Revision #11 Authors: Dmitry Gavinsky, Or Meir, Omri Weinstein, Avi Wigderson

Accepted on: 23rd October 2017 19:33

Downloads: 197

Keywords:

One of the major open problems in complexity theory is proving super-logarithmic lower bounds on the depth of circuits (i.e., $\mathbf{P} \subseteq \mathbf{NC}^1$). This problem is interesting for two reasons: first, it is tightly related to understanding the power of parallel computation and of small-space computation; second, it is one of the first milestones toward proving super-polynomial circuit lower bounds.

Karchmer, Raz, and Wigderson [KRW95] suggested to approach this problem by proving the following conjecture: given two Boolean

functions $f$ and $g$, the depth complexity of the composed function $g\diamond f$ is roughly the sum of the depth complexities of $f$ and $g$. They showed that the validity of this conjecture would imply that $\mathbf{P} \subseteq \mathbf{NC}^1$.

As a starting point for studying the composition of functions, they introduced a relation called ``the universal relation'', and suggested to study the composition of universal relations. This suggestion proved fruitful, and an analogue of the KRW conjecture for the universal relation was proved by Edmonds et. al. [EIRS01]. An alternative proof was given later by Hastad and Wigderson [HW93]. However, studying the composition of functions seems more difficult, and the KRW conjecture is still wide open.

In this work, we make a natural step in this direction, which lies between what is known and the original conjecture: we show that an analogue of the conjecture holds for the composition of a function with a universal relation.

Now gives proper credit to Hastad for the efficient randomized protocol that solves KW relations with an unbounded number of rounds.

Revision #10 Authors: Dmitry Gavinsky, Or Meir, Omri Weinstein, Avi Wigderson

Accepted on: 20th August 2016 00:30

Downloads: 356

Keywords:

One of the major open problems in complexity theory is proving super-logarithmic lower bounds on the depth of circuits (i.e., $\P\not\subseteq\NCo$). This problem is interesting for two reasons: first, it is tightly related to understanding the power of parallel computation and of small-space computation; second, it is one of the first milestones toward proving super-polynomial circuit lower bounds.

Karchmer, Raz, and Wigderson [KRW95] suggested to approach this problem by proving the following conjecture: given two Boolean

functions $f$ and~$g$, the depth complexity of the composed function $g\diamond f$ is roughly the sum of the depth complexities of $f$ and $g$. They showed that the validity of this conjecture would imply that $\P\not\subseteq\NCo$.

As a starting point for studying the composition of functions, they introduced a relation called ``the universal relation'', and suggested

to study the composition of universal relations. This suggestion proved fruitful, and an analogue of the KRW conjecture for the universal relation was proved by Edmonds et. al. [EIRS01]. An alternative proof was given later by H{\aa}stad and Wigderson [HW93]. However, studying the composition of functions seems more difficult, and the KRW conjecture is still wide open.

In this work, we make a natural step in this direction, which lies between what is known and the original conjecture: we show that an analogue of the conjecture holds for the composition of a function

with a universal relation.

Fixes following the journal version.

Revision #9 Authors: Dmitry Gavinsky, Or Meir, Omri Weinstein, Avi Wigderson

Accepted on: 23rd April 2015 21:36

Downloads: 576

Keywords:

One of the major open problems in complexity theory is proving super-polynomial lower bounds for circuits with logarithmic depth (i.e., $\mathbf{P}\not\subseteq\mathbf{NC}^1~$). This problem is interesting for two reasons: first, it is tightly related to understanding the power of parallel computation and of small-space computation; second, it is one of the first milestones toward proving super-polynomial circuit lower bounds.

Karchmer, Raz, and Wigderson [KRW95] suggested to approach this problem by proving the following conjecture: given two boolean functions $f$ and $g$, the depth complexity of the composed function $g\circ f$ is roughly the sum of the depth complexities of $f$ and $g$. They showed that the validity of this conjecture would imply that $\mathbf{P}\not\subseteq\mathbf{NC}^1~$.

As a starting point for studying the composition of functions, they introduced a relation called “the universal relation”, and suggested to study the composition of universal relations. This suggestion proved fruitful, and an analogue of the KRW conjecture for the universal relation was proved by Edmonds et. al. [EIRS01]. An alternative proof was given later by Hastad and Wigderson [HW93]. However, studying the composition of functions seems more difficult, and the KRW conjecture is still wide open.

