A Casual Tour Around a Circuit Complexity Bound

I will discuss the recent proof that the complexity class NEXP (nondeterministic exponential time) lacks nonuniform ACC circuits of polynomial size. The proof will be described from the perspective of someone trying to discover it.Comment: 21 pages, 2 figures. An earlier version appeared in SIGACT News, September 201

Algorithms and Lower Bounds for Threshold Circuits

学位記番号:工博甲50

Quantum Branching Programs and Space-Bounded Nonuniform Quantum Complexity

In this paper, the space complexity of nonuniform quantum computations is investigated. The model chosen for this are quantum branching programs, which provide a graphic description of sequential quantum algorithms. In the first part of the paper, simulations between quantum branching programs and nonuniform quantum Turing machines are presented which allow to transfer lower and upper bound results between the two models. In the second part of the paper, different variants of quantum OBDDs are compared with their deterministic and randomized counterparts. In the third part, quantum branching programs are considered where the performed unitary operation may depend on the result of a previous measurement. For this model a simulation of randomized OBDDs and exponential lower bounds are presented.Comment: 45 pages, 3 Postscript figures. Proofs rearranged, typos correcte

Functional Lower Bounds for Restricted Arithmetic Circuits of Depth Four

Recently, Forbes, Kumar and Saptharishi [CCC, 2016] proved that there exists an explicit $d^{O(1)}$-variate and degree $d$ polynomial $P_{d}\in VNP$ such that if any depth four circuit $C$ of bounded formal degree $d$ which computes a polynomial of bounded individual degree $O(1)$, that is functionally equivalent to $P_d$, then $C$ must have size $2^{\Omega(\sqrt{d}\log{d})}$. The motivation for their work comes from Boolean Circuit Complexity. Based on a characterization for $ACC^0$ circuits by Yao [FOCS, 1985] and Beigel and Tarui [CC, 1994], Forbes, Kumar and Saptharishi [CCC, 2016] observed that functions in $ACC^0$ can also be computed by algebraic $\Sigma\mathord{\wedge}\Sigma\Pi$ circuits (i.e., circuits of the form -- sums of powers of polynomials) of $2^{\log^{O(1)}n}$ size. Thus they argued that a $2^{\omega(\log^{O(1)}{n})}$ "functional" lower bound for an explicit polynomial $Q$ against $\Sigma\mathord{\wedge}\Sigma\Pi$ circuits would imply a lower bound for the "corresponding Boolean function" of $Q$ against non-uniform $ACC^0$. In their work, they ask if their lower bound be extended to $\Sigma\mathord{\wedge}\Sigma\Pi$ circuits. In this paper, for large integers $n$ and $d$ such that $\omega(\log^2n)\leq d\leq n^{0.01}$, we show that any $\Sigma\mathord{\wedge}\Sigma\Pi$ circuit of bounded individual degree at most $O\left(\frac{d}{k^2}\right)$ that functionally computes Iterated Matrix Multiplication polynomial $IMM_{n,d}$ ($\in VP$) over $\{0,1\}^{n^2d}$ must have size $n^{\Omega(k)}$. Since Iterated Matrix Multiplication $IMM_{n,d}$ over $\{0,1\}^{n^2d}$ is functionally in $GapL$, improvement of the afore mentioned lower bound to hold for quasipolynomially large values of individual degree would imply a fine-grained separation of $ACC^0$ from $GapL$

Circuit Size Lower Bounds and #SAT Upper Bounds Through a General Framework

Most of the known lower bounds for binary Boolean circuits with unrestricted depth are proved by the gate elimination method. The most efficient known algorithms for the #SAT problem on binary Boolean circuits use similar case analyses to the ones in gate elimination. Chen and Kabanets recently showed that the known case analyses can also be used to prove average case circuit lower bounds, that is, lower bounds on the size of approximations of an explicit function. In this paper, we provide a general framework for proving worst/average case lower bounds for circuits and upper bounds for #SAT that is built on ideas of Chen and Kabanets. A proof in such a framework goes as follows. One starts by fixing three parameters: a class of circuits, a circuit complexity measure, and a set of allowed substitutions. The main ingredient of a proof goes as follows: by going through a number of cases, one shows that for any circuit from the given class, one can find an allowed substitution such that the given measure of the circuit reduces by a sufficient amount. This case analysis immediately implies an upper bound for #SAT. To~obtain worst/average case circuit complexity lower bounds one needs to present an explicit construction of a function that is a disperser/extractor for the class of sources defined by the set of substitutions under consideration. We show that many known proofs (of circuit size lower bounds and upper bounds for #SAT) fall into this framework. Using this framework, we prove the following new bounds: average case lower bounds of 3.24n and 2.59n for circuits over U_2 and B_2, respectively (though the lower bound for the basis B_2 is given for a quadratic disperser whose explicit construction is not currently known), and faster than 2^n #SAT-algorithms for circuits over U_2 and B_2 of size at most 3.24n and 2.99n, respectively. Here by B_2 we mean the set of all bivariate Boolean functions, and by U_2 the set of all bivariate Boolean functions except for parity and its complement