On Nondeterminism, Enumeration Reducibility and Polynomial Bounds

Mathematical Logic Quarterly 43 (3):287-310 (1997)
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Abstract

Enumeration reducibility is a notion of relative computability between sets of natural numbers where only positive information about the sets is used or produced. Extending e‐reducibility to partial functions characterises relative computability between partial functions. We define a polynomial time enumeration reducibility that retains the character of enumeration reducibility and show that it is equivalent to conjunctive non‐deterministic polynomial time reducibility. We define the polynomial time e‐degrees as the equivalence classes under this reducibility and investigate their structure on the recursive sets, showing in particular that the pe‐degrees of the computable sets are dense and do not form a lattice, but that minimal pairs exist. We define a jump operator and use it to produce a characterisation of the polynomial hierarchy.

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References found in this work

Theory of Recursive Functions and Effective Computability.Hartley Rogers - 1971 - Journal of Symbolic Logic 36 (1):141-146.
On minimal pairs of enumeration degrees.Kevin McEvoy & S. Barry Cooper - 1985 - Journal of Symbolic Logic 50 (4):983-1001.
Some applications of forcing to hierarchy problems in arithmetic.Peter G. Hinman - 1969 - Mathematical Logic Quarterly 15 (20-22):341-352.
Arithmetical Reducibilities I.Alan L. Selman - 1971 - Mathematical Logic Quarterly 17 (1):335-350.

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