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Uniform ConstantDepth Threshold Circuits for Division and Iterated Multiplication
, 2002
"... this paper. 2.1. Circuit Classes We begin by formally defining the three circuit complexity classes that will concern us here. These are given by combinatorial restrictions on the circuits of the family. We will then define the uniformity restrictions we will use. Finally, we will give the equival ..."
Abstract

Cited by 41 (8 self)
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this paper. 2.1. Circuit Classes We begin by formally defining the three circuit complexity classes that will concern us here. These are given by combinatorial restrictions on the circuits of the family. We will then define the uniformity restrictions we will use. Finally, we will give the equivalent formulations of uniform circuit complexity classes in terms of descriptive complexity classes
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 18 (2 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
Arithmetic Circuits and Polynomial Replacement Systems
 the Proceedings of the 2000 FSTTCS conference
, 1999
"... This paper addresses the problems of counting proof trees (as introduced by Venkateswaran and Tompa) and counting proof circuits, a related but seemingly more natural question. These problems lead to a common generalization of straightline programs which we call polynomial replacement systems. W ..."
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Cited by 5 (2 self)
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This paper addresses the problems of counting proof trees (as introduced by Venkateswaran and Tompa) and counting proof circuits, a related but seemingly more natural question. These problems lead to a common generalization of straightline programs which we call polynomial replacement systems. We contribute a classication of these systems and we investigate their complexity. Diverse problems falling in the scope of this study include, for example, counting proof circuits, and evaluating f[; +gcircuits over the natural numbers. A number of complexity results are obtained, e.g., the former problem is shown #Pcomplete, while the latter is shown to be equivalent to a particular type of replacement systems. 1 Introduction 1.1 Motivation ^ x1 x2 g2 g1 _ g3 _ _ g4 When + and replace _ and ^ in the adjacent gure, the gate g 1 on input x 1 = x 2 = 1 evaluates to 9. Equivalently, the treelike Boolean circuit T obtained from the circuit drawn has 9 proof trees [VT89],...
On the power of algebraic branching programs of width two
"... Abstract. We show that there are families of polynomials having small depthtwo arithmetic circuits that cannot be expressed by algebraic branching programs of width two. This clarifies the complexity of the problem of computing the product of a sequence of twobytwo matrices, which arises in sever ..."
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Cited by 3 (1 self)
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Abstract. We show that there are families of polynomials having small depthtwo arithmetic circuits that cannot be expressed by algebraic branching programs of width two. This clarifies the complexity of the problem of computing the product of a sequence of twobytwo matrices, which arises in several settings. 1
Semantical Counting Circuits
, 1998
"... . Counting functions can be defined syntactically or semantically depending on whether they count the number of witnesses in a nondeterministic or in a deterministic computation on the input. In the Turing machine based model, these two ways of defining counting were proven to be equivalent for ..."
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Cited by 1 (0 self)
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. Counting functions can be defined syntactically or semantically depending on whether they count the number of witnesses in a nondeterministic or in a deterministic computation on the input. In the Turing machine based model, these two ways of defining counting were proven to be equivalent for many important complexity classes. In the circuit based model, it was done for #P and #L, but for lowlevel complexity classes such as #AC 0 and #NC 1 only the syntactical definitions were considered. We give appropriate semantical definitions for these two classes and prove them to be equivalent to the syntactical ones. This enables us to show that #AC 0 is included in the family of counting functions computed by polynomial size and constant width counting branching programs, therefore completing a result of Caussinus et al [CMTV98]. We also consider semantically defined probabilistic complexity classes corresponding to AC 0 and NC 1 and prove that in the case of unboun...
COMPLEXITY THEORETIC ASPECTS OF PLANAR RESTRICTIONS AND OBLIVIOUSNESS
, 2006
"... In this thesis, we deal largely with complexity theoretic aspects in planar restrictions and obliviousness. Our main motivation was to identify problems for which the planar restriction is much easier, computationally, than the unrestricted version. First, we study constant width polynomialsized ci ..."
