuzzy relational data bases. © 1994 John Wiley & Sons, Inc.     

Cataloguing/analogizing: A nonmonotonic view

Reasoning deductively under incomplete information is nonmonotonic in nature since the arrival of additional information may invalidate or reverse previously obtained conclusions. It amounts to apply generic default rules in an appropriate way to a particular (partially described) situation. This type of nonmonotonic reasoning can only provide plausible conclusions. Analogical reasoning is another form of commonly used reasoning that yields brittle conclusions. It is nondeductive in nature and proceeds by putting particular situations in parallel. Analogical reasoning also exhibits nonmonotonic features, as investigated in this paper when particular situations may be incompletely stated. The paper reconsiders the pattern of plausible reasoning proposed by Polya, “a and b are analogous, a is true, then b true is more credible,'' from a nonmonotonic reasoning point of view. A representation of the statement “a and b are analogous” in terms of nonmonotonic consequences relations is presented. This representation is then related to a logical definition of analogical proportions, i.e. statements of the form “a is to b as c is to d” that has been recently proposed and extended to other types of proportions. Remarkably enough, semantic equivalence between conditional objects of the form “b given a,” which have been shown as being at the root of nonmonotonic reasoning, constitutes another type of noticeable proportions. By offering a parallel between two important forms of commonsense reasoning, this paper enriches the comparison between nonmonotonic reasoning and analogical reasoning that is not often made. © 2011 Wiley Periodicals, Inc.     

On the power of fuzzy logic

In this article we address the issues brought up by Elkan in his article, “The paradoxical success of fuzzy logic,” [IEEE Expert, 3–8 (1994)]. Elkan's work has caused concern since it purportedly reveals a Fuzzy Logic weakness regarding its theoretical foundations. A further investigation of Elkan's theorem (“Theorem 1”) revealed that its conclusion is not correct. After indicating the points where we disagree with Elkan, we reformulate Theorem 1, calling this new version “Theorem 2.” Theorems 1 and 2 have the same hypotheses but different conclusions. According to Theorem 2 there is a region of points that do hold the equivalence in the hypotheses of Theorem 1. In other words, one does not need to change the definition of logical equivalence in Theorem 1 in order to prove that Fuzzy Logic does not collapse to a two-valued logic. In a further analysis of Theorem 2 we show that Elkan's work does not affect the power of Fuzzy Logic to model vagueness. © 1996 John Wiley & Sons, Inc.     

On contra‐symmetry and MPT conditionality in fuzzy logic

This article deals with the N‐contrapositive symmetry of fuzzy implication operators J verifying either Modus Ponens or Modus Tollens inequalities, in a similar and complementary framework to the one in which Fodor (“Contrapositive symmetry of fuzzy implications.” Fuzzy Set Syst 1995;69:141–156) did begin with the subject in fuzzy logic, that is, with the verification of J(a, b) = J(N(b), N(a)) for all a, b in [0,1] and some strong‐negation function N. This property corresponds to the classical p → q = ¬q → ¬p. The aim of this article is to study that property in relation to either Modus Ponens or Modus Tollens meta‐rules of inference when the functions J are taken among those that belong to the usual families of implications in fuzzy logic. That is, the contra‐positive of S implications, R implications, Q implications, and Mamdani–Larsen operators, verifying either Modus Ponens or Modus Tollens inequalities or both, the conditionality's aspect on which lies the complementarity with Fodor. Within this study new types of implication functions are introduced and analyzed. © 2005 Wiley Periodicals, Inc. Int J Int Syst 20: 313–326, 2005.     

Averaging linguistic truth values in fuzzy approximate reasoning

The properties of a new rule for fuzzy conditional inference are presented and discussed. The rule is based on the extended mean operator defined on fuzzy numbers. The related propositions have the form “X is A is τ,” where τ is an element of the term set of the linguistic variable truth. The results obtained via the rule match with Fukami's and with the critical analysis carried out by Mizumoto and Zimmermann about the generalized modus ponens. © 1998 John Wiley & Sons, Inc.     

Contrary‐to‐duty reasoning with preference‐based dyadic obligations

In this paper we introduce Prohairetic Deontic Logic (PDL), a preference‐based dyadic deontic logic. In our preference‐based interpretation of obligations “α should be (done) if β is (done)” is true if (1) no ¬α ∧ β state is as preferable as an α ∧ β state and (2) the preferred β states are α states. We show that this representation solves different problems of deontic logic. The first part of the definition is used to formalize contrary‐to‐duty reasoning, which, for example, occurs in Chisholm’s and Forrester’s notorious deontic paradoxes. The second part is used to make deontic dilemmas inconsistent.     

