Keywords
- Polynomial Time
- Turing Machine
- Partial Function
- Minimal Cover
- Recursive Operator
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
S. Ahmad, Some results on the structure of the Σ 2 enumeration degrees, Recursive Function Theory Newsletter 38 (July, 1989), item 373.
S. Ahmad, Embedding the diamond in the Σ 2 enumeration degrees, to appear.
K. Ambos-Spies, Minimal pairs for polynomial time reducibilities, to appear.
L. Adleman and K. Manders, Reducibility, randomness, and intractibility, Proc. 9th ACM STOC (1977), 151–163.
T. Baker, J. Gill and R. Solovay, Relativizations of the P=NP? question, SIAM J. Comput. 4 (1975), 431–442.
R. Beigel, W. Gasarch and J. Owings, Terse sets and verbose sets, Recursive Function Theory Newsletter 36 (1987), item 367.
P. Casalegno, On the T-degrees of partial functions, J. Symbolic Logic 50 (1985), 580–588.
J. Case, Enumeration reducibility and partial degrees, Ann. Math. Log. 2 (1971), 419–439.
J. Case, Maximal arithmetical reducibilities, Z. Math. Logik Grundlag. Math. 20 (1974), 261–270.
S. Cook, The complexity of theorem proving procedures, Proc. 3rd ACM STOC (1971), 151–158.
S.B. Cooper, Degrees of unsolvability complementary between recursively enumerable degrees, Part I, Ann. Math. Logic 4 (1972), 31–73.
S.B. Cooper, Minimal degrees and the jump operator, J. Symbolic Logic 38 (1973), 249–271.
S.B. Cooper, Minimal pairs and high recursively enumerable degrees, J. Symbolic Logic 39 (1974), 655–660.
S.B. Cooper, Partial degrees and the density problem, J. Symbolic Logic 47 (1982), 854–859.
S.B. Cooper, Partial degrees and the density problem. Part 2: The enumeration degrees of the Σ 2 sets are dense, J. Symbolic Logic 49 (1984), 503–513.
S.B. Cooper, Enumeration reducibility using bounded information: Counting minimal covers, Z. Math. Logik Grundlag. Math. 33 (1987), 537–560.
S.B. Cooper, The jump is definable in the structure of the degrees of unsolvability, to appear Bull. Amer. Math. Soc. (1990).
S.B. Cooper, The enumeration degrees are not dense, to appear.
S.B. Cooper and C.S. Copestake, Properly Σ 2 enumeration degrees, Z. Math. Logik Grundlag. Math. 34 (1988), 491–522.
S.B. Cooper, S. Lempp and P. Watson, On the degrees of d-r.e. sets, Israel J. Math. 67 (1989), 137–152.
S.B. Cooper and A. Sorbi, Initial segments of the enumeration degrees, to appear.
C.S. Copestake, The Enumeration Degrees of Σ 2 Sets, Ph.D. Thesis, Leeds University, 1987.
C.S. Copestake, 1-genericity in the enumeration degrees, J. Symbolic Logic 53 (1988), 878–887.
C.S. Copestake, 1-generic enumeration degrees below 0' e , to appear in the proceedings of Heyting '88, Bulgaria, 1988 (Plenum Press).
C.S. Copestake, A 1-generic enumeration degree which bounds no minimal pair, to appear.
C.S. Copestake, Nondeterminacy, enumeration reducibility and polynomial bounds, to appear.
W. Craig, On axiomatizability within a system, J. Symbolic Logic 18 (1953), 30–32.
M. Davis, “Computability and Unsolvability,” McGraw-Hill, New York, 1958.
E.Z. Dyment, Certain properties of the Medvedev lattice, Mat. Sb. (new series) 101 (143) (1976), 360–379 (Russian).
E.Z. Dyment, Exact bounds of denumerable collections of degrees of difficulty, Mat. Zametki 28 (1980), 899–910 (Russian).
S. Feferman, Degrees of unsolvability associated with classes of formalized theories, J. Symbolic Logic 22 (1957), 161–175.
S. Feferman, Some applications of the notions of forcing and generic sets, Fund. Math. 56 (1965), 325–345.
R.M. Friedberg and H. Rogers, Jr., Reducibility and completeness for sets of integers, Z. Math. Logik Grundlag. Math. 5 (1959), 117–125.
L. Gutteridge, Some Results on Enumeration Reducibility, Ph.D. Dissertation, Simon Frazer University, 1971.
C.A. Haught, The degrees below a 1-generic degree and less than 0', J. Symbolic Logic 51 (1986), 770–777.
F. Hebeisen, Masstheoretische Ergebnisse fur WT-Grade, Z. Math. Logik Grundlag. Math. 25 (1979), 33–36.
F. Hebeisen, Uber Halbordnungen von WT-Graden in e-Graden, Z. Math. Logik Grundlag. Math. 25 (1979), 209–212.
P.G. Hinman, Some applications of forcing to hierarchy problems in arithmetic, Z. Math. Logik Grundlag. Math. 15 (1969), 341–352.
