1993-The Paradoxical Success of Fuzzy Logic
From: AAAI-93 Proceedings. Copyright © 1993, AAAI (www.aaai.org). All rights reserved.
The Paradoxical Success of Fuzzy Logic*
Charles Elkan Department of Computer Science and Engineering University of California, San Diego La Jolla, California 92093-0114
Abstract the use of fuzzy logic. Finally, Section 5 shows how cur- rent work on fuzzy control is encountering dilemmas that This paper investigates the question of which are already well-known from work in other areas of ar- aspects of fuzzy logic are essential to its prac- tificial intelligence, and Section 6 provides some overall tical usefulness. We show that as a formal sys- conclusions. tem, a standard version of fuzzy logic collapses mathematically to two-valued logic, while em- 2 A paradox in fuzzy logic pirically, fuzzy logic is not adequate for rea- soning about uncertain evidence in expert sys- As is natural in a research area as active as fuzzy logic, tems. Nevertheless, applications of fuzzy logic theoreticians have investigated many different formal in heuristic control have been highly success- systems, and applications have also used a variety of ful. We argue that the inconsistencies of fuzzy systems. Nevertheless, the basic intuitions are relatively logic have not been harmful in practice because constant. At its simplest, fuzzy logic is a generaliza- current fuzzy controllers are far simpler than tion of standard propositional logic from two truth val- other knowledge-based systems. In the future, ues false and true to degrees of truth between 0 and 1. the technical limitations of fuzzy logic can be Formally, let A denote an assertion. In fuzzy logic, expected to become important in practice, and A is assigned a numerical value t(A), called the degree work on fuzzy controllers will also encounter of truth of A, such that 0 < t(A) 5 1. For a sentence several problems of scale already known for composed from simple assertions and logical connectives other knowledge-based systems. “and” (A), “or” (V), and “not” (-A or A), degree of truth is defined as follows: Definition 1: 1 Introduction t(A A B) = min{t(A), t(B)} Fuzzy logic methods have been applied successfully in many real-world systems, but the coherence of the foun- t(A V B) = max{t(A), t(B)} dations of fuzzy logic remains under attack. Taken to- +A) = 1 - t(A) gether, these two facts constitute a paradox, which this t(A) = t(B) if A and B are paper attempts to resolve. More concretely, the aim of logically equivalent. H this paper is to identify which aspects of fuzzy logic ren- der it so useful in practice and which aspects are inessen- In the last case of this definition, let “logically equiva- tial. Our conclusions are based on a new mathematical lent” mean equivalent according to the rules of classical result, on a survey of the literature on the use of fuzzy two-valued propositional calculus. The use of alternative logic in heuristic control, and on our own practical ex- definitions of logical equivalence is discussed at the end perience developing two large-scale expert systems. of this section. This paper is organized as follows. First, Section Fuzzy logic is intended to allow an indefinite variety 2 proves and discusses the theorem mentioned above, of numerical truth values. The result proved here is that which is that only two truth values are possible inside only two different truth values are in fact possible in the a standard system of fuzzy logic. In an attempt to un- formal system of Definition 1. derstand how fuzzy logic can be useful despite this para- Theorem 1: For any two assertions A and B, either dox, Sections 3 and 4 examine the main practical uses t(B) = t(A) or t(B) = 1 -t(A). of fuzzy logic, in expert systems and heuristic control. Proof: Let A and B be arbitrary assertions. Consider Our tentative conclusion is that successful applications of fuzzy logic are successful because of factors other than the two sentences A A B and B V (z A B>. These are logically equivalent, so *This work was supported in part by the National Science Foundation under Award No. IRI-9110813. t(A A B> = t(B v (x A B).
