Uncertainty in Knowledge-Based Systems

Our decisions reflect uncertainty in various ways. We take account of the uncertainty embodied in the roll of the die; we less often take account of the uncertainty of our belief that the die is fair. We need to take account of both uncertain knowledge and our knowledge of uncertainty.. “Evidence” itself has been regarded as uncertain. We argue that point-valued probabilities are a poor representation of uncertainty; that we need not be concerned with uncertain evidence; that interval-valued probabilities that result from knowledge of convex sets of distribution functions in reference classes (properly) include Shafer's mass functions as a special case; that these probabilities yield a plausible non-monotonic form of inference (uncertain inference, inductive inference, statistical inference); and finally that this framework provides a very nearly classical decision theory— so far as it goes. It is unclear how global the principles (such as minimax) that go beyond the principle of maximizing expected utility are.

We discuss the problems involved in representing default knowledge. We suggest that default knowledge can be captured by using possibility qualified statements. We investigate the representation of possibility qualified statements via the theory of approximate reasoning. We apply our representation scheme to some examples. In particular, we show that this scheme gives reasonable results for a knowledge base in which we have "normally Quakers are pacifists" and "normally republicans are not pacifists".

We consider the representation of data through linguistic variables characterized by means of attributes, eventually transformed by so-called modifiers. We study several fuzzy implications and combination operators used to define a generalized modus ponens in the case where observations are slightly different from the premise of rules of the knowledge base, but a conclusion is required. We compare the properties of these tools to provide indications on their behavior in different situations. We study the stability of particular modifiers when used to describe the difference between an observation and the fact involved in the antecedent of a rule.

The framework of evidence theory is used to represent uncertainty pervading a set of statements which refer to subsets of a universe. Grades of credibility and plausibility attached to statements specify a class of bodies of evidence. Using newly appeared measures of specificity, a principle is stated in order to select, among these bodies of evidence, the one which suitably represents the available information in the least arbitrary way. It is shown that this principle, which is similar to the maximum entropy principle, leads to a deductive reasoning approach under uncertainty, and also provides a rule of combination which does not presuppose any independence assumption. Particularly, it is more general than Dempster's.

This work presents results of research to develop of approximate inference formulas in the context of the calculus of evidence of Dempster-Shafer and the theory of interval probabilities. These formulas generalize the probabilistic domain the well-known relations between the truth values of the antecedent and the consequent propositions of a classical implication P → Q.Approximate conditional knowledge is assumed to be expressed either as sets of possible values (actually numeric intervals) of conditional probabilities, or as the Shafer belief functions induced by certain restricted multivalued mappings — the nature of the restriction representing the truth of the antecedent in an implication — between a space representing possible states of the real world and another space used to represent propositional truth.A notion of consistence between unconditional and conditional probability intervals is introduced to represent agreement of the constraints induced by both types of distributions. The integration of conditional and unconditional knowledge is accordingly described as the refinement of probability interval estimates for unconditional propositions so as to achieve consistence with conditional constraints. This refinement always results in interval-valued distributions which are more specific than the corresponding unmodified estimates.Formulas for conditional knowledge integration are discussed, together with the computational characteristics of the methods derived from them. Of particular importance is one such evidence-integration formalism produced under a belief function interpretation — generalizing both modus ponens and modus tollens inferential mechanisms — which integrates conditional and unconditional knowledge without resorting to iterative or sequential approximations. Further, this formalism produces Shafer mass distributions as output using similar distributions as input.

Monotone capacities are characterized by properties of their Möbius inverses. A necessary property of probabilities dominating a given capacity is given. It is shown to be also sufficient if and only if the capacity is monotone of infinite order. A characterization of dominating probabilities specific to capacities or order 2 is also proved.

To develop an approach to utilizing continuous statistical information within the Dempster-Shafer framework, we combine methods proposed by Strat and by Shafer. We first derive continuous possibility and mass functions from probability-density functions. Then we propose a rule for combining such evidence that is simpler and can be computed more efficiently than Dempster's rule. We discuss the relationship between Dempster's rule and our proposed rule for combining evidence over continuous frames.

A hybrid uncertainty theory is developed to bridge the gap between fuzzy set theory and Dempster-Shafer theory. Its basis is the Dempster-Shafer formalism, which is extended to include a complete set of basic operations for handling uncertainties in a set-theoretic framework. The new operator theory, O-Theory, retains the probabilistic flavor of Dempster's original point-to-set mappings but includes the potential for defining a range of operators like those found in fuzzy set theory.

