Optimization and NP_R-completeness of certain fewnomials
Proceedings of the 2009 conference on Symbolic numeric computation
Computational Aspects of Equilibria
SAGT '09 Proceedings of the 2nd International Symposium on Algorithmic Game Theory
Optimizing n-variate (n+k)-nomials for small k
Theoretical Computer Science
Computing the Least Fixed Point of Positive Polynomial Systems
SIAM Journal on Computing
On the Complexity of Nash Equilibria and Other Fixed Points
SIAM Journal on Computing
The complexity of nash equilibria in limit-average games
CONCUR'11 Proceedings of the 22nd international conference on Concurrency theory
Model Checking of Recursive Probabilistic Systems
ACM Transactions on Computational Logic (TOCL)
Polynomial time algorithms for multi-type branching processesand stochastic context-free grammars
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
On the complexity of the equivalence problem for probabilistic automata
FOSSACS'12 Proceedings of the 15th international conference on Foundations of Software Science and Computational Structures
Computing bits of algebraic numbers
TAMC'12 Proceedings of the 9th Annual international conference on Theory and Applications of Models of Computation
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
On the Sum of Square Roots of Polynomials and Related Problems
ACM Transactions on Computation Theory (TOCT)
On the Computational Complexity of Stochastic Controller Optimization in POMDPs
ACM Transactions on Computation Theory (TOCT)
On the complexity of model checking interval-valued discrete time Markov chains
Information Processing Letters
LTL model checking of interval markov chains
TACAS'13 Proceedings of the 19th international conference on Tools and Algorithms for the Construction and Analysis of Systems
The orbit problem in higher dimensions
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
CAV'13 Proceedings of the 25th international conference on Computer Aided Verification
Analyzing probabilistic pushdown automata
Formal Methods in System Design
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We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis: (a) the Blum-Shub-Smale model of computation over the reals; and (b) 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 $N0$. 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 the arithmetic circuit identity testing (ACIT) problem. In particular, we show that if $n!$ is not ultimately easy, then ACIT has subexponential complexity.