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

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 ...

In this work a complete problem for an unambiguous logspace class is presented. This is surprising since unambiguity is a `promise' or `semantic' concept. These usually lead to classes apparently without complete problems. 1 Introduction One of the most central questions of complexity theory is to compare determinism with nondeterminism. Our inability to exhibit the precise relationship between these two features motivates the investigation of intermediate features such as symmetry or unambiguity. In this paper we will concentrate on the notion of unambiguity. Unfortunately, unambiguity of a device or of a language is in general an undecidable property. Unambiguous classes are not defined by a `syntactical' machine property but rather by a `semantical' restriction. A nasty consequence is the apparent lack of complete sets. In the case of time bounded computations there are relativizations of unambiguity which provably have no complete problem ([10]). For space bounded computations t...

We show that the st-connectivity problem for directed planar graphs can be decided in unambiguous logarithmic space. 1.

In the exponential case circuits of bounded depth characterize the polynomial hierachy. Using the notion of an unambiguous circuit we give a uniform framework to relate the various types of unambiguous polynomial hierarchies and to explain their differences.

We show that the complexity class LogFew is contained in NL " SPL. Previously, this was known only to hold in the nonuniform setting. Key Words: Nondeterministic Logspace Computation, Nonuniform Complexity, Derandomization, ffl-biased Sample Space. 1 Introduction In [RA97], a probabilistic construction was used to show that the complexity classes NL/poly and UL/poly coincide. That is, in the context of nonuniform complexity, nonuniform logspace is no more powerful than unambiguous logspace. It was observed in [AR98] that the equality NL=UL holds also in the uniform setting, under a plausible hypothesis concerning pseudorandom number generators. However, it remains an important open question whether NL=UL can be established without resorting to unproved assumptions. The results and techniques of [RA97] were extended in [AR98] in a number of ways. One extension involves the class LogFew, defined in [BDHM92]. (Formal definitions appear below.) No inclusion relation was known between...

Abstract. There has been a great deal of progress in the fifteen years that have elapsed since Wigderson published his survey on the complexity of the graph connectivity problem [Wig92]. Most significantly, Reingold solved the longstanding question of the complexity of the s-t connectivity problem in undirected graphs, showing that this is complete for logspace (L) [Rei05]. This survey talk will focus on some of the remaining open questions dealing with graph reachability problems. Particular attention will be paid to these topics: – Reachability in planar directed graphs (and more generally, in graphs of low genus) [ADR05,BTV07]. – Reachability in different classes of grid graphs [ABC + 06]. – Reachability in mangroves [AL98]. The problem of finding a path from one vertex to another in a graph is the first problem that was identified as being complete for a natural subclass of P; it was shown to be complete for nondeterministic logspace (NL) by Jones [Jon75]. Restricted versions of this problem were subsequently shown to be complete for other natural complexity

We prove space hierarchy and separation results for randomized and other semantic models of computation with advice. Previous works on hierarchy and separation theorems for such models focused on time as the resource. We obtain tighter results with space as the resource. Our main theorems are the following. Let s(n) be any space-constructible function that is Ω(log n) and such that s(an) = O(s(n)) for all constants a, and let s ′ (n) be any function that is ω(s(n)). There exists a language computable by two-sided error randomized machines using s ′ (n) space and one bit of advice that is not computable by two-sided error randomized machines using s(n) space and min(s(n), n) bits of advice. There exists a language computable by zero-sided error randomized machines in space s ′ (n) with one bit of advice that is not computable by one-sided error randomized machines using s(n) space and min(s(n), n) bits of advice. The condition that s(an) = O(s(n)) is a technical condition satisfied by typical space bounds that are at most linear. We also obtain weaker results that apply to generic semantic models of computation. 1

Abstract. We prove space hierarchy and separation results for randomized and other semantic models of computation with advice. Previous works on hierarchy and separation theorems for such models focused on time as the resource. We obtain tighter results with space as the resource. Our main theorems are the following. Let s(n) be any spaceconstructible function that is Ω(log n) and such that s(an) = O(s(n)) for all constants a, and let s ′ (n) be any function that is ω(s(n)). There exists a language computable by two-sided error randomized machines using s ′ (n) space and one bit of advice that is not computable by two-sided error randomized machines using s(n) space and min(s(n), n) bits of advice. There exists a language computable by zero-sided error randomized machines in space s ′ (n) with one bit of advice that is not computable by one-sided error randomized machines using s(n) space and min(s(n), n) bits of advice. The condition that s(an) = O(s(n)) is a technical condition satisfied by typical space bounds that are at most linear. We also obtain weaker results that apply to generic semantic models of computation. 1.

Abstract. We prove space hierarchy and separation results for randomized and other semantic models of computation with advice. Previous works on hierarchy and separation theorems for such models focused on time as the resource. We obtain tighter results with space as the resource. Our main theorems are the following. Let s(n) be any spaceconstructible function that is Ω(log n) and such that s(an) = O(s(n)) for all constants a, and let s ′ (n) be any function that is ω(s(n)). There exists a language computable by two-sided error randomized machines using s ′ (n) space and one bit of advice that is not computable by two-sided error randomized machines using s(n) space and min(s(n), n) bits of advice. There exists a language computable by zero-sided error randomized machines in space s ′ (n) with one bit of advice that is not computable by one-sided error randomized machines using s(n) space and min(s(n), n) bits of advice. The condition that s(an) = O(s(n)) is a technical condition satisfied by typical space bounds that are at most linear. We also obtain weaker results that apply to generic semantic models of computation. 1.