The generalized modus ponens is a fuzzy logic pattern of reasoning that permits to make inferences with rules having imprecision both in their antecedent and consequent parts. Though it is a very powerful approximate reasoning tool (from a theoretical point of view), this technique may result in unacceptably slow executions if inappropriately implemented. There are several ways to avoid the inefficiency bottleneck. One of them, that is the object of this paper, consists in introducing an approximation technique focussing only of what is semantically important. This approximation technique is conceived so as to be used in situations where the dependency between two given variables is described via a collection of rules. Moreover, this paper addresses the problem in the setting having the main features that follow:-the possibility distributions involved in facts and rules are continuous (the referential is the real line), normalized, unimodal and expressed by parametrized functions;-only single antecedent rules are considered;-the rules are consistent and it is assumed that their antecedents and consequents do not overlap too much;-the deduction process is based on the ‘min’ conjunction and Gödel implication operators.The ultimate goal of this work is to render the generalized modus ponens technique usable in practical deduction systems.

The Matrix Controlled Inference Engine (MACIE) style expert system uses a knowledge base automatically generated from a set of crisp training examples. However, when viewed from a fuzziness perspective, it is seen that the Pocket Algorithm which generates the knowledge base operates nondeterministically. Thus we have the fuzzy generation of a crisp expert system, rather than the usual crisp generation of a fuzzy expert system. It is also shown how MACIE can directly implement fuzzy expert systems.

In this paper, Orlovsky's concept of decision making with fuzzy preference relation is studied. On the one hand, the special significance of max-min transitivity inside the family of max-⋆ transitivities is stablished. On the other hand, a necessary and sufficient condition for the existence of a non empty set of unfuzzy nondominated alternatives is proved. Moreover, other alternative methods are proposed in order to solve some practical difficulties.

The majority of the current knowledge based systems/expert systems (KBS/ES), decision support systems (DSS), and management information systems (MIS) have followed the traditional pattern of dealing only with crisply defined, non-fuzzy ("hard") problem situations.This paper presents an approach to developing an overall intelligent problem solving environment, to be used by those who are faced with "soft", fuzzy, unstructured executive and administrative problems.A suggested methodology approaches the commercial/managerial environments as human activity systems, and thus it considers the problem environment as "soft" (i.e. ill-defined). Personal, as well as consensus models of the problem situations provide the basis for problem understanding and decision support, The theory of fuzzy sets and conceptual modelling/cognitive mapping provide the framework.An overall architecture of such an environment consists of :a)-an interface module for :i)human-computer communication. This interface has the task of transforming the input information which is expressed mainly in : natural language (text, descriptions, beliefs etc.); and in the form of specific domain data. This input is transformed into conceptual models and cognitive maps according to appropriate fuzzy relational data bases.ii)system-user interface for communicating the advice and general support to the decision-makers.b)-a knowledge base including an array of cognitive maps and conceptual models of various problem-owners. These represent beliefs and thinking about a range of problems specific to their organization. These models could be merged if the decision makers want to reach a consensus decision, or they could be used individually. The knowledge base can also include knowledge representation modules consisting of rules and facts.c)-an inference engine to form advice on the basis of comparing the models of the last two sections above, as well as by processing input data via standard DSS techniques if appropriate.

This paper discusses a framework for assigning probabilities to rules in an expert system to deal with uncertainty knowledge processing. The objective is to enable a system to respond to environmental or uncontrollable factors in a strategic manner. We define guards for representing probabilities and then discuss operators and axioms, as part of a problem solver, for guiding probability assignments. We then demonstrate the capabilities of this problem solver by applying to an example taken from game theory.

This paper describes a method for performing knowledge base (re)organization in Expert Systems oriented to classification, interpretation and diagnosis problems. The methodology can be applied either to the input descriptions of a set of samples, giving thus a preliminary characterization of groups of samples, or to a set of intermediate level descriptions, supplied by a human expert or previously automatically learned. An example of application is also given.

A good inference engine must be expected to deal effectively with contradictions arising from uncertain, incomplete and contradictory information.Systems based on classiscal logics obviously cannot cope with that.Usual non-monotonic logics. on the other hand, have some particularyly undesirable features: they don't tend toward stabilization, in the sense that their picture of the world can change any moment in any point; they don't learn from experience, that is never draw conclusions about the reliability of the source or the state of the subject under examination; they cannot sensibly have a fixed "action point", that is a point of belief at which actions are taken, even if a sequence of degrees of belief is present; finally, to have a logical model, they must have infinitely increasing levels of belief, which is very unnatural. All this brings about several absurd responses.We outline here a system which, while being non-classical, has none of the features above. Its model is based on a many-valued extension of the Kripke model for intuitionistic logic. The main idea is that if contradictions are discovered with a certain frequency, the system tries to identify the source, context, or both, responsible, and the then decreases the appropriate reliabilities. This action is balanced by an opposite set of functions which increases such reliability if no contradictions are discovered and not enough "believe knowledge" has been produced.The problem of having a finited number of degrees of belief, and a fixed point of action, is tackled by defining a meta-model, inside which the object models can collapse and be reconstructed when contradictions are discovered at the top truth value. This corresponds to expressing a (human) decision about formal logic in the formalism itself.In the end we briefly present some more features, interesting but not essential.

