Bulletin of the American Mathematical Society

Recursively enumerable sets and degrees

Robert I. Soare

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Source
Bull. Amer. Math. Soc., Volume 84, Number 6 (1978), 1149-1181.

Dates
First available in Project Euclid: 4 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.bams/1183541465

Mathematical Reviews number (MathSciNet)
MR508451

Zentralblatt MATH identifier
0401.03018

Subjects
Primary: 02F25 02F30
Secondary: 02F47 02G05 02G10 06A20 02F50

Citation

Soare, Robert I. Recursively enumerable sets and degrees. Bull. Amer. Math. Soc. 84 (1978), no. 6, 1149--1181. https://projecteuclid.org/euclid.bams/1183541465


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References

  • [Al] D. A. Alton, Uniformities in recursively enumerable sets, Doctoral Dissertation, Cornell Univ., Ithaca, N. Y., 1970.
  • [A2] D. A. Alton, Recursively enumerable sets which are uniform for finite extensions, J. Symbolic Logic 36 (1971), 271-287.
  • [BeSo] V. L. Bennison and R. I. Soare, Some lowness properties and computational complexity sequences, Theor. Comput Sci. (to appear).
  • [Bi1] G. Birkhoff, On the combination of subalgebras, Proc. Cambridge Philos. Soc. 29 (1933), 441-464.
  • [Bi2] G. Birkhoff, Lattice theory, 3rd ed., Amer. Math. Soc. Colloq. Publ., vol. 25, Amer. Math. Soc., Providence, R I., 1967.
  • [BIMq] M. Blum and I. Marques, On complexity properties of recursively enumerable sets, J. Symbolic Logic 38 (1973), 579-593.
  • [Bol] W. W. Boone, Certain simple unsolvable problems of group theory, Indag. Math. 16 (1954), 231-237, 492-497; ibid. 17 (1955), 252-256, 571-577; ibid. 19 (1957), 22-27, 227-232.
  • [Bo2] W. W. Boone, The word problem, Ann. of Math. (2) 70 (1959), 207-265.
  • [Bo3] W. W. Boone, Word problems and recursively enumerable degrees of unsolvability, A sequel on finitely presented groups, Ann. of Math. (2) 84 (1966), 49-84.
  • [CoJ] P. Cohen and C. G. Jockusch, Jr., A lattice property of Post's simple set, Illinois J. Math. 19 (1975), 450-453.
  • [Ch] A. Church, An unsolvable problem of elementary number theory, Amer. J. Math. 58 (1936), 345-363.
  • [Cp1] S. B. Cooper, Sets recursively enumerable in high degrees, Notices Amer. Math. Soc. 19 (1972), A-20.
  • [Cp2] S. B. Cooper, Minimal upper bounds for sequences of recursively enumerable degrees, J. London Math. Soc. 5 (1972), 445-450.
  • [Cp3] S. B. Cooper, Degrees of unsolvability complementary between recursively enumerable degrees I, Ann. Math. Logic 4 (1972), 31-73.
  • [Cp4] S. B. Cooper, Jump equivalence of the Δ20 hyperhyperimmune sets, J. Symbolic Logic 37 (1972) 598-600.
  • [Cp5] S. B. Cooper, An annotated bibliography for the structure of the degrees below 0' with special reference to that of the recursively enumerable degrees, Recursive Function Theory Newsletter 5 (1974), 1-15.
  • [Cp6] S. B. Cooper, Minimal pairs and high recursively enumerable degrees, J. Symbolic Logic 39 (1974), 655-660.
  • [Cp7] S. B. Cooper, On a theorem of C. E. M. Yates (preprint), 1974.
  • [Da1] M. Davis, Computability and unsolvability, McGraw-Hill, New York, 1958.
  • [Da2] M. Davis, (Ed.), The undecidable. Basic papers on undecidable propositions, unsolvable problems and computable functions, Raven Press, Hewlitt, N. Y., 1965.
  • [Da3] M. Davis, Hilbert's tenth problem is unsolvable, Amer. Math. Monthly 80 (1973), 233-269.
  • [De] A. N. Degtev, Hypersimple sets with retraceable complements, Algebra i Logika 10 (1971), 235-246.
