We give a general complexity classification scheme for monotone computation, including monotone space-bounded and Turing machine models not previously considered. We propose monotone complexity classes including mAC i , mNC i , mLOGCFL, mBWBP , mL, mNL, mP , mBPP and mNP . We define a simple notion of monotone reducibility and exhibit complete problems. This provides a framework for stating existing results and asking new questions. We show that mNL (monotone nondeterministic log-space) is not closed under complementation, in contrast to Immerman's and Szelepcs 'enyi's nonmonotone result [Imm88, Sze87] that NL = co-NL; this is a simple extension of the monotone circuit depth lower bound of Karchmer and Wigderson [KW90] for st-connectivity. We also consider mBWBP (monotone bounded width branching programs) and study the question of whether mBWBP is properly contained in mNC 1 , motivated by Barrington's result [Bar89] that BWBP = NC 1 . Although we cannot answer t...

In order to study circuit complexity classes within NC¹ in a uniform setting, we need a uniformity condition which is more restrictive than those in common use. Two such conditions, stricter than NC¹ uniformity [Ru81,Co85], have appeared in recent research: Immerman's families of circuits defined by first-order formulas [Im87a,Im87b] and a uniformity corresponding to Buss' deterministic log-time reductions [Bu87]. We show that these two notions are equivalent, leading to a natural notion of uniformity for low-level circuit complexity classes. We show that recent results on the structure of NC¹ [Ba89] still hold true in this very uniform setting. Finally, we investigate a parallel notion of uniformity, still more restrictive, based on the regular languages. Here we give characterizations of subclasses of the regular languages based on their logical expressibility, extending recent work of Straubing, Th'erien, and Thomas [STT88]. A preliminary version of this work appeared as [BIS88].

. Define the MODm -degree of a boolean function F to be the smallest degree of any polynomial P , over the ring of integers modulo m, such that for all 0-1 assignments ~x, F (~x) = 0 iff P (~x) = 0. We obtain the unexpected result that the MODm -degree of the OR of N variables is O( r p N ), where r is the number of distinct prime factors of m. This is optimal in the case of representation by symmetric polynomials. The MOD n function is 0 if the number of input ones is a multiple of n and is one otherwise. We show that the MODm -degree of both the MOD n and :MOD n functions is N\Omega\Gamma1/ exactly when there is a prime dividing n but not m. The MODm -degree of the MODm function is 1; we show that the MODm -degree of :MODm is N\Omega\Gamma30 if m is not a power of a prime, O(1) otherwise. A corollary is that there exists an oracle relative to which the MODmP classes (such as \PhiP) have this structure: MODmP is closed under complementation and union iff m is a prime power, and...

The 1980's saw rapid and exciting development of techniques for proving lower bounds in circuit complexity. This pace has slowed recently, and there has even been work indicating that quite different proof techniques must be employed to advance beyond the current frontier of circuit lower bounds. Although this has engendered pessimism in some quarters, there have in fact been many positive developments in the past few years showing that significant progress is possible on many fronts. This paper is a (necessarily incomplete) survey of the state of circuit complexity as we await the dawn of the new millennium.

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.

We show that uniform families of ACC circuits of subexponential size cannot compute the permanent function. This also implies similar lower bounds for certain sets in PP. This is one of the very few examples of a lower bound in circuit complexity whose proof hinges on the uniformity condition; it is still unknown if there is any set in Ntime #2 n O#1# # that does not have nonuniform ACC circuits.

We show that searching a width /' maze is complete for II, i.e., for the /"th level of the AC hierarchy. Equivalently, st-connectivity for width /' grid graphs is complete for II. As an application, we show that there is a data structure solving dynamic st-connectivity for constant width grid graphs with time bound O (log log n) per operation on a random access machine. The dynamic algorithm is derived from the parallel one in an indirect way using algebraic tools.

The computation tree of a nondeterministic machine M with input x gives rise to a leaf string formed by concatenating the outcomes of all the computations in the tree in lexicographical order. We may characterize problems by considering, for a particular "leaf language" Y , the set of all x for which the leaf string of M is contained in Y . In this way, in the context of polynomial time computation, leaf languages were shown to capture many complexity classes. In this paper, we study the expressibility of the leaf language mechanism in the contexts of logarithmic space and of logarithmic time computation. We show that logspace leaf languages yield a much finer classification scheme for complexity classes than polynomial time leaf languages, capturing also many classes within P. In contrast, logtime leaf languages basically behave like logtime reducibilities. Both cases are more subtle to handle than the polynomial time case. We also raise the issue of balanced versus non-balanced comp...

We use algebraic techniques to obtain superlinear lower bounds on the size of boundedwidth branching programs to solve a number of problems. In particular, we show that any bounded-width branching program computing a nonconstant threshold function has length \Omega\Gamma n log log n); improving on the previous lower bounds known to apply to all such threshold functions. We also show that any program over a finite solvable monoid computing products in a nonsolvable group has length\Omega\Gamma n log log n): This result is a step toward proving the conjecture that the circuit complexity class ACC 0 is properly contained in NC 1 : A preliminary version of this paper appeared in the Proceedings of the 1991 Structure in Complexity Theory Symposium. 1. The Main Results In this paper we describe a general algebraic technique for obtaining superlinear lower bounds on the length of bounded-width branching programs to solve certain problems. Our method is based on the interpretation, ...

Let M be a fixed finite monoid. We consider the problem of implementing a data type containing a vector x = (x1 ; x2 ; : : : ; xn) 2 M n , initially (1; 1; : : : ; 1), with two kinds of operations, for each i 2 f1; : : : ; ng and a 2 M , an operation change i;a which changes x i to a and a single operation product returning Q n i=1 x i . This is the dynamic word problem for M . If we in addition for each j 2 f1; : : : ; ng have an operation prefix j returning Q j i=1 x i , we get the dynamic prefix problem for M . We analyze the complexity of these problems in the cell probe or decision assignment tree model for two natural cell sizes, 1 bit and log n bits. We obtain a partial classification of the complexity based on algebraic properties of M .