C.G. Jockusch Jr., Reducibilities in Recursive Function Theory, Ph.D. Dissertation, MIT, Cambridge, Mass.,1966.
C.G. Jockusch Jr., Semirecursive sets and positive reducibility, Trans. Amer. Math. Soc. 131 (1968), 420–436.
C.G. Jockusch Jr., Upward closure and cohesive degrees, Israel J. Math. 15 (1973), 332–335.
C.G. Jockusch Jr., Degrees of generic sets, in F.R. Drake and S.S. Wainer, eds., Recursion Theory: its Generalisations and Applications, Proc. of Logic Colloquium, Leeds, 1979 (Cambridge University Press, 1980), 110–139.
N.D. Jones, Reducibility among combinatorial problems in log n space, in Proc. of Seventh Annual Princeton Conference on Information Sciences and Systems, 1973.
R. Karp, Reducibility among combinatorial problems, in Miller and Thatcher, eds., Complexity of Computer Computations (Plenum Press, New York, 1972), 85–103.
S.C. Kleene, “Introduction to Metamathematics,” Van Nostrand, New York, 1952.
S.C. Kleene and E.L. Post, The upper semi-lattice of degrees of recursive unsolvability, Ann. of Math. (2) 59 (1954), 379–407.
A.H. Lachlan, Some notions of reducibility and productiveness, Z. Math. Logik Grundlag. Math. 11 (1965), 17–44.
A.H. Lachlan, Lower bounds for pairs of recursively enumerable degrees, Proc. London Math. Soc. 16 (1966), 537–569.
A.H. Lachlan, Initial segments of many-one degrees, Canad. J. Math. 22 (1970), 75–85.
A.H. Lachlan, Embedding nondistributive lattices in the recursively enumerable degrees, in W. Hodges, ed., Conference in Mathematical Logic, London, 1970 (Springer-Verlag, Berlin, 1972), 149–177.
A.H. Lachlan, Bounding minimal pairs, J. Symbolic Logic 44 (1979), 626–642.
R. Ladner, N. Lynch and A. Selman, A comparison of polynomial time reducibilities, Theoret. Comput. Sci. 1 (1975), 103–123.
J. Lagemann, Embedding Theorems in the Reducibility Ordering of the Partial Degrees, Ph.D. Dissertation, MIT, 1972.
M. Lerman, “Degrees of Unsolvability,” Perspectives in Mathematical Logic, Omega Series, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1983.
T.J. Long, Strong nondeterministic polynomial-time reducibilities, Theoret. Comput. Sci. 21 (1982), 1–25.
K. McEvoy, The Structure of the Enumeration Degrees, Ph.D. Thesis, Leeds University, 1984.
K. McEvoy, Jumps of quasi-minimal enumeration degrees, J. Symbolic Logic 50 (1985), 839–848.
K. McEvoy and S.B. Cooper, On minimal pairs of enumeration degrees, J. Symbolic Logic 50 (1985), 983–1001.
Yu.T. Medvedev, Degrees of difficulty of the mass problem, Dokl. Akad. Nauk SSSR, N.S. 104 (1955), 501–504 (Russian).
A.R. Meyer and L.J. Stockmeyer, The equivalence of regular expressions with squaring requires exponential space, in Proc. Thirteenth Annual IEEE Symposium on Switching and Automata Theory (1972), 125–129.
A.R. Meyer and L.J. Stockmeyer, Word problems requiring exponential time, in Proc. Fifth Annual Symposium on Theory of Computing (1973).
B.B. Moore, Structure of the Degrees of Enumeration Reducibility, Ph.D. Dissertation, Syracuse University, 1974.
J. Myhill, A note on degrees of partial functions, Proc. Amer. Math. Soc. 12 (1961), 519–521.
J. Myhill and J.C. Shepherdson, Effective operations on partial recursive functions, Z. Math. Logik Grundlag. Math. 1 (1955), 310–317.
A. Nerode and R.A. Shore, Second order logic and first order theories of reducibility orderings, in J. Barwise et al, eds., The Kleene symposium (North-Holland, Amsterdam, 1980), 181–200.
P. Odifreddi, Strong reducibilities, Bull. Amer. Math. Soc. (N.S.) 4 (1981), 37–86.
P. Odifreddi, “Classical Recursion Theory,” North-Holland, Amsterdam, New York, Oxford, 1989.
P. Odifreddi, “Classical Recursion Theory, Vol. II,” North-Holland, Amsterdam, New York, Oxford, to appear.
G.D. Plotkin, A set-theoretical definition of application, Memo. MIP-R-95, School of Artificial Intelligence, University of Edinburgh, 1972.
E.A. Polyakov and M.G. Rozinas, Enumeration reducibilities, Sib. Math. J. 18 (1977), 594–599.
E.A. Polyakov and M.G. Rozinas, Relationships between different forms of relative computability, Math. USSR-Sbornik 35 (1979), 425–436.
