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60
On the complexity of numerical analysis
 IN PROC. 21ST ANN. IEEE CONF. ON COMPUTATIONAL COMPLEXITY (CCC ’06
, 2006
"... We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis: • The BlumShubSmale model of computation over the reals. • A problem we call the “Generic Task of Numerical Computation, ” which captures an aspect of doing numerical computation ..."
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Cited by 48 (7 self)
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We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis: • The BlumShubSmale 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 divisionfree straightline program producing an integer N, decide whether N> 0. • In the BlumShubSmale model, polynomial time computation over the reals (on discrete inputs) is polynomialtime 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 polynomialtime 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.
On the Complexity of Nash Equilibria and Other Fixed Points (Extended Abstract)
 IN PROC. FOCS
, 2007
"... We reexamine what it means to compute Nash equilibria and, more generally, what it means to compute a fixed point of a given Brouwer function, and we investigate the complexity of the associated problems. Specifically, we study the complexity of the following problem: given a finite game, Γ, with 3 ..."
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Cited by 39 (4 self)
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We reexamine what it means to compute Nash equilibria and, more generally, what it means to compute a fixed point of a given Brouwer function, and we investigate the complexity of the associated problems. Specifically, we study the complexity of the following problem: given a finite game, Γ, with 3 or more players, and given ɛ> 0, compute an approximation within ɛ of some (actual) Nash equilibrium. We show that approximation of an actual Nash Equilibrium, even to within any nontrivial constant additive factor ɛ < 1/2 in just one desired coordinate, is at least as hard as the long standing squareroot sum problem, as well as a more general arithmetic circuit decision problem that characterizes Ptime in a unitcost model of computation with arbitrary precision rational arithmetic; thus placing the approximation problem in P, or even NP, would resolve major open problems in the complexity of numerical computation. We show similar results for market equilibria: it is hard to estimate with any nontrivial accuracy the equilibrium prices in an exchange economy with a unique equilibrium, where the economy is given by explicit algebraic formulas for the excess demand functions. We define a class, FIXP, which captures search problems that can be cast as fixed point
Algorithmic verification of recursive probabilistic state machines
 In Proc. 11th TACAS
, 2005
"... Abstract. Recursive Markov Chains (RMCs) ([EY04]) are a natural abstract model of procedural probabilistic programs and related systems involving recursion and probability. They succinctly define a class of denumerable Markov chains that generalize multitype branching (stochastic) processes. In thi ..."
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Cited by 37 (7 self)
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Abstract. Recursive Markov Chains (RMCs) ([EY04]) are a natural abstract model of procedural probabilistic programs and related systems involving recursion and probability. They succinctly define a class of denumerable Markov chains that generalize multitype branching (stochastic) processes. In this paper, we study the problem of model checking an RMC against a given ωregular specification. Namely, given an RMC A and a Büchi automaton B, we wish to know the probability that an execution of A is accepted by B. We establish a number of strong upper bounds, as well as lower bounds, both for qualitative problems (is the probability = 1, or = 0?), and for quantitative problems (is the probability ≥ p?, or, approximate the probability to within a desired precision). Among these, we show that qualitative model checking for general RMCs can be decided in PSPACE in A  and EXPTIME in B, and when A is either a singleexit RMC or when the total number of entries and exits in A is bounded, it can be decided in polynomial time in A. We then show that quantitative model checking can also be done in PSPACE in A, and in EXPSPACE in B. When B is deterministic, all our complexities in B  come down by one exponential. For lower bounds, we show that the qualitative model checking problem, even for a fixed RMC, is already EXPTIMEcomplete. On the other hand, even for simple reachability analysis, we showed in [EY04] that our PSPACE upper bounds in A can not be improved upon without a breakthrough on a wellknown open problem in the complexity of numerical computation. 1
Recursive Markov decision processes and recursive stochastic games
 In Proc. of 32nd Int. Coll. on Automata, Languages, and Programming (ICALP’05
, 2005
"... Abstract. We introduce Recursive Markov Decision Processes (RMDPs) and Recursive Simple Stochastic Games (RSSGs), and study the decidability and complexity of algorithms for their analysis and verification. These models extend Recursive Markov Chains (RMCs), introduced in [EY05a,EY05b] as a natural ..."
