I 1 i ng i on scheme n 2 before describing our method

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Unformatted text preview: paraconsistent relation h ft 2 (?n) j for some t0 2 R+; t( i) = t0 (Ai); for all i; 1 i ng; ft 2 (?n) j for some t0 2 R?; t( i ) = t0(Ai ); for all i; 1 i ng i on scheme ?n. 2 Before describing our method to convert the given database P into a set of de nitions for the predicate symbols in P , let us look at an example. Suppose the following are the only clauses with the predicate symbol p in their heads: p(X) p(Y) r(X,Y), s(Y,a) :p(Y) From these clauses the algebraic de nition constructed for the symbol p is the following: __ p = ( _ {X} (r(X; Y) 1 ?p(Y))) X] _ (s(Y; Z)))) Y] Such a conversion exploits the close connection between attribute names in relation schemes and variables in clauses, as pointed out in 24]. The expression thus constructed can be used to arrive at a better approximation of the paraconsistent relation p from some approximations of p, r and s. The algebraic expression for the predicate symbol p is a union ( _ ) of the expressions obtained from each clause containing the symbol p in its head. It therefore su ces to give an algorithm for converting one clause into an expression. 15 Algorithm CONVERT Input: A general deductive database clause l0 l1 ; : : : ; l m . Let l0 be of the form p0(A01 ; : : : ; A0k0 ), and each li, 1 i m, be either of the form pi(Ai1 ; : : : ; Aik ), or of the form :pi (Ai1; : : : ; Aik ). For any i, 0 i m, let Vi be the set of all variables occurring in li . Output: An algebraic expression involving paraconsistent relations. Method: The expression is constructed by the following steps: i i 1. For each argument Aij of literal li , construct argument Bij and condition Cij as follows: (a) If Aij is a constant a, then Bij is any brand new variable and Cij is Bij = a. (b) If Aij is a variable, such that for each k, 1 k < j , Aik 6= Aij , then Bij is Aij and Cij is true. (c) If Aij is a variable, such that for some k, 1 k < j , Aik = Aij , then Bij is a brand new variable and Cij is Aij = Bij . 2. Let ^i be the atom pi(Bi1 ; : : : ; Bik ), and Fi be the conjunction Ci1 ^ ^ Cik . If li l is a positive literal, then let Qi be the expression _ V ( _ F (^i)). Otherwise, let Qi be l _ the expression ? _ V ( _ F (^i)). l As a syntactic optimisation, if all conjuncts of Fi are true (i.e. all arguments of li are distinct variables), then both _ F and _ V are reduced to identity operations, and are hence dropped from the expression. For example, if li = :p(X,Y), then _ Qi = ?p(X,Y). _ 3. Let E be the natural join (1) of the Qi 's thus obtained, 1 i m. The output expression is ( _ F0 ( _ V (E ))) B01 ; : : : ; B0k0 ], where V is the set of variables occurring in ^0. l As in step 2, if all conjuncts in F0 are true, then _ F0 is dropped from the output expression. However, _ V is never dropped, as the clause body may contain variables not in V . 2 i i i i i i i i From the algebraic expressions obtained by Algorithm CONVERT for clauses in the given general deductive database, we construct a system of equations de ning par...
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This note was uploaded on 09/28/2013 for the course CSC 8710 taught by Professor Staff during the Fall '08 term at Georgia State University, Atlanta.

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