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By Joseph R. Shoenfield (Eds.)

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W e can If R, i s ( l ~ , we ~ )can , show as i n t h e proof i n t h a t a n o r a c l e f o r a r e l a t i o n of d e g r e e 2 0' s u f f i c e s . $8 If Rs i s ( 2 n ) , we can f i n d m; b u t we need a n o r a c l e t o t e l l u s t h e common v a l u e of F ( m ) and G(n!. Thus a n o r a c l e for e i t h e r F or G s u f f i c e s . We have shown t h a t t h e d e g r e e of t h e c o n s t r u c t i o n i s and I, a u 0 ' struction, b' = a.

Now l e t o ( cb above, t h e r e i s a Since 8 and y on T such t h a t with I 8 c6 O( C A and 6 C 6 on T and [ I ] b ( m on T such t h a t C% C y = 6 . and 6 C k. By t h e and [ I ] I( m ' = \ I ] * ( m ) . A do n o t I - s p l i t d , [ I ] ( m ) = k . D. and T ( d (I), , a r e defined, they I - s p l i t T ( a ) . Lemma 2 . I f T i s I - s p l i t t i n g and A i s on T , t h e n A is I -minimal. Proof. We assume t h a t F A = [ I I A i s t o t a l and prove t h a t ip, F. Let B be t h e s e t o f d such t h a t T(o<'l i s d e f i n e d and MININAL DEGREES 51 T(d) CA.

S e l e c t o r f o r B, t h e n A i s t h e domain of F . Corollary 2. D. B i s RE and A i s d e f i n e d by x E A-~Y( E B), t h e n A i s FiE. P r o o f . S i n c e A i s t h e domain of any s e l e c t o r f o r B, t h i s f o l l o w s from t h e theorem and C o r o l l a r y 1. As F i s RE. D. an example, t h e range A of a r e c u r s i v e p a r t i a l f u n c t i o n F o r if I i s a n a l g o r i t h m f o r F, t h e n x E BCJjy([I](y) = x) and [ I ] ( y ) = x i s a n RF, r e l a t i o n of x,y by ( 2 ) .

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