In this work, we make a natural step in this direction, which lies between what is known and the original conjecture: we show that an analogue of the conjecture holds for the composition of a function with a universal relation. We also suggest a candidate for the next step and provide initial results toward it.

Our main technical contribution is developing an approach based on the notion of information complexity for analyzing KW relations - communication problems that are closely related to questions on circuit depth and formula complexity. Recently, information complexity has proved to be a powerful tool, and underlined some major progress on several long-standing open problems in communication complexity. In this work, we develop general tools for analyzing the information complexity of KW relations, which may be of independent interest.

Revision #8 Authors: Dmitry Gavinsky, Or Meir, Omri Weinstein, Avi Wigderson

Accepted on: 29th May 2014 23:57

Downloads: 694

Keywords:

One of the major open problems in complexity theory is proving super-polynomial lower bounds for circuits with logarithmic depth (i.e., $\mathbf{P}\not\subseteq\mathbf{NC}_1~$). This problem is interesting for two reasons: first, it is tightly related to understanding the power of parallel computation and of small-space computation; second, it is one of the first milestones toward proving super-polynomial circuit lower bounds.

One of the major open problems in complexity theory is proving super-logarithmic lower bounds on the depth of circuits (i.e., $\mathbf{P}\not\subseteq\mathbf{NC}_1~$). This problem is interesting for two reasons: first, it is tightly related to understanding the power of parallel computation and of small-space computation; second, it is one of the first milestones toward proving super-polynomial circuit lower bounds.

Karchmer, Raz, and Wigderson [KRW95] suggested to approach this problem by proving the following conjecture: given two boolean functions $f$ and $g$, the depth complexity of the composed function $g\circ f$ is roughly the sum of the depth complexities of $f$ and $g$. They showed that the validity of this conjecture would imply that $\mathbf{P}\not\subseteq\mathbf{NC}_1~$.

As a starting point for studying the composition of functions, they introduced a relation called “the universal relation”, and suggested to study the composition of universal relations. This suggestion proved fruitful, and an analogue of the KRW conjecture for the universal relation was proved by Edmonds et. al. [EIRS01]. An alternative proof was given later by H{\aa}stad and Wigderson [HW93]. However, studying the composition of functions seems more difficult, and the KRW conjecture is still wide open.

In this work, we make a natural step in this direction, which lies between what is known and the original conjecture: we show that an analogue of the conjecture holds for the composition of a function with a universal relation. We also suggest a candidate for the next step and provide initial results toward it.

Our main technical contribution is developing an approach based on the notion of information complexity for analyzing KW relations -- communication problems that are closely related to questions on circuit depth and formula complexity. Recently, information complexity has proved to be a powerful tool, and underlined some major progress on several long-standing open problems in communication complexity. In this work, we develop general tools for analyzing the information complexity of KW relations, which may be of independent interest.

Updated an acknowledgement.

Revision #7 Authors: Dmitry Gavinsky, Or Meir, Omri Weinstein, Avi Wigderson

Accepted on: 28th April 2014 21:13

Downloads: 598

Keywords:

One of the major open problems in complexity theory is proving super-logarithmic

lower bounds on the depth of circuits (i.e., $\mathbf{P} \not\subseteq \mathbf{NC}^1$).

This problem is interesting for two reasons: first, it is tightly

related to understanding the power of parallel computation and of

small-space computation; second, it is one of the first milestones

toward proving super-polynomial circuit lower bounds.

Karchmer, Raz, and Wigderson [KRW95] suggested to approach this problem by proving the following conjecture: given two boolean

functions $f$ and~$g$, the depth complexity of the composed function

$g \circ f$ is roughly the sum of the depth complexities of $f$ and

$g$. They showed that the validity of this conjecture would imply

that $\mathbf{P} \not\subseteq \mathbf{NC}^1$.

As a starting point for studying the composition of functions, they

introduced a relation called ``the universal relation'', and suggested

to study the composition of universal relations. This suggestion proved

fruitful, and an analogue of the KRW conjecture for the universal

relation was proved by Edmonds et. al. [EIRS01]. An alternative proof was given later by H{\aa}stad and Wigderson [HW93].

However, studying the composition of functions seems more difficult,

and the KRW conjecture is still wide open.

In this work, we make a natural step in this direction, which lies

between what is known and the original conjecture: we show that an

analogue of the conjecture holds for the composition of a function

with a universal relation. We also suggest a candidate for the next

step and provide initial results toward it.

