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43
Recursive Markov chains, stochastic grammars, and monotone systems of nonlinear equations
 IN STACS
, 2005
"... We define Recursive Markov Chains (RMCs), a class of finitely presented denumerable Markov chains, and we study algorithms for their analysis. Informally, an RMC consists of a collection of finitestate Markov chains with the ability to invoke each other in a potentially recursive manner. RMCs offer ..."
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Cited by 67 (11 self)
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We define Recursive Markov Chains (RMCs), a class of finitely presented denumerable Markov chains, and we study algorithms for their analysis. Informally, an RMC consists of a collection of finitestate Markov chains with the ability to invoke each other in a potentially recursive manner. RMCs offer a natural abstract model for probabilistic programs with procedures. They generalize, in a precise sense, a number of well studied stochastic models, including Stochastic ContextFree Grammars (SCFG) and MultiType Branching Processes (MTBP). We focus on algorithms for reachability and termination analysis for RMCs: what is the probability that an RMC started from a given state reaches another target state, or that it terminates? These probabilities are in general irrational, and they arise as (least) fixed point solutions to certain (monotone) systems of nonlinear equations associated with RMCs. We address both the qualitative problem of determining whether the probabilities are 0, 1 or inbetween, and
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
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
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 complexity of real functions
, 2005
"... We establish a new connection between the two most common traditions in the theory of real computation, the BlumShubSmale model and the Computable Analysis approach. We then use the connection to develop a notion of computability and complexity of functions over the reals that can be viewed as an ..."
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Cited by 15 (5 self)
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We establish a new connection between the two most common traditions in the theory of real computation, the BlumShubSmale model and the Computable Analysis approach. We then use the connection to develop a notion of computability and complexity of functions over the reals that can be viewed as an extension of both models. We argue that this notion is very natural when one tries to determine just how “difficult ” a certain function is for a very rich class of functions. 1
Shallow circuits with highpowered inputs
 Proceedings of the Second Symposium on Innovations in Computer Science
, 2011
"... Abstract: A polynomial identity testing algorithm must determine whether an input polynomial (given for instance by an arithmetic circuit) is identically equal to 0. In this paper, we show that a deterministic blackbox identity testing algorithm for (highdegree) univariate polynomials would imply ..."
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Cited by 8 (5 self)
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Abstract: A polynomial identity testing algorithm must determine whether an input polynomial (given for instance by an arithmetic circuit) is identically equal to 0. In this paper, we show that a deterministic blackbox identity testing algorithm for (highdegree) univariate polynomials would imply a lower bound on the arithmetic complexity of the permanent. The lower bounds that are known to follow from derandomization of (lowdegree) multivariate identity testing are weaker. To obtain a lower bound for the permanent it would be sufficient to derandomize identity testing for polynomials of a very specific norm: sums of products of sparse polynomials with sparse coefficients. This observation leads to new versions of the ShubSmale τconjecture on integer roots of univariate polynomials. In particular, we show that a lower bound for the permanent would follow if one could give a polynomial upper bound on the number of real roots of sums of products of sparse polynomials (Descartes ’ rule of signs gives such a bound for sparse polynomials and products thereof). In fact the same lower bound would follow even if one could only prove a slightly superpolynomial upper bound on the number of real roots. This is a consequence of a new result on reduction to depth 4 for arithmetic circuits which we establish in a companion paper. We also show that an even weaker bound on the number of real roots would suffice to obtain a lower bound on the size of depth 4 circuits computing the permanent. These results suggest the intriguing possibility that tools from real analysis might be brought to bear on a longstanding open problem: what is the arithmetic complexity of the permanent polynomial?
On defining integers and proving arithmetic circuit lower bounds
 Computational Complexity
"... Abstract. Let τ(n) denote the minimum number of arithmetic operations sufficient to build the integer n from the constant 1. We prove that if there are arithmetic circuits of size polynomial in n for computing the permanent of n by n matrices, then τ(n!) is polynomially bounded in log n. Under the s ..."
