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Small Degrees in Ordinary Recursion Theory
SMALL DEGREES IN ORDINARY RECURSION THEORY A.N. DEGTEV Tymen, U.S.S.R.
The 1- and the reducibilities between m- and tt-reducibility are considered in this report. If an r- is such a reducibility, let L, be the upper semilattice of recursively enumerable (r. e.) r-degrees and Th(L,) the elementary theory of L,. We call an m-degree undissolvable if it contains only one 1-degree (and, consequently, consists only of cylinders). YOUNG(1966) noticed that every m-degree is undissolvable or contains an infinite chain of 1-degrees. This result easily follows from the fact that if A is a non-cylinder, then A $ A is also a non-cylinder and A < A $A. If A is not a cylinder but an r. e. nonrecursive set, let L ( A ) be a partial order of the 1-degrees contained in the m-degree of A. It is shown in DEGTEV (1976a) that the structures of L(A) are of great variety. In particular, L ( A ) has an infinite antichain (i.e. infinitely many pair-incomparable elements) and two incomparable elements, whose least upper bound is the greatest element of L(A). If A is a simple set, then L ( A ) is not an upper or lower semilattice and (as Dekker (1976a) it is also proved: remarked) has no minimal elements. In DEGTEV (a) there is an r.e. set A such that L(A) is a dense lattice with least element; (b) for each n there is an r. e. set A such that L(A) has the least element 0 and exactly n elements a1,a,, ...,a,, such that (i) i # j * ai$a,&a,$ai; (ii) (Vi)(VaEL(A))(a
238
A. N. DEGTEV
We call an q-dosed non-recursive set A ideal if the equivalence q is such that there are only two q-closed recursive sets: 0 and N, and we call A q-maximal if A is r.e. and for all q-closed r.e. sets B, BA \ or NU?contains only finitely many equivalence classes with respect to 1. It is obvious that an ?-maximal set has a minimal m-degree. ERSHOV (1971) showed that the m-degrees of any ideal set are undissolvable and if A has a recursively inseparable r.e. set B E N U (or A is simple non-hypersimple), then "A = {n: D, n A # 01 is an ideal set (for a suitable q), where D, is a finite set with canonical index n. In particular (JOCKUSCH, 1969), every r.e. nonrecursive T-degree has an r. e. undissolvable m-degree, because every such T-degree has some simple non-hypersimple set (YATES,1965). Note that there are r.e. sets which are not ideal whose m-degrees are undissolvable (DEGTEV,1973; DENISOV,1974). On the other hand, LACHLAN(1972) proved that every r.e. non-recursive T-degree has an q-maximal set. It may be proved (DEGTEV, 1976b) that every such T-degree has an q-maximal set whose m-degree is undissolvable and, consequently, has a minimal r.e. 1-degree. An example of a r.e. set which is not q-maximal (for all q) but has a minimal m-degree is also constructed in DEGTEV(1976b). Let r- be a reducibility between m- and tt-reducibility. The main ones of these reducibilities are: m-, bc-, bd-, btt-, c-, d- and tt-. Recall that A<,B
--
* (Wr.r.f.)(Vx)(x~A
A G d B e,(3qt.r.f.)(Vx)(xE A
Dv(X)~B),
Dvp(x) n B # 0).
The bc- and bd- (bounded c- and d-) reducibilities are defined in the natural way. It is known (DEGTEV, 1973) that L, is not a lattice, has minimal elements and an a # 0 such that for any b
(va)(vb)(a # 1 & a $ b
-
* (~c)(a~c&c&a&b
For r- = m- LACHLAN(1966) proved that A @ B r-complete
A r-completev B r-complete.
This result holds for r- E {bc-, c->, but not for r- E {bd-, d-, btt-, tt-}. Hence Th(L,) # Th(LR)where r- E {m-, bc-, c-} and R E {bd-, d-, btt-, tt-}. MARCHENKOV (1976, 1977) proved that Th(L,) # Th(L,) where r- E {bd-, btt-} and R- E {d-, tt-}. The author has received the new theorems:
SMALL DEGREES IN RECURSlON THEORY
239
Th(L,) # Th(L,) where r- E (bc-, c-}, and Th(L,) # Th(L,) where r- E (btt-, tt-} and R-E (bd-, d-}. The following question arises: Th(Lbc) = Th(L,)?
