Nondiamond Theorems for Polynomial Time Reducibility

Abstract We investigate the structure of recursive sets under polynomial time Turing reducibility. In particular, we solve a question of Ambos-Spies by constructing a polynomial time degree that is not the supremum of a minimal pair. The proof is of some technical interest as it uses π2 priority arguments and the speedup phenomenon.

[1]  Richard E. Ladner,et al.  On the Structure of Polynomial Time Reducibility , 1975, JACM.

[2]  Uwe Schöning.  A Uniform Approach to Obtain Diagonal Sets in Complexity Classes , 1982, Theor. Comput. Sci..

[3]  Jr. Hartley Rogers.  Theory of Recursive Functions and Effective Computability , 1969 .

[4]  Stephen A. Cook,et al.  The complexity of theorem-proving procedures , 1971, STOC.

[5]  Robert I. Soare,et al.  Minimal Pairs and Complete Problems , 1990, STACS.

[6]  Robert I. Soare,et al.  Minimal Pairs and Complete Problems , 1994, Theor. Comput. Sci..

[7]  Klaus Ambos-Spies.  Three theorems on polynomial degrees of NP-sets , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[8]  Kurt Mehlhorn.  The "Almost All" Theory of Subrecursive Degrees is Decidable , 1974, ICALP.

[9]  P. Odifreddi.  Classical recursion theory , 1989 .

[10]  Richard A. Shore,et al.  The p-T Degrees of the Recursive Sets: Lattice Embeddings, Extensions of Embeddings and the Two-Quantifier Theory , 1992, Theor. Comput. Sci..

[11]  Theodore A. Slaman,et al.  On the Theory of the PTIME Degrees of the Recursive Sets , 1990, J. Comput. Syst. Sci..

[12]  Richard J. Lipton,et al.  On the Structure of Sets in NP and Other Complexity Classes , 1981, Theor. Comput. Sci..

[13]  Richard E. Ladner.  Polynominal time reducibility , 1973, STOC '73.

[14]  Kurt Mehlhorn.  Polynomial and Abstract Subrecursive Classes , 1976, J. Comput. Syst. Sci..

[15]  Paul Young,et al.  Some structural properties of polynomial reducibilities and sets in NP , 1983, STOC.

[16]  R. Soare.  Recursively enumerable sets and degrees , 1987 .

[17]  Rodney G. Downey.  Lattice Nonembeddings and Initial Segments of the Recursively Enumerable Degrees , 1990, Ann. Pure Appl. Log..

[18]  Rodney G. Downey,et al.  On honest polynomial reductions, relativizations, and P=NP , 1989, [1989] Proceedings. Structure in Complexity Theory Fourth Annual Conference.

[19]  Richard A. Shore,et al.  The P-T-degrees of the recursive sets: lattice embeddings, extensions of embeddings and the two quantifier theory , 1989, [1989] Proceedings. Structure in Complexity Theory Fourth Annual Conference.

[20]  Klaus Ambos-Spies.  On the structure of the polynomial time degrees of recursive sets , 1985, Forschungsberichte.

[21]  PAUL CHEW,et al.  A Note on Structure and Looking Back Applied to the Complexity of Computable Functions , 1981, J. Comput. Syst. Sci..

[22]  Uwe Schöning.  Minimal pairs for P , 1984, Theor. Comput. Sci..