  • [Dk1] J. Dekker, Two notes on recursively enumerable sets, Proc. Amer. Math. Soc. 4 (1953), 495-501.
  • [Dk2] J. Dekker, A theorem on hypersimple sets, Proc. Amer. Math. Soc. 5 (1954), 791-796.
  • [Dk3] J. Dekker, Productive sets, Trans. Amer. Math. Soc. 78 (1955), 129-149.
  • [DkMy1] J. C. E. Dekker and J. Myhill, Some theorems on classes of recursively enumerable sets, Trans. Amer. Math. Soc. 89 (1958), 25-29.
  • [DkMy2] J. C. E. Dekker and J. Myhill, Retraceable sets, Canad. J. Math. 10 (1958), 357-373.
  • [E1] Y. L. Ershov, Decidability of the elementary theory of relatively complemented distributive lattices and of the theory of filters, Algebra i Logika 3 (1964), 17-38. (Russian)
  • [E2] Y. L. Ershov, A hierarchy of sets, Parts I, II, III, Algebra and Logic 7 (1968), 25-43, 212-232; ibid. 9 (1970), 20-31.
  • [Fe] S. Feferman, Degrees of unsolvability associated with classes of formalized theories, J. Symbolic Logic 22 (1957) 161-175.
  • [Fn] L. Feiner, Orderings and Boolean algebras not isomorphic to recursive ones, Doctoral Dissertation, M.I.T., Cambridge, Mass., 1967.
  • [Fr1] R. M. Friedberg, The fine structure of degrees of unsolvability of recursively enumerable sets, Seminars of Cornell Institute for Symbolic Logic, 1957, pp. 404-406.
  • [Fr2] R. M. Friedberg, Two recursively enumerable sets of incomparable degrees of unsolvability, Proc. Nat Acad. Sci. USA. 43 (1957), 236-238. MR 18 #867.
  • [Fr3] R. M. Friedberg, A criterion for completeness of degrees of unsolvability, J. Symbolic Logic 22 (1957), 159-160.
  • [Fr4] R. M. Friedberg, Three theorems on recursive enumeration: I. Decomposition, II. Maximal set, III. Enumeration without duplication, J. Symbolic Logic 23 (1958), 309-316. MR 22 #13.
  • [FrRg] R. M. Friedberg and H. Rogers, Jr., Reducibility and completeness for sets of integers, Z. Math. Logik Grundlagen Math. 5 (1959), 117-125.
  • [GiMr] J. Gill and P. Morris, On subcreative sets and S-reducibility, J. Symbolic Logic 39 (1974), 669-677.
  • [Gö1] K. Gödel, Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme. I, Monatsh. Math. Phys. 38 (1931), 173-198.
  • [Gö2] K. Gödel, On undecidable propositions of formal mathematical systems, Notes by S. C. Kleene and Barkley Rosser on lectures at the Institute for Advanced Study, Princeton, N. J., 1934. Reprinted in Davis, [Da2].
  • [Ha] W. Hanf, Model theoretic methods in the study of elementary logic, Sympos. Theory of Models (Berkeley), North-Holland, Amsterdam, 1965, pp. 132-145.
  • [Hy1] L. Hay, The class of recursively enumerable subsets of a recursively enumerable set, Pacific J. Math. 46 (1973), 167-183.
  • [Hy2] L. Hay, The halting problem relativized to complements, Proc. Amer. Math. Soc. 41 (1973), 583-587.
  • [J1] C. G. Jockusch, Jr., The degrees of hyperhyperimmune sets, J. Symbolic Logic 34 (1969), 489-493.
  • [J2] C. G. Jockusch, Jr., Relationships between reducibilities, Trans. Amer. Math. Soc. 142 (1969), 229-237.
  • [J3] C. G. Jockusch, Jr., Degrees in which the recursive sets are uniformly recursive, Canad. J. Math. 24 (1972), 1092-1099.
  • [J4] C. G. Jockusch, Jr., Review of Lerman [Le3], MR 45 #3200.
  • [J5] C. G. Jockusch, Jr., II10 classes and Boolean combinations of recursively enumerable sets, J. Symbolic Logic 39 (1974), 95-96.
