. The first result concerns branching programs having width (log n) O(1) . We give an \Omega\Gamma n log n= log log n) lower bound for the size of such branching programs computing almost any symmetric Boolean function and in particular the following explicit function: "the sum of the input variables is a quadratic residue mod p" where p is any given prime between n 1=4 and n 1=3 . This is a strengthening of previous nonlinear lower bounds obtained by Chandra, Furst, Lipton and by Pudl'ak. We mention that by iterating our method the result can be further strengthened to \Omega\Gamma n log n). The second result is a C n lower bound for read-once-only branching programs computing an explicit Boolean function. For n = \Gamma v 2 \Delta , the function computes the parity of the number of triangles in a graph on v vertices. This improves previous exp(c p n) lower bounds for other graph functions by Wegener and Z'ak. The result implies a linear lower bound for the space comp...

We define a first-order extension LK(CP) of the cutting planes proof system CP as the first-order sequent calculus LK whose atomic formulas are CP-inequalities P i a i \Delta x i b (x i 's variables, a i 's and b constants). We prove an interpolation theorem for LK(CP) yielding as a corollary a conditional lower bound for LK(CP)-proofs. For a subsystem R(CP) of LK(CP), essentially resolution working with clauses formed by CPinequalities, we prove a monotone interpolation theorem obtaining thus an unconditional lower bound (depending on the maximum size of coefficients in proofs and on the maximum number of CP-inequalities in clauses). We also give an interpolation theorem for polynomial calculus working with sparse polynomials. The proof relies on a universal interpolation theorem for semantic derivations [16, Thm. 5.1]. LK(CP) can be viewed as a two-sorted first-order theory of Z considered itself as a discretely ordered Z-module. One sort of variables are module ele...

We propose a symmetric version of Razborov's method of approximation to prove lower bounds for monotone circuit complexity. Traditionally, only DNF formulas have been used as approximators, whereas we use both CNF and DNF formulas. As a consequence we no longer need the Sunflower lemma that has been essential for the method of approximation. The new approximation...

Both the bottleneck counting argument [7, 8] and Razborov's approximation method [1, 4, 12] have been used to prove exponential lower bounds for monotone circuits. We show that under the monotone circuit model for every proof by the approximation method, there is a bottleneck counting proof and vice versa. We also illustrate the elegance of the bottleneck counting technique with a simple self-explained example: the proof of a (previously known) lower bound for the 3-CLIQUEn problem by the bottleneck counting argument. Keywords: Computational complexity; Circuit complexity; Monotone circuit complexity 1 Introduction Razborov's proof of an exponential lower bound on the size of monotone Boolean circuits to detect cliques in a graph [12] [1], represented a breakthrough in the theory of monotone circuit complexity. The proof introduced the method of approximation. The method is roughly as follows. Consider two sets of test inputs, a positive (the output is 1) and a negative one. Gi...

This paper coins the term "The Fusion Method" to a recent approach for proving circuit lower bounds. It describes the method, and surveys its achievements, potential and challenges. 1 Introduction In a recent paper, Karchmer [6] suggested an elegant way in which one can view at the same time both the "approximation method" of Razborov [13] and the "topological approach" of Sipser [15] for proving circuit lower bounds. In Karchmer's setting the lower bound prover shows that a given circuit C is too small for computing a given function f by contradiction, in the following way. She tries to combine (or 'fuse', as we propose calling it) correct accepting computations of inputs in f \Gamma1 (1) by C into an incorrect accepting computation of an input in f \Gamma1 (0). It turns out that this "Fusion Method" reduces the dynamic computation of f by C into a static combinatorial cover problem, which provides the lower bound. Moreover, different restrictions on how we can fuse computations ...

this paper. References

Our main result is a combinatorial lower bounds criterion for a general model of monotone circuits, where we allow as gates: (i) arbitrary monotone Boolean functions whose minterms or maxterms (or both) have length 6 d, and (ii) arbitrary real-valued non-decreasing functions on 6 d variables. This resolves a problem, raised by Razborov in 1986, and yields, in a uniform and easy way, non-trivial lower bounds for circuits computing explicit functions even when d !1. The proof is relatively simple and direct, and combines the bottlenecks counting method of Haken with the idea of finite limit due to Sipser. We demonstrate the criterion by super-polynomial lower bounds for explicit Boolean functions, associated with bipartite Paley graphs and partial t-designs. We then derive exponential lower bounds for clique-like graph functions of Tardos, thus establishing an exponential gap between the monotone real and non-monotone Boolean circuit complexities. Since we allow real gates, the criterion...

We prove a lower bound of 2\Omega i ( n log n ) 1 3 j on the monotone size of an explicit function in monotone-NP (where n is the number of input variables). This is higher than any previous lower bound on the monotone size of a function. The previous best being a lower bound of about 2 \Omega\Gamma n 1 4 ) for Andreev's function, proved in [AlBo87]. Our lower bound is proved by the symmetric version of Razborov's method of approximations. However, we present this method in a new and simpler way: Rather than building approximator functions for all the gates in a circuit, we use a gate elimination argument that is based on a Monotone Switching Lemma. The bound applies for a family of functions, each defined by a construction of a small probability space of c-wise independent random variables. 1 Introduction 1.1 Background and Previous Work A monotone function is one that can be computed by a monotone circuit i.e., a circuit with only AND and OR gates. The monoton...

Our main result is a combinatorial lower bounds criterion for monotone circuits over the reals. We allow any unbounded fanin non-decreasing real-valued functions as gates. The only requirement is their "locality ". Unbounded fanin AND and OR gates, as well as any threshold gate T m s (x 1 ; : : : ; xm ) with small enough threshold value minfs; m \Gamma s + 1g, are simplest examples of local gates. The proof is relatively simple and direct, and combines the bottlenecks counting approach of Haken with the idea of finite limit due to Sipser. Apparently this is the first combinatorial lower bounds criterion for monotone computations. It is symmetric and yields (in a uniform and easy way) exponential lower bounds. 1. Introduction The question of determining how much economy the universal non-monotone basis f; ; :g provides over the monotone basis f; g has been a long standing open problem in Boolean circuit complexity. The The work was supported by a DFG grant Me 1077/10-1. Preliminary...

In this paper we study the lower bounds problem for monotone circuits. The main goal is to extend and simplify the well known method of approximations proposed by A. Razborov in 1985. The main result is the following combinatorial criterion for the monotone circuit complexity: a monotone Boolean function f(X) of n variables X = fx 1 ; : : : ; x n g requires monotone circuits of size exp(\Omega\Gamma t= log t)) if there is a family F ` 2 X such that: (i) each set in F is either a minterm or a maxterm of f; and (ii) D k (F)=D k+1 (F) t for every k = 0; 1; : : : ; t \Gamma 1: Here D k (F) is the k-th degree of F , i.e. maximum cardinality of a subfamily H ` F with j " Hj k: 1 Introduction The question of determining how much economy the universal non-monotone basis f; ; :g provides over the monotone basis f; g has been a long standing open problem in Boolean circuit complexity. In 1985, Razborov [10, 11] achieved a major development in this direction. He worked out the, so-called,...