In this lecture we prove a lower bound on the size of a constant depth circuit which computes the XOR of n bits. Before we talk about bounds on the size of a circuit, let us first clarify what we mean by circuit depth and circuit size. The depth of a circuit is defined as the length of the longest path from the input to output. The size of a circuit is the number of AND and OR gates in the circuit. Note that, for our purpose, we assume all the gates have unlimited fan-in and fan-out. We define AC 0 to be the class of decision problems solvable by circuits of polynomial size and constant depth. We want to prove the result that PARITY is not in AC 0. There are two known techniques to prove this result. In this class, we will talk about a proo...
We study depth 3 circuits of the form OR-AND-XOR, or equivalently -- DNF of parities. This model was...
. We prove that constant depth circuits of size n log O(1) n over the basis AND, OR, PARITY are ...
International audienceAn Algebraic Circuit for a polynomial P is a computational model for construct...
Boolean circuits were introduced in complexity theory to provide a model for parallel computation. A...
AbstractAlmost everything is known on the complexity of the parity function in fan-in 2 circuits ove...
Boolean circuits were introduced in complexity theory to provide a model for parallel computation. A...
This dissertation presents some circuit complexity results and techniques. Circuit complexity is a b...
[[abstract]]We consider small depth boolean circuits with basis {AND, OR, NOT}. We obtain lower boun...
We give a deterministic algorithm for counting the number of satisfying assignments of any AC^0[oplu...
We show average-case lower bounds for explicit Boolean functions against bounded-depth threshold cir...
We study the complexity of computing symmetric and threshold functions by constant-depth circuits wi...
Abstract: "This report provides a complete exposition of the main proof in Johan Håstad's thesis [...
Minimum Circuit Size Problem (MCSP) asks to decide if a given truth table of an n-variate boolean fu...
The boolean circuit complexity classes AC 0 ⊆ AC 0[m] ⊆ TC 0 ⊆ NC 1 have been studied intensely. Oth...
AbstractExponential size lower bounds are obtained for some depth three circuits computing conjuncti...
We study depth 3 circuits of the form OR-AND-XOR, or equivalently -- DNF of parities. This model was...
. We prove that constant depth circuits of size n log O(1) n over the basis AND, OR, PARITY are ...
International audienceAn Algebraic Circuit for a polynomial P is a computational model for construct...
Boolean circuits were introduced in complexity theory to provide a model for parallel computation. A...
AbstractAlmost everything is known on the complexity of the parity function in fan-in 2 circuits ove...
Boolean circuits were introduced in complexity theory to provide a model for parallel computation. A...
This dissertation presents some circuit complexity results and techniques. Circuit complexity is a b...
[[abstract]]We consider small depth boolean circuits with basis {AND, OR, NOT}. We obtain lower boun...
We give a deterministic algorithm for counting the number of satisfying assignments of any AC^0[oplu...
We show average-case lower bounds for explicit Boolean functions against bounded-depth threshold cir...
We study the complexity of computing symmetric and threshold functions by constant-depth circuits wi...
Abstract: "This report provides a complete exposition of the main proof in Johan Håstad's thesis [...
Minimum Circuit Size Problem (MCSP) asks to decide if a given truth table of an n-variate boolean fu...
The boolean circuit complexity classes AC 0 ⊆ AC 0[m] ⊆ TC 0 ⊆ NC 1 have been studied intensely. Oth...
AbstractExponential size lower bounds are obtained for some depth three circuits computing conjuncti...
We study depth 3 circuits of the form OR-AND-XOR, or equivalently -- DNF of parities. This model was...
. We prove that constant depth circuits of size n log O(1) n over the basis AND, OR, PARITY are ...
International audienceAn Algebraic Circuit for a polynomial P is a computational model for construct...