RFuzzy: Syntax, semantics and implementation details of a simple and expressive fuzzy tool over Prolog

We present the RFuzzy framework, a Prolog-based tool for representing and reasoning with fuzzy information. The advantages of our framework in comparison to previous tools along this line of research are its easy, user-friendly syntax, and its expressivity through the availability of default values and types.In this approach we describe the formal syntax, the operational semantics and the declarative semantics of RFuzzy (based on a lattice). A least model semantics, a least fixpoint semantics and an operational semantics are introduced and their equivalence is proven. We provide a real implementation that is free and available. (It can be downloaded from http://babel.ls.fi.upm.es/software/rfuzzy/.) Besides implementation details, we also discuss some actual applications using RFuzzy.     

The dynamics of default reasoning

In this paper we study default reasoning from a dynamic, agent-oriented, semantics-based point of view. In a formal framework used to specify and to reason about rational agents, we introduce actions that model the (attempted) jumping to conclusions that is a fundamental part of reasoning by default. Application of such an action consists of three parts. First it is checked whether the formula that the agent tries to jump to is a default, thereafter it is checked whether the default formula can consistently be incorporated by the agent, and if this is the case the formula is included in the agent's beliefs. As for all actions in our framework, we define the ability and opportunity of agents to apply these actions, and the states of affairs following application. To formalise formulae being defaults, we introduce the modality of common possibility. This modality is related to, but not reducible to, the notions of common knowledge and ‘everybody knows’-knowledge. To model the qualitative difference that exists between hard, factual knowledge and beliefs derived by default, we employ different modalities to represent these concepts, thus combining knowledge, beliefs, and defaults in one framework. Based on the concepts used to model the default reasoning of agents, we look into the dynamics of the supernormal fragment of default logic. We show in particular that by sequences of jumps to conclusions agents can end up with extensions in the sense of default logic of their belief.     

Reasoning with Sets of Defaults in Default Logic

We present a general approach for representing and reasoning with sets of defaults in default logic, focusing on reasoning about preferences among sets of defaults. First, we consider how to control the application of a set of defaults so that either all apply (if possible) or none do (if not). From this, an approach to dealing with preferences among sets of default rules is developed. We begin with an ordered default theory , consisting of a standard default theory, but with possible preferences on sets of rules. This theory is transformed into a second, standard default theory wherein the preferences are respected. The approach differs from other work, in that we obtain standard default theories and do not rely on prioritized versions of default logic. In practical terms this means we can immediately use existing default logic theorem provers for an implementation. Also, we directly generate just those extensions containing the most preferred applied rules; in contrast, most previous approaches generate all extensions, then select the most preferred. In a major application of the approach, we show how semimonotonic default theories can be encoded so that reasoning can be carried out at the object level. With this, we can reason about default extensions from within the framework of a standard default logic. Hence one can encode notions such as skeptical and credulous conclusions, and can reason about such conclusions within a single extension.     

What Should Default Reasoning be,by Default?

This is a position paper concerning the role of empirical studies of human default reasoning in the formalization of AI theories of default reasoning. We note that AI motivates its theoretical enterprise by reference to human skill at default reasoning, but that the actual research does not make any use of this sort of information and instead relies on intuitions of individual investigators. We discuss two reasons theorists might not consider human performance relevant to formalizing default reasoning: (a) that intuitions are sufficient to describe a model, and (b) that human performance in this arena is irrelevant to a competence model of the phenomenon. We provide arguments against both these reasons. We then bring forward three further considerations against the use of intuitions in this arena: (a) it leads to an unawareness of predicate ambiguity, (b) it presumes an understanding of ordinary language statements of typicality, and (c) it is similar to discredited views in other fields. We advocate empirical investigation of the range of human phenomena that intuitively embody default reasoning. Gathering such information would provide data with which to generate formal default theories and against which to test the claims of proposed theories. Our position is that such data are the very phenomena that default theories are supposed to explain.     

A model of legal reasoning with cases incorporating theories and values

Reasoning with cases has been a primary focus of those working in AI and law who have attempted to model legal reasoning. In this paper we put forward a formal model of reasoning with cases which captures many of the insights from that previous work. We begin by stating our view of reasoning with cases as a process of constructing, evaluating and applying a theory. Central to our model is a view of the relationship between cases, rules based on cases, and the social values which justify those rules. Having given our view of these relationships, we present our formal model of them, and explain how theories can be constructed, compared and evaluated. We then show how previous work can be described in terms of our model, and discuss extensions to the basic model to accommodate particular features of previous work. We conclude by identifying some directions for future work.     