698 Elkan Now degree of membership for fuzzy set intersections, unions, and complements. t(A) = 1 - min(t(A), 1 - t(B)} On the other hand, the theorem does not necessarily = 1 + max{-t(A), -1 + t(B)} apply to any version of fuzzy logic that modifies or rejects = max{ 1 - t(A), t(B)} any of the four postulates of Definition 1. It is however possible to carry through the proof of the theorem in and many variant systems of fuzzy logic. In particular, the theorem remains true when negation is modelled by any t(B V (x A B)) = max{t(B),min{l - t(A), 1 - t(B)}}. operator in the Sugeno class [Sugeno, 19771, and when The numerical expressions above are different if disjunction or conjunction are modelled by operators in the Yager classes [Yager, 19801.’ t(B) < 1 -t(B) < 1 - t(A), Of course, the last postulate of Definition 1 is the most that is if t(B) < l-t(B) and t(A) < t(B), which happens controversial one, and the postulate that one naturally if t(A) < t(B) < 0.5. So it cannot be true that t(A) < first wants to modify in order to preserve a continuum of t(B) < 0.5. degrees of truth. Unfortunately, it is not clear which sub- set of classical tautologies and equivalences should be, or Now note that the sentences A A B and B V (x A can be, required to hold in a system of fuzzy logic. What B) are both re-expressions of the material implication all formal fuzzy logics have in common is that they re- A + B. One by one, consider the seven other material ject at least one classical tautology, namely the law of implication sentences involving A and B excluded middle (the assertion XVA). Intuitionistic logic Z-B [van Dalen, 19831 a 1so rejects this law, but rejects in ad- dition De Morgan’s laws, which are entailed by the first A+B three postulates of Definition 1. One could hope that X--+B fuzzy logic is therefore a formal system whose tautologies B-A are a subset of the classical tautologies, and a superset of B+A the intuitionistic tautologies. However, Theorem 1 can still be proved even if logical equivalence is restricted to B---+X mean intuitionistic equivalence.’ It is an open question B + -A. how to choose a notion of logical equivalence that si- multaneously (i) remains philosophically justifiable, (ii) BY the same reasoning as before, none of the following allows useful inferences in practice, and (iii) removes the can be true: opportunity to prove results similar to Theorem 1. H 1 -t(A) < t(B) < 0.5 t(A) < 1 -t(B) < 0.5 3 Fuzzy logic in expert systems
l-t(A) Rule-Based Reasoning 699 naturally either true or false. Rather, uncertainty of The difficulties identified here with fuzzy logic and some sort is attached to them. Fuzzy logic is an attempt probability theory as formalisms for reasoning about un- to capture valid patterns of reasoning about uncertainty. certainty do occur in practice. We have recently de- The notion is now well accepted that there exist many signed, implemented, and deployed at IBM two large- different types of uncertainty, vagueness, and ignorance scale expert systems [Hekmatpour and Elkan, 1993; [Smets, 19911. H owever, there is still debate as to what Hekmatpour and Elkan, 19921. One system, CHATKB, types of uncertainty are captured by fuzzy logic.3 Many solves problems encountered by engineers while using papers have discussed at a high level of mathematical VLSI design tools. The other system, WESDA, diagnoses abstraction the question of whether fuzzy logic provides faults in machines that polish semiconductor wafers. suitable laws of thought for reasoning about probabilistic The knowledge possessed by each system consists of a uncertainty. Our conclusion from practical experience library of cases and a deep domain theory which is rep- in the construction of expert systems is that fuzzy logic resented as a decision tree where each node corresponds is not uniformly suitable for reasoning about uncertain to a fact about the state of the tool being diagnosed. evidence. A simple example shows what the difficulty is. Relevant cases are attached to each leaf of the decision Suppose the universe of discourse is a collection of mel- tree. Roughly, the children of each node represent evi- ons, and there are two predicates red and watermelon, dence in favour of the parent node, or potential causes where red and green refer to the colour of the flesh of of the parent node. CHATKB or WESDA retrieves an old a melon. For some not very well-known melon 2, sup- case to solve a new problem by choosing a path through pose that t(red(x)) = 0.5 and t(watermelon(x)) = 0.8, its decision tree. A path from the root to a leaf is chosen meaning that the evidence that 2 is red inside has by combining a priori child node likelihoods with evi- strength 0.5 and the evidence that x is a watermelon dence acquired through questioning