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Undecidability and Definability for Parametrized Polynomial Time m-Reducibilities

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Logical Methods

Part of the book series: Progress in Computer Science and Applied Logic ((PCS,volume 12))

Abstract

In the setting of the parametrized reducibilities introduced by the second author and Mike Fellows, we prove a number of decidability and definability results. In particular the undecidability of the relevant m-degree structures is proven. The relationship with classical notions is analyzed, and this leads to a number of observations about classical constructions in the PTIME degrees. Methods include 0″, 0″′ and 0 (4) priority arguments combined with speedup type arguments.

Partially supported by a Victoria University Postdoctoral Fellowship, and by an N.S.F. Postdoctoral Fellowship.

Partially supported by the Victoria University IGC, the U.S.-New Zealand cooperative foundation grant INT 90-20558, Cornell University and the MSI at Cornell.

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References

  1. Abrahamson, K., R. Downey and M. Fellows, Fixed Parameter Tractability and Completeness IV: W[P] and PSPACE. To appear.

    Google Scholar 

  2. Abrahamson, K., R. Downey and M. Fellows, Fixed Parameter Intractability II. To appear in STACS’ 93.

    Google Scholar 

  3. Ambos-Spies, K., S. Homer and R. Soare [1990], Minimal pairs and complete problems. Proc. STACS’90, Lecture Notes in Computer Science, 415, Springer-Verlag, 24–36.

    Google Scholar 

  4. Ambos-Spies, K. [1984], On the structure of polynomial time degrees. STACS 84, Lecture Notes in Computer Science, 166, Springer-Verlag, 198–208.

    Google Scholar 

  5. Ambos-Spies, K. [1985], On the structure of the polynomial time degrees of recursive sets. Technical report 206, Abteilung Informatik, Universität Dortmund.

    Google Scholar 

  6. Ambos-Spies, K. [1987], Minimal pairs for polynomial time reducibilities. Computation and Proof Theory, Lecture Notes in Computer Science, 270, Springer-Verlag, 1–13.

    Google Scholar 

  7. Ambos-Spies, K. and A. Nies, The Theory of The Polynomial Time Many-One Degrees is Undecidable. To appear.

    Google Scholar 

  8. Ambos-Spies, K., A. Nies and R.A. Shore [1992], The Theory of the Recursively Enumerable Weak Truth Table Degrees is Undecidable. J. Symb. Logic 57, 864–874.

    Article  MathSciNet  MATH  Google Scholar 

  9. Aoki, K., J. Shinoda and T. Tsuda, On  2 Theories of hp-T Degrees of Low Sets. To appear.

    Google Scholar 

  10. Balcazaar, J., J. Diaz and J. Gabarro [1987, 1989], Structural Complexity, Vols. 1 and 2. Springer-Verlag.

    Google Scholar 

  11. Buss, J.F. and J. Goldsmith, Nondeterminism Within P. To appear in SIAM J. of Computing.

    Google Scholar 

  12. Bodlaender, H.L. [1990], On Disjoint Cycles. Technical Report RUU-CS-90-29, Dept. of Computer Science, Utrecht University, Utrecht, The Netherlands.

    Google Scholar 

  13. Downey, R., Nondiamond Theorems for Polynomial Time Reducibility. To appear in J.C.S.S.

    Google Scholar 

  14. Downey, R. and M. Fellows [1992], Fixed Parameter Tractability and Completeness. Congressus Numerantium 87, 161–187.

    MathSciNet  Google Scholar 

  15. Downey, R. and M. Fellows, Fixed Parameter Tractability and Completeness I: Basic Results. To appear.

    Google Scholar 

  16. Downey, R. and M. Fellows, Fixed Parameter Tractability and Completeness II: On Completeness for W[1], To appear.

    Google Scholar 

  17. Downey, R. and M. Fellows [1992], Fixed Parameter Intractability. Proc. Structure in Complexity Theory, 7th Annual Conference, 36–49.

    Google Scholar 

  18. Downey, R. and M. Fellows, Fixed Parameter Tractability and Intractability: a Survey. To appear in Annals of Pure and Appl. Logic.

    Google Scholar 

  19. Downey, R. and M. Fellows, Parametrized Computational Feasibility. To appear in Feasible Mathematics II (eds. P. Clote and J.B. Remmel).

    Google Scholar 

  20. Fellows, M.R. and M.A. Längsten [1989], On Search, Decision and the Efficiency of Polynomial-Time Algorithms. Proc. Symp. on Theory of Computing (STOC), 501–512.

    Google Scholar 

  21. Fellows, M.R. and M.A. Längsten [1989], An Analogue of the Myhill-Nerode Theorem and Its Use in Computing Finite Basis Characterizations. Proc. Symp. Foundations of Comp. Sci. (FOCS), 520–525.