It is known that every r.e. non-recursive T-degree contains an infinite antichain of r.e. tt-degrees (DEGTEV,1972b), every r. e. non-recursive tt-degree contains an undissolvable m-degree and every r. e. non-recursive btt-degree has no a maximal m-degree of all the m-degrees which it contains (KOBZEV,1975). JOCKUSCH(1969) showed that every non-recursive tt-degree has an infinite chain of m-degrees. The author proved (DEGTEV, 1978) the following: (i) every non-recursive tt -degree contains at least two btt -degrees; (ii) i f a T-degree a is such that a’>O”, then it has no minimal r.e. tt-degrees. The author’s latest result in this trend is: there exists a non-recursive tt-degree which has no undissolvable m-degree. Is there a btt -degree, which contains only one m-degree? The proofs of these results may also be found in DEGTEV (1979) and in the author’s paper Some results on upper semilattices and m-degrees (forthcoming). References Dmmv G4jlITEB A. H.), 1972a, 0 6 m-cmenenm npocmux .unoxecma, Anre6pa M noruria,
11, Ne 2, np. 130-139. ( ~ E I T E B , A. H.), 1972by Hac/rednnsewue rWnoxecma u m a 6 ~ z u w accrodu~ mcmb, Anre6pa II norma, T. 11, M 3, crp. 257-268. DEGTEV (H&TEBA. H.), 1973, 0 tt u m-cmenewx, h e 6 p a II norma, T. 12, Ne 2, Crp. 143-161. Dmmv (.J~&TEB A. H.), 1976a, 0 uacrnuuno ynopm3ouennu.x .unoxermdOT l-cmened, codepwazquxcx 8 p . n. m-cmenemx, Anre6pa EI norma, T. 15, N! 3, crp. 249-266. DmTEv (8iirPE~A. H.), 1976by 0 Mum.uaib?zblx 1-cmenensx u ma6nuuxoti c8oduMocmuY c6. MPT. XyPHZl‘l, T. XVIIy N! 5, Crp. 1014-1022 Dmmv (,D&TB A. H.), 1978, Tpu meopem o tt-cmenenm, AnreGpa M norma, T. 17, N! 3, crp. 270-281 Dmmv (,I&~TEB A. H.), 1979, 0 ceo6odu~ocrn~x ma6/ruunoro m u m 8 meopuu aaopumMOB, YClIeXU MaT. H a p y T. 34, N! 3, CTp. 137-168 D m o v ( ~ B H U C O B C . a.), 1974, Tpu meopem 06 3ne~enmapnuxm e o p w u tt-ceodurwocmu, h e 6 p a II norma, T. 13, N! 1, crp. 5-8 ERSHOV (EPIIIOBIO. n.), 1969, runeprunepnpocmue m-menenu, h e 6 p a M norma, T. 5, Crp. 523-552. ERSHOV(EPJDOBIo. JI.), 1971, lIo3urnuenbre ~ K B U B ~ ~ ~ H ~AnreBpa O C ~ U EI, n o r m T. 10, N ! 6, crp. 620-650. T.
Dmmv
240
A. N. DEGTEV
JOCK~SCH, C., 1969, Relationships between reducibilities, Transactions of the American Mathemetical Society, vol. 142, 1, pp. 229-237 KOBZBV (KOKJEB,I-.H.), 1975, btt-C6OdUAiOC??lb,~ C C e p THoBoCE~~~~CK, ~ , crp. 3-69. A. H., 1966, A note on universal sets, The Journal of Symbolic Logic, vol. LACHLAN, 31, 4, pp. 573-574 A. H., 1972, Two theorems on many-one degrees of r. e. sets, Algebra and Logic, LACHLAN, vol. 11, 2, pp. 216-229 S. S. (MAFWWKOB, C. C.), 1976, K cpaimenuw sepxnux nonypeulemorc p. n. MARCHENKOV, d ~ u u i i a r xcmeneneil I( m-meneiiefi, MaT. 3 m m , T. 20, N2 1, cTp. 19-25 MARCHENKOV, s. S . ( f i P - O B , c. c.), 1977, 0 p. n. fiU%UfizZfibHbZX btt-cmeneiim, MaT. C6OpAEm, T. 103, JV? 4, np. 550-562. YAW, C. E. M.,1965, Three theorems on the degrees of r. e. sets, Duke Mathematical Journal, vol. 32.3, pp. 461-468 YOUNG, P. P., 1966, Linear ordering under one-one reducibility, The Journal of Symbolic Logic, vol. 31.1, pp. 70-85
