Given a successor relation S (i.e., a directed line graph), and given two distinguished points s and t, the problem ORD is to determine whether s precedes t in the unique ordering defined by S. We show that ORD is L-complete (via quantifier-free projections). We then show that first-order logic with counting quantifiers, a logic that captures TC 0 ([BIS90]) over structures with a built-in total-ordering, can not express ORD. Our original proof of this in the conference version of this paper ([Ete95]) employed an Ehrenfeucht-Fraiss'e Game for first-order logic with counting ([IL90]). Here we show how the result follows from a more general one obtained independently by Nurmonen, [Nur96]. We then show that an appropriately modified version of the EF game is "complete" for the logic with counting in the sense that it provides a necessary and sufficient condition for expressibility in the logic. We observe that the L-complete problem ORD is essentially sparse if we ignore reorderings of v...

For a monoid G, the iterated multiplication problem is the computation of the product of n elements from G. By refining known completeness arguments, we show that as G varies over a natural series of important groups and monoids, the iterated multiplication problems are complete for most natural, low-level complexity classes. The completeness is with respect to "first-order projections" -- low-level reductions that do not obscure the algebraic nature of these problems.

We show that all sets that arecomplete for NP under non-uniform AC are isomorphic under non-uniform AC -computable isomorphisms. Furthermore, these sets remain NP-complete even under non-uniform NC reductions.

We show that the permanent cannot be computed by uniform constant-depth threshold circuits of size T (n) for any function T such that for all k, T (k) (n) = o(2 n ). More generally, we show that any problem that is hard for the complexity class C=P requires circuits of this size (on the uniform constant-depth threshold circuit model). In particular, this lower bound applies to any problem that is hard for the complexity classes PP or #P.

The many types of resource bounded reductions that are both object of study and research tool in structural complexity theory have given rise to a large variety of completeness notions. A complete set in a complexity class is a manageable object that represents the structure of the entire class. The study of its structure can reveal properties that are general in that complexity class, and the study of the structure of complete sets in different classes can reveal secrets about the relation between these classes. The research into all sorts of aspects and properties of complete sets has been and will be a major topic in structural complexity theory. In this expository paper we review the progress that has been made in recent years on selected topics of the study of complete sets.

In this expository paper, we investigate the structure of complexity classes and the structure of complete sets therein. We give an overview of recent results on both set structure and class structure induced by various notions of reductions. 1 Introduction After the demonstration of the completeness of several problems for NP by Cook [Coo71] and Levin [Lev73] and for many other problems by Karp [Kar72], the interest in completeness notions in complexity classes has tremendously increased. Virtually every form of reduction known in computability theory has found its way to complexity theory. This is usually done by imposing time and/or space bounds on the computational power of the device representing the reduction. Early on, Ladner et al. [LLS75] categorized the then known types of reductions and made a comparison between these by constructing sets that are reducible to each other via one type of reduction and not reducible via the other. They however were interested just in the rela...

problem for a given constraint language is either in P or is NPcomplete, and identified all tractable cases. Schaefer’s dichotomy theorem actually shows that there are at most two constraint satisfaction problems, up to polynomial-time isomorphism (and these isomorphism types are distinct if and only if P ̸ = NP). We show that if one considers AC 0 isomorphisms, then there are exactly six isomorphism types (assuming that the complexity classes NP, P, ⊕L, NL, and L are all distinct). A similar classification holds for quantified constraint satisfaction problems.

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.

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