52 research outputs found
Convergence Thresholds of Newton's Method for Monotone Polynomial Equations
Monotone systems of polynomial equations (MSPEs) are systems of fixed-point
equations where
each is a polynomial with positive real coefficients. The question of
computing the least non-negative solution of a given MSPE arises naturally in the analysis of stochastic models such as stochastic
context-free grammars, probabilistic pushdown automata, and back-button
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 for strongly connected MSPEs, such that after 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 as a function of the minimal component
of the least fixed-point of . Using this result we
show that is at most single exponential resp. linear for strongly
connected MSPEs derived from probabilistic pushdown automata resp. from
back-button processes. Further, we prove the existence of a threshold for
arbitrary MSPEs after which each new iteration computes at least new
bits of the solution, where and are the width and height of the DAG of
strongly connected components.Comment: version 2 deposited February 29, after the end of the STACS
conference. Two minor mistakes correcte
PReMo : An Analyzer for P robabilistic Re cursive Mo dels
This paper describes PReMo, a tool for analyzing Recursive Markov Chains, and their controlled/game extensions: (1-exit) Recursive Markov Decision Processes and Recursive Simple Stochastic Games
On the Complexity of the Equivalence Problem for Probabilistic Automata
Checking two probabilistic automata for equivalence has been shown to be a
key problem for efficiently establishing various behavioural and anonymity
properties of probabilistic systems. In recent experiments a randomised
equivalence test based on polynomial identity testing outperformed
deterministic algorithms. In this paper we show that polynomial identity
testing yields efficient algorithms for various generalisations of the
equivalence problem. First, we provide a randomized NC procedure that also
outputs a counterexample trace in case of inequivalence. Second, we show how to
check for equivalence two probabilistic automata with (cumulative) rewards. Our
algorithm runs in deterministic polynomial time, if the number of reward
counters is fixed. Finally we show that the equivalence problem for
probabilistic visibly pushdown automata is logspace equivalent to the
Arithmetic Circuit Identity Testing problem, which is to decide whether a
polynomial represented by an arithmetic circuit is identically zero.Comment: technical report for a FoSSaCS'12 pape
Optimal Strategies in Infinite-state Stochastic Reachability Games
We consider perfect-information reachability stochastic games for 2 players
on infinite graphs. We identify a subclass of such games, and prove two
interesting properties of it: first, Player Max always has optimal strategies
in games from this subclass, and second, these games are strongly determined.
The subclass is defined by the property that the set of all values can only
have one accumulation point -- 0. Our results nicely mirror recent results for
finitely-branching games, where, on the contrary, Player Min always has optimal
strategies. However, our proof methods are substantially different, because the
roles of the players are not symmetric. We also do not restrict the branching
of the games. Finally, we apply our results in the context of recently studied
One-Counter stochastic games
On the Metric-Based Approximate Minimization of Markov Chains
We address the behavioral metric-based approximate minimization problem of Markov Chains (MCs), i.e., given a finite MC and a positive integer k, we are interested in finding a k-state MC of minimal distance to the original. By considering as metric the bisimilarity distance of Desharnais at al., we show that optimal approximations always exist; show that the problem can be solved as a bilinear program; and prove that its threshold problem is in PSPACE and NP-hard. Finally, we present an approach inspired by expectation maximization techniques that provides suboptimal solutions. Experiments suggest that our method gives a practical approach that outperforms the bilinear program implementation run on state-of-the-art bilinear solvers
Computation of distances for regular and context-free probabilistic languages
Several mathematical distances between probabilistic languages have been investigated in the literature, motivated by applications in language modeling, computational biology, syntactic pattern matching and machine learning. In most cases, only pairs of probabilistic regular languages were considered. In this paper we extend the previous results to pairs of languages generated by a probabilistic context-free grammar and a probabilistic finite automaton.PostprintPeer reviewe
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