We show that non-deterministic time NT IME(n) is not contained in deterministic time n # 2-# and polylogarithmic space, for any # > 0. This implies that (infinitely often) satisfiability cannot be solved in time O(n # 2-# ) and poly-logarithmic space. A similar result is presented for uniform circuits.

Finite-model theory is a study of the logical properties of finite mathematical structures. This paper is a very personalized view of finite-model theory, where the author focuses on his own personal history, and results and problems of interest to him, especially those springing from work in his Ph.D. thesis. Among the topics discussed are:

this paper, if T is time-constructible, then

In 1985, Goldwasser, Micali and Rackoff formulated interactive proof systems as a tool for developing cryptographic protocols. Indeed, many exciting cryptographic results followed from studying interactive proof systems and the related concept of zero-knowledge. Interactive proof systems also have an important part in complexity theory merging the well established concepts of probabilistic and nondeterministic computation. This thesis will study the complexity of various models of interactive proof systems. A perfect zero-knowledge interactive protocol convinces a verifier that a string is in a language without revealing any additional knowledge in an information theoretic sense. This thesis will show that for any language that has a perfect zero-knowledge proof system, its complement has a short interactive protocol. This result implies that there are not any perfect zero-knowledge protocols for NP-complete languages unless the polynomial-time hierarchy collapses. Thus knowledge comp...

: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 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 : : : : : : : : : : : : : : : : : : : : ...

We prove tight hierarchy theorems for bounded error probabilistic quasi-polynomial time classes, under several hardness assumptions. We show that assuming either (1) the Permanent does not have a subexponential time BP algorithm, or (2) some function in EXPTIME does not have subexponential size circuits, then for every 1 ff ! fi, BPTIME(2 (log n) ff ) $ BPTIME(2 (log n) fi ): Department of Computer Science, State University of New York at Buffalo, Buffalo, NY 14260. Email: cai@cs.buffalo.edu. Research supported in part by NSF grant CCR-9634665 and a J. S. Guggenheim Fellowship. y Department of Computer Science, State University of New York at Buffalo, Buffalo, NY 14260. Email: apn@cs.buffalo.edu. Research supported in part by NSF grant CCR-9634665. z Department of Computer Science, University of Houston, Houston, TX 77204. Email: siva@cs.uh.edu. Research supported in part by NSF CAREER award CCR-9734164. 1 Introduction It has been observed for some time now that the ...

this article mention all of the amazing research in computational complexity theory. We survey various areas in complexity choosing papers more for their historical value than necessarily the importance of the results. We hope that this gives an insight into the richness and depth of this still quite young eld

The authors of [MT05] proposed a declarative constraint programming framework based on classical logic extended with non-monotone inductive definitions. In the framework, a problem instance is a finite structure, and a problem specification is a formula defining the relationship between an instance and its solutions. Thus, problem solving amounts to expanding a finite structure with new relations, to satisfy the formula. We present here the complexities of model expansion for a number of logics, alongside those of satisfiability and model checking. As the task is equivalent to witnessing the existential quantifiers in ∃SO model checking, the paper is in large part of a survey of this area, together with some new results. In particular, we describe the combined and data complexity of FO(ID), first-order logic extended with inductive definitions [DT04] and the guarded and k-guarded logics of [AvBN98] and [GLS01]. 1

We show that for any constant a, ZPP/b(n) strictly contains ZPTIME(n a)/b(n) for some b(n) = O(log n log log n). Our techniques are very general and give the same hierarchy for

We propose a framework for modelling and solving search problems using logic, and describe a project whose goal is to produce practically effective, general purpose tools for representing and solving search problems based on this framework. The mathematical foundation lies in the areas of finite model theory and descriptive complexity, which provide us with many classical results, as well as powerful techniques, not available to many other approaches with similar goals. We describe the mathematical foundations; explain an extension to classical logic with inductive definitions that we consider central; give a summary of complexity and expressiveness properties; describe an approach to implementing solvers based on grounding; present grounding algorithms based on an extension of the relational algebra; describe an implementation of our framework which includes use of inductive definitions, sorts and order; and give experimental results comparing the performance of our implementation with ASP solvers and another solver based on the same framework. 1.