Circuit Complexity before the Dawn of the New Millennium

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 di erent 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 signi cant 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. 1 Superpolynomial Size Lower Bounds Complexity theory long ago achieved its goal of presenting interesting and important computational problems that, although computable, nonetheless require such huge circuits to compute that they are computationally intractable. In fact, in Stockmeyer's thesis, the unusual step was taken of translating an asymptotic result into concrete terms: Theorem1. [Sto74] Any circuit that takes as input a formula (in the language of WS1S) with up to 616 symbols and produces as output a correct answer saying whether the formula is valid or not, requires at least 10 gates. To quote from [Sto87]: Even if gates were the size of a proton and were connected by in nitely thin wires, the network would densely ll the known universe. In the intervening years complexity theory has made some progress proving that other problems A require circuits of superpolynomial size (in symbols: A 62 P/poly), but no such A has been shown to exist in nondeterministic exponential ? Supported in part by NSF grant CCR-9509603. time (NTIME(2 O(1) )) or even in the potentially larger class DTIME(2 O(1) )NP. Where can we nd sets that are not in P/poly? A straightforward diagonalization shows that for any superpolynomial time-bound T , there is a problem in DSPACE(T (n)) P/poly. Recall that deterministic space complexity is roughly the same as alternating time complexity [CKS81]. It turns out that the full power of alternation is not needed to obtain sets outside of P/poly { two alternations su ce, as can be shown using techniques of [Kan82] (see also [BH92]). Combined with Toda's theorem [Tod91] we obtain the following. Theorem2. [Kan82, BH92, Tod91] Let T be a time-constructible superpolynomial function. Then { NTIME(T (n))NP 6 P/poly. { DTIME(T (n)) 6 P/poly. A further improvement was reported by Kobler and Watanabe, who showed that even ZPTIME(T (n)) is not contained in P/poly [KW]. (Here, ZPTIME(T (n)) is zero-error probabilistic time T (n).) Is this the best that we can do? To the best of my knowledge, it is not known if the classes PrTIME(2 O(1)n ) (unbounded error probabilistic quasipolynomial time) and DTIME(2 O(1) )= are contained in P/poly (even relative to an oracle). There are oracles relative to which DTIME(2 O(1) ) has polynomial-size circuits [Hel86, Wil85], thus showing that relativizable techniques cannot be used to present superpolynomial circuit size bounds for NTIME(2 O(1) ). Note, however that nonrelativizing techniques have been used on closely-related problems [BFNW93]. More to the point, as reported in [KW], Buhrman and Fortnow and also Thierauf have shown that the exponential-time version of the complexity class MA contains problems outside of P/poly, although this is false relative to some oracles. (In particular, this shows that PrTIME(2 O(1) ) is not contained P/poly.) One can hope that further insights will lead to more progress on this front. In the mean time, it has turned out to be very worthwhile to consider some important subclasses of P/poly. 2 Smaller Circuit Classes We will focus our attention on ve important circuit complexity classes: 1. AC is the class of problems solvable by polynomial-size, constant-depth circuits of AND, OR, and NOT gates of unbounded fan-in. AC corresponds to O(1)-time computation on a parallel computer, and it also consists exactly of the languages that can be speci ed in rst-order logic [Imm89, BIS90]. AC circuits are powerful enough to add and subtract n-bit numbers. 2 Thus this survey will ignore the large body of beautiful work on the circuit complexity of larger subclasses of P and NC. 2. NC1 is the class of problems solvable by circuits of AND, OR, and NOT gates of fan-in two and depth O(logn). NC circuits capture exactly the circuit complexity required to evaluate a Boolean formula [Bus93], and to recognize a regular set [Bar89]. There are deep connections between circuit complexity and algebra, and NC1 corresponds to computation over any non-solvable algebra [Bar89]. 3. ACC0 is the class of problems solvable by polynomial-size, constant-depth circuits of unbounded fan-in AND, OR, NOT, and MODm gates. (A MODm gate takes inputs x1; : : : ; xn and determines if the number of 1's among these inputs is a multiple of m.) To be more precise, AC0(m) is the class of problems solvable by polynomial-size, constant-depth circuits of unbounded fan-in AND, OR, NOT, and MODm gates, and ACC0 = S m AC 0(m). In the algebraic theory mentioned above, ACC corresponds to computation over any solvable algebra [BT88]. Thus in the algebraic theory, ACC is the most natural subclass of NC1. 4. TC is the class of problems solvable by polynomial-size, constant-depth threshold circuits. TC captures exactly the complexity of integer multiplication and division, and sorting [CSV84]. Also, TC is a good complexitytheoretic model for \neural net" computation [PS88, PS89]. 5. NC is the class of problems solvable by circuits of AND, OR, and NOT gates of fan-in two and depth O(1). Note that each output bit can only depend on O(1) input bits in such a circuit. Thus any function in NC is computed by depth two AC circuits, merely using DNF or CNF expansion. NC is obviously extremely limited; such circuits cannot even compute the logical OR of n input bits. One of the surprises of circuit complexity is that, in spite of its severe limitations, NC is in some sense quite \close" to AC in computational power. Quite a few powerful techniques are known for proving lower bounds for AC circuits; it is known that AC is properly contained in ACC. It is not hard to see that ACC TC NC. As we shall see below, weak lower bounds have been proven for ACC and TC, whereas almost nothing is known for NC.