the user. We have has strength 0.8. According to the rules of fuzzy logic, found that this process of combining evidence is a type t(red(x) A watermelon(x)) = 0.5. This is not reason- of reasoning about uncertainty that cannot be modelled able, because watermelons are normally red inside. Red- adequately by the axioms of fuzzy logic, or by those of ness and being a watermelon are mutually reinforcing probability theory. facts, so intuitively, x is a red watermelon with certainty Methods for reasoning about uncertain evidence are an greater than 0.5. active research area in artificial intelligence, and the con- The deep issue here is that the degree of uncertainty clusions reached in this section are not new. Our prac- of a conjunction is not in general determined uniquely by tical experience does, however, independently confirm the degree of uncertainty of the assertions entering into previous arguments about the inadequacy of systems for the conjunction. There does not exist a function f such reasoning about uncertainty that propagate numerical that the rule t(A A B) = f(t(A), t(B)) is always valid, factors according only to which connectives appear in when t represents the degree of certainty of fragments assertions [Pearl, 1988]. of evidence. The certainty of A A B depends on the content of the assertions A and B as well as on their 4 Fuzzy logic in heuristic control numerical certainty. This fact is recognized implicitly in Heuristic control is the area of application in which fuzzy probabilistic reasoning, since probability theory does not logic has been the most successful. There is a wide con- assign unique probability values to conjunctions. What sensus that the techniques of traditional mathematical probability theory says is that control theory are often inadequate. The reasons for this include the reliance of traditional methods on lin- l- (l-Pr(A)+l-Pr(B)) 700 Elkan ues to meet resistance. For example, at the 1991 Inter- The question which naturally arises is which of the national Joint Conference on Artificial Intelligence (IJ- features of fuzzy controllers identified above are essential CAI’91, Sydney, Australia) Takeo Kanade gave an in- to their success. It appears that the first four shared vited talk on computer vision in which he described at properties are vital to practical success, because they length Matsushita’s camcorder image stabilizing system make the celebrated credit assignment problem solvable, [Uomori et al., 19901, without mentioning that it uses while the use of fuzzy logic is not essential. fuzzy logic. In a nutshell, the credit assignment problem is to dis- Almost all currently deployed heuristic controllers us- cover how to modify part of a complex system in order to ing fuzzy logic are similar in five important aspects. A improve it, given only an evaluation of its overall perfor- good description of a prototypical example of this stan- mance. In general, solving the credit assignment prob- dard architecture appears in [Sugeno et al., 19891. lem is impossible: the task is tantamount to generating many bits of information (a change to the internals of a o First, the knowledge base of a typical fuzzy con- complex system) from just a few bits of information (the troller consists of under 100 rules; often under 20 input/output performance of the system). However, the rules are used. Fuzzy controllers are orders of first four shared features of fuzzy controllers make the magnitude smaller than systems built using tradi- credit assignment problem solvable for them. tional artificial intelligence formalisms: the knowl- First, since it consists of only a small number of rules, edge base of CHATKB, for example, occupies many the knowledge base of a fuzzy controller is a small system megabytes. to modify. Second, the short paths between the inputs o Second, the knowledge entering into fuzzy con- and outputis of a fuzzy controller mean that the effect of trollers is structurally shallow, both statically and any change in the controller is localized, so it is easier to dynamically. It is not the case that some rules pro- discover a change that has a desired effect without hav- duce conclusions which are then used as premises in ing other undesired consequences. Third, the iterative other rules. Statically, rules are organized in a flat way in which fuzzy controllers are refined allows a large list, and dynamically, there is no run-time chaining number of observations of input/output performance to of inferences. be used for system improvement. Fourth, the continu- Third, the knowledge recorded in a fuzzy controller ous nature of the many parameters of a fuzzy controller typically reflects immediate correlations between allows small quantities of performance information to be the inputs and outputs of