By Abe J M (eds )
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Additional info for Advances in logic, artificial intelligence and robotics Laptec 2002
Now we define a different mapping TN that maps an interpretation of annotated formulas into itself. [Definition 4] Let J// be a Herbrand interpretation of the annotated logic. Then. 4 and J// satisfies the formula^ }. where U denotes the least upper bound. We define a special interpretation A to be an interpretation which assigns the value / to all members of B^. |r/ < A}. [Theorem 1] Let A be a ground atom and P a logic program. A is a logical consequence of P <=> P^ \= ( A : s ) . [Proof] Since "A is a logical consequence of P is equivalent to A G Tp t ^ this theorem is proved by induction on the least integer d such that A € Tp t d and TN t d \= (A : s).
Therefore, we give an automated theorem proving method for the many-sorted theory SEn as it is. G. and Winograd,T. : An Overview of KRL. a Knowledge Representation Language. L. and Lee. E. : Symbolic Logic and Mechanical Theorem Proving. J. : Automatic Deduction and Equality. Proc. of the 1979 Annual Conf. ACM. E. : On the Relation between Free Description Theories and Standard Quantification Theory. NDFJL, 17 (1976)  Hayes,P. J. : The Logic of Frames, in Frame Conceptions and Text Understanding.
T m ) ) = I . iff ) A Vyi(Ei(yi) -+ (B * (yt) = x, = yt))V (~ (Eifa) A Vyi(E(yi) -)• (5 * (^) = x, = yi))) A x, - a,) A P(t\ . . , xit . . , t m ))) - 1 iff if €i is the unique element of Gi such that V^-B * (yi)) = 1 and Vfa) = e"j, or there is no such element and Vfa) = K(OJ) € Gi, then F(P(^, . . , xt, . . , t m ) ) = 1. Since B(XI) contains no description, from [Case 1] and by induction, U(Bfa)) = l iff V ( B * ( V i ) ) = l. Hence, U(P(tl, . . , ix&fa), . . , t m )) = 1 iff V((P(t\ .