Daniel G. Schwartz

Fuzzy logic flowers in Japan

Fuzzy logic is briefly explained, and its use in controllers is described. Further theoretical development and commercial implementation of these ideas in Japan is surveyed. Work progress in domains other than control is indicated.<<ETX>>

The case for an interval-based representation of linguistic truth

Abstract This paper develops an interval-based approach to the concept of linguistic truth. A special-purpose interval logic is defined, and it is argued that, for many applications, this logic provides a potentially useful alternative to the conventional fuzzy logic. The key idea is to interpret the numerical truth value v(p) of a proposition p as a degree of belief in the logical certainty of p , in which case p is regarded as true , for example, if v(p) falls within a certain range, say, the interval [0.7, 1]. This leads to a logic which, although being only a special case of fuzzy logic, appears to be no less linguistically correct and at the same time offers definite advantages in terms of mathematical simplicity and computational speed. It is also shown that this same interval logic can be generalized to a lattice-based logic having the capacity to accommodate propositions p which employ fuzzy predicates of type 2.

Dynamic reasoning with qualified syllogisms

Abstract A qualified syllogism is a classical Aristotelean syllogism that has been “qualified” through the use of fuzzy quantifiers, likelihood modifiers, and usuality modifiers, e.g., “ Most birds can fly; Tweety is a bird; therefore, it is likely that Tweety can fly.” This paper introduces a formal logic Q of such syllogisms and shows how this may be employed in a system of nonmonotonic reasoning. In process are defined the notions of path logic and dynamic reasoning system (DRS). The former is an adaptation of the conventional formal system which explicitly portrays reasoning as an activity that takes place in time. The latter consists of a path logic together with a multiple-inheritance hierarchy. The hierarchy duplicates some of the information recorded in the path logic, but additionally provides an extralogical specificity relation. The system uses typed predicates to formally distinguish between properties and kinds of things. The effectiveness of the approach is demonstrated through analysis of several “puzzles” that have appeared previously in the literature, e.g., Tweety the Bird, Clyde the Elephant, and the Nixon Diamond. It is also outlined how the DRS framework accommodates other reasoning techniques—in particular, predicate circumscription, a “localized” version of default logic, a variant of nonmonotonic logic, and reason maintenance. Furthermore it is seen that the same framework accomodates a new formulation of the notion of unless . A concluding section discusses the relevance of these systems to the well-known frame problem .

A system for reasoning with imprecise linguistic information

Abstract This writing lays the foundation for a model of natural human reasoning with imprecise linguistic information. Key to the model is a collection of abstraction mechanisms based on the concept of a linguistic variable, which was first introduced for this purpose within the context of a semantics based on fuzzy sets. The present approach differs from the earlier one, however, in that (1) it doe s not require the use of fuzzy sets for the interpretation of linguistic terms and (2) the meanings of logical inferences are given as algorithms that act directly on terms themselves rather than on their underlying interpretations. Thus this work constitutes a return to the more purely symbolic or axiomatic representations of logical deduction, whereas the fuzzy-sets model concerns denotational or semantic representations. The new model should not be viewed as a negation of the earlier approaches, however, but as an augmentation of them. The present work is intended as the beginning of a larger system that encompasses both styles of reasoning. Two distinct types of logical inference are proposed, together with two associated modes of evidence combination. Final sections sketch the design of a backward-chaining algorithm through which the various inference types can be employed in diagnostic or, more generally, classificational reasoning systems. The algorithm is expected to be computationally manageable and therefore amenable to implementation in a functioning inference engine.

Axioms for a theory of semantic equivalence

Abstract Godel-type semantic completeness theorems are established for a formal theory of semantic equivalence based on L.A. Zadehs concept of a linguistic variable. The linguistics that is employed allows for the expression of propositions such as “it is not the case that ‘young’ is semantically equivalent with ‘not old’”, or, in symbols (young(x) ≅ ∼old(x)). The result is a two-leveled semantics which, at the lower level, is a multivalent interpretation of fuzzy assertions (e.g., ∼old(x)) in terms of fuzzy subsets of a given universe and, at the upper level, is a two-valued (classical) interpretation based on the fact that two closed fuzzy assertions either do or do not have the same truth value. The main proof is of the Henkin variety, employing adaptations of the algebraic methods found in Rasiowa [9] and Rasiowa and Sikorski [10].

