Representing fuzzy decision tables in a fuzzy relational database environment.

In this paper the representation of decision tables in a relational database environment is discussed. First, crisp decision tables are defined. Afterwards a technique to represent decision tables in a relational system is presented. Next, fuzzy extensions are made to crisp decision tables in order to deal with imprecision and uncertainty. As a result, with crisp decision tables as special cases fuzzy decision tables are defined which include fuzziness in the conditions as well as in the actions. Analogous to the crisp case, it is demonstrated how fuzzy decision tables can be stored in a fuzzy relational database environment. Furthermore, consultation of these tables is discussed using fuzzy queries.Decision making;

Verification and validation of knowledge-based systems with an example from site selection.

In this paper, the verification and validation of Knowledge-Based Systems (KBS) using decision tables (DTs) is one of the central issues. It is illustrated using real-market data taken from industrial site selection problems.One of the main problems of KBS is that often there remain a lot of anomalies after the knowledge has been elicited. As a consequence, the quality of the KBS will degrade. This evaluation consists mainly of two parts: verification and validation (V&V). To make a distinction between verification and validation, the following phrase is regularly used: Verification deals with 'building the system right', while validation involves 'building the right system'. In the context of DTs, it has been claimed from the early years of DT research onwards that DTs are very suited for V&V purposes. Therefore, it will be explained how V&V of the modelled knowledge can be performed. In this respect, use is made of stated response modelling designs techniques to select decision rules from a DT. Our approach is illustrated using a case-study dealing with the locational problem of a (petro)chemical company in a port environment. The KBS developed has been named Matisse, which is an acronym of Matching Algorithm, a Technique for Industrial Site Selection and Evaluation.Selection; Systems;

Modelling decision tables from data.

On most datasets induction algorithms can generate very accurate classifiers. Sometimes, however, these classifiers are very hard to understand for humans. Therefore, in this paper it is investigated how we can present the extracted knowledge to the user by means of decision tables. Decision tables are very easy to understand. Furthermore, decision tables provide interesting facilities to check the extracted knowledge on consistency and completeness. In this paper, it is demonstrated how a consistent and complete DT can be modelled starting from raw data. The proposed method is empirically validated on several benchmarking datasets. It is shown that the modelling decision tables are sufficiently small. This allows easy consultation of the represented knowledge.Data;

A synthesis of fuzzy rule-based system verification.

The verification of fuzzy rule bases for anomalies has received increasing attention these last few years. Many different approaches have been suggested and many are still under investigation. In this paper, we give a synthesis of methods proposed in literature that try to extend the verification of clasical rule bases to the case of fuzzy knowledge modelling, without needing a set of representative input. Within this area of fyzzy V&V we identify two dual lines of thought respectively leading to what is identified as static and dynamic anomaly detection methods. Static anomaly detection essentially tries to use similarity, affinity or matching measures to identify anomalies wihin a fuzzy rule base. It is assumed that the detection methods can be the same as those used in a non-fuzzy environment, except that the formerly mentioned measures indicate the degree of matching of two fuzzy expressions. Dynamic anomaly detection starts from the basic idea that any anomaly within a knowledge representation formalism, i.c. fuzzy if-then rules, can be identified by performing a dynamic analysis of the knowledge system, even without providing special input to the system. By imposing a constraint on the results of inference for an anomaly not to occur, one creates definitions of the anomalies that can only be verified if the inference pocess, and thereby the fuzzy inference operator is involved in the analysis. The major outcome of the confrontation between both approaches is that their results, stated in terms of necessary and/or sufficient conditions for anomaly detection within a particular situation, are difficult to reconcile. The duality between approaces seems to have translated into a duality in results. This article addresses precisely this issue by presenting a theoretical framework which anables us to effectively evaluate the results of both static and dynamic verification theories.

Restructuring and simplifying rule bases.

