This paper surveys investigations into how strong these commonalities are. More concretely, we are concerned with: What do NP-complete sets look like? To what extent are the properties of particular NP-complete sets, e.g., SAT, shared by all NP-complete sets? If there are are structural differences between NP-complete sets, what are they and what explains the differences? We make these questions, and the analogous questions for other complexity classes, more precise below. We need first to formalize NP-completeness. There are a number of competing definitions of NP-completeness. (See [Har78a, p. 7] for a discussion.) The most common, and the one we use, is based on the notion of m-reduction, also known as polynomial-time manyone reduction and Karp reduction. A set A is m-reducible to B if and only if there is a (total) polynomial-time computable function f such that for all x, x 2 A () f(x) 2 B: (1) 1

We demonstrate differences between reducibilities and corresponding completeness notions for nondeterministic time and space classes. For time classes the studied completeness notions under polynomial-time bounded (even logarithmic space bounded) reducibilities turn out to be different for any class containing NE . For space classes the completeness notions under logspace reducibilities can be separated for any class properly containing LOGSPACE . Key observation in obtaining the separations is the honesty property of reductions, which was recently observed to hold for the time classes and can be shown to hold for space classes. 1 Introduction Efficient reducibilities and completeness are two of the central concepts of complexity theory. Since the first use of polynomial time bounded Turing reductions by Cook [4] and the introduction of polynomial time bounded many-one reductions by Karp[9], considerable effort has been put in the investigation of properties and the relative strengt...

Abstract. We examine the computational complexity of context-free languages, mainly concentrating on two well-known structural properties—immunity and pseudorandomness. An infinite language is REG-immune (resp., CFL-immune) if it contains no infinite subset that is a regular (resp., context-free) language. We prove that (i) there is a context-free REG-immune language outside REG/n and (ii) there is a REG-bi-immune language that can be computed deterministically using logarithmic space. We also show that (iii) there is a CFL-simple set, where a CFL-simple language is an infinite context-free language whose complement is CFL-immune. Similar to the REG-immunity, a REG-primeimmune language has no polynomially dense subsets that are also regular. We further prove that (iv) there is a context-free language that is REG/n-bi-primeimmune but not even REG-immune. Concerning pseudorandomness of contextfree languages, we show that (v) CFL contains REG/n-pseudorandom languages. Finally, we prove that (vi) against REG/n, there exists an almost 1-1 pseudorandom generator computable in nondeterministic pushdown automata equipped with a write-only output tape and (vii) against REG, there is no almost 1-1 weak pseudorandom generator computable deterministically in linear time by a single-tape Turing machine.