    Google Scholar 

  22. Garey, M.R. and D.S. Johnson [1979], Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco.

    MATH  Google Scholar 

  23. Herrmann, E. [1984], The undecidability of the elementary theory of the lattice of recursively enumerable sets (abstract). Proc. of Second Frege Conference, Schwerin, DDR, 1984 (G. Wechsung, ed.). Akademie-Verlag, Berlin, 66–72.

    Google Scholar 

  24. Ladner, R. [1975], On the Structure of Polynomial Time Reducibility J.A.C.M, 22, 155–171.

    MathSciNet  MATH  Google Scholar 

  25. Manaster, A. and A. Nerode [1970], A Universal Embedding Property of the RETs. J. Symb. Logic, 35, 51–59.

    Article  MathSciNet  MATH  Google Scholar 

  26. Melhorn, K. [1976], Polynomial and Abstract Subrecursive Classes. J.C.S.S, 12, 147–178.

    Article  Google Scholar 

  27. Metakides, G. and A. Nerode [1977], Recursively Enumerable Vector Spaces. Ann. Math. Logic, 11, 141–171.

    Article  MathSciNet  Google Scholar 

  28. Nerode, A. and J.B. Remmel [1982], Recursion Theory on Matroids. In: Patras Logic Symposion ((ed. G. Metakides), North-Holland, 41–65.

    Google Scholar 

  29. Nerode, A. and J.B. Remmel [1987], Complexity Theoretic Algebra I: Vector Spaces over Finite Fields (Extended Abstract). Proc. Structure in Complexity Theory, 2nd Annual Conference, 218–241.

    Google Scholar 

  30. Nerode, A., J.B. Remmel and A. Scedrov [1989], Polynomially Graded Logic: a Graded Version of Gödel’s System T. LICS.

    Google Scholar 

  31. Nerode, A. and R. Shore [1980], Reducibility Orderings: Theories, Definability and Automorphisms. Ann. Math. Logic, 18, 61–89.

    Article  MathSciNet  MATH  Google Scholar 

  32. Nerode, A. and R. Smith [1980], Undecidability of the Lattice of Recursively Enumerable Subspaces. Proc. Third Brazilian Conf. on Math. Logic, Soc. Brasil. Lógica, São Paulo 1980, ed. A.I. Arruda, N.C.A. da Costa, A-M Sette, 245–252.

    Google Scholar 

  33. Robertson, N. and P.D. Seymour, Graph Minors XIII. The Disjoint Paths Problem. To appear.

    Google Scholar 

  34. Robertson, N. and P.D. Seymour, Graph Minors XV. Wagner’s Conjecture. To appear.

    Google Scholar 

  35. Shinoda, J. [1991], Personal Communication.

    Google Scholar 

  36. Shinoda, J. and T. Slaman [1990], On the Theory of PTIME Degrees of Recursive Sets. J.C.S.S, 41, 321–366.

    Article  MathSciNet  MATH  Google Scholar 

  37. Shore, R. and T. Slaman [1992], The P-T-Degrees of Recursive Sets; Lattice Embeddings, Extensions of Embeddings, and the Two Quantifier Theory. Theoretical Computer Science, 97, 263–284.

    Article  MathSciNet  MATH  Google Scholar 

  38. Stockmeyer, L. [1973], Planar 3-Colourability is W-Complete. S1GACT News, 5, 19–25.

    Article  Google Scholar 

  39. Valiant, L.G. and V.V. Vazirani [1986], NP is as easy as detecting unique solutions. Theoretical Computer Science, 47, 85–93.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Mathematics Department, Victoria University, Wellington, New Zealand

    Peter Cholak & Rod Downey

  2. Mathematics Department and Mathematical Sciences Institute, Cornell University, Ithaca, N.Y., 14853, USA

    Rod Downey

Authors
  1. Peter Cholak
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  2. Rod Downey
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Editor information

Editors and Affiliations

  1. Dept. of Mathematics / Comp. Science, Monash University, Clayton, Victoria, 3168, Australia

    John N. Crossley

  2. Dept. of Mathematics, University of California at San Diego, La Jolla, CA, 92093, USA

    Jeffrey B. Remmel

  3. Mathematical Sciences Institute, Cornell University, Ithaca, NY, 14853, USA

    Richard A. Shore  & Moss E. Sweedler  & 

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© 1993 Springer Science+Business Media New York

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Cholak, P., Downey, R. (1993). Undecidability and Definability for Parametrized Polynomial Time m-Reducibilities. In: Crossley, J.N., Remmel, J.B., Shore, R.A., Sweedler, M.E. (eds) Logical Methods. Progress in Computer Science and Applied Logic, vol 12. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0325-4_7

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  • DOI: https://doi.org/10.1007/978-1-4612-0325-4_7

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6708-9

  • Online ISBN: 978-1-4612-0325-4

  • eBook Packages: Springer Book Archive

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