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A*)(i = 1, . . , n). (A -> b}* = A* -> B*. 2 From FDn to SEn Given a FDn-model M =< Glt . . ,G n ,Gi, . . ,G n , U >, a S£n-model M D\. . , Z)n, V7 > can be defined by the *-translation. For each i = 1 ..... n : - Di = G, U Gt • - for any variable Xi of the z'-th sort, V'(x,) = U(xt) ; - for a particular constant a; of the z-th sort, - for the other constants C{ of the z-th sort, - for any function symbol /, of the z-th sort, V(a t ) 6 G, : V (Q) = ^' (c t ) : I '(/,•) = U(f,) : - for a particular predicate Ei not occurring in FDn.

For an atomic formula P t 1 , . . , t m , 22 K. Nakamatsu cmd A. Suzuki / Automated Theorem Proving (i) if none of tl, . . , tm contains a term of the form LXiA, (i) let P(ix il >li(xi 1 ),. ,ixlniAm(xlm)) stand for P(t1,. tm). where ix^A^x^) ..... LX im^m(xim} are all outermost descriptions occurring in tl ..... tm left to right. then where each ij(l < j < m) is one of 1. . , n ; 6. let A and B be any formulas, (~ A)* =~ A*. ) -+ A*)(i = 1, . . , n). (A -> b}* = A* -> B*. 2 From FDn to SEn Given a FDn-model M =< Glt .

Nakamatsu and A. Suzuki /Annotated Logic and Negation as Failure 35 logic. , Pf |= (A:s). Induction Step d = a + 1. In this case, there is a clause, BI A • • • A Bm -> A, in ground(P) such that ; for any integer z(l < i < m), 5t € TP t a. s) A - - - A ( 5 m : s ) . ,t'n), in P and P+ |= 3s/i • • • 3j/ fc (ti = *; A • • • A < n = ^ A (B! :s) A • • • A (B m :s)). Hence, P+MU,- ••,*«):*. Proof of 4= The proof is by the induction on the least integer d such that a Herbrand interpretation TN t d (= (A : s) whenever TN | d is a model of P+.

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