Using crucial literals to select better theories

When Horn clause theories are combined with integrity constraints to produce potentially refutable theories, Seki and Takeuchi have shown how crucial literals can be used to discriminate two mutually incompatible theories. A literal is crucial with respect to two theories if only one of the two theories supports the derivation of that literal. In other words, actually determining the truth value of the crucial literal will refute one of the two incompatible theories. This paper presents an integration of the idea of crucial literal with Theorist, a logic-based system for hypothetical reasoning. Theorist is a goal-directed nonmonotonic reasoning system that classifies logical formulas as possible hypotheses, facts, and observations. As Theorist uses full clausal logic, it does not require Seki and Takeuchi's notion of integrity constraint to define refutable theories. In attempting to deduce observation sentences, Theorist identifies instances of possible hypotheses as nomological explanations: consistent sets of hypothesis instances required to deduce observations. As multiple and mutually incompatible explanations are possible, the notion of crucial literal provides the basis for proposing experiments that distinguish competing explanations. We attempt to make three contributions. First, we adapt Seki and Takeuchi's method for Theorist. To do so, we incrementally use crucial literals as experiments, whose results are used to reduce the total number of explanations generated for a given set of observations. Next, we specify an extension which incrementally constructs a table of all possible crucial literals for any pair of theories. This extension is more efficient and provides the user with greater opportunity to conduct experiments to eliminate falsifiable theories. A prototype is implemented in CProlog, and several examples of diagnosis are considered to show its empirical efficiency. Finally, we point out that assumption-based truth maintenance systems (ATMS), as used in the multiple fault diagnosis system of de Kleer and Williams, are interesting special cases of this more general method of distinguishing explanatory theories.     

Yet some more complexity results for default logic

We identify several new tractable subsets and several new intractable simple cases for reasoning in the propositional version of Reiter's default logic. The majority of our findings are related to brave reasoning. By making some intuitive observations, most classes that we identify can be derived quite easily from some subsets of default logic already known in the literature. Some of the subsets we discuss are subclasses of the so-called “extended logic programs”. All the tractable subsets presented in this paper can be recognized in linear time.     

CASE-BASED REASONING FOR MANAGING NONCOMPLIANCE WITH CLINICAL GUIDELINES

Despite the recognized advantages that can be obtained in clinical practice when following clinical guidelines (GL), situations of noncompliance with them may emerge. Keeping track of such deviations from the default GL execution, and documenting the physician's motivations, would clearly be an added value. Moreover, repeated alterations of GL actions (or flow) may highlight the need for an adaptation of the GL itself to the local reality, or may even indicate an improper or weak initial GL definition.
In this article, we propose an approach for managing noncompliance with GL, based on the case-based reasoning methodology. In front of a new noncompliance case, our tool allows the physician to retrieve past situations similar to the current one, and to decide whether to reapply the same GL modifications adopted in them. Moreover, the tool is able to learn indications from the ground noncompliance cases that can be deployed for local adaptation, and possibly, for suggesting more formal GL revisions to be carried out by a committee of expert physicians.     

Fuzzy methods for case-based recommendation and decision support

The paper proposes two case-based methods for recommending decisions to users on the basis of information stored in a database. In both approaches, fuzzy sets and related (approximate) reasoning techniques are used for modeling user preferences and decision principles in a flexible manner. The first approach, case-based decision making, can principally be seen as a case-based counterpart to classical decision principles well-known from statistical decision theory. The second approach, called case-based elicitation, combines aspects from flexible querying of databases and case-based prediction. Roughly, imagine a user who aims at choosing an optimal alternative among a given set of options. The preferences with respect to these alternatives are formalized in terms of flexible constraints, the expression of which refers to cases stored in a database. As both types of decision support might provide useful tools for recommender systems, we also place the methods in a broader context and discuss the role of fuzzy set theory in some related fields.     

Alternative approaches to default logic

Reiter's default logic has proven to be an enduring and versatile approach to non-monotonic reasoning. Subsequent work in default logic has concentrated in two major areas. First, modifications have been developed to extend and augment the approach. Second, there has been ongoing interest in semantic foundations for default logic. In this paper, a number of variants of default logic are developed to address differing intuitions arising from the original and subsequent formulations. First, we modify the manner in which consistency is used in the definition of a default extension. The idea is that a global rather than local notion of consistency is employed in the formation of a default extension. Second, we argue that in some situations the requirement of proving the antecedent of a default is too strong. A second variant of default logic is developed where this requirement is dropped; subsequently these approaches are combined, leading to a final variant. These variants then lead to default systems which conform to alternative intuitions regarding default reasoning. For all of these approaches, a fixed-point and a pseudo-iterative definition are given; as well a semantic characterisation of these approaches is provided. In the combined approach we argue also that one can now reason about a set of defaults and can determine, for example, if a particular default in a set is redundant. We show the relation of this work to that of ukaszewicz and Brewka, and to the Theorist system.     