The 1- and the reducibilities between m- and tt-reducibility are considered in this report. If an r- is such a reducibility, let L, be the upper semilattice of recursively enumerable (r. e.) r-degrees and Th(L,) the elementary theory of L,. We call an m-degree undissolvable if it contains only one 1-degree (and, consequently, consists only of cylinders). YOUNG(1966) noticed that every m-degree is undissolvable or contains an infinite chain of 1-degrees. This result easily follows from the fact that if A is a non-cylinder, then A $ A is also a non-cylinder and A < A $A. If A is not a cylinder but an r. e. nonrecursive set, let L ( A ) be a partial order of the 1-degrees contained in the m-degree of A. It is shown in DEGTEV (1976a) that the structures of L(A) are of great variety. In particular, L ( A ) has an infinite antichain (i.e. infinitely many pair-incomparable elements) and two incomparable elements, whose least upper bound is the greatest element of L(A). If A is a simple set, then L ( A ) is not an upper or lower semilattice and (as Dekker (1976a) it is also proved: remarked) has no minimal elements. In DEGTEV (a) there is an r.e. set A such that L(A) is a dense lattice with least element; (b) for each n there is an r. e. set A such that L(A) has the least element 0 and exactly n elements a1,a,, ...,a,, such that (i) i # j * ai$a,&a,$ai; (ii) (Vi)(VaEL(A))(a
238
A. N. DEGTEV
We call an q-dosed non-recursive set A ideal if the equivalence q is such that there are only two q-closed recursive sets: 0 and N, and we call A q-maximal if A is r.e. and for all q-closed r.e. sets B, BA \ or NU?contains only finitely many equivalence classes with respect to 1. It is obvious that an ?-maximal set has a minimal m-degree. ERSHOV (1971) showed that the m-degrees of any ideal set are undissolvable and if A has a recursively inseparable r.e. set B E N U (or A is simple non-hypersimple), then "A = {n: D, n A # 01 is an ideal set (for a suitable q), where D, is a finite set with canonical index n. In particular (JOCKUSCH, 1969), every r.e. nonrecursive T-degree has an r. e. undissolvable m-degree, because every such T-degree has some simple non-hypersimple set (YATES,1965). Note that there are r.e. sets which are not ideal whose m-degrees are undissolvable (DEGTEV,1973; DENISOV,1974). On the other hand, LACHLAN(1972) proved that every r.e. non-recursive T-degree has an q-maximal set. It may be proved (DEGTEV, 1976b) that every such T-degree has an q-maximal set whose m-degree is undissolvable and, consequently, has a minimal r.e. 1-degree. An example of a r.e. set which is not q-maximal (for all q) but has a minimal m-degree is also constructed in DEGTEV(1976b). Let r- be a reducibility between m- and tt-reducibility. The main ones of these reducibilities are: m-, bc-, bd-, btt-, c-, d- and tt-. Recall that A<,B
--
* (Wr.r.f.)(Vx)(x~A
A G d B e,(3qt.r.f.)(Vx)(xE A
Dv(X)~B),
Dvp(x) n B # 0).
The bc- and bd- (bounded c- and d-) reducibilities are defined in the natural way. It is known (DEGTEV, 1973) that L, is not a lattice, has minimal elements and an a # 0 such that for any b
(va)(vb)(a # 1 & a $ b
-
* (~c)(a~c&c&a&b
For r- = m- LACHLAN(1966) proved that A @ B r-complete
A r-completev B r-complete.
This result holds for r- E {bc-, c->, but not for r- E {bd-, d-, btt-, tt-}. Hence Th(L,) # Th(LR)where r- E {m-, bc-, c-} and R E {bd-, d-, btt-, tt-}. MARCHENKOV (1976, 1977) proved that Th(L,) # Th(L,) where r- E {bd-, btt-} and R- E {d-, tt-}. The author has received the new theorems:
SMALL DEGREES IN RECURSlON THEORY
239
Th(L,) # Th(L,) where r- E (bc-, c-}, and Th(L,) # Th(L,) where r- E (btt-, tt-} and R-E (bd-, d-}. The following question arises: Th(Lbc) = Th(L,)?