  • [JSo1] C. G. Jockusch, Jr. and R. I. Soare, A minimal pair of II10 classes, J. Symbolic Logic 36 (1971), 66-78.
  • [JSo2] C. G. Jockusch, Jr. and R. I. Soare, Degrees of members ofII10 classes, Pacific J. Math. 40 (1972), 605-616.
  • [JSo3] C. G. Jockusch, Jr., II10 classes and degrees of theories, Trans. Amer. Math. Soc. 173 (1972), 33-56.
  • [JSo4] C. G. Jockusch, Jr., Post's problem and his hypersimple set, J. Symbolic Logic 38 (1973), 446-452.
  • [Ka] S. Kallibekov, Index sets of degrees of unsolvability, Algebra i Logika 10 (1971), 316-326. (Russian)
  • [Ke] C. F. Kent, Constructive analogues of the group of permutations of the natural numbers, Trans. Amer. Math. Soc. 104 (1962), 347-362.
  • [Kl1] S. C. Kleene, A theory of positive integers in formal logic, Amer. J. Math. 57 (1935), 153-173, 219-244.
  • [K12] S. C. Kleene, General recursive functions of natural numbers, Math. Ann. 112 (1936), 727-742.
  • [K13] S. C. Kleene, Introduction to metamathematics, Van Nostrand, New York, 1952. MR 14 #525.
  • [KlPo] S. C. Kleene and E. L. Post, The upper semi-lattice of degrees of recursive unsolvability, Ann. of Math. (2) 59 (1954), 379-407. MR 15 #772.
  • [Ku] A. V. Kuznecov, On primitive recursive functions of large oscillation, Dokl. Akad. Nauk SSSR, 71 (1950), 233-236. (Russian)
  • [La1] A. H. Lachlan, Some notions of reducibility and productiveness, Z. Math. Logik Grundlagen Math. 11 (1965), 17-44.
  • [La2] A. H. Lachlan, On a problem of G. E. Sacks, Proc. Amer. Math. Soc. 16 (1965), 972-979.
  • [La3] A. H. Lachlan, A note on universal sets, J. Symbolic Logic 31 (1966), 573-574.
  • [La4] A. H. Lachlan, The impossibility of finding relative complements for recursively enumerable degrees, J. Symbolic Logic 31 (1966), 434-454. MR 34 #5673.
  • [La5] A. H. Lachlan, Lower bounds for pairs of r.e. degrees, Proc. London Math. Soc. 16 (1966), 537-569. MR 34 #4126.
  • [La6] A. H. Lachlan, The priority method. I, Z. Math. Logik Grundlagen Math. 13 (1967), 1-10.
  • [La7] A. H. Lachlan, Complete recursively enumerable sets, Proc. Amer. Math. Soc. 19 (1968), 99-102.
  • [La8] A. H. Lachlan, On the lattice of recursively enumerable sets, Trans. Amer. Math. Soc. 130 (1968), 1-37. MR 37 #2594.
  • [La9] A. H. Lachlan, The elementary theory of recursively enumerable sets, Duke Math. J. 35 (1968), 123-146. MR 37 #2593.
  • [La10] A. H. Lachlan, Distributive initial segments of the degrees of unsolvability, Z. Math. Logik Grundlagen Math. 14 (1968), 457-472. MR 38 #5620.
  • [La11] A. H. Lachlan, Degrees of recursively enumerable sets which have no maximal superset, J. Symbolic Logic 33 (1968), 431-443. MR 38 #4314.
  • [La12] A. H. Lachlan, On some games which are relevant to the theory of recursively enumerable sets, Ann. of Math (2) 91 (1970), 291-310. MR 44 # 1652.
  • [La13] A. H. Lachlan, Embedding nondistributive lattices in the recursively enumerable degrees, Conf. Mathematical Logic, London 1970, Lecture Notes in Math., no. 255, Springer-Verlag, Berlin and New York, 1972, pp. 149-177.
  • [La14] A. H. Lachlan, The priority method for the construction of recursively enumerable sets, Proc. Cambridge Summer School in Logic, 1971, Lecture Notes in Math., no. 337, Springer-Verlag, Berlin and New York, 1973.