Our main technical contribution is developing an approach based on

the notion of information complexity for analyzing KW relations - communication problems that are closely related to questions on

circuit depth and formula complexity. Recently, information complexity

has proved to be a powerful tool, and underlined some major progress

on several long-standing open problems in communication complexity.

In this work, we develop general tools for analyzing the information

complexity of KW relations, which may be of independent interest.

Revision #6 Authors: Dmitry Gavinsky, Or Meir, Omri Weinstein, Avi Wigderson

Accepted on: 20th March 2014 23:13

Downloads: 584

Keywords:

One of the major open problems in complexity theory is proving super-logarithmic

lower bounds on the depth of circuits (i.e., $\mathbf{P}\not\subseteq\mathbf{NC}^1~$).

This problem is interesting for two reasons: first, it is tightly

related to understanding the power of parallel computation and of

small-space computation; second, it is one of the first milestones

toward proving super-polynomial circuit lower bounds.

Karchmer, Raz, and Wigderson [KRW95] suggested to approach this problem by proving the following conjecture: given two boolean

functions $f$ and~$g$, the depth complexity of the composed function

$g\circ f$ is roughly the sum of the depth complexities of $f$ and

$g$. They showed that the validity of this conjecture would imply

that $\mathbf{P}\not\subseteq\mathbf{NC}^1~$.

As a starting point for studying the composition of functions, they

introduced a relation called ``the universal relation'', and suggested

to study the composition of universal relations. This suggestion proved

fruitful, and an analogue of the KRW conjecture for the universal

relation was proved by Edmonds et. al. [EIRS01]. An alternative proof was given later by H{\aa}stad and Wigderson [HW93].

However, studying the composition of functions seems more difficult,

and the KRW conjecture is still wide open.

between what is known and the original conjecture: we show that an

analogue of the conjecture holds for the composition of a function

with a universal relation. We also suggest a candidate for the next

step and provide initial results toward it.

Our main technical contribution is developing an approach based on

the notion of information complexity for analyzing KW relations - communication problems that are closely related to questions on

circuit depth and formula complexity. Recently, information complexity

has proved to be a powerful tool, and underlined some major progress

on several long-standing open problems in communication complexity.

In this work, we develop general tools for analyzing the information

complexity of KW relations, which may be of independent interest.

Revision #5 Authors: Dmitry Gavinsky, Or Meir, Omri Weinstein, Avi Wigderson

Accepted on: 19th March 2014 22:26

Downloads: 619

Keywords:

One of the major open problems in complexity theory is proving super-logarithmic lower bounds on the depth of circuits (i.e., $\mathbf{P}\not\subseteq\mathbf{NC}^1~$). This problem is interesting for two reasons: first, it is tightly related to understanding the power of parallel computation and of small-space computation; second, it is one of the first milestones toward proving super-polynomial circuit lower bounds.

Abstract Karchmer, Raz, and Wigderson [KRW95] suggested to approach this problem by proving the following conjecture: given two boolean functions $f$ and $g$, the depth complexity of the composed function $g\circ f$ is roughly the sum of the depth complexities of $f$ and $g$. They showed that the validity of this conjecture would imply that $\mathbf{P}\not\subseteq\mathbf{NC}^1~$.

As a starting point for studying the composition of functions, they introduced a relation called “the universal relation”, and suggested to study the composition of universal relations. This suggestion proved fruitful, and an analogue of the KRW conjecture for the universal relation was proved by Edmonds et. al. [EIRS01]. An alternative proof was given later by H{\aa}stad and Wigderson [HW93]. However, studying the composition of functions seems more difficult, and the KRW conjecture is still wide open.

Our main technical contribution is developing an approach based on the notion of information complexity for analyzing KW relations -- communication problems that are closely related to questions on circuit depth and formula complexity. Recently, information complexity has proved to be a powerful tool, and underlined some major progress on several long-standing open problems in communication complexity. In this work, we develop general tools for analyzing the information complexity of KW relations, which may be of independent interest.

Revision #4 Authors: Dmitry Gavinsky, Or Meir, Omri Weinstein, Avi Wigderson

Accepted on: 19th March 2014 22:22

Downloads: 563

Keywords:

One of the major open problems in complexity theory is proving super-polynomial lower bounds for circuits with logarithmic depth (i.e., $\mathbf{P}\not\subseteq\mathbf{NC}_1~$). This problem is interesting for two reasons: first, it is tightly related to understanding the power of parallel computation and of small-space computation; second, it is one of the first milestones toward proving super-polynomial circuit lower bounds.