Calculus of fuzzy hierarchical censored production rules (FHCPRs)

The article addresses the problem of reasoning under time constraints with incomplete, vague, and uncertain information. It is based on the idea of Variable Precision Logic (VPL), introduced by Michalski and Winston, which deals with both the problem of reasoning with incomplete information subject to time constraints and the problem of reasoning efficiently with exceptions. It offers mechanisms for handling trade-offs between the precision of inferences and the computational efficiency of deriving them. As an extension of Censored Production Rules (CPRs) that exhibit variable precision in which certainty varies while specificity stays constant, a Hierarchical Censored Production Rules (HCPRs) system of Knowledge Representation proposed by Bharadwaj and Jain exhibits both variable certainty as well as variable specificity. Fuzzy Censored Production Rules (FCPRs) are obtained by augmenting ordinary fuzzy conditional statement: “if X is A then Y is B” (or A(x) ⇒ B(y) for short) with an exception condition and are written in the form: “if X is A then Y is B unless Z is C” (or A(x) ⇒ B(y) ∥ C(z)). Such rules are employed in situations in which the fuzzy conditional statement “if X is A then Y is B” holds frequently and the exception condition “Z is C” holds rarely. Thus, using a rule of this type we are free to ignore the exception condition, when the resources needed to establish its presence are tight or there simply is no information available as to whether it holds or does not hold. Thus if … then part of the FCPR expresses important information while the unless part acts only as a switch that changes the polarity of “Y is B” to “Y is not B” when the assertion “Z is C” holds. Our aim is to show how an ordinary fuzzy production rule on suitable modifications and augmentation with relevant information becomes a Fuzzy Hierarchical Censored Production Rules (FHCPRs), which in turn enables to resolve many of the problems associated with usual fuzzy production rules system. Examples are given to demonstrate the behavior of the proposed schemes. © 1996 John Wiley & Sons, Inc.     

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. © 1996 John Wiley & Sons, Inc.     

Structuring Metatheory on Inductive Definitions

We examine a problem for machine supported metatheory. There are true statements about a theory that are true of some (but only some) extensions; however, standard theory-structuring facilities do not support selective inheritance. We use the example of the deduction theorem for modal logic and show how a statement about a theory can explicitly formalize the closure conditions extensions should satisfy for it to remain true. We show how metatheories based on inductive definitions allow theories and general metatheorems to be organized this way and report on a case study using the theory FS₀.     

Strongly transitive fuzzy relations: An alternative way to describe similarity

The notion of a transitive closure of a fuzzy relation is very useful for clustering in pattern recognition, for fuzzy databases, etc. It is based on translating the standard definition of transitivity and transitive closure into fuzzy terms. This definition works fine, but to some extent it does not fully capture our understanding of transitivity. the reason is that this definition is based on fuzzifying only the positive side of transitivity: if R(a, b) and R(b, c), then R(a, c); but transitivity also includes a negative side: if R(a, b) and not R(a, c), then not R(b, c). In classical logic, this negative statement follows from the standard “positive” definition of transitivity. In fuzzy logic, this negative part of the transitivity has to be formulated as an additional demand. In the present article, we define a strongly transitive fuzzy relation as the one that satisfies both the positive and the negative parts of the transitivity demands, prove the existence of strong transitive closure, and find the relationship between strongly transitive similarity and clustering. © 1995 John Wiley & Sons, Inc.     

Fuzzy Logic: Misconceptions and Clarifications

Some commonly accepted statements concerning the basic fuzzy logicproposed by Lotfi Zadeh in 1965, have led to suggestions that fuzzy logicis not a logic in the same sense as classical bivalent logic. Thoseconsidered herein are: fuzzy logic generates results that contradictclassical logic, fuzzy logic collapses to classical logic, there can be no prooftheory for fuzzy logic, fuzzy logic is inconsistent, fuzzy logic producesresults that no human can accept, fuzzy logic is not proof-theoreticcomplete, fuzzy logic is too complex for practical use, and, finally, fuzzylogic is not needed. It is either proved or argued herein that all of the thesestatements are false and are, hence, misconceptions. A fuzzy logic withtruth values specified as subintervals of the real unit interval [0.0, 1.0] isintroduced. Proofs of the correctness, consistency, and proof theoreticcompleteness of the truth interval fuzzy logic are either summarized orcited. It is concluded that fuzzy logics deserve the accolade of logic tothe same degree that the term applies to classical logics.     

A Qualitative Approach to Syllogistic Reasoning

In this paper we present a new approach to a symbolic treatment of quantified statements having the following form Q A's are B's, knowing that A and B are labels denoting sets, and Q is a linguistic quantifier interpreted as a proportion evaluated in a qualitative way. Our model can be viewed as a symbolic generalization of statistical conditional probability notions as well as a symbolic generalization of the classical probabilistic operators. Our approach is founded on a symbolic finite M-valued logic in which the graduation scale of M symbolic quantifiers is translated in terms of truth degrees. Moreover, we propose symbolic inference rules allowing us to manage quantified statements.     

