Nick Reingold

PP is closed under intersection

In this seminal paper on probabilistic Turing machines, Gill asked whether the class PP is closed under intersection and union. We give a positive answer to this question. We also show that PP is closed under a variety of polynomial-time truth-table reductions. Consequences in complexity theory include the definite collapse and (assuming P ? PP) separation of certain query hierarchies over PP. Similar techniques allow us to combine several threshold gates into a single threshold gate. Consequences in the study of circuits include the simulation of circuits with a small number of threshold gates by circuits having only a single threshold gate at the root (perceptrons) and a lower bound on the number of threshold gates that are needed to compute the parity function.

Randomized competitive algorithms for the list update problem

We prove upper and lower bounds on the competitiveness of randomized algorithms for the list update problem of Sleator and Tarjan. We give a simple and elegant randomized algorithm that is more competitive than the best previous randomized algorithm due to Irani. Our algorithm uses randomness only during an initialization phase, and from then on runs completely deterministically. It is the first randomized competitive algorithm with this property to beat the deterministic lower bound. We generalize our approach to a model in which access costs are fixed but update costs are scaled by an arbitrary constantd. We prove lower bounds for deterministic list update algorithms and for randomized algorithms against oblivious and adaptive on-line adversaries. In particular, we show that for this problem adaptive on-line and adaptive off-line adversaries are equally powerful.

The perceptron strikes back

It is shown that every AC/sup 0/ predicate is computed by a low-degree probabilistic polynomial over the reals. It is demonstrated that circuits composed of a symmetric gate at the root with AND-OR subcircuits of constant depth can be simulated by probabilistic depth-2 circuits with essentially the same symmetric gate at the root and AND gates of small fanin at the bottom. In particular, every language recognized by a depth-d AC/sup 0/ circuit is decidable by a probabilistic perceptron of size 2 to the power O(log/sup 4d/ n) and of order O(log/sup 4d/ n) that uses O(log/sup 3/ n) probabilistic bits. As a corollary, the authors present a new proof that depth-d AND-OR circuits computing the parity of n binary inputs require size 2 to the power n/sup Omega (1/d)/.<<ETX>>

PP is closed under truth-table reductions

R. Beigel et al. (1991) showed that PP is closed under intersection and a variety of special cases of truth-table closure. In the present work, the authors extend the techniques of Beigel et al. to show that PP is closed under general polynomial-time truth-time reductions.<<ETX>>

PP Is Closed under Truth-Table Reductions

Beigel, Reingold, and Spielman (J. Comput. System Sci.50, 191?202 (1995)) showed that PP is closed under intersection and a variety of special cases of polynomial-time truth-table closure. We extend their techniques to show that PP is closed under general polynomial-time truth-table reductions. We also show that PP is closed under constant-round truth-table reductions.

Games I/O automata play

We introduce a game approach for specifying reactive systems. In particular, we define a simple two-player game between System and Environment, and consider the outcomes of such a game as a specification of a reactive system. We introduce six classes of game languages. We then show that the class of languages generated by I/O automata equals one of our game classes. An immediate corollary to the proof is that the fairness condition of I/O automata, which is defined as an extrinsic property by Lynch and Tuttle, can be incorporated as an intrinsic part of the automata. We also show closure properties of the six game classes. For example, we show that the class of languages defined by I/O automata is closed under union and hiding but not under intersection or complementation. The closure results are obtained by reasoning directly about games, thus demonstrating the advantage of the game-based approach.