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On the complexity of numerical analysis
 IN PROC. 21ST ANN. IEEE CONF. ON COMPUTATIONAL COMPLEXITY (CCC ’06
, 2006
"... We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis: • The BlumShubSmale model of computation over the reals. • A problem we call the “Generic Task of Numerical Computation, ” which captures an aspect of doing numerical computation ..."
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Cited by 45 (5 self)
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We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis: • The BlumShubSmale model of computation over the reals. • A problem we call the “Generic Task of Numerical Computation, ” which captures an aspect of doing numerical computation in floating point, similar to the “long exponent model ” that has been studied in the numerical computing community. We show that both of these approaches hinge on the question of understanding the complexity of the following problem, which we call PosSLP: Given a divisionfree straightline program producing an integer N, decide whether N> 0. • In the BlumShubSmale model, polynomial time computation over the reals (on discrete inputs) is polynomialtime equivalent to PosSLP, when there are only algebraic constants. We conjecture that using transcendental constants provides no additional power, beyond nonuniform reductions to PosSLP, and we present some preliminary results supporting this conjecture. • The Generic Task of Numerical Computation is also polynomialtime equivalent to PosSLP. We prove that PosSLP lies in the counting hierarchy. Combining this with work of Tiwari, we obtain that the Euclidean Traveling Salesman Problem lies in the counting hierarchy – the previous best upper bound for this important problem (in terms of classical complexity classes) being PSPACE. In the course of developing the context for our results on arithmetic circuits, we present some new observations on the complexity of ACIT: the Arithmetic Circuit Identity Testing problem. In particular, we show that if n! is not ultimately easy, then ACIT has subexponential complexity.
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 ..."
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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
SpaceEfficient Deterministic Simulation of Probabilistic Automata
, 1993
"... Given a description of a probabilistic automaton (onehead probabilistic finite automaton or probabilistic Turing machine) and an input string x of length n, we ask how much space does a deterministic Turing machine need in order to decide the acceptance of the input string by that automaton? The q ..."
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Cited by 17 (4 self)
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Given a description of a probabilistic automaton (onehead probabilistic finite automaton or probabilistic Turing machine) and an input string x of length n, we ask how much space does a deterministic Turing machine need in order to decide the acceptance of the input string by that automaton? The question is interesting even in the case of onehead oneway probabilistic finite automata (PFA). We call (rational) stochastic languages (S ? rat ) the class of languages recognized by PFA's whose transition probabilities and cutpoints (i.e. recognition thresholds) are rational numbers. The class S ? rat contains contextsensitive languages that are not context free, but on the other hand there are contextfree languages not included in S ? rat . Our main results are as follows: ffl The (proper) inclusion of S ? rat in Dspace(log n), which is optimal (i.e. S ? rat 6ae Dspace(o(log n))). The previous upper bounds were Dspace(n) [Dieu 1972], [Wang 1992] and Dspace(log n log log n)...
Nondeterministic NC¹ computation
"... We define the counting classes #NC¹, GapNC¹, PNC¹ and C=NC¹. We prove that boolean circuits, algebraic circuits, programs over nondeterministic finite automata, and programs over constant integer matrices yield equivalent definitions of the latter three classes. We investigate closure properties. We ..."
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Cited by 16 (4 self)
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We define the counting classes #NC¹, GapNC¹, PNC¹ and C=NC¹. We prove that boolean circuits, algebraic circuits, programs over nondeterministic finite automata, and programs over constant integer matrices yield equivalent definitions of the latter three classes. We investigate closure properties. We observe that #NC¹ ` #L and that C=NC¹ ` L. Then we exploit our finite automaton model and extend the padding techniques used to investigate leaf languages. Finally, we draw some consequences from the resulting body of leaf language characterizations of complexity classes, including the unconditional separation of ACC⁰ from MODPH as well as that of TC⁰ from the counting hierarchy. Moreover we obtain that dlogtimeuniformity and logspaceuniformity for AC⁰ coincide if and only if the polynomial time hierarchy equals PSPACE .
Uniform Circuits for Division: Consequences and Problems
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY 7:065
, 2000
"... Integer division has been known to lie in Puniform TC 0 since the mid1980's, and recently this was improved to L uniform TC 0 . At the time that the results in this paper were proved and submitted for conference presentation, it was unknown whether division lay in DLOGTIMEuniform TC 0 ( ..."
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Cited by 14 (6 self)
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Integer division has been known to lie in Puniform TC 0 since the mid1980's, and recently this was improved to L uniform TC 0 . At the time that the results in this paper were proved and submitted for conference presentation, it was unknown whether division lay in DLOGTIMEuniform TC 0 (also known as FOM). We obtain tight bounds on the uniformity required for division, by showing that division is complete for the complexity class FOM + POW obtained by augmenting FOM with a predicate for powering modulo small primes. We also show that, under a wellknown numbertheoretic conjecture (that there are many "smooth" primes), POW (and hence division) lies in FOM. Building on this work, Hesse has shown recently that division is in FOM [17]. The essential
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 ..."
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Cited by 13 (3 self)
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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
Electronic Colloquium on Computational Complexity, Report No. 65 (2000) Uniform Circuits for Division: Consequences and Problems
, 2000
"... The essential idea in the fast parallel computation of division and related problems is that of Chinese remainder representation (CRR) – storing a number in the form of its residues modulo many small primes. Integer division provides one of the few natural examples of problems for which all currentl ..."
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The essential idea in the fast parallel computation of division and related problems is that of Chinese remainder representation (CRR) – storing a number in the form of its residues modulo many small primes. Integer division provides one of the few natural examples of problems for which all currentlyknown constructions of efficient circuits rely on some sort of extra information or nonuniformity; the major stumbling block has seemed to be the difficulty of converting from CRR to binary. We give new bounds on the nonuniformity required for division; it is necessary and sufficient to be able to compute discrete logarithms modulo an O(log n) bit number. In particular, we show that the necessary uniformity predicates lie in a class that (provably) does not contain L. The fact that CRR operations can be carried out in log space has interesting implications for small space classes. We define two versions of s(n) space for s(n) = o(log n): dspace(s(n)) as the traditional version where the worktape begins blank, and DSPACE(s(n)) where the space bound is established by endmarkers before the computation starts. We present a new translational lemma, and derive as a consequence that (for example), if one can improve the result of [14] that {0 n: n is prime} � ∈ dspace(log log n) to show that {0 n: n is prime} � ∈ DSPACE(log log n), it would follow that L � = NP. 1