Graded tableaux for Rational Pavelka Logic

In this article, a sound and complete tableau system for Rational Pavelka Logic (RPL) is introduced. Extended formulas are used as the counterpart of the graded formulas. In this calculus, if we want to show that the graded formula (x, r) is tableau provable (in the finite fuzzy theory F, respectively), we develop a tableau for the extended formula [r, x] (for the set of extended formulas {[r, x], [x₁, a₁],…, [x_n, a_n] }, respectively). If this tableau closes we claim that (x, r) is tableau provable (in the fuzzy theory F, respectively). We claim also that x is valid at the degree equal to the l.u.b. that allows the closure of the tableaux. Our tableaux are a first step toward efficient procedures of automated deduction in narrow fuzzy logic with truth constants. © 2005 Wiley Periodicals, Inc. Int J Int Syst 20: 1273–1285, 2005.     

The probability that tweety is able to fly

In this note we examine the question of assigning a probabilistic valuation to a statement as “Tweety (a particular bird) is able to fly.” Namely, we suggest that a natural way to proceed is to rewrite it as “a (randomly chosen) bird with the same observable properties of Tweety is able to fly,” and consequently to assume that the probability of “Tweety is able to fly” is equal to the percentage of the past observed birds similar to Tweety that are able to fly. © 1994 John Wiley & Sons, Inc.     

Temporal Languages for Epistemic Programs

This paper adds temporal logic to public announcement logic (PAL) and dynamic epistemic logic (DEL). By adding a previous-time operator to PAL, we express in the language statements concerning the muddy children puzzle and sum and product. We also express a true statement that an agent’s beliefs about another agent’s knowledge flipped twice, and use a sound proof system to prove this statement. Adding a next-time operator to PAL, we provide formulas that express that belief revision does not take place in PAL. We also discuss relationships between announcements and the new knowledge agents thus acquire; such relationships are related to learning and to Fitch’s paradox. We also show how inverse programs and hybrid logic each can be used to help determine whether or not an arbitrary structure represents the play of a game. We then add a past-time operator to DEL, and discuss the importance of adding yet another component to the language in order to prove completeness.     

Active logic semantics for a single agent in a static world

For some time we have been developing, and have had significant practical success with, a time-sensitive, contradiction-tolerant logical reasoning engine called the active logic machine (ALMA). The current paper details a semantics for a general version of the underlying logical formalism, active logic. Central to active logic are special rules controlling the inheritance of beliefs in general (and of beliefs about the current time in particular), very tight controls on what can be derived from direct contradictions ($P\mathrm{\&}\neg P$), and mechanisms allowing an agent to represent and reason about its own beliefs and past reasoning. Furthermore, inspired by the notion that until an agent notices that a set of beliefs is contradictory, that set seems consistent (and the agent therefore reasons with it as if it were consistent), we introduce an “apperception function” that represents an agent's limited awareness of its own beliefs, and serves to modify inconsistent belief sets so as to yield consistent sets. Using these ideas, we introduce a new definition of logical consequence in the context of active logic, as well as a new definition of soundness such that, when reasoning with consistent premises, all classically sound rules remain sound in our new sense. However, not everything that is classically sound remains sound in our sense, for by classical definitions, all rules with contradictory premises are vacuously sound, whereas in active logic not everything follows from a contradiction.     

How to fully represent expert information about imprecise properties in a computer system: random sets,fuzzy sets,and beyond: an overview

To help computers make better decisions, it is desirable to describe all our knowledge in computer-understandable terms. This is easy for knowledge described in terms on numerical values: we simply store the corresponding numbers in the computer. This is also easy for knowledge about precise (well-defined) properties which are either true or false for each object: we simply store the corresponding “true” and “false” values in the computer. The challenge is how to store information about imprecise properties. In this paper, we overview different ways to fully store the expert information about imprecise properties. We show that in the simplest case, when the only source of imprecision is disagreement between different experts, a natural way to store all the expert information is to use random sets; we also show how fuzzy sets naturally appear in such random set representation. We then show how the random set representation can be extended to the general (“fuzzy”) case when, in addition to disagreements, experts are also unsure whether some objects satisfy certain properties or not.     

Attribute dependencies for data with grades I*,†

This paper examines attribute dependencies in data that involve grades, such as a grade to which an object is red or a grade to which two objects are similar. We thus extend the classical agenda by allowing graded, or “fuzzy”, attributes instead of Boolean, yes-or-no attributes in case of attribute implications, and allowing approximate match based on degrees of similarity instead of exact match based on equality in case of functional dependencies. In a sense, we move from bivalence, inherently present in the now-available theories of dependencies, to a more flexible setting that involves grades. Such a shift has far-reaching consequences. We argue that a reasonable theory of dependencies may be developed by making use of mathematical fuzzy logic, a recently developed many-valued logic. Namely, the theory of dependencies is then based on a solid logic calculus the same way classical dependencies are based on classical logic. For instance, rather than handling degrees of similarity in an ad hoc manner, we consistently treat them as truth values, the same way as true (match) and false (mismatch) are treated in classical theories. In addition, several notions intuitively embraced in the presence of grades, such as a degree of validity of a particular dependence or a degree of entailment, naturally emerge and receive a conceptually clean treatment in the presented approach. In the first part of this two-part paper, we discuss motivations, provide basic notions of syntax and semantics and develop basic results which include entailment of dependencies, associated closure structures and a logic of dependencies with two versions of completeness theorem.