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Cited by 37 (9 self)
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Abstract. We introduce Recursive Markov Decision Processes (RMDPs) and Recursive Simple Stochastic Games (RSSGs), and study the decidability and complexity of algorithms for their analysis and verification. These models extend Recursive Markov Chains (RMCs), introduced in [EY05a,EY05b] as a natural model for verification of probabilistic procedural programs and related systems involving both recursion and probabilistic behavior. RMCs define a class of denumerable Markov chains with a rich theory generalizing that of stochastic contextfree grammars and multitype branching processes, and they are also intimately related to probabilistic pushdown systems. RMDPs & RSSGs extend RMCs with one controller or two adversarial players, respectively. Such extensions are useful for modeling nondeterministic and concurrent behavior, as well as modeling a system’s interactions with an environment. We provide a number of upper and lower bounds for deciding, given an RMDP (or RSSG) A and probability p, whether player 1 has a strategy to force termination at a desired exit with probability at least p. We also address “qualitative ” termination questions, where p = 1, and model checking questions. 1
On the decidability of temporal properties of probabilistic pushdown automata
 In Proc. of STACS’05
, 2005
"... Abstract. We consider qualitative and quantitative modelchecking problems for probabilistic pushdown automata (pPDA) and various temporal logics. We prove that the qualitative and quantitative modelchecking problem for ωregular properties and pPDA is in 2EXPSPACE and 3EXPTIME, respectively. We ..."
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Cited by 30 (9 self)
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Abstract. We consider qualitative and quantitative modelchecking problems for probabilistic pushdown automata (pPDA) and various temporal logics. We prove that the qualitative and quantitative modelchecking problem for ωregular properties and pPDA is in 2EXPSPACE and 3EXPTIME, respectively. We also prove that modelchecking the qualitative fragment of the logic PECTL ∗ for pPDA is in 2EXPSPACE, and modelchecking the qualitative fragment of PCTL for pPDA is in EXPSPACE. Furthermore, modelchecking the qualitative fragment of PCTL is shown to be EXPTIMEhard even for stateless pPDA. Finally, we show that PCTL modelchecking is undecidable for pPDA, and PCTL + modelchecking is undecidable even for stateless pPDA. 1
Recursive concurrent stochastic games
 In Proc. of 33rd Int. Coll. on Automata, Languages, and Programming (ICALP’06
, 2006
"... Abstract. We study Recursive Concurrent Stochastic Games (RCSGs), extending our recent analysis of recursive simple stochastic games [16, 17] to a concurrent setting where the two players choose moves simultaneously and independently at each state. For multiexit games, our earlier work already show ..."
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Cited by 22 (4 self)
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Abstract. We study Recursive Concurrent Stochastic Games (RCSGs), extending our recent analysis of recursive simple stochastic games [16, 17] to a concurrent setting where the two players choose moves simultaneously and independently at each state. For multiexit games, our earlier work already showed undecidability for basic questions like termination, thus we focus on the important case of singleexit RCSGs (1RCSGs). We first characterize the value of a 1RCSG termination game as the least fixed point solution of a system of nonlinear minimax functional equations, and use it to show PSPACE decidability for the quantitative termination problem. We then give a strategy improvement technique, which we use to show that player 1 (maximizer) has ǫoptimal randomized Stackless & Memoryless (rSM) strategies for all ǫ> 0, while player 2 (minimizer) has optimal rSM strategies. Thus, such games are rSMdetermined. These results mirror and generalize in a strong sense the randomized memoryless determinacy results for finite stochastic games, and extend the classic HoffmanKarp [22] strategy improvement approach from the finite to an infinite state setting. The proofs in our infinitestate setting are very different however, relying on subtle analytic properties of certain power series that arise from studying 1RCSGs. We show that our upper bounds, even for qualitative (probability 1) termination, can not be improved, even to NP, without a major breakthrough, by giving two reductions: first a Ptime reduction from the longstanding squareroot sum problem to the quantitative termination decision problem for finite concurrent stochastic games, and then a Ptime reduction from the latter problem to the qualitative termination problem for 1RCSGs. 1.