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In this thesis, we deal largely with complexity theoretic aspects in planar restrictions and obliviousness. Our main motivation was to identify problems for which the planar restriction is much easier, computationally, than the unrestricted version. First, we study constant width polynomialsized circuits of low (polylogarithmic) genus; we show how such circuits characterize exactly the wellknown circuit complexity class ACC0 (given that the unrestricted version captures the whole of NC1). We also give a new circuit characterization of the class NC1. Shifting our focus from circuits to graphs, we look at different notions of connectivity. We investigate the directed planar graph reachability problem, as a possibly more tractable special case of the arbitrary graph reachability problem (which is NLcomplete). We prove that this problem logspacereduces to its complement, and also that reachability questions on genus 1 graphs reduce to that in planar graphs. We also prove that reachability in a particularly simple class of planar graphs (namely, grid graphs) is no easier than the general directed planar reachability question. We then proceed to isolate to several large classes of planar graphs for which the reachability questions are solvable in deterministic logspace. Counting the number of spanning trees in a graph is a useful extension of the task of determining
Arithmetic Versions of Constant Depth Circuit Complexity Classes
"... The boolean circuit complexity classes AC 0 # AC 0 [m] # TC 0 # NC 1 have been studied intensely. Other than NC 1 , they are defined by constantdepth circuits of polynomial size and unbounded fanin over some set of allowed gates. One reason for interest in these classes is that they ..."
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The boolean circuit complexity classes AC 0 # AC 0 [m] # TC 0 # NC 1 have been studied intensely. Other than NC 1 , they are defined by constantdepth circuits of polynomial size and unbounded fanin over some set of allowed gates. One reason for interest in these classes is that they contain the boundary marking the limits of current lower bound technology: such technology exists for AC 0 and some of the classes AC 0 [m], while the other classes AC 0 [m] as well as TC 0 lack such technology. Continuing a line of research originating from Valiant's work on the counting class #P , the arithmetic circuit complexity classes #AC 0 and #NC 1 have recently been studied. In this paper, we define and investigate the classes #AC 0 [m] and #TC 0 . Just as the boolean classes AC 0 [m] and TC 0 give a refined view of NC 1 , our new arithmetic classes, which fall into the inclusion chain #AC 0 # #AC 0 [m] # #TC 0 # #NC 1 , refine #NC 1 . These new classes (along with #AC 0 ) are also defined by constantdepth circuits, but the allowed gates compute arithmetic functions. We also introduce the classes Diff AC 0 [m] (differences of two AC 0 [m] functions), which generalize the class Diff AC 0 studied in previous work. We study the structure of three hierarchies: the #AC 0 [m] hierarchy, the Diff AC 0 [m] hierarchy, and a hierarchy of language classes. We prove class separations and containments where possible, and demonstrate relationships among the various hierarchies. For instance, we prove that the hierarchy of classes #AC 0 [m] has exactly the same structure as the hierarchy of classes AC 0 [m]: AC 0 [m] # AC 0 [m # ] iff #AC 0 [m] # #AC 0 [m # ] We also investigate closure properties of our new classe...
Arithmetic circuits, . . . limitations of skew formulae
, 2008
"... Functions in arithmetic NC¹ are known to have equivalent constant width polynomial degree circuits, but the converse containment is unknown. In a partial answer to this question, we show that syntactic multilinear circuits of constant width and polynomial degree can be depthreduced, though the re ..."
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Functions in arithmetic NC¹ are known to have equivalent constant width polynomial degree circuits, but the converse containment is unknown. In a partial answer to this question, we show that syntactic multilinear circuits of constant width and polynomial degree can be depthreduced, though the resulting circuits need not be syntactic multilinear. We then focus specifically on polynomialsize syntactic multilinear circuits, and study relationships between classes of functions obtained by imposing various resource (width, depth, degree) restrictions on these circuits. Along the way, we obtain a characterisation of NC 1 (and its arithmetic counterparts) in terms of log width restricted planar branching programs. We also study the power of skew formulae, and show that even exponential sums of these are unlikely to suffice to express the determinant function.
The Computational Complexity Column
"... Sometimes the simplest problems lead to the most interesting theory. Considerthe division problem, given a and b in binary, output b a ..."
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Sometimes the simplest problems lead to the most interesting theory. Considerthe division problem, given a and b in binary, output b a