the system to be con- used to make small system changes. trolled, as opposed to a deep, causal model of the Thus, what makes fuzzy controllers useful in practice system. The premises of rules refer to sensor ob- is the combination of a rule-based formalism with nu- servations and rule conclusions refer to actuator merical factors qualifying rules and the premises enter- settings.4 ing into rules. The principal advantage of rule-based formalisms is that knowledge can be acquired from ex- m The fourth important feature that deployed fuzzy perts or from experience incrementally: individual rules controllers share is that the numerical parameters and premises can be refined independently, or at least of their rules and of their qualitative input and more independently than items of knowledge in other output modules are tuned in a learning process. formalisms. Numerical factors have two main advan- Many different learning algorithms have been used tages. They allow a heuristic control system to interface for this purpose, and neural network learning mech- smoothly with the continuous outside world, and they al- anisms have been especially successful [Keller and low it to be tuned gradually: small changes in numerical Tahani, 1992; Yager, 19921. What the algorithms factor values cause small changes in behaviour. used for tuning fuzzy controllers themselves have None of these features contributing to the success of in common is that they are gradient-descent “hill- systems based on fuzzy logic is unique to fuzzy logic. It climbing” algorithms that learn by local optimiza- seems that most current applications of fuzzy logic could tion [Burkhardt and Bonissone, 19921. use other numerical rule-based formalisms instead, if a o Last but not least, by definition fuzzy controllers use learning algorithm was used to tune numerical values for the operators of fuzzy logic. Typically “minimum” those formalisms, as is customary when using fuzzy logic. and “maximum” are used, as are explicit possibility Several knowledge representation formalisms that are distributions (usually trapezoidal), and some fuzzy rule-based and numerical have been proposed besides implication operator. fuzzy logic. For example, well-developed systems are presented in [Sandewall, 19891 and [Collins and Michal- 4Rule premises refer to qualitative (“linguistic” in the ter- ski, 1989; Dontas and Zemankova, 19901. To the extent minology of fuzzy logic) sensor observations and rule con- that numerical qualification factors can be tuned in these clusions refer to qualitative actuator settings, whereas out- puts and inputs of sensors and actuators are typically real- formalisms, we expect that they would be equally use- valued. This means that two controller components usually ful for constructing heuristic controllers. Indeed, at least exist which map between numerical values and qualitative one has already been so used [Sammut and Michie, 19911. values. In fuzzy logic terminology, these components are said to defuzzify outputs and implement membership functions 5 eeapitulating mainstream AI respectively. Their behaviour is not itself describable using fuzzy logic, and typically they are implemented procedurally. Several research grou ps are attempting to scale up sys- terns based on fuzzy logic, and to lift the architectural Rule-Based Reasoning 701 limitations of current fuzzy controllers. For example, a strutting fuzzy controllers more intricate than current methodology for designing block-structured controllers ones. For a second example of relevant previous artifi- with guaranteed stability properties is studied in [Tanaka cial intelligence work, consider controllers that can carry and Sugeno, 19921, and methodological problems in con- state information from one moment to the next. These structing models of complex systems based on deep are mentioned as a topic for future research in [von Al- knowledge are considered in [Pedrycz, 19911. Controllers track et al., 19921. Symbolic AI formalisms for repre- with intermediate variables, thus with chaining of infer- senting systems whose behaviour depends on their his- ences, are investigated in [von Altrock et al., 19921. tory have been available since the 1960s [McCarthy and However, the designers of larger systems based on Hayes, 19691. Neural networks with similar properties fuzzy logic are encountering all the problems of scale al- (called recurrent networks) have been available for sev- ready identified in traditional knowledge-based systems. eral years [Elman, 19901, and have already been used It appears that the history of research in fuzzy logic is in control applications [Karim and Rivera, 19921. It re- recapitulating the history of research in other areas of mains to be seen whether research from a fuzzy logic