A set A is m-reducible (or Karp-reducible) to B iff there is a polynomial-time computable function f such that, for all x, x 2 A () f(x) 2 B. Two sets are: ffl 1-equivalent iff each is m-reducible to the other by one-one reductions; ffl p-invertible equivalent iff each is m-reducible to the other by one-one, polynomial-time invertible reductions; and ffl p-isomorphic iff there is an m-reduction from one set to the other that is one-one, onto, and polynomial-time invertible. In this paper we show the following characterization. Theorem The following are equivalent: (a) P = PSPACE. (b) Every two 1-equivalent sets are p-isomorphic. (c) Every two p-invertible equivalent sets are p-isomorphic. 2 1. Overview If A is m-reducible to B, we usually interpret this to mean that A is computationally no more difficult than B, since a procedure for computing B is easily converted into a procedure for computing A of comparable complexity. In fact, this interpretation is supported by muc...

For reducibilities r and r 0 such that r is weaker than r 0 , we say that the r-degree of A, i.e., the class of sets which are r-equivalent to A, collapses to the r 0 -degree of A if both degrees coincide. We investigate for the polynomial-time bounded many-one, bounded truth-table, truthtable, and Turing reducibilities whether and under which conditions such collapses can occur. While we show that such collapses do not occur for sets which are hard for exponential time, we have been able to construct a recursive set such that its bounded truth-table degree collapses to its many-one degree. The question whether there is a set such that its Turing degree collapses to its many-one degree is still open; however, we show that such a set -- if it exists -- must be recursive.

It is shown that the following are equivalent. 1. DSPACE(n) = NSPACE(n). 2. There is a non-trivial ≤1−NL m-degree that coincides with a ≤1−L m-degree. 3. For every class C closed under log-lin reductions, the ≤1−NL m coincides with the ≤1−L m-complete degree of C. 1

The degree structure of complete sets under 2DFA reductions is investigated. It is shown that, for any class C that is closed under log-lin reductions: -- All complete sets for the class C under 2DFA reductions are also complete under one-one, length-increasing 2DFA reductions and are first-order isomorphic. -- The 2DFA-isomorphism conjecture is false, i.e., the complete sets under 2DFA reductions are not isomorphic to each other via 2DFA reductions.

It is shown that for any class C closed under linear-time reductions, the complete sets for C under sublogarithmic reductions are also complete under 2DFA reductions, and thus are isomorphic under first-order reductions.

Abstract—The Berman-Hartmanis conjecture states that all NP-complete sets are P-isomorphic each other. On this conjecture, we first improve the result of [3] and show that all NP-complete sets are ≤ p/poly li,1-1-reducible to each other based on the assumption that there exist regular one-way functions that cannot be inverted by randomized polynomial-time algorithms. Secondly, we show that, besides the above assumption, if all one-way functions have some easy part to invert, then all NP-complete sets are P/polyisomorphic to each other. Index Terms—average-case one-way function; P-isomorophism conjecture; P/poly-isomorophism in NP; one-way function with easy cylinder I.

Sparse hard sets for complexity classes has been a central topic for two decades. The area is motivated by the desire to clarify relationships between completeness/hardness and density of languages and studies the existence of sparse complete/hard sets for various complexity classes under various reducibilities. Very recently, we have seen remarkable progress in this area for low-level complexity classes. In particular, the Hartmanis' sparseness conjectures for P and NL have been resolved. This article overviews the history of sparse hard set problems and exposes some of the recent results.

We survey some of the history of the most famous open question in computing: the P versus NP question. We summarize some of the progress that has been made to date, and assess the current situation.

this article. First, however, we will need to make a digression, while we discuss some recent progress on the isomorphism conjecture.

. For reducibilities r and r 0 such that r is weaker than r 0 , we say that the r-degree of A, i.e., the class of sets which are r-equivalent to A, collapses to the r 0 -degree of A if both degrees coincide. We investigate for the polynomial-time bounded many-one, bounded truth-table, truthtable, and Turing reducibilities whether and under which conditions such collapses can occur. While we show that such collapses do not occur for sets which are hard for exponential time, we have been able to construct a recursive set such that its bounded truth-table degree collapses to its many-one degree. The question whether there is a set such that its Turing degree collapses to its many-one degree is still open; however, we show that such a set -- if it exists -- must be recursive. 1 Introduction and Notation 1.1 Introduction Ladner, Lynch and Selman [12] first compared the strength of the polynomialtime reducibilities. For the most common reducibilities -- namely Turing (p-T)...