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 investigate the phenomenon of depth-reduction in commutativeand non-commutative arithmetic circuits. We prove that in the commutative setting, uniform semi-unbounded arithmetic circuits of logarithmic depth are as powerful as uniform arithmetic circuits of polynomial degree (and unrestricted depth); earlier proofs did not work in the uniform setting. This also provides a unified proof of the circuit characterizations of the class LOGCFL and its counting variant #LOGCFL. We show that AC 1 has no more power than arithmetic circuits of polynomial size and degree n O(log log n) (improving the trivial bound of n O(logn) ). Connections are drawn between TC 1 and arithmetic circuits of polynomial size and degree. Then we consider non-commutative computation. We show that over the algebra (\Sigma ; max, concat), arithmetic circuits of polynomial size and polynomial degree can be reduced to O(log 2 n) depth (and even to O(log n) depth if unbounded-fanin gates are allowed) . This...

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 for most complexity classes of interest, all sets complete under first-order projections are isomorphic under first-order isomorphisms. That is, a very restricted version of the Berman-Hartmanis Conjecture holds.

We prove in this paper that it is much harder to evaluate depth--2, size--N circuits with MOD m gates than with MOD p gates by k--party communication protocols: we show a k--party protocol which communicates O(1) bits to evaluate circuits with MOD p gates, while evaluating circuits with MOD m gates needs\Omega\Gamma N) bits, where p denotes a prime, and m a composite, non-prime power number. As a corollary, for all m, we show a function, computable with a depth--2 circuit with MODm gates, but not with any depth--2 circuit with MOD p gates. Obviously, the k--party protocols are not weaker than the k 0 --party protocols, for k 0 ? k. Our results imply that if there is a prime p between k and k 0 : k ! p k 0 , then there exists a function which can be computed by a k 0 --party protocol with a constant number of communicated bits, while any k--party protocol needs linearly many bits of communication. This result gives a hierarchy theorem for multi--party protocols. 1 1.

We define the counting classes #NC 1 , GapNC 1 , PNC 1 and C=NC 1 . We prove that boolean circuits, algebraic circuits, programs over nondeterministic finite automata, and programs over constant integer matrices yield equivalent definitions of the latter three classes. We investigate closure properties. We observe that #NC 1 ` #L and that C=NC 1 ` L. Then we exploit our finite automaton model and extend the padding techniques used to investigate leaf languages. Finally, we draw some consequences from the resulting body of leaf language characterizations of complexity classes, including the unconditional separation of ACC 0 from MOD-PH as well as that of TC 0 from the counting hierarchy. Moreover we obtain that dlogtimeuniformity and logspace-uniformity for AC 0 coincide if and only if the polynomial time hierarchy equals PSPACE .

We describe three orthogonal complexity measures: parallel time, amount of hardware, and degree of nonuniformity, which together parametrize most complexity classes. We show that the descriptive complexity framework neatly captures these measures using the parameters: quantifier depth, number of variable bits, and type of numeric predicates respectively. A fairly simple picture arises in which the basic questions in complexity theory -- solved and unsolved -- can be understood as questions about tradeoffs among these three dimensions. 1 Introduction An initial presentation of complexity theory usually makes the implicit assumption that problems, and hence complexity classes, are linearly ordered by "difficulty ". In the Chomsky Hierarchy each new type of automaton can decide more languages, and the Time Hierarchy Theorem tells us adding more time allows a Turing machine to decide more languages. Indeed the word "complexity" is often used (e.g., in the study of algorithms) to mean "wo...

: We prove that any depth--3 circuit with MOD m gates of unbounded fan-in on the lowest level, AND gates on the second, and a weighted threshold gate on the top needs either exponential size or exponential weights to compute the inner product of two vectors of length n over GF(2). More exactly we prove that log(wM ) = \Omega\Gamma n), where w is the sum of the absolute values of the weights, and M is the maximum fan--in of the AND gates on level 2. Setting all weights to 1, we have got a trade--off between the numbers of the MOD m gates and the AND gates. By our knowledge, this is the first trade--off result involving hard--to--handle MOD m gates. In contrast, with n AND gates at the bottom and a single MOD 2 gate at the top one can compute the inner product function. The lower--bound proof does not use any monotonicity or uniformity assumptions, and all of our gates have unbounded fan--in. The key step in the proof is a random evaluation protocol of a circuit with MOD m gates. ...

Arithmetic circuits are the focus of renewed attention in the complexity theory community. It is easy to list a few of the reasons for the increased interest: • Innovative work by Kabanets and Impagliazzo [KI03] shows that, in

We prove the first time-space tradeoffs for counting the number of solutions to an NP problem modulo small integers, and also improve upon known time-space tradeoffs for Sat. Let m> 0 be an integer, and define MODm-Sat to be the problem of determining if a given Boolean formula has exactly km satisfying assignments, for some integer k. We show for all primes p except for possibly one of them, and for all c < 2cos(π/7) ≈ 1.801, there is a d> 0 such that MODp-Sat is not solvable in n c time and n d space by general algorithms. That is, there is at most one prime p that does not satisfy the tradeoff. We prove that the same limitation holds for Sat and MOD6-Sat, as well as MODm-Sat for any composite m that is not a prime power. Our main tool is a general method for rapidly simulating deterministic computations with restricted space, by counting the number of solutions to NP predicates modulo integers. The simulation converts an ordinary algorithm into a “canonical ” one that consumes roughly the same amount of time and space, yet canonical algorithms have nice properties suitable for counting.