It is known that every r.e. non-recursive T-degree contains an infinite antichain of r.e. tt-degrees (DEGTEV,1972b), every r. e. non-recursive tt-degree contains an undissolvable m-degree and every r. e. non-recursive btt-degree has no a maximal m-degree of all the m-degrees which it contains (KOBZEV,1975). JOCKUSCH(1969) showed that every non-recursive tt-degree has an infinite chain of m-degrees. The author proved (DEGTEV, 1978) the following: (i) every non-recursive tt -degree contains at least two btt -degrees; (ii) i f a T-degree a is such that a’>O”, then it has no minimal r.e. tt-degrees. The author’s latest result in this trend is: there exists a non-recursive tt-degree which has no undissolvable m-degree. Is there a btt -degree, which contains only one m-degree? The proofs of these results may also be found in DEGTEV (1979) and in the author’s paper Some results on upper semilattices and m-degrees (forthcoming). References Dmmv G4jlITEB A. H.), 1972a, 0 6 m-cmenenm npocmux .unoxecma, Anre6pa M noruria,
11, Ne 2, np. 130-139. ( ~ E I T E B , A. H.), 1972by Hac/rednnsewue rWnoxecma u m a 6 ~ z u w accrodu~ mcmb, Anre6pa II norma, T. 11, M 3, crp. 257-268. DEGTEV (H&TEBA. H.), 1973, 0 tt u m-cmenewx, h e 6 p a II norma, T. 12, Ne 2, Crp. 143-161. Dmmv (.J~&TEB A. H.), 1976a, 0 uacrnuuno ynopm3ouennu.x .unoxermdOT l-cmened, codepwazquxcx 8 p . n. m-cmenemx, Anre6pa EI norma, T. 15, N! 3, crp. 249-266. DmTEv (8iirPE~A. H.), 1976by 0 Mum.uaib?zblx 1-cmenensx u ma6nuuxoti c8oduMocmuY c6. MPT. XyPHZl‘l, T. XVIIy N! 5, Crp. 1014-1022 Dmmv (,D&TB A. H.), 1978, Tpu meopem o tt-cmenenm, AnreGpa M norma, T. 17, N! 3, crp. 270-281 Dmmv (,I&~TEB A. H.), 1979, 0 ceo6odu~ocrn~x ma6/ruunoro m u m 8 meopuu aaopumMOB, YClIeXU MaT. H a p y T. 34, N! 3, CTp. 137-168 D m o v ( ~ B H U C O B C . a.), 1974, Tpu meopem 06 3ne~enmapnuxm e o p w u tt-ceodurwocmu, h e 6 p a II norma, T. 13, N! 1, crp. 5-8 ERSHOV (EPIIIOBIO. n.), 1969, runeprunepnpocmue m-menenu, h e 6 p a M norma, T. 5, Crp. 523-552. ERSHOV(EPJDOBIo. JI.), 1971, lIo3urnuenbre ~ K B U B ~ ~ ~ H ~AnreBpa O C ~ U EI, n o r m T. 10, N ! 6, crp. 620-650. T.
Dmmv
240
A. N. DEGTEV
JOCK~SCH, C., 1969, Relationships between reducibilities, Transactions of the American Mathemetical Society, vol. 142, 1, pp. 229-237 KOBZBV (KOKJEB,I-.H.), 1975, btt-C6OdUAiOC??lb,~ C C e p THoBoCE~~~~CK, ~ , crp. 3-69. A. H., 1966, A note on universal sets, The Journal of Symbolic Logic, vol. LACHLAN, 31, 4, pp. 573-574 A. H., 1972, Two theorems on many-one degrees of r. e. sets, Algebra and Logic, LACHLAN, vol. 11, 2, pp. 216-229 S. S. (MAFWWKOB, C. C.), 1976, K cpaimenuw sepxnux nonypeulemorc p. n. MARCHENKOV, d ~ u u i i a r xcmeneneil I( m-meneiiefi, MaT. 3 m m , T. 20, N2 1, cTp. 19-25 MARCHENKOV, s. S . ( f i P - O B , c. c.), 1977, 0 p. n. fiU%UfizZfibHbZX btt-cmeneiim, MaT. C6OpAEm, T. 103, JV? 4, np. 550-562. YAW, C. E. M.,1965, Three theorems on the degrees of r. e. sets, Duke Mathematical Journal, vol. 32.3, pp. 461-468 YOUNG, P. P., 1966, Linear ordering under one-one reducibility, The Journal of Symbolic Logic, vol. 31.1, pp. 70-85