: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : vii Chapter 1. INTRODUCTION : : : : : : : : : : : : : : : : : : : : : : : : : 1 2. PRELIMINARIES : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.1 Basic Definitions : : : : : : : : : : : : : : : : : : : : : : : 4 2.1.1 Basics : : : : : : : : : : : : : : : : : : : : : : : : 4 2.1.2 Boolean Formulas : : : : : : : : : : : : : : : : : 4 2.1.3 Arithmetic Formulas and Expressions : : : : : : 5 2.2 Computational Models : : : : : : : : : : : : : : : : : : : : 9 2.2.1 Deterministic Computation : : : : : : : : : : : : 9 2.2.2 Probabilistic Computation : : : : : : : : : : : : 11 2.2.3 Non-Deterministic Computation : : : : : : : : : 12 2.2.4 Alternating Computations : : : : : : : : : : : : 13 2.2.5 Interactive Proof Systems : : : : : : : : : : : : : 13 2.2.6 Multiple Prover Interactive Proof Systems : : : 15 2.2.7 Computation relative to an Oracle : : : : : : : : 15 2.3 Complexity Classes : : : : : : : : : : : : : : : : : : : : ...

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

For some fixed alphabet A with jAj 2, a language L ` A is in the class L 1=2 of the Straubing-Therien hierarchy if and only if it can be expressed as a finite union of languages A a 1 A a 2 A \Delta \Delta \Delta A anA , where a i 2 A and n 0. The class L 1 is defined as the boolean closure of L 1=2 . It is known that the classes L 1=2 and L 1 are decidable. We give a membership criterion for the single classes of the boolean hierarchy over L 1=2 . From this criterion we can conclude that this boolean hierarchy is proper and that its classes are decidable. In finite model theory the latter implies the decidability of the classes of the boolean hierarchy over the class \Sigma 1 of the FO[!]-logic. Moreover we prove a "forbidden-pattern" characterization of L 1 of the type: L 2 L 1 if and only if a certain pattern does not appear in the transition graph of a deterministic finite automaton accepting L. We discuss complexity theoretical consequences of our results. C...

en a graph G and a pair of vertices s; t, this reduction produces a polynomial number of graphs G 1 ; : : : ; G k of polynomial size, together with distinguished vertex-pairs (s 1 ; t 1 ); : : : ; (s k ; t k ), that satisfy the following conditions. If there is no path from s to t in G, then no G i has a path from s i to t i ; if there is a path from s to t in G, then with high probability, at least one of the G i 's has a unique path from s i to t i . This reduction is due to Avi Wigderson [Wig94], and it exploits the "isolation lemma" of Mulmuley, Vazirani and Vazira

We demonstrate differences between reducibilities and corresponding completeness notions for nondeterministic time and space classes. For time classes the studied completeness notions under polynomial-time bounded (even logarithmic space bounded) reducibilities turn out to be different for any class containing NE . For space classes the completeness notions under logspace reducibilities can be separated for any class properly containing LOGSPACE . Key observation in obtaining the separations is the honesty property of reductions, which was recently observed to hold for the time classes and can be shown to hold for space classes. 1 Introduction Efficient reducibilities and completeness are two of the central concepts of complexity theory. Since the first use of polynomial time bounded Turing reductions by Cook [4] and the introduction of polynomial time bounded many-one reductions by Karp[9], considerable effort has been put in the investigation of properties and the relative strengt...

We prove time-space tradeoffs for traversing undi-rected graphs. One of these is a quadratic lower bound on a deterministic model that closely matches the recent probabilistic upper bound of Broder, Karlin, Raghavan, and Upfal. The models used are variants of Cook and Rackoff’s “Jumping Automata for Graphs".

Abstract. Directed st-connectivity is the problem of deciding whether or not there exists a path from a distinguished node s to a distinguished node t in a directed graph. We prove a time– space lower bound on the probabilistic NNJAG model of Poon [Proc. 34th Annual Symposium on Foundations of Computer Science, Palo Alto, CA, 1993, pp. 218–227]. Let n be the number of nodes in the input graph and S and T be the space and time used by the NNJAG, respectively. We show that, for any δ>0, if an NNJAG uses space S ∈ O(n1−δ), then T ∈ 2Ω(log2 (n/S)) ; otherwise n log n) / log log n) S

On the complexity of the st-connectivity problem Chung Keung Poon Doctor of Philosophy 1996 Department of Computer Science University of Toronto The directed st-connectivity problem is fundamental to computer science. There are many applications which require algorithms to solve the problem in small space and preferably in small time as well. Furthermore, its space and time-space complexities are related to several long-standing open problems in complexity theory. Depth- and breadth-first search are well known algorithms that solve the problem in optimal (i.e., O(n m)) time while using O(n log n) space where n and m are the number of nodes and edges in the graph respectively. It can also be solved in O(log 2 n) space and 2 O(log 2 n) time by Savitch's algorithm. For space S between \Theta(log 2 n) and \Theta(n log n), the best running time is T = 2 O(log 2 (n log n=S)) \Theta mn due to Barnes et al.. Establishing matching lower bounds on the Turing machine model ha...

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We discuss several versions of a set theoretic \Delta-language as a reasonable prototype for "nested" data base query language where data base states and queries are considered, respectively, as hereditarily-finite sets and set theoretic operations. In a previous work such a language exactly corresponding to PTIME-computability was introduced. It is supposed that HF-sets are naturally presented by vertices of acyclic graphs. Here we consider a number of languages for Sub-PTIME computable set operations via corresponding graph transformers. Two such languages lead to a notion of NLOGSPACE and, respectively, DLOGSPACE computable queries over HF which appear the most natural, at our present knowledge, among others considered here. Unlike the "flat" relational data bases the problem of finding sufficiently good corresponding approach for HF proves to be more intricate and, furthermore, gives rise to some interesting questions in finite model theory (cf. Section 13). 1 General Introduction ...