Circuit Complexity b efore the Dawn of the New Millenniu m ? Eric Allender Department of Computer Science Rutgers University P.O. Box 1179 Piscataway, NJ 08855-1179 USA [email protected] http://www.cs.rutgers.edu/ allender Abstract. The 1980's saw rapid and exciting development of techniques for proving lower b ounds in circuit complexity. This pace has slowed re- cently, and there has even b een work indicating that quite di erent pro of techniques must b e employed to advance b eyond the current frontier of circuit lower b ounds. Although this has engendered p essimism in some quarters, there have in fact b een many p ositive developments in the past few years showing that signi cant progress is p ossible on many fronts. This pap er is a necessarily incomplete survey of the state of circuit complexityasweawait the dawn of the new millenni um. 1 Sup erp olynomial Size Lower Bounds Complexity theory long ago achieved its goal of presenting interesting and im- p ortant computational problems that, although computable, nonetheless require suchhuge circuits to compute that they are computationally intractable. In fact, in Sto ckmeyer's thesis, the unusual step was taken of translating an asymptotic result into concrete terms: Theorem 1. [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 123 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 sup erp olynomial size in symb ols: A 62 P/p oly, but no such A has b een shown to exist in nondeterministic exp onential ? Supp orted in part by NSF grant CCR-9509603. O 1 O 1 n n NP time NTIME 2 or even in the p otentially larger class DTIME2 . Where can we nd sets that are not in P/p oly? A straightforward diagonaliza- tion shows that for any sup erp olynomial time-b ound T , there is a problem in DSPACET n P/p oly. Recall that deterministic space complexity is roughly the same as alternating time complexity [CKS81]. It turns out that the full p ower of alternation is not needed to obtain sets outside of P/p oly { two alternations suce, as can b e shown using techniques of [Kan82] see also [BH92]. Combined with To da's theorem [To d91]we obtain the following. Theorem 2. [Kan82 , BH92, Tod91] Let T be a time-constructible superpolyno- mial function. Then NP { NTIMET n 6 P/p oly. PP { DTIMET n 6 P/p oly. A further improvementwas rep orted byKobler and Watanab e, who showed NP that even ZPTIME T n is not contained in P/p oly [KW]. Here, ZPTIME T n is zero-error probabilistic time T n. Is this the b est that we can do? To the b est of my knowledge, it is not known O 1n log if the classes PrTIME 2 unb ounded error probabilistic quasip olynomial O 1 n C P = time and DTIME2 are contained in P/p oly even relative to an ora- O 1 n NP cle. There are oracles relative to which DTIME2 has p olynomial-size circuits [Hel86, Wil85], thus showing that relativizable techniques cannot b e used O 1 n to present sup erp olynomial circuit size b ounds for NTIME 2 . Note, how- ever that nonrelativizing techniques have b een used on closely-related problems [BFNW93]. More to the p oint, as rep orted in [KW], Buhrman and Fortnow and also Thierauf have shown that the exp onential-time version of the complexity class MA contains problems outside of P/p oly, although this is false relative to O 1 n some oracles. In particular, this shows that PrTIME2 is not contained P/p oly. One can hop e that further insights will lead to more progress on this front. In the mean time, it has turned out to b e very worthwhile to consider some imp ortant sub classes of P/p oly. 2 Smaller Circuit Classes 2 We will fo cus our attention on ve imp ortant circuit complexity classes: 0 1. AC is the class of problems solvable by p olynomial-size, constant-depth 0 circuits of AND, OR, and NOT gates of unb ounded fan-in. AC corresp onds to O 1-time computation on a parallel computer, and it also consists exactly of the languages that can b e sp eci ed in rst-order logic [Imm89, BIS90]. 0 AC circuits are p owerful enough to add and subtract n-bit numb ers. 2 Thus this survey will ignore the large b o dy of b eautiful work on the circuit complexity of larger sub classes of P and NC. 1 2. NC is the class of problems solvable by circuits of AND, OR, and NOT gates 1 of fan-in two and depth O log n. NC circuits capture exactly the circuit complexity required to evaluate a Bo olean formula [Bus93], and to recognize a regular set [Bar89]. There are deep connections b etween circuit complexity 1 and algebra, and NC corresp onds to computation over any non-solvable algebra [Bar89]. 0 3. ACC is the class of problems solvable by p olynomial-size, constant-depth circuits of unb ounded fan-in AND, OR, NOT, and MODm gates. A MODm gate takes inputs x ;:::;x and determines if the numb er of 1's among 1 n 0 these inputs is a multiple of m. To b e more precise, AC m is the class of problems solvable by p olynomial-size, constant-depth circuits of unb ounded S 0 0 AC m. In fan-in AND, OR, NOT, and MODm gates, and ACC = m 0 the algebraic theory mentioned ab ove, ACC corresp onds to computation 0 over any solvable algebra [BT88]. Thus in the algebraic theory,ACC is the 1 most natural sub class of NC . 0 4. TC is the class of problems solvable by p olynomial-size, constant-depth 0 threshold circuits. TC captures exactly the complexityofinteger multipli- 0 cation and division, and sorting [CSV84]. Also, TC is a go o d complexity- theoretic mo del for \neural net" computation [PS88, PS89]. 0 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 dep end on 0 O 1 input bits in such a circuit. Thus any function in NC is computed by 0 depth twoAC circuits, merely using DNF or CNF expansion. 0 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 0 0 its severe limitations, NC is in some sense quite \close" to AC in computational power. 0 Quite a few p owerful techniques are known for proving lower b ounds for AC 0 0 circuits; it is known that AC is prop erly contained in ACC . It is not hard to 0 0 1 see that ACC TC NC .Aswe shall see b elow, weak lower b ounds have 0 0 1 b een proven for ACC and TC , whereas almost nothing is known for NC . 0 3 AC A dramatic series of pap ers in the 1980's [Ajt83, FSS84, Cai89,Yao85,Has87] 0 gave us a pro of that AC circuits require exp onential size even to determine if the numb er of 1's in the input is o dd or even. See also the excellent tuto- 0 rial [BS90]. The main to ol in proving this and other lower b ounds for AC is Hastad's Switching Lemma, one version of which states that most of the \sub- 0 0 functions" of anyAC function f are in NC . A sub-function of f is obtained by setting most of the n input bits to 0 or 1, leaving a function of the n remaining unset bits. Such a sub-function is called a restriction of f . An interesting new pro of of the Switching Lemma was presented by [Raz95] see also [FL95, AAR], and further extensions were presented by [Bea], the latter motivated in partic- ular by the usefulness of the Switching Lemma as a to ol in proving b ounds on the length of prop ositional pro ofs. Although the switching lemma is the most p owerful to ol wehave for prov- 0 ing lower b ounds for AC , it is not the only one. Lower b ound arguments were presented in [Rad94, HJP93] for depth three circuits, and a notion of determin- istic restriction was presented in [CR96] that is useful for proving nonlinear size b ounds. It is imp ortant to note that, although the Switching Lemma tells us that 0 any function f in AC is \close to" functions computed by depth two circuits since most restrictions of f are computed in depth two, it also provides the to ols to show that for all k , there are depth k + 1 circuits of linear size that require exp onential size to simulate with depth k circuits [Has87]. This is in sharp contrast to the class of circuits considered in the next section, where ecient depth reduction is p ossible.