Extending the Maximum Entropy Approach to Variable Strength Defaults

A generalisation of the maximum entropy (ME) approach to default reasoning [7,8] to cater for variable strength defaults is presented. The assumptions on which the original work was based are reviewed and revised. A new algorithm is presented that is shown to compute the ME-ranking under these more general conditions. The limitations of the revised approach are discussed and a test for the uniqueness of the ME-solution is given. The ME-solutions to several illustrative examples of default reasoning are given, and the approach is shown to handle them appropriately. The conclusion is that the ME-approach can be regarded as providing a benchmark theory of default reasoning against which default intuitions and other default systems may be assessed.     

An Interpretation of Default Logic in Minimal Temporal Epistemic Logic

When reasoning about complex domains, where information available is usually only partial, nonmonotonic reasoning can be an important tool. One of the formalisms introduced in this area is Reiter's Default Logic (1980). A characteristic of this formalism is that the applicability of default (inference) rules can only be verified in the future of the reasoning process. We describe an interpretation of default logic in temporal epistemic logic which makes this characteristic explicit. It is shown that this interpretation yields a semantics for default logic based on temporal epistemic models. A comparison between the various semantics for default logic will show the differences and similarities of these approaches and ours.     

Default reasoning by deductive planning

This paper deals with the automation of reasoning from incomplete information by means of default logics. We provide proof procedures for default logics' major reasoning modes, namely, credulous and skeptical reasoning. We start by reformulating the task of credulous reasoning in default logics as deductive planning problems. This interpretation supplies us with several interesting and valuable insights into the proof theory of default logics. Foremost, it allows us to take advantage of the large number of available methods, algorithms, and implementations for solving deductive planning problems. As an example, we demonstrate how credulous reasoning in certain variants of default logic is implementable by means of a planning method based on equational logic programming. In addition, our interpretation allows us to transfer theoretical results, such as complexity results, from the field of planning to that of default logics. In this way, we have isolated two yet unknown classes of default theories for which deciding credulous entailment is polynomial.Our approach to skeptical reasoning relies on an arbitrary method for credulous reasoning. It does not strictly require rather the inspection of all extensions, nor does it strictly require the computation of entire extensions to decide whether a formula is skeptically entailed. Notably, our approach abstracts from an underlying credulous reasoner. In this way, it can be used to extend existing formalisms for credulous reasoning to skeptical reasoning.This author was a visiting professor at the University of Darmstadt while parts of this work were being carried out. This author also acknowledges support from the Commission of the European Communities under grant no. ERB4001GT922433.     

AN APPROACH TO REASONING ABOUT HYBRID SYSTEMS

A formal approach is presented for proving temporal properties of dynamic systems. Its main advantage is that it can be used to prove properties of hybrid systems, i.e. those whose state contains both discrete and continuous parameters. In contrast, most current temporal reasoning techniques are restricted either to purely discrete systems or to purely continuous systems. Our approach is based upon a new modeling technique called DMOD. A DMOD model of a system defines the causality relation between events in the system, using definite clauses, i.e. logic programs. Thereby, the problem of reasoning about hybrid systems is reduced to one of reasoning about the behavior of definite clauses. As these possess a simple proof theory, reasoning is substantially simplified.     

An approach to fuzzy default reasoning for function approximation

This paper discusses fuzzy reasoning for approximately realizing nonlinear functions by a small number of fuzzy if-then rules with different specificity levels. Our fuzzy rule base is a mixture of general and specific rules, which overlap with each other in the input space. General rules work as default rules in our fuzzy rule base. First, we briefly describe existing approaches to the handling of default rules in the framework of possibility theory. Next, we show that standard interpolation-based fuzzy reasoning leads to counterintuitive results when general rules include specific rules with different consequents. Then, we demonstrate that intuitively acceptable results are obtained from a non-standard inclusion-based fuzzy reasoning method. Our approach is based on the preference for more specific rules, which is a commonly used idea in the field of default reasoning. When a general rule includes a specific rule and they are both compatible with an input vector, the weight of the general rule is discounted in fuzzy reasoning. We also discuss the case where general rules do not perfectly but partially include specific rules. Then we propose a genetics-based machine learning (GBML) algorithm for extracting a small number of fuzzy if-then rules with different specificity levels from numerical data using our inclusion-based fuzzy reasoning method. Finally, we describe how our approach can be applied to the approximate realization of fuzzy number-valued nonlinear functions