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 combi-natorial criterion for the monotone circuit complexity: a monotone Boolean

A parameterized problem 〈L, k〉 belongs to W [t] if there exists k′ computed from k such that 〈L, k〉 reduces to the weight-k′ satisfiability problem for weft-t circuits. We relate the fundamental question of whether the W [t] hierarchy is proper to parameterized problems for constant-depth circuits. We define classes G[t] as the analogues of AC depth-t for parameterized problems, and N [t] by we...

We study circuit expressions of logarithmic and poly-logarithmic polynomial-time Kolmogorov complexity, focusing on their complexity-theoretic characterizations and learnability properties. They provide a nontrivial circuit-like characterization for a natural nonuniform complexity class that lacked it up to now. We show that circuit expressions of this kind can be learned with membership querie...

In contrast to machine models like Turing machines or random access machines, circuits are a static computational model. The internal information ow of a computation is xed in advance, independent of the actual input. Therefore, the size and the depth are natural and simple measures for circuits and provide a worst case measure. We consider a model where an internal gate can be evaluated when t...

1. Basic properties of Turing Machines (TMs), Circuits & Complexity 2. P, NP, NP-completeness, Cook-Levin Theorem. 3. Hierarchy theorems, Circuit lower bounds. 4. Space complexity: PSPACE, PSPACE-completeness, L, NL, Closure properties 5. Polynomial-time hierarchy 6. #P and counting problems 7. Randomized complexity 8. Circuit lower bounds 9. Interactive proofs & PCP Theorem Hardness of approxi...

This dissertation presents the results of my research in two areas: parallel algorithms/circuit complexity, and algorithmic motion planning. The chapters on circuit complexity examine the parallel complexity of several fundamental problems (such as integer division) in the model of small depth circuits. In the later chapters on motion planning, we turn to the computationally intensive problem o...

The paper aims at tight upper bounds for the size of pattern classification circuits that can be used for a priori parameter settings in a machine learning context. The upper bounds relate the circuit size S(C) to nL := log2 mL , where mL is the number of training samples. In particular, we show that there exist unbounded fanin threshold circuits with less than (a) SR cc := 2 · √ 2nL + 3 gates ...

It is already shown that a Boolean function for a NP-complete problem can be computed by a polynomial-sized circuit if its variables have enough number of automorphisms. Looking at this previous study from the different perspective gives us the idea that the small number of automorphisms might be a barrier for a polynomial time solution for NP-complete problems. Here I show that by interpreting...

Since the publication of Furst, Saxe, and Sipser's seminal paper connecting AC 0 with the polynomial hierarchy FSS84], it has been well known that circuit lower bounds allow you to construct oracles that separate complexity classes. We will show that similar circuit lower bounds allow you to construct oracles that collapse complexity classes. For example, based on H astad's parity lower bound, ...