NDLI: Query answering over fact bases in fuzzy propositional logic

Content Provider	IEEE Xplore Digital Library
Author	Plesniewicz, G.S.
Copyright Year	2013
Description	Author affiliation: Appl. Math. Dept., Moscow Power Eng. Inst., Moscow, Russia (Plesniewicz, G.S.)
Abstract	Let L be a fuzzy propositional logic based on triangular norm min(x,y) (Zadeh's logic). A fact in L is an expression of the form r ≤ φ ≤ s where φ ∈ L and 0 ≤ r ≤ s ≤ 1. In fuzzy interpretation I of L every fact is true or false, and I(r ≤ φ ≤ s) = 1 if and only if the two-side inequality r ≤ I(φ) ≤ s is satisfied. Thus, the set FL of all facts defines a crisp logic with fuzzy interpretations. Logical consequence “\|=” in the logic FL is defined as usual: for any set E of facts and any fact α, E \|= α if there are no an interpretation I and a fact β ε E such that I(α) = 1 and I(β) = 0.. But in the logic FL there is also strong logical consequence \|=: E \|= r ≤ φ ≤ s if E \|= r ≤ φ ≤ s and it is not true that E \|= r' ≤ φ ≤ s with r'> r and not true E \|= r ≤ φ ≤ s' with s'<;s. A fact base is a finite set F of facts: F $=\{r_{i}$ ≤ $φ_{i}$ <; $s_{i}$ \| 1≤ i ≤ n}. One can consider the set K $=\{φ_{i}$ \| 1≤ i ≤ n} as a fuzzy knowledge base and F as an instance of K. A query is an expression of the form ?ψ where ψ ε L. The answer to the query to the fact base F is the fact r ≤ ψ ≤ s such that F \|=* r ≤ ψ ≤ s. The problem of query answering over fact bases in FL can be solved by analytical tableaux method. The method results in an algorithm with the exponential worst-case estimate (relatively to the size of F V {φ} where φ is a query). However, consider the situation when the knowledge base K and the query κ are fixed but fact bases F are arbitrary instances of K. Then, it is possible to answer query κ to fact bases F quickly. But preliminary we should deal with the parametric fact base associated with K. Under a parametric fact we mean an expression of the form a ≤ φ b where a and b are not numbers but parameters - variables with values in [0,1]. The parametric fact base associated with K is $P=\{a_{i}≤φ_{i}≤bi$ \| 1≤ i ≤ n} where $a_{i},$ bi are different parameters. Thus, if we replace the parameters by specific numbers from [0,1] (with adherence to corresponding inequality) we obtains a specific fact base which is an instance of the parametric fact base P. One can also consider query answering over parametric fact bases. Let P be a parametric fact base and ?ψ be a query. The answer to the query is the expression g ≤ ψ ≤ h such that Kλ \|=* gλ ≤ ψ ≤ hλ for any substitution λ of numbers from [0,1] for the parameters from K. Here g and h are appropriate expressions with the parameters from K. Using the analytical tableaux method, we show how to design the algorithm AL for finding the expressions g and h for a given knowledge base K. So let a knowledge base K = $\{φ_{i}$ \| 1≤ i ≤ n} and a query κ: ?ψ be fixed. Suppose we need to the parametric fact base P = $\{a_{i}$ ≤ $φ_{i}$ ≤ bi \| 1≤ i ≤ n} and obtain the expressions g and h; (2) apply the substitution λ = {ai / ri, bi / si \| 1≤ i ≤ n} to g and h; thus, we obtain the answer gλ ≤ ψ ≤ hλ.
Sponsorship	IEEE
Starting Page	1252
Ending Page	1255
File Size	370144
Page Count	4
File Format	PDF
ISBN	9781479903481
DOI	10.1109/IFSA-NAFIPS.2013.6608581
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2013-06-24
Publisher Place	Canada
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Knowledge based systems Fuzzy logic Algorithm design and analysis Syntactics Power engineering Abstracts
Content Type	Text
Resource Type	Article

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