  • [La15] A. H. Lachlan, Recursively enumerable degrees, Handwritten notes from lectures in Warsaw, April, 1973.
  • [La16] A. H. Lachlan, A recursively enumerable degree which will not split over all lesser ones, Ann. Math. Logic 9 (1975), 307-365.
  • [La17] A. H. Lachlan, Uniform enumeration operations, J. Symbolic Logic 40 (1975), 401-409.
  • [La18] A. H. Lachlan, wtt-complete sets are not necessarily tt-complete, Proc. Amer. Math. Soc. 48 (1975), 429-434.
  • [La19] A. H. Lachlan, Bounding minimal pairs (to appear).
  • [Ld1] R. E. Ladner, A completely mitotic nonrecursive r.e. degree, Trans. Amer. Math. Soc. 184 (1973), 479-507.
  • [Ld2] R. E. Ladner, Mitotic recursively enumerable sets, J. Symbolic Logic 38 (1973), 199-211.
  • [LdSs] R. E. Ladner and L. P. Sasso, The weak truth table degrees of recursively enumerable sets, Ann. Math. Logic 4 (1975), 429-448.
  • [Le1] M. Lerman, Recursive functions modulo co-r-maximal sets, Trans. Amer. Math. Soc. 148 (1970), 429-444.
  • [Le2] M. Lerman, Turing degrees and many-one degrees of maximal sets, J. Symbolic Logic 35 (1970), 29-40.
  • [Le3] M. Lerman, Some theorems on r-maximal sets and major subsets of recursively enumerable sets, J. Symbolic Logic 36 (1971), 193-215. MR 45 #3200.
  • [Le4] M. Lerman, Congruence relations, filters, ideals, and definability in lattices of α-recursively enumerable sets, J. Symbolic Logic 41 (1976), 405-418.
  • [Le5] M. Lerman, Admissible ordinals and priority arguments, Proc. Cambridge Summer School in Logic, 1971, Lecture Notes in Math., no. 337, Springer-Verlag, Berlin and New York, 1973. MR 52 #59.
  • [Le6] M. Lerman, Lattices of α-recursivefy enumerable sets, Oslo Conf. Generalized Recursion Theory, June, 1976 (to appear).
  • [Le7] M. Lerman, On elementary theories of some lattices of α-recursively enumerable sets (to appear).
  • [Le8] M. Lerman, Automorphism bases for the semilattice of recursively enumerable degrees, Notices Amer. Math. Soc. 24 (1977), A-251. Abstract #77T-E10
  • [LeSo1] M. Lerman and R. I. Soare, A decidable fragment of the elementary theory of the lattice of recursively enumerable sets, (to appear).
  • [LeSo2] M. Lerman and R. I. Soare, d-simple sets, small sets, and degree classes (to appear).
  • [LeShSo] M. Lerman, R. A. Shore and R. I. Soare, r-maximal major subsets, Israel. J. Math, (to appear).
  • [Ma1] D. A. Martin, A theorem on hyperhypersimple sets, J. Symbolic Logic 28 (1963), 273-278.
  • [Ma2] D. A. Martin, Completeness, the recursion theorem, and effectively simple sets, Proc. Amer. Math. Soc. 17 (1966), 838-842.
  • [Ma3] D. A. Martin, Classes of recursively enumerable sets and degrees of unsolvability, Z. Math. Logik Grundlagen Math. 12 (1966), 295-310. MR 37 #68.
  • [Ma4] D. A. Martin, On a question of G. E. Sacks, J. Symbolic Logic 31 (1966), 66-69.
  • [Ma5] D. A. Martin, The priority method of Sacks, mimeographed notes, 1966.
  • [MaMi] D. A. Martin and W. Miller, The degrees of hyperimmune sets, Z. Math. Logik Grundlagen Math. 14 (1968), 159-166.
  • [MaPr] D. A. Martin and M. B. Pour-El, Axiomatizable theories with few axiomatizable extensions, J. Symbolic Logic 35 (1970), 205-209.