On the convergence of Newton’s method for monotone systems of polynomial equations
 In Proceedings of STOC
, 2007
"... kiefersn, luttenml, esparza o ..."
Efficient qualitative analysis of classes of recursive markov decision processes and simple stochastic games
 In Proc. STACS’06
, 2006
"... Abstract. Recursive Markov Decision Processes (RMDPs) and Recursive Simple Stochastic Games (RSSGs) are natural models for recursive systems involving both probabilistic and nonprobabilistic actions. As shown recently [10], fundamental problems about such models, e.g., termination, are undecidable ..."
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Cited by 13 (7 self)
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Abstract. Recursive Markov Decision Processes (RMDPs) and Recursive Simple Stochastic Games (RSSGs) are natural models for recursive systems involving both probabilistic and nonprobabilistic actions. As shown recently [10], fundamental problems about such models, e.g., termination, are undecidable in general, but decidable for the important class of 1exit RMDPs and RSSGs. These capture controlled and game versions of multitype Branching Processes, an important and wellstudied class of stochastic processes. In this paper we provide efficient algorithms for the qualitative termination problem for these models: does the process terminate almost surely when the players use their optimal strategies? Polynomial time algorithms are given for both maximizing and minimizing 1exit RMDPs (the two cases are not symmetric). For 1exit RSSGs the problem is in NP∩coNP, and furthermore, it is at least as hard as other wellknown NP∩coNP problems on games, e.g., Condon’s quantitative termination problem for finite SSGs ([3]). For the class of linearlyrecursive 1exit RSSGs, we show that the problem can be solved in polynomial time.
Verifying Probabilistic Procedural Programs
, 2004
"... Monolithic nitestate probabilistic programs have been abstractly modeled by nite Markov chains, and the algorithmic veri  cation problems for them have been investigated very extensively. In this paper we survey recent work conducted by the authors together with colleagues on the algorithmi ..."
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Cited by 12 (3 self)
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Monolithic nitestate probabilistic programs have been abstractly modeled by nite Markov chains, and the algorithmic veri  cation problems for them have been investigated very extensively. In this paper we survey recent work conducted by the authors together with colleagues on the algorithmic veri cation of probabilistic procedural programs ([BKS,EKM04,EY04]). Probabilistic procedural programs can more naturally be modeled by recursive Markov chains ([EY04]), or equivalently, probabilistic pushdown automata ([EKM04]). A very rich theory emerges for these models. While our recent work solves a number of veri cation problems for these models, many intriguing questions remain open.
CONVERGENCE THRESHOLDS OF NEWTON’S METHOD FOR MONOTONE POLYNOMIAL EQUATIONS
, 2008
"... Monotone systems of polynomial equations (MSPEs) are systems of fixedpoint equations X1 = f1(X1,..., Xn),..., Xn = fn(X1,..., Xn) where each fi is a polynomial with positive real coefficients. The question of computing the least nonnegative solution of a given MSPE X = f(X) arises naturally in the ..."
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Cited by 12 (8 self)
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Monotone systems of polynomial equations (MSPEs) are systems of fixedpoint equations X1 = f1(X1,..., Xn),..., Xn = fn(X1,..., Xn) where each fi is a polynomial with positive real coefficients. The question of computing the least nonnegative solution of a given MSPE X = f(X) arises naturally in the analysis of stochastic models such as stochastic contextfree grammars, probabilistic pushdown automata, and backbutton processes. Etessami and Yannakakis have recently adapted Newton’s iterative method to MSPEs. In a previous paper we have proved the existence of a threshold kf for strongly connected MSPEs, such that after kf iterations of Newton’s method each new iteration computes at least 1 new bit of the solution. However, the proof was purely existential. In this paper we give an upper bound for kf as a function of the minimal component of the least fixedpoint µf of f(X). Using this result we show that kf is at most single exponential resp. linear for strongly connected MSPEs derived from probabilistic pushdown automata resp. from backbutton processes. Further, we prove the existence of a threshold for arbitrary MSPEs after which each new iteration computes at least 1/w2 h new bits of the solution, where w and h are the width and height of the DAG of strongly connected components.