Results 1  10
of
38
Arithmetic Circuits: A Chasm at Depth Four
, 2008
"... We show that proving exponential lower bounds on depth four arithmetic circuits imply exponential lower bounds for unrestricted depth arithmetic circuits. In other words, for exponential sized circuits additional depth beyond four does not help. We then show that a complete blackbox derandomization ..."
Abstract

Cited by 71 (5 self)
 Add to MetaCart
(Show Context)
We show that proving exponential lower bounds on depth four arithmetic circuits imply exponential lower bounds for unrestricted depth arithmetic circuits. In other words, for exponential sized circuits additional depth beyond four does not help. We then show that a complete blackbox derandomization of Identity Testing problem for depth four circuits with multiplication gates of small fanin implies a nearly complete derandomization of general Identity Testing. 1
Arithmetic circuits: the chasm at depth four gets wider
"... In their paper on the “chasm at depth four”, Agrawal and Vinay have shown that polynomials in m variables of degree O(m) which admit arithmetic circuits of size 2 o(m) also admit arithmetic circuits of depth four and size 2 o(m). This theorem shows that for problems such as arithmetic circuit lower ..."
Abstract

Cited by 36 (4 self)
 Add to MetaCart
(Show Context)
In their paper on the “chasm at depth four”, Agrawal and Vinay have shown that polynomials in m variables of degree O(m) which admit arithmetic circuits of size 2 o(m) also admit arithmetic circuits of depth four and size 2 o(m). This theorem shows that for problems such as arithmetic circuit lower bounds or blackbox derandomization of identity testing, the case of depth four circuits is in a certain sense the general case. In this paper we show that smaller depth four circuits can be obtained if we start from polynomial size arithmetic circuits. For instance, we show that if the permanent of n×n matrices has circuits of size polynomial inn, then it also has depth 4 circuits of sizen O( √ nlogn) If the original circuit uses only integer constants of polynomial size, then the same is true of the resulting depth four circuit. These results have potential applications to lower bounds and deterministic identity testing, in particular for sums of products of sparse univariate polynomials. We also use our techniques to reprove two results on: The existence of nontrivial boolean circuits of constant depth for languages in LOGCFL. Reduction to polylogarithmic depth for arithmetic circuits of polynomial size and polynomially bounded degree.
Arithmetic circuits and counting complexity classes
 In Complexity of Computations and Proofs,J.Krajíček, Ed. Quaderni di Matematica
"... Arithmetic circuits are the focus of renewed attention in the complexity theory community. It is easy to list a few of the reasons for the increased interest: • Innovative work by Kabanets and Impagliazzo [KI03] shows that, in ..."
Abstract

Cited by 27 (5 self)
 Add to MetaCart
(Show Context)
Arithmetic circuits are the focus of renewed attention in the complexity theory community. It is easy to list a few of the reasons for the increased interest: • Innovative work by Kabanets and Impagliazzo [KI03] shows that, in
Progress on Polynomial Identity Testing
"... Polynomial identity testing (PIT) is the problem of checking whether a given arithmetic circuit is the zero circuit. PIT ranks as one of the most important open problems in the intersection of algebra and computational complexity. In the last few years, there has been an impressive progress on this ..."
Abstract

Cited by 19 (5 self)
 Add to MetaCart
(Show Context)
Polynomial identity testing (PIT) is the problem of checking whether a given arithmetic circuit is the zero circuit. PIT ranks as one of the most important open problems in the intersection of algebra and computational complexity. In the last few years, there has been an impressive progress on this problem but a complete solution might take a while. In this article we give a soft survey exhibiting the ideas that have been useful. 1
The complexity of membership problems for circuits over sets of natural numbers
, 2007
"... The problem of testing membership in the subset of the natural numbers produced at the output gate of a {∪, ∩, − , +, ×} combinational circuit is shown to capture a wide range of complexity classes. Although the general problem remains open, the case {∪, ∩, +, ×} is shown NEXPTIMEcomplete, the cas ..."
Abstract