  • [Mtl] Y. Matiyasevič, Diophantine representation of enumerable predicates, Izv. Akad Nauk. SSSR Ser. Math. 35 (1971), 3-30. (Russian)
  • [Mt2] Y. Matiyasevič, Enumerable sets are diophantine, Dokl. Akad. Nauk SSSR 191 (1970), 279-282. (Russian)
  • [Mc] T. G. McLaughlin, On a class of complete simple sets, Canad. Math. Bull. 8 (1965), 33-37.
  • [Md] Y. T. Medvedev, On nonisomorphic recursively enumerable sets, Dokl. Akad. Nauk SSSR 102 (1955), 211-214. (Russian)
  • [MkNe] G. Metakides and A. Nerode, Recursively enumerable vector spaces, Ann. Math. Logic, 1978 (to appear).
  • [Mr] C. F. Miller, On group-theoretic decision problems and their classification, Ann. of Math. Studies, no. 68, Princeton Univ. Press, Princeton, N. J., 1971.
  • [MoSo] M. D. Morley and R. I. Soare, Boolean algebras, splitting theorems and Δ20 sets, Fund. Math. 90 (1975), 45-52.
  • [Mu] A. A. Muchnik, On the unsolvability of the problem of reducibility in the theory of algorithms, Dokl. Akad. Nauk SSSR 108 (1956), 194-197. (Russian).
  • [My1] J. Myhill, Creative sets, Z. Math. Logik Grundlagen Math. 1 (1955), 97-108.
  • [My2] J. Myhill, The lattice of recursively enumerable sets, J. Symbolic Logic 21 (1956), 220 (abstract).
  • [No] P. S. Novikov, On the algorithmic unsolvability of the word problem in groups, Trudy Mat. Inst. Steklov no. 44 Izdat Akad. Nauk SSSR, Moscow, (1955). (Russian)
  • [O1] J. C. Owings, Jr., Recursion, metarecursion and inclusion, J. Symbolic Logic 32 (1967), 173-178.
  • [O2] J. C. Owings, Jr., Review of Lachlan [La8], [La9] and Robinson [Ro3], [Ro4], J. Symbolic Logic 35 (1970), 153-155.
  • [Po] E. L. Post, Recursively enumerable sets of positive integers and their decision problems, Bull. Amer. Math. Soc. 50 (1944), 284-316. MR 6 #29.
  • [Ri1] H. G. Rice, Classes of recursively enumerable sets and their decision problems, Trans. Amer. Math. Soc. 74 (1953), 358-366.
  • [Ri2] H. G. Rice, On completely recursively enumerable classes and their key arrays, J. Symbolic Logic 21 (1956), 304-308.
  • [Ro1] R. W. Robinson, The inclusion lattice and degrees of unsolvability of the recursively enumerable sets, Doctoral Dissertation, Cornell Univ., Ithaca, N. Y., 1966.
  • [Ro2] R. W. Robinson, Recursively enumerable sets not contained in any maximal set, Notices Amer. Math. Soc. 13 (1966), 325. Abstract #632-4.
  • [Ro3] R. W. Robinson, Two theorems on hyperhypersimple sets, Trans. Amer. Math. Soc. 128 (1967), 531-538. MR 35 #6549.
  • [Ro4] R. W. Robinson, Simplicity of recursively enumerable sets, J. Symbolic Logic 32 (1967), 162-172. MR 36 #1323.
  • [Ro5] R. W. Robinson, A dichotomy of the recursively enumerable sets, Z. Math. Logik Grundlagen Math. 14 (1968), 339-356. MR 38 #5623.
  • [Ro6] R. W. Robinson, Interpolation and embedding in the recursively enumerable degrees, Ann. of Math. (2) 93 (1971), 285-314.
  • [Ro7] R. W. Robinson, Jump restricted interpolation in the r.e. degrees, Ann. of Math. (2) 93 (1971) 586-596.
  • [Rg1] H. Rogers, Jr., Computing degrees of unsolvability, Math. Ann. 138 (1959), 125-140.
  • [Rg2] H. Rogers, Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967. MR 37 #61.
  • [Rg3] H. Rogers, Jr., Some problems of definability in recursive function theory, Sets, Models and Recursion Theory, John Crossley, ed., North-Holland, Amsterdam, 1967, pp. 183-201.
