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.

We show that the graph isomorphism problem is hard under DLOGTIME uniform AC0 many-one reductions for the complexity classes NL, PL (probabilistic logarithmic space) for every logarithmic space modular class ModkL and for the class DET of problems NC¹ reducible to the determinant. These are the strongest known hardness results for the graph isomorphism problem and imply a randomized logarithmic space reduction from the perfect matching problem to graph isomorphism. We also investigate hardness results for the graph automorphism problem.

A partially-observable Markov decision process (POMDP) is a generalization of a Markov decision process that allows for incomplete information regarding the state of the system. POMDPs are used to model controlled stochastic processes, from health care to manufacturing control processes (see [19] for more examples). We consider several flavors of finite-horizon POMDPs. Our results concern the complexity of the policy evaluation and policy existence problems, which are characterized in terms of completeness for complexity classes. Although a large body of literature in mathematics, operations research, and engineering deals with optimization and approximation strategies for POMDPs, there has been little work aimed at characterizing the complexity of these problems and proving lower bounds. We prove a new upper bound of the policy evaluation problem for POMDPs, showing it is Probabilistic Logspace complete. From this, we prove policy existence problems for several variants of unobservabl...

Introduction The classes MOD k L for k 2 consist of those languages A, for which there exists a nondeterministic Turing machine M working on logarithmically bounded space such that, for all inputs x, x belongs to set A if and only if the number of accepting paths in the computation of M on x is not divisible by k. These classes were defined and examined in [BDHM92]. Their importance stems from the fact that the complexity of a number of problems from linear algebra over Z=kZ is given by these classes (in the sense that they are complete in the respective classes); among these problems are singularity of matrices, inversion of matrices, iterated matrix product, etc. Buntrock et al. also examined structural properties of these classes. E.g., it was shown in [BDHM92] that for prime<

Markov Decision Processes (MDPs) model controlled stochastic systems. Like Markov chains, an MDP consists of states and probabilistic transitions; unlike Markov chains, there is assumed to be an outside controller who chooses an action (with its associated transition matrix) at each step of the process, according to some strategy or policy. In addition, each state and action pair has an associated reward. The goal of the controller is to maximize the expected reward. MDPs are used in applications as diverse as wildlife management and robot navigation control. Optimization and approximation strategies for these models constitute a major body of literature in mathematics, operations research, and engineering. We consider the complexity of the following decision problem: for a given MDP and type of policy, is there such a policy for that MDP with positive expected reward? The complexity of this problem depends on at least half a dozen factors, including the information available to the co...

Tensor calculus over semirings is shown relevant to complexity theory in unexpected ways. First, evaluating well-formed tensor formulas with explicit tensor entries is shown complete for \PhiP, for NP, and for #P as the semiring varies. Indeed the permanent of a matrix is shown expressible as the value of a tensor formula in much the same way that Berkowitz' theorem expresses its determinant. Second, restricted tensor formulas are shown to capture the classes LOGCFL and NL, their parity counterparts \PhiLOGCFL and \PhiL, and several other counting classes. Finally, the known inclusions NP=poly ` \PhiP=poly, LOGCFL=poly ` \PhiLOGCFL=poly, and NL=poly ` \PhiL=poly, which have scattered proofs in the literature [21, 39], are shown to follow from the new characterizations in a single blow. 1 Introduction Consider an algebraic structure S with certain operations. The following problem is sometimes called the word problem of S: given a reasonable encoding of a well-formed expression T ove...

. Counting functions can be defined syntactically or semantically depending on whether they count the number of witnesses in a non-deterministic or in a deterministic computation on the input. In the Turing machine based model, these two ways of defining counting were proven to be equivalent for many important complexity classes. In the circuit based model, it was done for #P and #L, but for low-level complexity classes such as #AC 0 and #NC 1 only the syntactical definitions were considered. We give appropriate semantical definitions for these two classes and prove them to be equivalent to the syntactical ones. This enables us to show that #AC 0 is included in the family of counting functions computed by polynomial size and constant width counting branching programs, therefore completing a result of Caussinus et al [CMTV98]. We also consider semantically defined probabilistic complexity classes corresponding to AC 0 and NC 1 and prove that in the case of unboun...

The computational complexity of finite horizon policy evaluation and policy existence problems are studied for several policy types and representations of Markov decision processes. In almost all cases, the problems are shown to be complete for their complexity classes; classes range from nondeterministic logarithmic space and probabilistic logarithmic space (highly parallelizable classes) to exponential space. In many cases, this work shows that problems that already were widely believed to be hard to compute are probably intractable (complete for NP, NP PP , or PSPACE), or provably intractable (EXPTIMEcomplete or worse). The major contributions of the paper are to pinpoint the complexity of these problems; to isolate the factors that make these problems computationally complex; to show that even problems such as median-policy or average-policy evaluation may be intractable; and the introduction of natural NP PP -complete problems. 1 Introduction Markov decision processes are us...

We investigate alternative notions of approximation for problems inside P (deter-ministic polynomial time), and show that even a slightly nontrivial information about a problem may be as hard to obtain as the solution itself. For example, we prove that if one could eliminate even a single possibility for the value of an arithmetic circuit on a given input, then this would imply that the class P has fast (polygarithmic time) parallel solutions. In other words, this would constitute a proof that there are no inherently sequential problems in P, which is quite unlikely. The result is robust with respect to eliminating procedures that are allowed to err (by excluding the correct value) with small probability. We also show that several fundamental linear algebra problems are hard in this sense. It turns out that it is as hard to substantially reduce the number of possible values for the determinant and rank as to compute them exactly. Finally, we show that (in some precise sense) randomness can be nontrivially substituted for nondeterminism in space. Although it is believed that randomness does not give more than a constant factor advantage in space over determinism, it is not even known whether it is no more powerful than nondeterminism. We will show that the latter is true for a restricted version of probabilistic logspace, where the error is potentially larger than what can be achieved by amplification.

We give two time- and space-efficient simulations of quantum computations with intermediate measurements, one by classical randomized computations with unbounded error and the other by quantum computations that use an arbitrary fixed universal set of gates. Specifically, our simulations show that every language solvable by a bounded-error quantum algorithm running in time t and space s is also solvable by an unbounded-error randomized algorithm running in time O(t · log t) and space O(s + log t), as well as by a bounded-error quantum algorithm restricted to use an arbitrary universal set and running in time O(t · polylog t) and space O(s + log t), provided the universal set is closed under adjoint. We also develop a quantum model that is particularly suitable for the study of general computations with simultaneous time and space bounds. As an application of our randomized simulation, we obtain the first nontrivial lower bound for general quantum algorithms solving problems related to satisfiability. Our bound applies to MajSAT and MajMajSAT, which are the problems of determining the truth value of a given Boolean formula whose variables are fully quantified by one or two majority quantifiers, respectively. We prove that for every real d and every positive real δ there exists a real c> 1 such that either • MajMajSAT does not have a bounded-error quantum algorithm running in time O(n c), or • MajSAT does not have a bounded-error quantum algorithm running in time O(n d) and space O(n 1−δ). In particular, MajMajSAT does not have a bounded-error quantum algorithm running in time O(n 1+o(1) ) and space O(n 1−δ) for any δ> 0. Our lower bounds hold for any reasonable uniform model of quantum computation, in particular for the model we develop. 1