Results 1 - 10
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95
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 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 ..."
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Cited by 73 (5 self)
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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.
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 68 (8 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 non-trivial constant additive factor ɛ < 1/2 in just one desired coordinate, is at least as hard as the long standing square-root sum problem, as well as a more general arithmetic circuit decision problem that characterizes P-time in a unit-cost 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
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 52 (11 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 context-free grammars and multi-type 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
"... We consider qualitative and quantitative model-checking problems for probabilistic pushdown automata (pPDA) and various temporal logics. We prove that the qualitative and quantitative model-checking problem for ω-regular properties and pPDA is in 2-EXPSPACE and 3-EXPTIME, respectively. We also pro ..."
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Cited by 43 (12 self)
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We consider qualitative and quantitative model-checking problems for probabilistic pushdown automata (pPDA) and various temporal logics. We prove that the qualitative and quantitative model-checking problem for ω-regular properties and pPDA is in 2-EXPSPACE and 3-EXPTIME, respectively. We also prove that model-checking the qualitative fragment of the logic PECTL ∗ for pPDA is in 2-EXPSPACE, and model-checking the qualitative fragment of PCTL for pPDA is in EXPSPACE. Furthermore, model-checking the qualitative fragment of PCTL is shown to be EXPTIME-hard even for stateless pPDA. Finally, we show that PCTL model-checking is undecidable for pPDA, and PCTL + model-checking is undecidable even for stateless pPDA.
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 multi-type branching (stochastic) processes. In thi ..."
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Cited by 43 (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 multi-type branching (stochastic) processes. In this paper, we study the problem of model checking an RMC against a given ω-regular 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 single-exit 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 EXPTIME-complete. 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 well-known open problem in the complexity of numerical computation. 1
Quantitative analysis of probabilistic pushdown automata: . . .
, 2005
"... Probabilistic pushdown automata (pPDA) have been identified as a natural model for probabilistic programs with rcursive procedure calls. Previous works considered the decidability and complexity of the model-checking problem for pPDA and various probabilistic temporal logics. In this paper we concen ..."
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Cited by 41 (14 self)
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Probabilistic pushdown automata (pPDA) have been identified as a natural model for probabilistic programs with rcursive procedure calls. Previous works considered the decidability and complexity of the model-checking problem for pPDA and various probabilistic temporal logics. In this paper we concentrate on computing the expected values and variances of various random variables defined over runs of a given probabilistic pushdown automaton. In particular, we show how to compute the expected accumulated reward and the expected gain for certain classes of reward functions. Using these results, we show how to analyze various quantitative properties of pPDA that are not expressible in conventional probabilistic temporal logics.
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 multi-exit games, our earlier work already show ..."
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Cited by 30 (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 multi-exit games, our earlier work already showed undecidability for basic questions like termination, thus we focus on the important case of single-exit RCSGs (1-RCSGs). We first characterize the value of a 1-RCSG 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 ǫ-optimal randomized Stackless & Memoryless (r-SM) strategies for all ǫ> 0, while player 2 (minimizer) has optimal r-SM strategies. Thus, such games are r-SM-determined. These results mirror and generalize in a strong sense the randomized memoryless determinacy results for finite stochastic games, and extend the classic Hoffman-Karp [22] strategy improvement approach from the finite to an infinite state setting. The proofs in our infinite-state setting are very different however, relying on subtle analytic properties of certain power series that arise from studying 1-RCSGs. 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 P-time reduction from the long-standing square-root sum problem to the quantitative termination decision problem for finite concurrent stochastic games, and then a P-time reduction from the latter problem to the qualitative termination problem for 1-RCSGs. 1.
On the convergence of Newton’s method for monotone systems of polynomial equations
- In Proceedings of STOC
, 2007
"... kiefersn, luttenml, esparza o ..."
Quasibirth-death processes, Tree-Like QBDs, probabilistic 1-counter automata, and pushdown systems
, 2008
"... We begin by observing that (discrete-time) Quasi-Birth-Death Processes (QBDs) are equivalent, in a precise sense, to (discrete-time) probabilistic 1-Counter Automata (p1CAs), and both Tree-Like QBDs (TL-QBDs) and Tree-Structured QBDs (TS-QBDs) are equivalent to both probabilistic Pushdown Systems ..."
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Cited by 23 (8 self)
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We begin by observing that (discrete-time) Quasi-Birth-Death Processes (QBDs) are equivalent, in a precise sense, to (discrete-time) probabilistic 1-Counter Automata (p1CAs), and both Tree-Like QBDs (TL-QBDs) and Tree-Structured QBDs (TS-QBDs) are equivalent to both probabilistic Pushdown Systems
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 non-probabilistic actions. As shown recently [10], fundamental problems about such models, e.g., termination, are undecidable ..."
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Cited by 22 (9 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 non-probabilistic actions. As shown recently [10], fundamental problems about such models, e.g., termination, are undecidable in general, but decidable for the important class of 1-exit RMDPs and RSSGs. These capture controlled and game versions of multi-type 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 1-exit 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 well-known NP∩coNP problems on games, e.g., Condon’s quantitative termination problem for finite SSGs ([3]). For the class of linearly-recursive 1-exit RSSGs, we show that the problem can be solved in polynomial time.
