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N) ; 2. for any constant Cj of the i-th sort, (Q)* = Cj(z = 1, . . , n) ; 3. , tm*) ; 4. for a description iXiA(xi)(i = 1, . . , n), i(yi) ^(A* (y,) = Xi = j/<))) A a:,- = a,)), where at is a particular constant in SEn such that t/(a,) = ^ an<^ ^ G Gz : 5. 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^) .....

Dn, V >. Hence, A* is valid and provable in SEn. Proof of only if-part. 4* is valid in SEn. 4* is valid in SEn iff V'(-4*) = 1 in every SDn-model M* =< DI, . . , Dn, V >. Then, from [Lemma 2], U(A) = 1 in the corresponding FDn-model M =< GI, . . , G n , GI, . . , Gn, U >. 4 is valid and provable in FDn. D. 4 Automated Deduction for FDn In this section, we provide an algorithm for theorem proving of the many-sorted free description theory FDn. We use the RUE and NRF by Diglicoli [3] that can deal with equality to construct the theorem prover for FDn* which is based on the *-translation.

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) [5] Hayes,P. J. : The Logic of Frames, in Frame Conceptions and Text Understanding. C. : D-script : A Computational Theory of Descriptions. IEEE Trans. 4 (1976) [7] Nakamatsu,K. and Suzuki,A. : Automatic Theorem Proving for Modal Predicate Logic. Trans. IECE Japan, E67 (1984) 203 210 [8] Nakamatsu,K.

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