We examine the computational complexity of testing and finding small plans in probabilistic planning domains with both flat and propositional representations. The complexity of plan evaluation and existence varies with the plan type sought; we examine totally ordered plans, acyclic plans, and looping plans, and partially ordered plans under three natural definitions of plan value. We show that problems of interest are complete for a variety of complexity classes: PL, P, NP, co-NP, PP, NP PP, co-NP PP , and PSPACE. In the process of proving that certain planning problems are complete for NP PP , we introduce a new basic NP PP -complete problem, E-Majsat, which generalizes the standard Boolean satisfiability problem to computations involving probabilistic quantities; our results suggest that the development of good heuristics for E-Majsat could be important for the creation of efficient algorithms for a wide variety of problems.

Any implementation of Carter-Wegman universal hashing from n-bit strings to m-bit strings requires a time-space tradeoff of TS = Ω(nm). The bound holds in the general boolean branching program model, and thus in essentially any model of computation. As a corollary, computing a+b*c in any field F requires a quadratic time-space tradeoff, and the bound holds for any representation of the elements of the field. Other lower bounds on the...

This is a survey of space-bounded probabilistic computation, summarizing the present state of knowledge about the relationships between the various complexity classes associated with such computation. The survey especially emphasizes recent progress in the construction of pseudorandom generators that fool probabilistic space-bounded computations, and the application of such generators to obtain deterministic simulations.

We present a fast parallel deterministic algorithm for testing multivariate integral polynomials for absolute irreducibility, that is irreducibility over the complex numbers. More precisely, we establish that the set of absolutely irreducible integral polynomials belongs to the complexity class NC of Boolean circuits of polynomial size and logarithmic depth. Therefore it also belongs to the class of sequentially polynomial-time problems. Our algorithm can be extended to compute in parallel one irreducible complex factor of a multivariate integral polynomial. However, the coefficients of the computed factor are only represented modulo a not necessarily irreducible polynomial specifying a splitting field. A consequence of our algoithm is that multivariate polynomials over finite fields can be tested for absolute irreducibility in deterministic sequential polynomial time in the size of the input. We also obtain a sharp bound for the last prime p for which, when taking an absolutely irreducible integral polynomial modulo p, the polynomial's irreducibility in the algebraic closure of the finite field order p is not preserved.

Abstract. Boolean circuits of polynomial size and poly-logarithmic depth are given for computing the Hermite and Smith normal forms of polynomial matrices over finite fields and the field of rational numbers. The circuits for the Smith normal form computation are probabilistic ones and also determine very efficient sequential algorithms. Furthermore, we give a polynomial-time deterministic sequential algorithm for the Smith normal form over the rationals. The Smith normal form algorithms are applied to the Rational canonical form of matrices over finite fields and the field of rational numbers. Ke ywords: Parallel algorithm, Hermite normal form, Smith normal form, polynomial-time complexity. 1.

Recent results byToda, Vinay, Damm, and Valianthave shown that the complexity of the determinantischaracterized by the complexity of counting the number of accepting computations of a nondeterministic logspace-bounded machine. #This class of functions is known as #L.# By using that characterization and by establishing a few elementary closure properties, we giveavery simple proof of a theorem of Jung, showing that probabilistic logspace-bounded #PL# machines lose none of their computational power if they are restricted to run in polynomial time.

In this paper we propose a definition for honest verifier quantum statistical zero-knowledge interactive proof systems and study the resulting complexity class, which we denote QSZK

Given a description of a probabilistic automaton (one-head probabilistic finite automaton or probabilistic Turing machine) and an input string x of length n, we ask how much space does a deterministic Turing machine need in order to decide the acceptance of the input string by that automaton? The question is interesting even in the case of one-head one-way probabilistic finite automata (PFA). We call (rational) stochastic languages (S ? rat ) the class of languages recognized by PFA's whose transition probabilities and cutpoints (i.e. recognition thresholds) are rational numbers. The class S ? rat contains context-sensitive languages that are not context free, but on the other hand there are context-free languages not included in S ? rat . Our main results are as follows: ffl The (proper) inclusion of S ? rat in Dspace(log n), which is optimal (i.e. S ? rat 6ae Dspace(o(log n))). The previous upper bounds were Dspace(n) [Dieu 1972], [Wang 1992] and Dspace(log n log log n)...

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 the "correct" definition of randomized space-bounded computation, the machine has access to a random coin. The coin can be flipped at will, but outcomes of previous coin flips cannot be recalled unless they are saved in the machine's limited memory. In contrast to this read-once mechanism of accessing the random source, one may consider Turing machines which have access to a random tape. Here, the random bits may be multiply accessed by the machine. In this note we show a very concrete sense in which multiple access to the random bits is better than read-once access to them: Every language accepted with bounded 2-sided error by a read-once-randomized Logspace machine, can be accepted with zero error by a randomized Logspace machine having multiple access to the random bits. Finally we characterize the class of languages that can be accepted with two-sided error by randomized Logspace machines with multiple access to the random bits as exactly the class of languages tha...