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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 ..."
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Cited by 28 (1 self)
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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
On TC^0, AC^0, and Arithmetic Circuits
 Journal of Computer and System Sciences
, 2000
"... Continuing a line of investigation that has studied the function classes #P [Val79b], #SAC [Val79a, Vin91, AJMV], #L [AJ93b, Vin91, AO94], and #NC [CMTV96], we study the class of functions . One way to define #AC is as the class of functions computed by constantdepth polynomialsize ..."
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Cited by 18 (6 self)
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Continuing a line of investigation that has studied the function classes #P [Val79b], #SAC [Val79a, Vin91, AJMV], #L [AJ93b, Vin91, AO94], and #NC [CMTV96], we study the class of functions . One way to define #AC is as the class of functions computed by constantdepth polynomialsize arithmetic circuits of unbounded fanin addition and multiplication gates. In contrast to the preceding # Part of this research was done while visiting the University of Ulm under an Alexander von Humboldt Fellowship.
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 ..."
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Cited by 17 (3 self)
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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
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 ..."
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Cited by 16 (0 self)
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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
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 ..."
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Cited by 9 (3 self)
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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] ..."
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Cited by 9 (0 self)
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this paper we show that the Integer Circuit problem is PSPACEcomplete, resolving an open problem posed by McKenzie, Vollmer and Wagner [7]
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 ..."
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Cited by 5 (1 self)
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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 Tensor Calculus
 In Proceedings of the 15th Conference on Computational Complexity
, 2000
"... Tensor calculus over semirings is shown relevant to complexity theory in unexpected ways. First, evaluating wellformed tensor formulas with explicit tensor entries is shown complete for \PhiP, for NP, and for #P as the semiring varies. Indeed the permanent of a matrix is shown expressible as the va ..."
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Cited by 4 (2 self)
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Tensor calculus over semirings is shown relevant to complexity theory in unexpected ways. First, evaluating wellformed tensor formulas with explicit tensor entries is shown complete for \PhiP, for NP, and for #P as the semiring varies. Indeed the permanent of a matrix is shown expressible as the value of a tensor formula in much the same way that Berkowitz' theorem expresses its determinant. Second, restricted tensor formulas are shown to capture the classes LOGCFL and NL, their parity counterparts \PhiLOGCFL and \PhiL, and several other counting classes. Finally, the known inclusions NP=poly ` \PhiP=poly, LOGCFL=poly ` \PhiLOGCFL=poly, and NL=poly ` \PhiL=poly, which have scattered proofs in the literature [21, 39], are shown to follow from the new characterizations in a single blow. 1 Introduction Consider an algebraic structure S with certain operations. The following problem is sometimes called the word problem of S: given a reasonable encoding of a wellformed expression T ove...