Distinguishing false from true alerts in Snort by data mining patterns of alerts

The Snort network intrusion detection system is well known for triggering large numbers of false alerts. In addition, it usually only warns of a potential attack without stating what kind of attack it might be. This paper presents a clustering approach for handling Snort alerts more effectively. Central to this approach is the representation of alerts using the Intrusion Detection Message Exchange Format, which is written in XML. All the alerts for each network session are assembled into a single XML document, thereby representing a pattern of alerts. A novel XML distance measure is proposed to obtain the distance between two such XML documents. A classical clustering algorithm, implemented based on this distance measure, is then applied to group the alert patterns into clusters. Our experiment with the MIT 1998 DARPA data sets demonstrates that the clustering algorithm can distinguish between normal sessions that give rise to false alerts and those sessions that contain real attacks, and in about half of the latter cases can effectively identify the name of the attack.

Agent-oriented epistemic reasoning

This paper introduces a formal system Σ of subjective epistemic reasoning that encodes a method of reasoning with conditions of knowledge and belief. The conditions are subjective in that they are taken from the perspective of an agents perception of his own state of knowledge or belief with respect to his observable world. Belief is measured along a series of linguistic degrees, e.g., strongly believes, fairly confidently believes, somewhat disbelieves, etc., and knowledge is taken as unequivocal belief. The system employs a novel, dual-leveled language that follows fuzzy logic by interpreting the logical or and and as the arithmetical max and min. Numerous properties of Σ, illustrating its intuitive appeal for the intended purpose, are derived.

A resolution-based system for symbolic approximate reasoning

Abstract The aim of this paper is twofold. First, it continues the development of a symbolic approach to approximate reasoning as an alternative to the well-known semantic approaches based on fuzzy sets. While this exacts a price in expressive power, it has the advantage of being computationally simpler. In addition, it accommodates formulation of certain aspects of approximate reasoning that are not easily expressed in terms of fuzzy sets, or where the notion of a fuzzy set might not naturally apply. Five different such forms of inference, or reasoning techniques, are discussed. Second, this work shows how the proposed symbolic approach may be implemented in a Prolog-like question-answering system, known as SAR . To illustrate, an automated bank loan advisor based on this system might be presented the query “ Suitability(Jim) ?” and respond with something like “ Suitability(Jim ; very_good).” To this end we develop SAR resolution, an adaptation of the well-known SLD resolution which underlies Prolog. SAR resolution differs from the earlier version in that (1) it requires generation of a resolution tree, rather than a single path, (2) it requires attaching a computational formula to each resolvent, reflecting the particular inferencing technique being employed at that step, and (3) it requires incorporating a means for (symbolic) evidence combination. In generating and traversing the resolution tree, SAR resolution behaves essentially as SLD resolution when moving in the downward direction (from the root), and applies computational formulas and evidence combination procedures when moving in the upward direction. Thus it is more complex than SLD resolution, but is nonetheless simple enough for many real-world applications. In effect SAR is a general purpose “fuzzy classifier” and accordingly should find use in many expert systems of the classification genre, e.g., for diagnosis, troubleshooting, monitoring, and multicriteria decision making. The SAR resolution technique easily accommodates forms of inference other than the five discussed here. As examples: one could adjoin the well-known “compositional rule of inference,” or a mode of inference whose underlying computation is provided by a neural net. Thus this paper implicitly provides a general methodology by which one may devise reasoning systems that present the user with a variety of inferencing techniques, from which one may then choose as the situation demands.

Fuzzy inference in a formal theory of semantic equivalence

Abstract This work shows how ten different fuzzy implication operators, together with appropriately corresponding versions of four different rules of inference, can be represented within a formal axiomatization of the theory of approximate reasoning. The same formalization methods can also be applied to Zadehs notion of semantic entailment, as well as to all of the familiar linguistic hedges.

Layman's probability theory: a calculus for reasoning with linguistic likelihood

Abstract This paper distinguishes between the task of modeling chance occurrences in the physical world and that of modeling natural human reasoning about such occurrences. It is proposed that everyday common sense often prescribes that the probabilistic and , which corresponds to the joint probability of physical events, should be expressed in terms of the arithmetic min operation, and that it is dual, the or , should be couched in terms of the arithmetic max . Thus, in effect, it is proposed that the logic normally associated with the theory of fuzzy sets can be adapted as a kind of “laymans probability theory”. The rationale for such an “erroneous” model of probability is that such seems to underly natural human decision making in the face of uncertainty. In the interests of modeling and, to whatever extent possible, replicating such decision processes, it is of interest to properly encode the underlying logics. The crux of the article it to present such an encoding as a new kind of formal logical system for reasoning with linguistic likelihood, i.e., with fuzzy terms such as likely , very likely , somewhat unlikely , etc. It is verified that the system allows for semantically coherent expression of the desired probablistic arguments.