artificial intelligence. This section discusses the knowl- perspective will provide new solutions to the fundamen- edge engineering dilemmas faced by developers of fuzzy tal issues of artificial intelligence. controllers, and then points to dealing with state infor- mation as another issue arising in research on fuzzy con- 6 Conclusions trollers that has also arisen previously. Applications of fuzzy logic in heuristic control have been The rules in the knowledge bases of current fuzzy con- highly successful, despite the collapse of fuzzy logic as a trollers are obtained directly by interviewing experts. formal system to two-valued logic, and despite the inad- Indeed, the original motivation for using fuzzy logic equacy of fuzzy logic for reasoning about uncertainty in in building heuristic controllers was that fuzzy logic is expert systems. The inconsistencies of fuzzy logic have designed to capture human statements involving vague not been harmful in practice because current fuzzy con- quantifiers such as “considerable.” More recently, a con- trollers are far simpler than other knowledge-based sys- sensus has developed that research must focus on ob- tems. First, long chains of inference are not performed taining “procedures for fuzzy controller design based in controllers based on fuzzy logic, so there is no op- on fuzzy models of the process” [Driankov and Ek- portunity for inconsistency between paths of reasoning lund, 19911. Mainstream work on knowledge engineer- that should be equivalent to manifest itself. Second, the ing, however, has already transcended the dichotomy be- knowledge recorded in a fuzzy controller is not a consis- tween rule-based and model-based reasoning. tent causal model of the process being controlled, but Expert systems whose knowledge consists of if- rather an assemblage of visible correlations between sen- then rules have at least two disadvantages. First, sor observations and actuator settings. Since this knowl- maintenance of a rule base becomes complex and edge is not itself consistent and probabilistic, the prob- time-consuming as the size of a system increases abilistic inadequacy of fuzzy logic is not an issue. More- [Newquist, 19881. S econd, rule-based systems tend to be over, the ability to refine the parameters of a fuzzy con- brittle: if an item of knowledge is missing from a rule, troller iteratively can compensate for the arbitrariness the system may fail to find a solution, or worse, may of the fuzzy logic operators as applied inside a limited draw an incorrect conclusion [Abbott, 1988]. The main domain. disadvantage of model-based approaches, on the other The common assumption that heuristic controllers hand, is that it is very difficult to construct sufficiently based on fuzzy logic are successful because they use fuzzy detailed and accurate models of complex systems. More- logic appears to be an instance of the post hoc, ergo over, models constructed tend to be highly application- propter hoc fallacy. The fact that using fuzzy logic is cor- specific and not generalizable [Bourne et al., 19911. related with success does not entail that using fuzzy logic Many recent expert systems, therefore, including causes success. In the future, the technical limitations CHATKB and WESDA, are neither rule-based nor model- of fuzzy logic identified in this paper can be expected based in the standard way. For these systems, the aim to become important in practice. Other general dilem- of the knowledge engineering process is not simply to mas of artificial intelligence work can also be expected acquire knowledge from human experts, whether this to become critical-in particular, the issue of designing knowledge is correlational as in present fuzzy controllers, learning mechanisms that can solve the credit assign- or deep as in model-based expert systems. Rather, the ment problem when the simplifying features of present aim is to develop a theory of the situated performance controllers are absent. of the experts. Concretely, under this view of knowledge Acknowledgements. The author is grateful to sev- engineering, knowledge bases are constructed to model eral colleagues for useful comments on earlier versions of the beliefs and practices of experts and not any “objec- this paper, and to John Lamping for asking if Theorem 1 tive” truth about underlying physical processes. An im- holds when equivalence is understood intuitionistically. portant benefit of this approach is that the organization of an expert’s beliefs provides an implicit organization eferences of knowledge about the external process with which the [Abbott, 1988] K. Abbott. Robust operative diagnosis as knowledge-based system is intended to interact. problem solving in a hypothesis space. 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