One crucial problem in Artificial Intelligence is succeeding in managing uncertain knowledge. By the present paper, a particular type of uncertainty is considered: that on the possible causes of observed effects. Said uncertainty will be formalized by means of the credibility of the causes themselves. Credibility that will be achieved by convolving, through an abductive paradigm: the evidence with which each cause is indicated by obtained observations; the plausibility of the same cause; and the clarity of the performed indication. The issue results as an outline of the matter developed in earlier papers; in it, intermediate passages and proofs are abridged.

Pattern classification is an important technique for medical decision making. In this paper, a method based on a new set of orthogonal polynomials is described. The method uses a modified form of supervised learning to assign data to the correct category from two or more possible choices. Results have been compared to standard nonlinear statistical discriminant analysis techniques, and have been shown to be consistently more accurate.

The paper outlines the role of Information Retrieval techniques in the construction of Knowledge-Based Systems. A Functional Communication Structure selects and communicates the relevant information by means of fuzzy logical and fuzzy relational requests between the individual knowledge acceptors and knowledge donors. The theory of fuzzy relational products developed by Bandler and Kohout qualifies as an especially adequate tool to handle fuzziness of selection as well as fuzziness of information contents.

This paper proposes a new approach to the indexation of documents by keywords, taking into account to what extent a given keyword may and must appear in an acceptable description of a considered document. Possibility (resp. necessity) measures are used to estimate the possible (resp. certain) relevance of a document with respect to a query.

This paper demonstrates how fuzzy relational information retrieval techniques are utilised in the context of clinical decision making by means of knowledge-based system. Our example of practical application outlines the use of the methods for retrieval of patterns describing dexterity assessment of neurological patients through handwriting. A partial view of the architecture of DEXTERON KBS unit is presented and the role of fuzzy information retrieval techniques in its design is outlined.

The use of uncertainty in a rule-based expert system for the analysis of chest pain is discussed. The system, EMERGE, has been evaluated retrospectively and prospectively and has been found to perform extremely well. The original system has been altered to handle degrees of presence of symptoms and variable contribution of antecedents. It also utilizes a logical construct which generalizes traditional AND/OR logic.

In order to build an expert system based not only on the opinion of a single expert but constructed using information collected from a group of experts, one may use techniques of knowledge elicitation in conjunction with methods coming from the theories of consensus and synthesis of judgements. In this paper we describe a knowledge elicitation methodology and we extend it in order to deal with judgements of several experts.

We have presented a method of collecting information from multiple experts and assembling it into a knowledge-base which has no known contradictions. We argued that this is one of the useful tools for building a large knowledge-based system. We described HOLMES-I, a Prolog-based system which has both forward and backward chaining, Reason Maintenance with Multiple Belief Spaces and Contradiction Resolution. Finally we described some of the limitations of the current version of HOLMES and indicated our goals for the next version.

Human perception of resemblance in spatio-temporal patterns is modeled with the procedural normal description schemata [5–11], which are used for gathering experiential knowledge embedded in sensory data. The representation and the quantification of uncertainty in human perception and in the formal schemata which model it are examined in this paper.

Logical fault trees are used for obtaining reliability- oriented representations of complex engineered systems. The construction of fault trees is time-consuming and may be affected by several sources of potential error. An expert system has been designed to reduce and control uncertainties that emerge during the construction process. Fuzzy algebra and multiple-valued logic provide the formal instruments for solving problems encountered and for organizing the information which may become available to the analyst. Consequently, a multiplevalued fuzzy logical tree is proposed as a general representation of the engineered system. The tree can be compressed and reduced to the crisp binary case to establish comparisons and perform evaluations as required.

We try to indicate what not to do in characterizations of information measures — because it makes little sense or because it already has been done. For this we have to summarize at least roughly what has been done already. We mention also some problems which we do think are worth working at.

Most of the well known measures like the Shannon entropy, the entropy of degree β, the generalized directed divergence (the information improvement), the weighted entropies etc. have many algebraic properties in common, in particular, the sum representation. They also possess some form of ‘additivity’. Characterization of information measures through these and other algebraic properties satisfied by them involve many functional equations, of which the fundamental equation of information theory and the ‘sum form’ play distinct roles. Here the focus is made on the ‘sum form’, illustrating however at least in one instance the interconnection between these two types of functional equations. Also emphasis is made on the generalized directed divergence.

An hereditary linear "observation operator" ℓ gives at instant t an image y(t) of the state x of a dynamical differential linear system whose initial state is known only through a probability distribution with information entropy H_x(t_o). The information entropy H_y(t) of the probability distribution of y(t) is equal to H_x(t_o) plus a "dynamical gain fo information entropy" and an "observational gain of information entropy". The dynamical gain involves the trace of the evolution matrix of the system. The observational gain involves ℓ and the fundamental matrix of the system. Special cases are presented, one involving a "generalized Laplace transform with matrix argument".