Possibilistic expert systems and process control utilizing fuzzy logic

Abstract

An explicit assumption of continuity is used to generate a fuzzy implication operator, which yields an envelope of possibility for the conclusion. A single fuzzy rule A B entails an infinite set of possible hypothese A′ B′ whose degree of consistency with the original rule is a function of the “distance” between A and A′ and the “distance” between B and B′. This distance may be measured geometrically or by set union/intersection. As the distance between A and A′ increases, the possibility distribution B* spreads further outside B somewhat like a bell curve, corresponding to common sense reasoning about a continuous process. The manner in which this spreading occurs is controlled by parameters encoding assumptions about (a) the maximum possible rate of change of B′ with respect to A′(b) the degree of conservatism or speculativeness desired for the reasoning process (c) the degree to which the process is continuous of chaotic.

The embodiments of the invention in which an exclusive property or privilege is claimed are defined as follows:  
     
         1 . A method of evaluating a confidence in an outcome of a fuzzy logic possibilistic system comprising the steps of 
 providng a rule that maps a given input to a predictable output;    selecting a plurality of inputs that differ from said given output;    establishing a relationship between said given input and said selected inputs;    assigning a degree of possibility to possible outcomes resulting from application of said selected ones of inputs to said rule;    said degree of possibility being correlated to said relationship established between said given input and said selected input;    establishing an envelope of possibility that encompasses each of said possible outcomes resulting from said selected inputs;    according to each of said selected inputs a credibility to establish an envelope of belief within said envelope of possibility;    comparing said envelope of belief and said envelope of possibility;    and determining confidence in an indicated outcome based on difference between said envelopes.    
     
     
         2 . A method according to  claim 1 , wherein a plurality of rules are provided and said envelope of possibility contains possible outcomes from each of said rules.  
     
     
         3 . A method according to  claim 1 , wherein said envelope of possibility is established by consideration of adjacent sets of outcomes.  
     
     
         4 . A method according to  claim 3 , wherein said envelope is established through interpolation between adjacent sets.  
     
     
         5 . A method according to  claim 3 , wherein said envelope is established through extrapolation between adjacent sets.  
     
     
         6 . A method according to  claim 1  including a set of examples associated with said rule to provide a plurality of possible outcomes for an input.  
     
     
         7 . A method according to  claim 1  wherein said relationship is established based on similarity between said selected inputs and said given input of said rule.  
     
     
         8 . A method according to any proceeding claim wherein a parameter is applied to limit said outcomes and thereby modify said envelope of possibility.  
     
     
         9 . A method according to any proceeding claim wherein a subset of said envelope of belief is established by applying a parameter to qualify said inputs.  
     
     
         10 . A method according to any proceeding claim wherein a subset of said envelope of possibility is established by applying a parameter to qualify said outputs.  
     
     
         11 . A possibilistic expert system utilizing fuzzy logic rule sets to determine an outcome from a set of inputs including: a set of parameters initially determined by an expert of said system; at least one set of rule inputs and a corresponding set of rule outputs; a plurality of predetermined functions to operate on selected ones of said parameters and said rule inputs; some of said predetermined functions being used to assign a degree of possibility to each of a number of possible outcomes; wherein each of said degree of possibility of each of said possible outcomes are used to establish at least one envelope of possibility containing allowable outcomes from said rule set.  
     
     
         12 . A possibilistic expert system according to  claim 11  further comprising at least one envelope of belief is established by applying a credibility to said inputs and a plurality of criteria with which said envelope of possibility and said envelope of belief are compared thereto.  
     
     
         13 . A possibilistic expert system according to  claim 11 , wherein said predetermined functions include interpolation and extrapolation to generate said envelope of possibility from at least two disjoint sets of said possible outcomes.  
     
     
         14 . A possibilistic expert system according to  claim 12 , further comprising a plurality of examples used in conjunction with said sets of said rules.  
     
     
         15 . A possibilistic expert system according to  claim 11 , further comprising a plurality of distance measures to calculate a degree of similarity between said disjoint sets.  
     
     
         16 . A possibilistic expert system according to  claim 15 , wherein said predetermined functions control a shape of said envelope of possibility.  
     
     
         17 . A possibilistic expert system according to  claim 16 , wherein said predetermined functions also control a rate of spreading of each of said envelopes.  
     
     
         18 . A possibilistic expert system according to  claim 11 , wherein said rate of spreading is a function of distance between said set of parameters and said rule input.  
     
     
         19 . A possibilistic expert system according to  claim 11 , further comprising a system of weighting for a plurality of multi-dimensional inputs to promote sensitivity of said output to specific dimensions of said multi-dimensional input.  
     
     
         20 . A possibilistic expert system according to  claim 19 , further comprising the use of at least one fuzzy implication operator to encode the degree of chaos versus continuity present in said system set up by said expert.  
     
     
         21 . A possibilistic expert system according to  claim 20 , wherein a plurality of fractal parameters are used to calculate said envelope of possibility for a fractal system.  
     
     
         22 . A method for determining an outcome from a set of inputs in an expert system, said method comprising the steps of: 
 a) determining a set of parameters by the expert for the system;    b) establishing at least one rule using at least two of said sets of parameters as input;    c) according a value to each of selected ones of sets of parameters;    d) computing an envelope of possibility by operating on inputs and said selected ones of parameters by applying a predetermined function thereto;    e) computing a belief function by applying a credibility factor to said inputs    f) comparing said envelope of possibility and belief function with predetermined criteria; and    g) producing an output indicative of a result of said comparison.    
     
     
         23 . A method for determining an outcome from a set of inputs in an expert system as defined in  claim 22 , said predetermined function including: a spreading function, interpolation and extrapolation.