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 context-free languages. In terms of complexity classes, this can be stated as: NL/poly = UL/poly LogCFL/poly = UAuxPDA(log n; n O(1) )/poly

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 logspace-bounded 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 logspace-bounded #PL# machines lose none of their computational power if they are restricted to run in polynomial time.

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 CCR-9509603 and CCR-9734918. z Supported in part by the ...

Wilson's [19] model of oracle gates provides a framework for considering reductions whose strength is intermediate between truth-table 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 n-T ` BF PP ` PrTIME(n O(log n) ). Because P PP log 2 n= log log n-T 6` PP relative to an oracle, we think it is unlikely that PP is closed under Boole...

Abstract. 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 NEXPTIME-complete, the cases {∪, ∩, − , ×}, {∪, ∩, ×}, {∪, ∩, +} are shown PSPACE-complete, the case {∪, +} is shown NP-complete, the case {∩, +} is shown C=L-complete, and several other cases are resolved. Interesting auxiliary problems are used, such as testing nonemptyness for union-intersection-concatenation 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

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 well-known in this formulation. The underlying...

Introduction The classes MOD k L for k 2 consist of those languages A, for which there exists a nondeterministic Turing machine M working on logarithmically bounded space such that, for all inputs x, x belongs to set A if and only if the number of accepting paths in the computation of M on x is not divisible by k. These classes were defined and examined in [BDHM92]. Their importance stems from the fact that the complexity of a number of problems from linear algebra over Z=kZ is given by these classes (in the sense that they are complete in the respective classes); among these problems are singularity of matrices, inversion of matrices, iterated matrix product, etc. Buntrock et al. also examined structural properties of these classes. E.g., it was shown in [BDHM92] that for prime<

this paper describes the complete problems that motivate interest in these classes, discusses some surprising recent discoveries, and points out open problems where progress can reasonably be expected. 3 Definitions

this paper describes the complete problems that motivate interest in these classes, discusses some surprising recent discoveries, and points out open problems where progress can reasonably be expected

We investigate alternative notions of approximation for problems inside P (deter-ministic 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.