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13
Making Nondeterminism Unambiguous
, 1997
"... We show that in the context of nonuniform complexity, nondeterministic logarithmic space bounded computation can be made unambiguous. An analogous result holds for the class of problems reducible to contextfree languages. In terms of complexity classes, this can be stated as: NL/poly = UL/poly Lo ..."
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Cited by 48 (13 self)
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We show that in the context of nonuniform complexity, nondeterministic logarithmic space bounded computation can be made unambiguous. An analogous result holds for the class of problems reducible to contextfree languages. In terms of complexity classes, this can be stated as: NL/poly = UL/poly LogCFL/poly = UAuxPDA(log n; n O(1) )/poly
Relationships Among PL, L, and the Determinant
, 1996
"... Recent results byToda, Vinay, Damm, and Valianthave shown that the complexity of the determinantischaracterized by the complexity of counting the number of accepting computations of a nondeterministic logspacebounded machine. #This class of functions is known as #L.# By using that characterizati ..."
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Cited by 37 (9 self)
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Recent results byToda, Vinay, Damm, and Valianthave shown that the complexity of the determinantischaracterized by the complexity of counting the number of accepting computations of a nondeterministic logspacebounded machine. #This class of functions is known as #L.# By using that characterization and by establishing a few elementary closure properties, we giveavery simple proof of a theorem of Jung, showing that probabilistic logspacebounded #PL# machines lose none of their computational power if they are restricted to run in polynomial time.
Isolation, Matching, and Counting: Uniform and Nonuniform Upper Bounds
 Journal of Computer and System Sciences
, 1998
"... We show that the perfect matching problem is in the complexity class SPL (in the nonuniform setting). This provides a better upper bound on the complexity of the matching problem, as well as providing motivation for studying the complexity class SPL. Using similar techniques, we show that counting t ..."
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Cited by 29 (6 self)
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We show that the perfect matching problem is in the complexity class SPL (in the nonuniform setting). This provides a better upper bound on the complexity of the matching problem, as well as providing motivation for studying the complexity class SPL. Using similar techniques, we show that counting the number of accepting paths of a nondeterministic logspace machine can be done in NL/poly, if the number of paths is small. This clarifies the complexity of the class LogFew (defined and studied in [BDHM91]). Using derandomization techniques, we then improve this to show that this counting problem is in NL. Determining if our other theorems hold in the uniform setting remains an The material in this paper appeared in preliminary form in papers in the Proceedings of the IEEE Conference on Computational Complexity, 1998, and in the Proceedings of the Workshop on Randomized Algorithms, Brno, 1998. y Supported in part by NSF grants CCR9509603 and CCR9734918. z Supported in part by the ...
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 15 (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
Circuits Over PP and PL
 In IEEE Conference on Computational Complexity
, 1997
"... Wilson's [19] model of oracle gates provides a framework for considering reductions whose strength is intermediate between truthtable and Turing. Improving on a stream of results by Beigel, Reingold, Spielman, Fortnow, and Ogihara [6, 9, 14], we prove that PL and PP are closed under NC 1 reduc ..."
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Cited by 11 (0 self)
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Wilson's [19] model of oracle gates provides a framework for considering reductions whose strength is intermediate between truthtable and Turing. Improving on a stream of results by Beigel, Reingold, Spielman, Fortnow, and Ogihara [6, 9, 14], we prove that PL and PP are closed under NC 1 reductions. This answers an open problem of Ogihara [14]. More generally, we show that NC PP k+1 = AC PP k and NC PL k+1 = AC PL k for all k 0. On the other hand, we construct an oracle A such that NC PP A k 6= NC PP A k+1 for all integers k 1. Slightly weaker than NC 1 reductions are Boolean formula reductions. We ask whether PL and PP are closed under Boolean formula reductions. This is a nontrivial question despite NC 1 = BF, because that equality is easily seen not to relativize. We prove that P PP log 2 n= log log nT ` BF PP ` PrTIME(n O(log n) ). Because P PP log 2 n= log log nT 6` PP relative to an oracle, we think it is unlikely that PP is closed under Boole...