Rule bases are commonly acquired, by expert and/or knowledge engineer, in a form which is well suited for acquisition purposes. When the knowledge base is executed, however, a different structure may be required. Moreover, since human experts normally do not provide the knowledge in compact chunks, rule bases often suffer from redundancy. This may considerably harm efficiency. In this paper a procedure is examined to transform rules that are specified in the knowledge acquisition process into an efficient rule base by way of decision tables. This transformation algorithms allows the generation of a minimal rule representation of the knowledge, and verification and optimization of rule bases and other specification (e.g. legal texts, procedural descriptions, ...). The proposed procedures are fully supported by the PROLOGA tool.

The behavior of fuzzy implications in a fuzzy knowledge base.

More and more companies today discover the advantages of using knowledge bases for their processes and services. Recently, fuzzy set theory has also captured the attention due to good performances within control systems. Therefore, it is very appealing to combine the advantages of these two areas into a fuzzy knowledge base. However, obtaining the results of control systems in a knowleg based environment is not so straightforward. This paper will investigate one aspect of the reasoning process, namely the behavior of the implication. From the different tests performed, four main behaviors of implications can be found. First of all, there are the implications not always resulting in a convex set. A second classs - the so-called impotent implications- doesn't change the predefined set at all. A third grouping reveals always a constant value portion, that rises or falls according to the changed input. A final divsion shifts the complete set in its whole conformably the intuition.Implications; Companies; Advantages; Knowledge; Processes; Theory; Performance; Systems; Value;

An eclectic quadrant of rule based system verification: work grounded in verification of fuzzy rule bases.

In this paper, we used a research approach based on grounded theory in order to classify methods proposed in literature that try to extend the verification of classical rule bases to the case of fuzzy knowledge modeling. Within this area of verification we identify two dual lines of thought respectively leading to what is termed respectively static and dynamic anomaly detection methods. The major outcome of the confrontation of both approaches is that their results, most often stated in terms of necessary and/or sufficient conditions are difficult to reconcile. This paper addresses precisely this issue by the construction of a theoretical framework, which enables to effectively evaluate the results of both static and dynamic verification theories. Things essentially go wrong when in the quest for a good affinity, matching or similarity measure, one neglects to take into account the effect of the implication operator, an issue that rises above and beyond the fuzzy setting that initiated the research. The findings can easily be generalized to verification issues in any knowledge coding setting.Systems;

A tool-supported approach to inter-tabular verification.

The use of decision tables to verify KBS has been advocated several times in the V&V literature. However, one of the main drawbacks of those system is that they fail to detect anomalies which occur over rule chains. In a decision table based context this means that anomalies which occur due to interactions between tables are neglected. These anomalies are called inter-tabular anomalies. In this paper we investigate an approach that deals with inter-tabular anomalies. One of the prerequisites for the approach was that it could be used by the knowledge engineer during the development of the KBS. This requires that the anomaly check can be performed on-line. As a result, the approach partly uses heuristics where exhaustive checks would be too inefficient. All detection facilities that will be described, have been implemented in a table-based development tool called PROLOGA. The use of this tool will be briefly illustrated. In addition, some experiences in verifying large knowledge bases are discussed.

On the Convergence in Distribution of Measurable Multifunctions, Normal Integrands, Stochastic Processes and Stochastic Infima

The concept of the distribution function of a closed-valued measurable multifunction is introduced and used to study the convergence in distribution of sequences of multifunctions and the epi-convergence in distribution of normal integrands; in particular various compactness criteria are exhibited. The connections with the convergence theory for stochastic processes is analyzed and for purposes of illustration we apply the theory to sketch out a modified derivation of Donsker's Theorem (Brownian motion as a limit of random walks). We also suggest the potential application of the theory to the study of the convergence of stochastic infima

Weak Convergence of Probability Measures Revisited

The hypo-convergence of upper semicontinuous functions provides a natural framework for the study of the convergence of probability measures. This approach also yields some further characterizations of weak convergence and tightness