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Cited by 8 (0 self)
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Abstract. Let τ(n) denote the minimum number of arithmetic operations sufficient to build the integer n from the constant 1. We prove that if there are arithmetic circuits of size polynomial in n for computing the permanent of n by n matrices, then τ(n!) is polynomially bounded in log n. Under the same assumption on the permanent, we conclude that the PochhammerWilkinson polynomials ∏n k=1 (X − k) and the Taylor approximations ∑n k=0 1 k! Xk and ∑n k=1 1 k Xk of exp and log, respectively, can be computed by arithmetic circuits of size polynomial in log n (allowing divisions). This connects several so far unrelated conjectures in algebraic complexity.
Computing equilibria: A computational complexity perspective
, 2009
"... Computational complexity is the subfield of computer science that rigorously studies the intrinsic difficulty of computational problems. This survey explains how complexity theory defines “hard problems”; applies these concepts to several equilibrium computation problems; and discusses implications ..."
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Cited by 7 (2 self)
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Computational complexity is the subfield of computer science that rigorously studies the intrinsic difficulty of computational problems. This survey explains how complexity theory defines “hard problems”; applies these concepts to several equilibrium computation problems; and discusses implications for computation, games, and behavior. We assume
Theory of real computation according to EGC
 In Proceedings of the Dagstuhl Seminar on Reliable Implementation of Real Number Algorithms: Theory and Practice, Lecture Notes in Computer Science
, 2006
"... The Exact Geometric Computation (EGC) mode of computation has been developed over the last decade in response to the widespread problem of numerical nonrobustness in geometric algorithms. Its technology has been encoded in libraries such as LEDA, CGAL and Core Library. The key feature of EGC is the ..."
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Cited by 7 (2 self)
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The Exact Geometric Computation (EGC) mode of computation has been developed over the last decade in response to the widespread problem of numerical nonrobustness in geometric algorithms. Its technology has been encoded in libraries such as LEDA, CGAL and Core Library. The key feature of EGC is the necessity to decide zero in its computation. This paper addresses the problem of providing a foundation for the EGC mode of computation. This requires a theory of real computation that properly addresses the Zero Problem. The two current approaches to real computation are represented by the analytic school and algebraic school. We propose a variant of the analytic approach based on real approximation. • To capture the issues of representation, we begin with a reworking of van der Waerden’s idea of explicit rings and fields. We introduce explicit sets and explicit algebraic structures. • Explicit rings serve as the foundation for real approximation: our starting point here is not R, but F ⊆ R, an explicit ordered ring extension of Z that is dense in R. We develop the approximability of real functions within standard Turing machine computability, and show its connection to the analytic approach. • Current discussions of real computation fail to address issues at the intersection of continuous and discrete computation. An appropriate computational model for this purpose is obtained by extending Schönhage’s pointer machines to support both algebraic and numerical computation. • Finally, we propose a synthesis wherein both the algebraic and the analytic models coexist to play complementary roles. Many fundamental questions can now be posed in this setting, including transfer theorems connecting algebraic computability with approximability. 1
The complexity of two problems on arithmetic circuits
, 2007
"... By using arithmetic circuits, encoding multivariate polynomials may be drastically more efficient than writing down the list of monomials. Via the study of two examples, we show however that such an encoding can be hard to handle with a Turing machine even if the degree of the polynomial is low. Nam ..."
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Cited by 6 (0 self)
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By using arithmetic circuits, encoding multivariate polynomials may be drastically more efficient than writing down the list of monomials. Via the study of two examples, we show however that such an encoding can be hard to handle with a Turing machine even if the degree of the polynomial is low. Namely we show that deciding whether the coefficient of a given monomial is zero is hard for P #P under strong nondeterministic Turing reductions. As a result, this problem does not belong to the polynomial hierarchy unless this hierarchy collapses. For polynomials over fields of characteristic k> 0, this problem is ModkPcomplete. This gives a coNP ModkP algorithm for deciding an upper bound on the degree of a polynomial given by a circuit in fields of characteristic k> 0.