D. Posner, High Degrees, Ph.D. Dissertation, University of California, Berkeley, 1977.
E.L. Post, Recursively enumerable sets of positive integers and their decision problems, Bull. Amer. Math. Soc. 50 (1944), 304–337.
R.W. Robinson, A dichotomy of the recursively enumerable sets, Z. Math. Logik Grundlag. Math. 14 (1968), 339–356.
H. Rogers Jr., “Theory of Recursive Functions and Effective Computability,” McGraw-Hill, New York, 1967.
M.G. Rozinas, Partial degrees and τ-degrees, Siberian Math. J. 15 (1974), 935–941.
M.G. Rozinas, The semilattice of e-degrees, Ivanov. Gos. Univ., Ivanovo (1978), 71–84 (Russian).
M.G. Rozinas, Partial degrees of immune and hyperimmune sets, Siberian Math. J. 19 (1978), 613–616.
G.E. Sacks, A minimal degree less than 0', Bull. Amer. Math. Soc. 67 (1961), 416–419.
G.E. Sacks, Recursive enumerability and the jump operator, Trans. Amer. Math. Soc. 108 (1963), 223–239.
L.E. Sanchis, Hyperenumeration reducibility, Notre Dame J. Formal Logic 19 (1978), 405–415.
L.E. Sanchis, Reducibilities in two models for combinatory logic, J. Symbolic Logic 44 (1979), 221–233.
L.P. Sasso Jr., Degrees of Unsolvability of Partial Functions, Ph.D. Dissertation, University of California, Berkeley, 1971.
L.P. Sasso Jr., A minimal partial degree ≤ 0', Proc. Amer. Math. Soc. 38 (1973), 388–392.
L.P. Sasso Jr., A survey of partial degrees, J. Symbolic Logic 40 (1975), 130–140.
D. Scott, Lambda calculus and recursion theory, in Kanger, ed., Proc. Third Scandinavian Logic Sympos. (North-Holland, Amsterdam, 1975), 154–193.
D. Scott, Data types as lattices, in Proc. Logic Conf., Kiel, SVLNM 499 (1975), 579–651.
A.L. Selman, Arithmetical reducibilities I, Z. Math. Logik Grundlag. Math. 17 (1971), 335–350.
A.L. Selman, Arithmetical reducibilities II, Z. Math. Logik Grundlag. Math. 18 (1972), 83–92.
A.L. Selman, Polynomial time enumeration reducibility, SIAM J. Comput. 7 (1978), 440–457.
J. Shinoda and T.A. Slaman, On the theory of the polynomial degrees of the recursive sets, to appear.
J.R. Shoenfield, On degrees of unsolvability, Ann. of Math. (2) 69 (1959), 644–653.
J.R. Shoenfield, A theorem on minimal degrees, J. Symbolic Logic 31 (1966), 539–544.
A. Silver, Polynomial time singleton enumeration reducibilities, to appear.
D.G. Skordev, On partial conjunctive reducibility, in Second All-Union Conference on Math. Logic, Inst. Prikl. Mat. (1972), 43–44.
T.A. Slaman and W.H. Woodin, Definability in the degrees, to appear.
B.Ya. Solon, e-powers of hyperimmune retraceable sets, Siberian Math. J. 19 (1978), 122–127.
R.I. Soare, “Recursively Enumerable Sets and Degrees,” Springer-Verlag, Berlin, New York, London, Tokyo, 1987.
A. Sorbi, On quasi-minimal e-degrees and total e-degrees, Proc. Amer. Math. Soc. 102 (1988), 1005–1008.
A. Sorbi, Some remarks on the algebraic structure of the Medvedev lattice, to appear.
A. Sorbi, On some filters and ideals of the Medvedev lattice, to appear.
A. Sorbi, Embedding Brouwer algebras in the Medvedev lattice, to appear.
C. Spector, On degrees of recursive unsolvability, Ann. of Math. (2) 64 (1956), 581–592.
V. Vuckovic, Almost recursivity and partial degrees, Z. Math. Logik Grundlag. Math. 20 (1974), 419–426.
P. Watson, On restricted forms of enumeration reducibility, to appear.
C.E.M. Yates, Three theorems on the degree of recursively enumerable sets, Duke Math. J. 32 (1965), 461–468.
S.D. Zakharov, e-and s-degrees, Algebra and Logic 23 (1984), 273–281.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1990 Springer-Verlag
About this paper
Cite this paper
Cooper, S.B. (1990). Enumeration reducibility, nondeterministic computations and relative computability of partial functions. In: Ambos-Spies, K., Müller, G.H., Sacks, G.E. (eds) Recursion Theory Week. Lecture Notes in Mathematics, vol 1432. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0086114
Download citation
DOI: https://doi.org/10.1007/BFb0086114
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-52772-5
Online ISBN: 978-3-540-47142-4
eBook Packages: Springer Book Archive