A Note on Closure Properties of Logspace MOD Classes
 INFORMATION PROCESSING LETTERS
, 1999
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Worlds To Die For
, 1995
"... We survey the background and challenges of a number of open problems in the theory of relativization. Among the topics covered are pseudorandom generators, time hierarchies, the potential collapse of the polynomial hierarchy, and the existence of complete sets. Relativization (i.e., oracle) theory h ..."
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Cited by 5 (2 self)
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We survey the background and challenges of a number of open problems in the theory of relativization. Among the topics covered are pseudorandom generators, time hierarchies, the potential collapse of the polynomial hierarchy, and the existence of complete sets. Relativization (i.e., oracle) theory has seen its share of ups and downs. Extensive surveys of current knowledge [Ver94] and debates as to relativization theory's merits [Har85,All90, HCC + 92,For94] can be found in the literature. However, in a nutshell, one could rather fairly say that as ups and downs go, relativization theory is on the mat. Still, that is not to say that relativization theory has no interesting open issues left with which to challenge theoretical computer scientists. It does, and here are a few such issues. Problem 1: Show that with probability one, the polynomial hierarchy is proper. The above statement is, to say the least, elliptic. However, the problem is wellknown in this formulation. The underlying...
Counting Classes and the Fine Structure between NC¹ and L
, 2011
"... The class NC¹ of problems solvable by bounded fanin circuit families of logarithmic depth is known to be contained in logarithmic space L, but not much about the converse is known. In this paper we examine the structure of classes in between NC¹ and L based on counting functions or, equivalently, b ..."
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The class NC¹ of problems solvable by bounded fanin circuit families of logarithmic depth is known to be contained in logarithmic space L, but not much about the converse is known. In this paper we examine the structure of classes in between NC¹ and L based on counting functions or, equivalently, based on arithmetic circuits. The classes PNC¹ and C=NC¹, defined by a test for positivity and a test for zero, respectively, of arithmetic circuit families of logarithmic depth, sit in this complexity interval. We study the landscape of Boolean hierarchies, constantdepth oracle hierarchies, and logarithmicdepth oracle hierarchies over PNC¹ and C=NC¹. We provide complete problems, obtain the upper bound L for all these hierarchies, and prove partial hierarchy collapses. In particular, the constantdepth oracle hierarchy over PNC¹ collapses to its first level PNC¹, and the constantdepth oracle hierarchy over C=NC¹ collapses to its second level.
and
"... provides a framework for considering reductions whose strength is intermediate between truthtable and Turing. Improving on a stream of previously published results, we prove that PL and PP are closed under NC 1 reductions. This answers an open problem of Ogihara (1996, ``Proc. 37th Ann. IEEE Symp. ..."
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provides a framework for considering reductions whose strength is intermediate between truthtable and Turing. Improving on a stream of previously published results, we prove that PL and PP are closed under NC 1 reductions. This answers an open problem of Ogihara (1996, ``Proc. 37th Ann. IEEE Symp. Found. Computer Sci.''). More generally, we show that NC PP k+1= AC PP k and NC PL k+1=AC PL oracle A such that NC PPA k k for all k 0. On the other hand, we construct an
Alternative notions of approximation and spacebounded computations
, 2003
"... We investigate alternative notions of approximation for problems inside P (deterministic polynomial time), and show that even a slightly nontrivial information about a problem may be as hard to obtain as the solution itself. For example, we prove that if one could eliminate even a single possibilit ..."
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We investigate alternative notions of approximation for problems inside P (deterministic polynomial time), and show that even a slightly nontrivial information about a problem may be as hard to obtain as the solution itself. For example, we prove that if one could eliminate even a single possibility for the value of an arithmetic circuit on a given input, then this would imply that the class P has fast (polygarithmic time) parallel solutions. In other words, this would constitute a proof that there are no inherently sequential problems in P, which is quite unlikely. The result is robust with respect to eliminating procedures that are allowed to err (by excluding the correct value) with small probability. We also show that several fundamental linear algebra problems are hard in this sense. It turns out that it is as hard to substantially reduce the number of possible values for the determinant and rank as to compute them exactly. Finally, we show that (in some precise sense) randomness can be nontrivially substituted for nondeterminism in space. Although it is believed that randomness does not give more than a constant factor advantage in space over determinism, it is not even known whether it is no more powerful than nondeterminism. We will show that the latter is true for a restricted version of probabilistic logspace, where the error is potentially larger than what can be achieved by amplification.