Smaller ACC0 Circuits for Symmetric Functions

What is the power of constant-depth circuits with $MOD_m$ gates, that can count modulo $m$? Can they efficiently compute MAJORITY and other symmetric functions? When $m$ is a constant prime power, the answer is well understood: Razborov and Smolensky proved in the 1980s that MAJORITY and $MOD_m$ require super-polynomial-size $MOD_q$ circuits, where $q$ is any prime power not dividing $m$. However, relatively little is known about the power of $MOD_m$ circuits for non-prime-power $m$. For example, it is still open whether every problem in $EXP$ can be computed by depth-$3$ circuits of polynomial size and only $MOD_6$ gates. We shed some light on the difficulty of proving lower bounds for $MOD_m$ circuits, by giving new upper bounds. We construct $MOD_m$ circuits computing symmetric functions with non-prime power $m$, with size-depth tradeoffs that beat the longstanding lower bounds for $AC^0[m]$ circuits for prime power $m$. Our size-depth tradeoff circuits have essentially optimal dependence on $m$ and $d$ in the exponent, under a natural circuit complexity hypothesis. For example, we show for every $\varepsilon > 0$ that every symmetric function can be computed with depth-3 $MOD_m$ circuits of $\exp(O(n^{\varepsilon}))$ size, for a constant $m$ depending only on $\varepsilon > 0$. That is, depth-$3$ $CC^0$ circuits can compute any symmetric function in \emph{subexponential} size. This demonstrates a significant difference in the power of depth-$3$ $CC^0$ circuits, compared to other models: for certain symmetric functions, depth-$3$ $AC^0$ circuits require $2^{\Omega(\sqrt{n})}$ size [H{\aa}stad 1986], and depth-$3$ $AC^0[p^k]$ circuits (for fixed prime power $p^k$) require $2^{\Omega(n^{1/6})}$ size [Smolensky 1987]. Even for depth-two $MOD_p \circ MOD_m$ circuits, $2^{\Omega(n)}$ lower bounds were known [Barrington Straubing Th\'erien 1990].Comment: 15 pages; abstract edited to fit arXiv requirement

New Lower Bounds and Derandomization for ACC, and a Derandomization-Centric View on the Algorithmic Method

In this paper, we obtain several new results on lower bounds and derandomization for ACC? circuits (constant-depth circuits consisting of AND/OR/MOD_m gates for a fixed constant m, a frontier class in circuit complexity): 1) We prove that any polynomial-time Merlin-Arthur proof system with an ACC? verifier (denoted by MA_{ACC?}) can be simulated by a nondeterministic proof system with quasi-polynomial running time and polynomial proof length, on infinitely many input lengths. This improves the previous simulation by [Chen, Lyu, and Williams, FOCS 2020], which requires both quasi-polynomial running time and proof length. 2) We show that MA_{ACC?} cannot be computed by fixed-polynomial-size ACC? circuits, and our hard languages are hard on a sufficiently dense set of input lengths. 3) We show that NEXP (nondeterministic exponential-time) does not have ACC? circuits of sub-half-exponential size, improving the previous sub-third-exponential size lower bound for NEXP against ACC? by [Williams, J. ACM 2014]. Combining our first and second results gives a conceptually simpler and derandomization-centric proof of the recent breakthrough result NQP := NTIME[2^polylog(n)] ? ? ACC? by [Murray and Williams, SICOMP 2020]: Instead of going through an easy witness lemma as they did, we first prove an ACC? lower bound for a subclass of MA, and then derandomize that subclass into NQP, while retaining its hardness against ACC?. Moreover, since our derandomization of MA_{ACC?} achieves a polynomial proof length, we indeed prove that nondeterministic quasi-polynomial-time with n^?(1) nondeterminism bits (denoted as NTIMEGUESS[2^polylog(n), n^?(1)]) has no poly(n)-size ACC? circuits, giving a new proof of a result by Vyas. Combining with a win-win argument based on randomized encodings from [Chen and Ren, STOC 2020], we also prove that NTIMEGUESS[2^polylog(n), n^?(1)] cannot be 1/2+1/poly(n)-approximated by poly(n)-size ACC? circuits, improving the recent strongly average-case lower bounds for NQP against ACC? by [Chen and Ren, STOC 2020]. One interesting technical ingredient behind our second result is the construction of a PSPACE-complete language that is paddable, downward self-reducible, same-length checkable, and weakly error correctable. Moreover, all its reducibility properties have corresponding AC?[2] non-adaptive oracle circuits. Our construction builds and improves upon similar constructions from [Trevisan and Vadhan, Complexity 2007] and [Chen, FOCS 2019], which all require at least TC? oracle circuits for implementing these properties

On circuit complexity classes and iterated matrix multiplication

In this thesis, we study small, yet important, circuit complexity classes within NC^1, such as ACC^0 and TC^0. We also investigate the power of a closely related problem called Iterated Matrix Multiplication and its implications in low levels of algebraic complexity theory. More concretely, 1. We show that extremely modest-sounding lower bounds for certain problems can lead to non-trivial derandomization results. a. If the word problem over S_5 requires constant-depth threshold circuits of size n^{1+epsilon} for some epsilon > 0, then any language accepted by uniform polynomial-size probabilistic threshold circuits can be solved in subexponential time (and more strongly, can be accepted by a uniform family of deterministic constant-depth threshold circuits of subexponential size.) b. If there are no constant-depth arithmetic circuits of size n^{1+epsilon} for the problem of multiplying a sequence of n 3-by-3 matrices, then for every constant d, black-box identity testing for depth-d arithmetic circuits with bounded individual degree can be performed in subexponential time (and even by a uniform family of deterministic constant-depth AC circuits of subexponential size). 2. ACC_m circuits are circuits consisting of unbounded fan-in AND, OR and MOD_m gates and unary NOT gates, where m is a fixed integer. We show that there exists a language in non-deterministic exponential time which can not be computed by any non-uniform family of ACC_m circuits of quasi-polynomial size and o(loglog n) depth, where $m$ is an arbitrarily chosen constant. 3. We show that there are families of polynomials having small depth-two arithmetic circuits that cannot be expressed by algebraic branching programs of width two. This clarifies the complexity of the problem of computing the product of a sequence of two-by-two matrices, which arises in several settings.Ph. D.Includes bibliographical referencesIncludes vitaby Fengming Wan