Cited by 19 (0 self)
 Add to MetaCart
The problem of testing membership in the subset of the natural numbers produced at the output gate of a {∪, ∩, − , +, ×} combinational circuit is shown to capture a wide range of complexity classes. Although the general problem remains open, the case {∪, ∩, +, ×} is shown NEXPTIMEcomplete, the cases {∪, ∩, − , ×}, {∪, ∩, ×}, {∪, ∩, +} are shown PSPACEcomplete, the case {∪, +} is shown NPcomplete, the case {∩, +} is shown C=Lcomplete, and several other cases are resolved. Interesting auxiliary problems are used, such as testing nonemptyness for unionintersectionconcatenation circuits, and expressing each integer, drawn from a set given as input, as powers of relatively prime integers of one’s choosing. Our results extend in nontrivial ways past work by
Deterministic identity testing of depth 4 multilinear circuits with bounded top fanin
, 2009
"... ..."
On TC⁰, AC⁰, and Arithmetic Circuits
 JOURNAL OF COMPUTER AND SYSTEM SCIENCES
, 2000
"... Continuing a line of investigation that has studied the function classes #P [Val79b], #SAC¹ [Val79a, Vin91, AJMV], #L [AJ93b, Vin91, AO94], and #NC¹ [CMTV96], we study the class of functions #AC⁰. One way to define #AC⁰ is as the class of functions computed by constantdepth polynomialsize arith ..."
Abstract

Cited by 13 (3 self)
 Add to MetaCart
Continuing a line of investigation that has studied the function classes #P [Val79b], #SAC¹ [Val79a, Vin91, AJMV], #L [AJ93b, Vin91, AO94], and #NC¹ [CMTV96], we study the class of functions #AC⁰. One way to define #AC⁰ is as the class of functions computed by constantdepth polynomialsize arithmetic circuits of unbounded fanin addition and multiplication gates. In contrast to the preceding
Bounded Depth Arithmetic Circuits: Counting and Closure
, 1999
"... Constantdepth arithmetic circuits have been defined and studied in [AAD97, ABL98]; these circuits yield the function classes #AC . These function classes in turn provide new characterizations of the computational power of threshold circuits, and provide a link between the circuit classes AC ..."
Abstract

Cited by 11 (4 self)
 Add to MetaCart
Constantdepth arithmetic circuits have been defined and studied in [AAD97, ABL98]; these circuits yield the function classes #AC . These function classes in turn provide new characterizations of the computational power of threshold circuits, and provide a link between the circuit classes AC (where many lower bounds are known) and TC (where essentially no lower bounds are known). In this paper, we resolve several questions regarding the closure properties of #AC .
Integer Circuit Evaluation is PSPACEcomplete
 Journal of Computer and System Sciences
"... this paper we show that the Integer Circuit problem is PSPACEcomplete, resolving an open problem posed by McKenzie, Vollmer and Wagner [7] ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
(Show Context)
this paper we show that the Integer Circuit problem is PSPACEcomplete, resolving an open problem posed by McKenzie, Vollmer and Wagner [7]
Classifying polynomials and identity testing
, 2009
"... email: One of the fundamental problems of computational algebra is to classify polynomials according to the hardness of computing them. Recently, this problem has been related to another important problem: Polynomial identity testing. Informally, the problem is to decide if a certain succinct repre ..."
Abstract

Cited by 9 (1 self)
 Add to MetaCart
(Show Context)
email: One of the fundamental problems of computational algebra is to classify polynomials according to the hardness of computing them. Recently, this problem has been related to another important problem: Polynomial identity testing. Informally, the problem is to decide if a certain succinct representation of a polynomial is zero or not. This problem has been extensively studied owing to its connections with various areas in theoretical computer science. Several efficient randomized algorithms have been proposed for the identity testing problem over the last few decades but an efficient deterministic algorithm is yet to be discovered. It is known that such an algorithm will imply hardness of computing an explicit polynomial. In the last few years, progress has been made in designing deterministic algorithms for restricted circuits, and also in understanding why the problem is hard even for small depth. In this article, we survey important results for the polynomial identity testing problem and its connection with classification of polynomials. 1.