Abstract. We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis:• The Blum-Shub-Smale model of computation over the reals.• A problem we call the "Generic Task of Numerical Computation," which captures an aspect of doing numerical computation in floating point, similar to the "long exponent model" that has been studied in the numerical computing community. We show that both of these approaches hinge on the question of understanding the complexity of the following problem, which we call PosSLP: Given a division-free straight-line program producing an integer N , decide whether N > 0.• In the Blum-Shub-Smale model, polynomial time computation over the reals (on discrete inputs) is polynomial-time equivalent to PosSLP, when there are only algebraic constants. We conjecture that using transcendental constants provides no additional power, beyond nonuniform reductions to PosSLP, and we present some preliminary results supporting this conjecture.• The Generic Task of Numerical Computation is also polynomial-time equivalent to PosSLP. We prove that PosSLP lies in the counting hierarchy. Combining this with work of Tiwari, we obtain that the Euclidean Traveling Salesman Problem lies in the counting hierarchy -the previous best upper bound for this important problem (in terms of classical complexity classes) being PSPACE.In the course of developing the context for our results on arithmetic circuits, we present some new observations on the complexity of ACIT: the Arithmetic Circuit Identity Testing problem. In particular, we show that if n! is not ultimately easy, then ACIT has subexponential complexity.
Controlled stochastic systems occur in science engineering, manufacturing, social sciences, and many other cntexts. If the systems is modeled as a Markov decision process (MDP) and will run
ad infinitum
, the optimal control policy can be computed in polynomial time using linear programming. The problems considered here assume that the time that the process will run is finite, and based on the size of the input. There are mny factors that compound the complexity of computing the optimal policy. For instance, there are many factors that compound the complexity of this computation. For instance, if the controller does not have complete information about the state of the system, or if the system is represented in some very succint manner, the optimal policy is provably not computable in time polynomial in the size of the input. We analyze the computational complexity of evaluating policies and of determining whether a sufficiently good policy exists for a MDP, based on a number of confounding factors, including the observability of the system state; the succinctness of the representation; the type of policy; even the number of actions relative to the number of states. In almost every case, we show that the decision problem is complete for some known complexity class. Some of these results are familiar from work by Papadimitriou and Tsitsiklis and others, but some, such as our PL-completeness proofs, are surprising. We include proofs of completeness for natural problems in the as yet little-studied classes NP
PP
.
We c haracterize the complexity of some natural and important problems in linear algebra. In particular, we identify natural complexity classes for which the problems of a determining if a system of linear equations is feasible and b computing the rank of an integer matrix, as well as other problems, are complete under logspace reductions.As an important part of presenting this classi cation, we show that the exact counting logspace hierarchy" collapses to near the bottom level. We review the de nition of this hierarchy below. We further show that this class is closed under NC 1 -reducibility, and that it consists of exactly those languages that have logspace uniform span programs introduced by Karchmer and Wigderson over the rationals.In addition, we contrast the complexity of these problems with the complexity o f determining if a system of linear equations has an integer solution.
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