Challenges in computational lower bounds

We draw two incomplete, biased maps of challenges in computational complexity lower bounds

Polynomials that Sign Represent Parity and Descartes' Rule of Signs

A real polynomial $P(X_1,..., X_n)$ sign represents $f: A^n \to \{0,1\}$ if for every $(a_1, ..., a_n) \in A^n$, the sign of $P(a_1,...,a_n)$ equals $(-1)^{f(a_1,...,a_n)}$. Such sign representations are well-studied in computer science and have applications to computational complexity and computational learning theory. In this work, we present a systematic study of tradeoffs between degree and sparsity of sign representations through the lens of the parity function. We attempt to prove bounds that hold for any choice of set $A$. We show that sign representing parity over $\{0,...,m-1\}^n$ with the degree in each variable at most $m-1$ requires sparsity at least $m^n$. We show that a tradeoff exists between sparsity and degree, by exhibiting a sign representation that has higher degree but lower sparsity. We show a lower bound of $n(m -2) + 1$ on the sparsity of polynomials of any degree representing parity over $\{0,..., m-1\}^n$. We prove exact bounds on the sparsity of such polynomials for any two element subset $A$. The main tool used is Descartes' Rule of Signs, a classical result in algebra, relating the sparsity of a polynomial to its number of real roots. As an application, we use bounds on sparsity to derive circuit lower bounds for depth-two AND-OR-NOT circuits with a Threshold Gate at the top. We use this to give a simple proof that such circuits need size $1.5^n$ to compute parity, which improves the previous bound of ${4/3}^{n/2}$ due to Goldmann (1997). We show a tight lower bound of $2^n$ for the inner product function over $\{0,1\}^n \times \{0, 1\}^n$.Comment: To appear in Computational Complexit

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