Karchmer, Raz, and Wigderson [KRW95] suggested to approach this problem by proving the following conjecture: given two boolean functions $f$ and $g$, the depth complexity of the composed function $g\circ f$ is roughly the sum of the depth complexities of $f$ and $g$. They showed that the validity of this conjecture would imply that $\mathbf{P}\not\subseteq\mathbf{NC}_1~$.

As a starting point for studying the composition of functions, they introduced a relation called “the universal relation”, and suggested to study the composition of universal relations. This suggestion proved fruitful, and an analogue of the KRW conjecture for the universal relation was proved by Edmonds et. al. [EIRS01]. An alternative proof was given later by Hastad and Wigderson [HW93]. However, studying the composition of functions seems more difficult, and the KRW conjecture is still wide open.

Minor fixes.

Revision #3 Authors: Dmitry Gavinsky, Or Meir, Omri Weinstein, Avi Wigderson

Accepted on: 19th March 2014 22:21

Downloads: 627

Keywords:

One of the major open problems in complexity theory is proving super-polynomial lower bounds for circuits with logarithmic depth (i.e., $\mathbf{P}\not\subseteq\mathbf{NC}_1~$). This problem is interesting for two reasons: first, it is tightly related to understanding the power of parallel computation and of small-space computation; second, it is one of the first milestones toward proving super-polynomial circuit lower bounds.

Karchmer, Raz, and Wigderson [KRW95] suggested to approach this problem by proving the following conjecture: given two boolean functions $f$ and $g$, the depth complexity of the composed function $g\circ f$ is roughly the sum of the depth complexities of $f$ and $g$. They showed that the validity of this conjecture would imply that $\mathbf{P}\not\subseteq\mathbf{NC}_1~$.

As a starting point for studying the composition of functions, they introduced a relation called “the universal relation”, and suggested to study the composition of universal relations. This suggestion proved fruitful, and an analogue of the KRW conjecture for the universal relation was proved by Edmonds et. al. [EIRS01]. An alternative proof was given later by Hastad and Wigderson [HW93]. However, studying the composition of functions seems more difficult, and the KRW conjecture is still wide open.

Revision #2 Authors: Dmitry Gavinsky, Or Meir, Omri Weinstein, Avi Wigderson

Accepted on: 19th March 2014 22:18

Downloads: 591

Keywords:

Abstract One of the major open problems in complexity theory is proving super-logarithmic lower bounds on the depth of circuits (i.e., $\mathbf{P}\not\subseteq\mathbf{NC}^1~$). This problem is interesting for two reasons: first, it is tightly related to understanding the power of parallel computation and of small-space computation; second, it is one of the first milestones toward proving super-polynomial circuit lower bounds.

Karchmer, Raz, and Wigderson [KRW95] suggested to approach this problem by proving the following conjecture: given two boolean functions $f$ and $g$, the depth complexity of the composed function $g\circ f$ is roughly the sum of the depth complexities of $f$ and $g$. They showed that the validity of this conjecture would imply that $\mathbf{P}\not\subseteq\mathbf{NC}^1~$.

As a starting point for studying the composition of functions, they introduced a relation called “the universal relation”, and suggested to study the composition of universal relations. This suggestion proved fruitful, and an analogue of the KRW conjecture for the universal relation was proved by Edmonds et. al. [EIRS01]. An alternative proof was given later by H{\aa}stad and Wigderson [HW93]. However, studying the composition of functions seems more difficult, and the KRW conjecture is still wide open.

Abstract In this work, we make a natural step in this direction, which lies between what is known and the original conjecture: we show that an analogue of the conjecture holds for the composition of a function with a universal relation. We also suggest a candidate for the next step and provide initial results toward it.

Abstract Our main technical contribution is developing an approach based on the notion of information complexity for analyzing KW relations -- communication problems that are closely related to questions on circuit depth and formula complexity. Recently, information complexity has proved to be a powerful tool, and underlined some major progress on several long-standing open problems in communication complexity. In this work, we develop general tools for analyzing the information complexity of KW relations, which may be of independent interest.

Minor fixes.

Revision #1 Authors: Dmitry Gavinsky, Or Meir, Omri Weinstein, Avi Wigderson

Accepted on: 31st December 2013 11:12

Downloads: 766

Keywords:

TR13-190 Authors: Dmitry Gavinsky, Or Meir, Omri Weinstein, Avi Wigderson

Publication: 28th December 2013 13:29

Downloads: 1808

Keywords: