On the power of homogeneous depth 4 arithmetic circuits

We prove exponential lower bounds on the size of homogeneous depth 4 arithmetic circuits computing an explicit polynomial in $VP$. Our results hold for the {\it Iterated Matrix Multiplication} polynomial - in particular we show that any homogeneous depth 4 circuit computing the $(1,1)$ entry in the product of $n$ generic matrices of dimension $n^{O(1)}$ must have size $n^{\Omega(\sqrt{n})}$. Our results strengthen previous works in two significant ways. Our lower bounds hold for a polynomial in $VP$. Prior to our work, Kayal et al [KLSS14] proved an exponential lower bound for homogeneous depth 4 circuits (over fields of characteristic zero) computing a poly in $VNP$. The best known lower bounds for a depth 4 homogeneous circuit computing a poly in $VP$ was the bound of $n^{\Omega(\log n)}$ by [LSS, KLSS14].Our exponential lower bounds also give the first exponential separation between general arithmetic circuits and homogeneous depth 4 arithmetic circuits. In particular they imply that the depth reduction results of Koiran [Koi12] and Tavenas [Tav13] are tight even for reductions to general homogeneous depth 4 circuits (without the restriction of bounded bottom fanin). Our lower bound holds over all fields. The lower bound of [KLSS14] worked only over fields of characteristic zero. Prior to our work, the best lower bound for homogeneous depth 4 circuits over fields of positive characteristic was $n^{\Omega(\log n)}$ [LSS, KLSS14]

Resolution over Linear Equations and Multilinear Proofs

We develop and study the complexity of propositional proof systems of varying strength extending resolution by allowing it to operate with disjunctions of linear equations instead of clauses. We demonstrate polynomial-size refutations for hard tautologies like the pigeonhole principle, Tseitin graph tautologies and the clique-coloring tautologies in these proof systems. Using the (monotone) interpolation by a communication game technique we establish an exponential-size lower bound on refutations in a certain, considerably strong, fragment of resolution over linear equations, as well as a general polynomial upper bound on (non-monotone) interpolants in this fragment. We then apply these results to extend and improve previous results on multilinear proofs (over fields of characteristic 0), as studied in [RazTzameret06]. Specifically, we show the following: 1. Proofs operating with depth-3 multilinear formulas polynomially simulate a certain, considerably strong, fragment of resolution over linear equations. 2. Proofs operating with depth-3 multilinear formulas admit polynomial-size refutations of the pigeonhole principle and Tseitin graph tautologies. The former improve over a previous result that established small multilinear proofs only for the \emph{functional} pigeonhole principle. The latter are different than previous proofs, and apply to multilinear proofs of Tseitin mod p graph tautologies over any field of characteristic 0. We conclude by connecting resolution over linear equations with extensions of the cutting planes proof system.Comment: 44 page

On Computing Multilinear Polynomials Using Multi-r-ic Depth Four Circuits

International audienceIn this paper, we are interested in understanding the complexity of computing multilinear polynomials using depth four circuits in which polynomial computed at every node has a bound on the individual degree of r (referred to as multi-r-ic circuits). The goal of this study is to make progress towards proving superpolynomial lower bounds for general depth four circuits computing multilinear polynomials, by proving better and better bounds as the value of r increases. Recently, Kayal, Saha and Tavenas (Theory of Computing, 2018) showed that any depth four arithmetic circuit of bounded individual degree r computing a multilinear polynomial on n^O(1) variables and degree d = o(n), must have size at least (n/r^1.1)^{\sqrt{d/r}} when r is o(d) and is strictly less than n^1/1.1. This bound however deteriorates with increasing r. It is a natural question to ask if we can prove a bound that does not deteriorate with increasing r or a bound that holds for a larger regime of r. We here prove a lower bound which does not deteriorate with r , however for a specific instance of d = d (n) but for a wider range of r. Formally, we show that there exists an explicit polynomial on n^{O(1)} variables and degree Θ(log^2(n)) such that any depth four circuit of bounded individual degree r < n^0.2 must have size at least exp(Ω (log^2 n)). This improvement is obtained by suitably adapting the complexity measure of Kayal et al. (Theory of Computing, 2018). This adaptation of the measure is inspired by the complexity measure used by Kayal et al. (SIAM J. Computing, 2017)

On the Limits of Depth Reduction at Depth 3 Over Small Finite Fields

Recently, Gupta et.al. [GKKS2013] proved that over Q any $n^{O(1)}$-variate and $n$-degree polynomial in VP can also be computed by a depth three $\Sigma\Pi\Sigma$ circuit of size $2^{O(\sqrt{n}\log^{3/2}n)}$. Over fixed-size finite fields, Grigoriev and Karpinski proved that any $\Sigma\Pi\Sigma$ circuit that computes $Det_n$ (or $Perm_n$) must be of size $2^{\Omega(n)}$ [GK1998]. In this paper, we prove that over fixed-size finite fields, any $\Sigma\Pi\Sigma$ circuit for computing the iterated matrix multiplication polynomial of $n$ generic matrices of size $n\times n$, must be of size $2^{\Omega(n\log n)}$. The importance of this result is that over fixed-size fields there is no depth reduction technique that can be used to compute all the $n^{O(1)}$-variate and $n$-degree polynomials in VP by depth 3 circuits of size $2^{o(n\log n)}$. The result [GK1998] can only rule out such a possibility for depth 3 circuits of size $2^{o(n)}$. We also give an example of an explicit polynomial ($NW_{n,\epsilon}(X)$) in VNP (not known to be in VP), for which any $\Sigma\Pi\Sigma$ circuit computing it (over fixed-size fields) must be of size $2^{\Omega(n\log n)}$. The polynomial we consider is constructed from the combinatorial design. An interesting feature of this result is that we get the first examples of two polynomials (one in VP and one in VNP) such that they have provably stronger circuit size lower bounds than Permanent in a reasonably strong model of computation. Next, we prove that any depth 4 $\Sigma\Pi^{[O(\sqrt{n})]}\Sigma\Pi^{[\sqrt{n}]}$ circuit computing $NW_{n,\epsilon}(X)$ (over any field) must be of size $2^{\Omega(\sqrt{n}\log n)}$. To the best of our knowledge, the polynomial $NW_{n,\epsilon}(X)$ is the first example of an explicit polynomial in VNP such that it requires $2^{\Omega(\sqrt{n}\log n)}$ size depth four circuits, but no known matching upper bound

Subclasses of Presburger Arithmetic and the Weak EXP Hierarchy

It is shown that for any fixed $i>0$, the $\Sigma_{i+1}$-fragment of Presburger arithmetic, i.e., its restriction to $i+1$ quantifier alternations beginning with an existential quantifier, is complete for $\mathsf{\Sigma}^{\mathsf{EXP}}_{i}$, the $i$-th level of the weak EXP hierarchy, an analogue to the polynomial-time hierarchy residing between $\mathsf{NEXP}$ and $\mathsf{EXPSPACE}$. This result completes the computational complexity landscape for Presburger arithmetic, a line of research which dates back to the seminal work by Fischer & Rabin in 1974. Moreover, we apply some of the techniques developed in the proof of the lower bound in order to establish bounds on sets of naturals definable in the $\Sigma_1$-fragment of Presburger arithmetic: given a $\Sigma_1$-formula $\Phi(x)$, it is shown that the set of non-negative solutions is an ultimately periodic set whose period is at most doubly-exponential and that this bound is tight.Comment: 10 pages, 2 figure