It is a known approach to translate propositional formulas into systems of polynomial inequalities and to consider proof systems for the latter ones. The well-studied proof systems of this kind are the Cutting Planes proof system (CP) utilizing linear inequalities and the Lovasz-Schrijver calculi (LS) utilizing quadratic inequalities. We introduce generalizations LS^d of LS that operate with polynomial inequalities of degree at most d. It turns out

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...

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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,...

here ¯(f) stands for the size of the largest minterm of f and Mn is the set of all monotone Boolean functions on n variables. Definition: A sequence of functions [f n ] = ff 1 ; f 2 ; : : :g is called to be (m; d)-local (with respect to polynomial-size monotone circuits) if there exists a sequence of circuits [Cn ] = fC 1 ; C 2 ; : : :g such that, for each n (i) Cn is over the basis M m;d , (ii) Cn computes f , and (iii) size(Cn ) n O(1) . Theorem 1 (Razborov [1,2]): Let k<F30.84

This dissertation looks at problems in complexity theory; specifically, we are interested in finding explicit Boolean functions which are provably hard to compute in some reasonable model of computation. Although functions requiring exponential resources are known to be plentiful by elementary counting arguments [55], the problem of actually proving such lower bounds for explicit functions, say functions computable in NP, is much more di#cult. For a function in NP, the best known lower bound on circuit size are only linear [2], the best known lower bounds on circuit depth are only logarithmic