Approximate Reasoning In F(S) And Continuity Of The Triple I Method

It is well known that mathematical logic is a formalized symbolictheory,it plays great attention to formal infernece but not to numberical computa-tion.On the contrary, numberical computation uses interpolation,iteration,differenceand probability method to study diverse kinds of questions:It pays attention to thequestion as well as error estimate,and ingores inference method.Hence the researchof mathematical loic and numberical differs greatly.A natural question is:wether theycan be combined to form a unified theory?Professor Wang guojun gave the affirma-tive answer in his monograph<>.For proposing the quantitative logic,the truth degree of a formula wasgiven,also,the similarity degree of two formulas and Pseudo-metric among formulaswere given.Based on those above,the divergency and consistency degree of a groupof formulas were proposed, and thus one kind of approximation reasoning theorybased on truth degree was established,also,the framework of three kinds of approx-imation reasoning and the theory of reasoning error was proposed.The paper mainlydeals with the inner link between the three kinds of reasoning error.In addition,along with the successful application of fuzzy control in productionpratice,fuzzy inference as its key content attract more and more scholars'attention.Theresearch on the CRI method and Triple-I method which are commonly used inthe process of fuzzy reasoning has acheived great acomplishment.But,most concen-trte the extention and improvement of Triple-I method.We hope that the reasoningmethod we use can minimize the error produced in the input of fuzzy reasoning soas not to have a large bear on the final rsults since the error can not be avoided com-pletely in the practce.The positive answer to the question wether Triple-I methodsatisfies the requirements above is given in this paper.The main results of the present paper is as following.(1)The equivalence of three kinds of approximation reasoning in Two valuedlogic systerm L is proved.It tell us that the reasoning error can be obtained by onlyconsidering the distance between one formula and the logic deduction of a groupof formulas in the process of approxomation reasoning,especially, the distance iszero in the case of the formula being the member of deduction set of the group of formulas.Obviously, the inference ability of this group of formulas promoting thisformula is stronger,the distance from it to the formula becomes nearer.(2)In fuzzy logic system L~*,the definition ofâ… -inference error and that ofâ…¡-inference error are equivalent,but not equivalent to theâ…¢-inference error.The innerlink between theâ… -inference error and theâ…¢-inference error is given,that is whattheorem 2.3.6 sys.â… -inference error and theâ…¢-inference error are mutually equivalentin the case that B is a 1/2~+-condition tautology.Furthermore, the relation between theâ… -inference error and theâ…¢-inference error when B is not the 1/2~+-tautology is given.(3) The concept of logic equation is proposed for the first time,and the corre-sponding existence theory of the solution to the logic equation has been given,theproperies of solution to logic equation were analized.It also pointed out that thesolution set of the equation is inconsistent.(4) The continuity of the Triple-I method has been discussed.It is pointed thatcorresponding Triple-Imethod is continuity with respect to the regular operator.Thisresult indicated that Triple-I algorithm is one ideal inference method.