Query answering over fact bases in fuzzy propositional logic

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 b- ses 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.