Circumscription and the closed world assumption with its variants are well-known nonmonotonic techniques for reasoning with incomplete knowledge. Their complexity in the propositional case has been studied in detail for fragments of propositional logic. One open problem is whether the deduction problem for arbitrary propositional theories under the extended closed world assumption or under circumscription is $\Pi^P_2$-complete, i.e., complete for a class of the second level of the polynomial hierarchy. We answer this question by proving these problems $\Pi^P_2$-complete, and we show how this result applies to other variants of closed world reasoning.

This paper is dedicated to the memory of Ronald V. Book, 1937-1997.

A set A is p-terse (p-superterse) if, for all q, it is not possible to answer q queries to A by making only q \Gamma 1 queries to A (any set X). Formally, let PF A q-tt be the class of functions reducible to A via a polynomial-time truthtable reduction of norm q, and let PF A q-T be the class of functions reducible to A via a polynomial-time Turing reduction that makes at most q queries. A set A is p-terse if PF A q-tt 6` PF A (q\Gamma1)-T for all constants q. A is p-superterse if PF A q-tt 6` PF X q-T for all constants q and sets X . We show that all NP-hard sets (under p tt -reductions) are p-superterse, unless it is possible to distinguish uniquely satisfiable formulas from satisfiable formulas in polynomial time. Consequently, all NP-complete sets are psuperterse unless P = UP (one-way functions fail to exist), R = NP (there exist randomized polynomial-time algorithms for all problems in NP), and the polynomial-time hierarchy collapses. This mostly solves the main open...

The process of reconstructing evolutionary trees can be viewed formally as an optimization problem. Recently, decision problems associated with the most commonly used approaches to reconstructing such trees have been shown to be NP-complete [Day87, DJS86, DS86, DS87, GF82, Kri88, KM86]. In this thesis, a framework is established that incorporates all such problems studied to date. Within this framework, the NP-completeness results for decision problems are extended by applying theorems from [CT91, Gas86, GKR92, JVV86, KST89, Kre88, Sel91] to derive bounds on the computational complexity of several functions associated with each of these problems, namely ffl evaluation functions, which return the cost of the optimal tree(s), ffl solution functions, which return an optimal tree, ffl spanning functions, which return the number of optimal trees, ffl enumeration functions, which systematically enumerate all optimal trees, and ffl random-selection functions, which return a random...

An oracle A is k-cheatable if there is a polynomial-time algorithm to determine the answers to 2 k parallel queries to A from the answers to only k queries to some other oracle B. It is known that 1-cheatable sets cannot be bi-immune for P. In contrast, we construct 2-cheatable sets that are bi-immune for arbitrary time complexity classes. In addition, for each k, we construct a set that is (k + 1)-cheatable, but not k-cheatable; we show that this separation does not hold with biimmunity. We show that if a recursive set A is bi-immune for P then there exists an infinite 1-cheatable set that is polynomial-time mreducible to A. Consequently if NP contains a set that is bi-immune for P then NP contains a set that is not polynomial-time Turingequivalent to a self-reducible set. 1. Introduction Complexity theory deals with how hard problems are. Time, space, and alternation have served as measures of difficulty. Recently, researchers have Research supported by a Fannie and John Hertz ...

The purpose of this work is to present an overview of the class of problems solvable in probabilistic polynomial time with double sided error (PP ). We explore the relationship of PP to other complexity classes, in particular NP and the polynomial hierarchy, and discuss closure under some standard operations such as intersection and complementation. New proofs are given of some results from the literature using techniques developed by the author. ii Acknowledgements Several people made possible the successful completion of this work. Express thanks must be given to some of them. ffl To my girlfriend, Claudia Iturriaga Vel'azquez, for her unconditional love. ffl To Prabhakar Ragde. His patience towards my uncommon research style which, in particular, comprises exponentially many queries and interruptions is praised. ffl To all my friends, for cheering me up during hard times. Particular thanks to Vladimir Estivill-Castro and Luke O'Connor. Lastly, financial support from the Institut...