  • [Rs] B. Rosser, Extensions of some theorems of Gödel and Church, J. Symbolic Logic 1 (1936), 87-91.
  • [Sa1] G. E. Sacks, A minimal degree less than 0', Bull. Amer. Math. Soc. 67 (1961), 416-419.
  • [Sa2] G. E. Sacks, On the degrees less than 0', Ann of Math. (2) 77 (1963), 211-231.
  • [Sa3] G. E. Sacks, Recursive enumerability and the jump operator, Trans. Amer. Math. Soc. 108 (1963), 223-239. MR 27 #5681.
  • [Sa4] G. E. Sacks, A maximal set which is not complete, Michigan Math. J. 11 (1964), 193-205. MR 29 #3368.
  • [Sa5] G. E. Sacks, The recursively enumerable degrees are dense, Ann. of Math. (2) 80 (1964), 300-312. MR 29 #3367.
  • [Sa6] G. E. Sacks, A simple set which is not effectively simple, Proc. Amer. Math. Soc. 15 (1964), 51-55.
  • [Sa7] G. E. Sacks, Degrees of unsolvability, rev. ed., Ann. of Math. Studies, no. 55, Princeton Univ. Press, Princeton, N. J., 1966.
  • [Sa8] G. E. Sacks, On a theorem of Lachlan and Martin, Proc. Amer. Math. Soc. 18 (1967), 140-141.
  • [Sa9] G. E. Sacks, RE sets higher up, Logic, Foundations of Mathematics and Computability Theory, 173-194, D. Reidel, Dordrecht-Holland, 1977.
  • [Ss] L. P. Sasso, Deficiency sets and bounded information reducibilities, Trans. Amer. Math. Soc. 200 (1974), 267-290.
  • [Sf1] J. R. Shoenfield, Quasicreative sets, Proc. Amer. Math. Soc. 8 (1957), 964-967.
  • [Sf2] J. R. Shoenfield, On degrees of unsolvability, Ann. of Math. (2) 69 (1959), 644-653.
  • [Sf3] J. R. Shoenfield, Undecidable and creative theories, Fund. Math. 49 (1961), 171-179.
  • [Sf4] J. R. Shoenfield, Application of model theory to degrees of unsolvability, Sympos. Theory of Models, North-Holland, Amsterdam, 1965, pp. 359-363. MR 34 #53.
  • [Sf5] J. R. Shoenfield, Degrees of unsolvability, North-Holland, Amsterdam, 1971. MR 49 #4768.
  • [Sf6] J. R. Shoenfield, The decision problem for recursively enumerable degrees, Bull. Amer. Math. Soc. 81 (1975), 973-977.
  • [Sf7] J. R. Shoenfield, Degrees of classes of r.e. sets, J. Symbolic Logic 41 (1976), 695-696.
  • [Sh1] R. A. Shore, A decidable class of two quantifier sentences in the theory of the recursively enumerable degrees, vol. 24 Notices Amer. Math. Soc. (1977), A-436. Abstract #77T-E48.
  • [Sh2] R. A. Shore, Nowhere simple sets and the lattice of recursively enumerable sets, J. Symbolic Logic 43 (1978), 322-330.
  • [Sh3] R. A. Shore, Determining automorphisms of the recursively enumerable sets, Proc. Amer. Math. Soc. 65 (1977), 318-325.
  • [Sh4] R. A. Shore, α-recursion theory, Handbook of Mathematical Logic, J. Barwise, ed., North-Holland, Amsterdam, 1978.
  • [Si] S. G. Simpson, Degrees of unsolvability: A survey of results, Handbook of Mathematical Logic, J. Barwise, ed., North-Holland, Amsterdam, 1978.
  • [Sm] R. M. Smullyan, Effectively simple sets, Proc. Amer. Math Soc. 15 (1964), 893-894.
  • [So1] R. I. Soare, The Friedberg-Muchnik Theorem re-examined, Canad. J. Math. 24 (1972), 1070-1078.
  • [So2] R. I. Soare, Automorphisms of the lattice of recursively enumerable sets, Bull. Amer. Math. Soc. 80 (1974), 53-58.
