Analogues of Semicursive Sets and Effective Reducibilities to the Study of NP Complexity

where R is a relation in P and, for some polynomial p , y ranges over words of length not exceeding p([x[). This same analogy to the classical characterization of the recursively enumerable sets extends to the polynomial hierarchy, which of course is the natural analogue of the arithmetical hierarchy. Even Wrathall's result (Wrathall, 1976) that Bo, (a well-known complete set in PSPACE, see Stockmeyer (1976) is not in PH, if the hierarchy is proper, is reminiscent of Tarski's theorem that truth is not arithmetic. Efficient reducibilities are used to classify decidable problems in much the same way that effective reducibilities are used to classify undecidable problems. The reducibilities formulated by Cook (1971) and by Karp (1973) are just the restrictions to polynomial time of Turing and many-one reducibilities, respectively. These are denoted ~<~and <em, where the superscript indicates the time bound. Surprisingly, the structure of NP <~-degrees is similar to the one of r.e. ~<r-degrees, see Ladner (1975). For example, they form a dense uppersemilattice. ~<m B for recursive Clearly, A ~e mB ~ A ~<~B. A ~<~B does not imply A e sets that are recognizable in time 2 n (Ladner, Lynch, and Selman, 1975). It

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