Computing Continuous-Time Markov Chains as Transformers of Unbounded Observables

  • Vincent Danos
  • Tobias Heindel
  • Ilias Garnier
  • Jakob Grue Simonsen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10203)

Abstract

The paper studies continuous-time Markov chains (CTMCs) as transformers of real-valued functions on their state space, considered as generalised predicates and called observables. Markov chains are assumed to take values in a countable state space \(\mathbf {S}\); observables \(f: \mathbf {S} \rightarrow {\mathbb {R}}\) may be unbounded. The interpretation of CTMCs as transformers of observables is via their transition function \( P_{t} \): each observable \(f\) is mapped to the observable \( P_{t} f\) that, in turn, maps each state \(x\) to the mean value of \(f\) at time \(t\) conditioned on being in state \(x\) at time \(0\).

The first result is computability of the time evolution of observables, i.e., maps of the form \((t,f)\,{\mapsto }\, P_{t} f\), under conditions that imply existence of a Banach sequence space of observables on which the transition function \( P_{t} \) of a fixed CTMC induces a family of bounded linear operators that vary continuously in time (w.r.t. the usual topology on bounded operators). The second result is PTIME-computability of the projections \(t\,{\mapsto }\,( P_{t} f)(x)\), for each state \(x\), provided that the rate matrix of the CTMC \(X_t\) is locally algebraic on a subspace containing the observable \(f\).

The results are flexible enough to accommodate unbounded observables; explicit examples feature the token counts in stochastic Petri nets and sub-string occurrences of stochastic string rewriting systems. The results provide a functional analytic alternative to Monte Carlo simulation as test bed for mean-field approximations, moment closure, and similar techniques that are fast, but lack absolute error guarantees.

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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Vincent Danos
    • 1
    • 3
  • Tobias Heindel
    • 2
  • Ilias Garnier
    • 1
    • 3
  • Jakob Grue Simonsen
    • 2
  1. 1.Département d’InformatiqueÉcole Normale SupérieureParisFrance
  2. 2.Department of Computer ScienceUniversity of CopenhagenCopenhagenDenmark
  3. 3.School of InformaticsUniversity of EdinburghEdinburghUK

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