  • [So3] R. I. Soare, Automorphisms of the lattice of recursively enumerable sets. Part I: Maximal sets, Ann. of Math. 100 (1974), 80-120. MR 50 #10058.
  • [So4] R. I. Soare, The infinite injury priority method, J. Symbolic Logic 41 (1976), 513-530.
  • [So5] R. I. Soare, Recursively enumerable sets, Omega Series, Springer-Verlag, Berlin and New York, (to appear).
  • [So6] R. I. Soare, Automorphisms of the lattice of recursively enumerable sets. Part II: Low sets (to appear).
  • [So7] R. I. Soare, Post's program and complete recursively enumerable sets (to appear).
  • [So8] R. I. Soare, Computational complexity, speedable and levelable sets, J. Symbolic Logic 42 (1977), 545-563.
  • [Sp1] C. Spector, On degrees of recursive unsolvability, Ann. of Math. (2) 64 (1956), 581-592. MR 18 #1118.
  • [Sp2] C. Spector, Inductively defined sets of natural numbers, Warsaw Symposium on Infinitistic Methods, New York, 1961, pp. 98-102.
  • [St] M. Stob, Doctoral Dissertation, Univ. of Chicago, Chicago, III., 1979.
  • [Te1] S. Tennenbaum, Degree of unsolvability and the rate of growth of functions, Notices Amer. Math. Soc. 8 (1961), 608.
  • [Te2] S. Tennenbaum, Degree of unsolvability and the rate of growth of functions, Proc. Sympos. Math. Theory of Automata, Microwave Res. Inst. Sympos. Ser. vol. 12, Polytechnic Press, Brooklyn, N. Y., 1962.
  • [Th] S. K. Thomason, Sublattices of the recursively enumerable degrees, Z. Math. Logik Grundlagen Math. 17 (1971), 273-280. MR 45 #8523.
  • [Ts] R. E. Tulloss, Some complexities of simplicity: concerning grades of simplicity of recursively enumerable sets, Doctoral Dissertation, Univ. of Calif., Berkeley, Calif., 1971.
  • [Tu] A. M. Turing, On computable numbers, with an application to the Entscheidungsproblem, Proc. London Math. Soc. 42 (1936), 230-265; ibid. 43 (1936), 544-546.
  • [U] V. A. Uspenskiĭ, Some notes on recursively enumerable sets, Z. Math. Logik Grundlagen Math. 3 (1957), 157-170; English transl., Amer. Math. Soc. Transl. (2) 23 (1963), 89-101.
  • [Y1] C. E. M. Yates, Recursively enumerable sets and retracing functions, Z. Math. Logik Grundlagen Math. 8 (1962), 331-345.
  • [Y2] C. E. M. Yates, Three theorems on the degree of recursively enumerable sets, Duke Math. J. 32 (1965), 461-468. MR 31 #4721.
  • [Y3] C. E. M. Yates, A minimal pair of r.e. degrees, J. Symbolic Logic 31 (1966), 159-168. MR 34 #5677.
  • [Y4] C. E. M. Yates, On the degrees of index sets, Trans. Amer. Math. Soc. 121 (1966), 309-328.
  • [Y5] C. E. M. Yates, Recursively enumerable degrees and the degrees less than0', Sets, Models and Recursion Theory, North-Holland, Amsterdam, 1967, pp. 264-271.
  • [Y6] C. E. M. Yates, On the degrees of index sets. II. Trans. Amer. Math. Soc. 135 (1969), 249-266.
  • [Y7] C. E. M. Yates, Initial segments of the degrees of unsolvability, Part I, Mathematical Logic and the Foundations of Set Theory (Jerusalem), North-Holland, Amsterdam, 1970, pp. 63-83.
  • [Y8] C. E. M. Yates, Initial segments and implications for the structure of degrees, Conference in Mathematical Logic-London 1970, Lecture Notes in Math., no. 255, Springer-Verlag, Berlin and New York, 1972, pp. 305-335.
  • [Y9] C. E. M. Yates, Prioric games and minimal degrees below0', Fund. Math. 82 (1974), 217-237.
  • [Y10] C. E. M. Yates, Banach-Mazur games, comeager sets, and degrees of unsolvability, Math. Proc. Cambridge Philos. Soc. (to appear).