Download Degrees of Unsolvability by Joseph R. Shoenfield (Eds.) PDF

By Joseph R. Shoenfield (Eds.)

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Then x E A f S 3y( E B ) where Now suppose t h a t A i s RE in t h e Let I be a n i n d e x of A i n B . By ( 3 ) o f 14, X E A - X E ~ t-,30((0( C B& x To show t h a t A is &+I, o< C B and x E WT a r e E Wy). i t s u f f i c e s by t h e r u l e s t o show t h a t The l a t t e r i s RE, hence El, EVALUATING DEGREGS hence 35 To t r e a t t h e f o r m e r , n o t e t h a t o ( c B < (b'n l h ( o < ) ) ( d ( n )= B ( n ) ) ; s o by t h e r u l e s , i t s u f f i c e s t o v e r i f y t h a t k = B ( n ) i s But Tr & n k = B ( n ) c, ( k = s o k = B(n) i s zn+l by lTn B) v ( k = Fa &.

And t a k e A to b e t h e unique s e t such However, t h e t r e e s a r e t r e a t e d d i f f e r - A t s t e p s , we d e f i n e t r e e s T ently. For i n o t be t o t a l ) . < S i for i 5 ks (which need S ks, Tf+l w i l l b e a s u b t r e e of T:; S S 6, w i l l b e on Tk is and hence on a l l t h e T i . and W e w i l l always S have Tg = I d . A t t h e end of t h e c o n s t r u c t i o n , we show t h a t converges i n a s u i t a b l e s e n s e t o a t r e e Ti. T! Now we d e s c r i b e s t e p s .

Any such I i s t h e n c a l l e d an i n d e x of A . Parameter Theorem. If A i s a n RE r e l a t i o n i n X X Y, t h e n 22 FZCURSIVE ENUMERABILITY t h e r e i s a r e c u r s i v e f u n c t i o n F from Y t o Alg(X,N) such t h a t X € WF(y)@ A f o r a l l x and y. P r o o f . L e t I be a n i n d e x of A , and l e t F(y) be t h e a l g o r i t h m J such t h a t t h e p r o c e s s of computing a c c o r d i n g t o J w i t h input x i s t h e same as t h e p r o c e s s 3f computing a c c o r d - i n g t o I w i t h i n p u t .

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