9 research outputs found
On the parity complexity measures of Boolean functions
The parity decision tree model extends the decision tree model by allowing
the computation of a parity function in one step. We prove that the
deterministic parity decision tree complexity of any Boolean function is
polynomially related to the non-deterministic complexity of the function or its
complement. We also show that they are polynomially related to an analogue of
the block sensitivity. We further study parity decision trees in their
relations with an intermediate variant of the decision trees, as well as with
communication complexity.Comment: submitted to TCS on 16-MAR-200
Efficient quantum protocols for XOR functions
We show that for any Boolean function f on {0,1}^n, the bounded-error quantum
communication complexity of XOR functions satisfies that
, where d is the F2-degree of f, and
.
This implies that the previous lower bound by Lee and Shraibman \cite{LS09} is tight
for f with low F2-degree. The result also confirms the quantum version of the
Log-rank Conjecture for low-degree XOR functions. In addition, we show that the
exact quantum communication complexity satisfies , where is the number of nonzero Fourier coefficients of
f. This matches the previous lower bound
by Buhrman and de Wolf \cite{BdW01} for low-degree XOR functions.Comment: 11 pages, no figur
Truth Table Minimization of Computational Models
Complexity theory offers a variety of concise computational models for
computing boolean functions - branching programs, circuits, decision trees and
ordered binary decision diagrams to name a few. A natural question that arises
in this context with respect to any such model is this:
Given a function f:{0,1}^n \to {0,1}, can we compute the optimal complexity
of computing f in the computational model in question? (according to some
desirable measure).
A critical issue regarding this question is how exactly is f given, since a
more elaborate description of f allows the algorithm to use more computational
resources. Among the possible representations are black-box access to f (such
as in computational learning theory), a representation of f in the desired
computational model or a representation of f in some other model. One might
conjecture that if f is given as its complete truth table (i.e., a list of f's
values on each of its 2^n possible inputs), the most elaborate description
conceivable, then any computational model can be efficiently computed, since
the algorithm computing it can run poly(2^n) time. Several recent studies show
that this is far from the truth - some models have efficient and simple
algorithms that yield the desired result, others are believed to be hard, and
for some models this problem remains open.
In this thesis we will discuss the computational complexity of this question
regarding several common types of computational models. We shall present
several new hardness results and efficient algorithms, as well as new proofs
and extensions for known theorems, for variants of decision trees, formulas and
branching programs