Technical Reports

A List by Author: Jan Křetínský

e-mail:
xkretins(a)fi.muni.cz

Probabilistic Bisimulation: Naturally on Distributions

by Holger Hermanns, Jan Krčál, Jan Křetínský, April 2014, 36 pages.

FIMU-RS-2014-03. Available as Postscript, PDF.

Abstract:

In contrast to the usual understanding of probabilistic systems as stochastic processes, recently these systems have also been regarded as transformers of probabilities. In this paper, we give a natural definition of strong bisimulation for probabilistic systems corresponding to this view that treats probability distributions as first-class citizens. Our definition applies in the same way to discrete systems as well as to systems with uncountable state and action spaces. Several examples demonstrate that our definition refines the understanding of behavioural equivalences of probabilistic systems. In particular, it solves a longstanding open problem concerning the representation of memoryless continuous time by memoryfull continuous time. Finally, we give algorithms for computing this bisimulation not only for finite but also for classes of uncountably infinite systems.

Verification of Open Interactive Markov Chains

by Tomáš Brázdil, Holger Hermanns, Jan Krčál, Jan Křetínský, Vojtěch Řehák, A full version of the paper presented at conference FSTTCS 2012. November 2012, 52 pages.

FIMU-RS-2012-04. Available as Postscript, PDF.

Abstract:

Interactive Markov chains (IMC) are compositional behavioral models extending both labeled transition systems and continuous-time Markov chains. IMC pair modeling convenience - owed to compositionality properties - with effective verification algorithms and tools - owed to Markov properties. Thus far however, IMC verification did not consider compositionality properties, but considered closed systems. This paper discusses the evaluation of IMC in an open and thus compositional interpretation. For this we embed the IMC into a game that is played with the environment. We devise algorithms that enable us to derive bounds on reachability probabilities that are assured to hold in any composition context.

Modal Process Rewrite Systems

by Nikola Beneš, Jan Křetínský, A full version of the paper presented at ICTAC 2012. June 2012, 25 pages.

FIMU-RS-2012-02. Available as Postscript, PDF.

Abstract:

We consider modal transition systems with infinite state space generated by finite sets of rules. In particular, we extend process rewrite systems to the modal setting and investigate decidability of the modal refinement relation between systems from various subclasses. Since already simulation is undecidable for most of the cases, we focus on the case where either the refined or the refining process is finite. Namely, we show decidability for pushdown automata extending the non-modal case and surprising undecidability for basic parallel processes. Further, we prove decidability when both systems are visibly pushdown automata. For the decidable cases, we also provide complexities. Finally, we discuss a notion of bisimulation over MTS.

Dual-Priced Modal Transition Systems with Time Durations

by Nikola Beneš, Jan Křetínský, Kim Guldstrand Larsen, Mikael Moeller, Jiří Srba, A full version of the paper presented at conference LPAR 2012. January 2012, 23 pages.

FIMU-RS-2012-01. Available as Postscript, PDF.

Abstract:

Modal transition systems are a well-established specification formalism for a high-level modelling of component-based software systems. We present a novel extension of the formalism called modal transition systems with durations where time durations are modelled as controllable or uncontrollable intervals. We further equip the model with two kinds of quantitative aspects: each action has its own running cost per time unit, and actions may require several hardware components of different costs. We ask the question, given a fixed budget for the hardware components, what is the implementation with the cheapest long-run average reward. We give an algorithm for computing such optimal implementations via a reduction to a new extension of mean payoff games with time durations and analyse the complexity of the algorithm.

Parametric Modal Transition Systems

by Nikola Beneš, Jan Křetínský, Kim Guldstrand Larsen, Mikael Moller, Jiří Srba, A full version of the paper presented at ATVA 2011 July 2011, 24 pages.

FIMU-RS-2011-03. Available as Postscript, PDF.

Abstract:

Modal transition systems (MTS) is a well-studied specification formalism of reactive systems supporting a step-wise refinement methodology. Despite its many advantages, the formalism as well as its currently known extensions are incapable of expressing some practically needed aspects in the refinement process like exclusive, conditional and persistent choices. We introduce a new model called parametric modal transition systems (PMTS) together with a general modal refinement notion that overcome many of the limitations and we investigate the computational complexity of modal refinement checking.

Disjunctive Modal Transition Systems and Generalized LTL Model Checking

by Nikola Beneš, Ivana Černá, Jan Křetínský, November 2010, 44 pages.

FIMU-RS-2010-12. Available as Postscript, PDF.

Abstract:

Modal transition systems (MTS) is a well established formalism used for specification and for abstract interpretation. We consider its disjunctive extension (DMTS) and we show that refinement problems for DMTS are not harder than in the case of MTS. There are two main results in the paper. Firstly, we give a solution to the common implementation and specification problems lowering the complexity from EXPTIME to PTIME. Secondly, we identify a fundamental error made in previous attempts at LTL model checking of MTS and provide algorithms for LTL model checking of MTS and DMTS. Moreover, we show how to apply this result to compositional verification and circumvent the general incompleteness of the MTS composition.

Process Algebra for Modal Transition Systemses

by Nikola Beneš, Jan Křetínský, A full version of the paper presented at MEMICS 2010. September 2010, 15 pages.

FIMU-RS-2010-11. Available as Postscript, PDF.

Abstract:

The formalism of modal transition systems (MTS) is a well established framework for systems specification as well as abstract interpretation. Nevertheless, due to incapability to capture some useful features, various extensions have been studied, such as e.g. mixed transition systems or disjunctive MTS. Thus a need to compare them has emerged. Therefore, we introduce transition systems with obligations as a general model encompassing all the aforementioned models, and equip it with a process algebra description. Using these instruments, we then compare the previously studied subclasses and characterize their relationships.

Stochastic Real-Time Games with Qualitative Timed Automata Objectives

by Tomáš Brázdil, Jan Krčál, Jan Křetínský, Antonín Kučera, Vojtěch Řehák, A full version of the paper presented at CONCUR 2010. August 2010, 39 pages.

FIMU-RS-2010-05. Available as Postscript, PDF.

Abstract:

We consider two-player stochastic games over real-time probabilistic processes where the winning objective is specified by a timed automaton. The goal of player I is to play in such a way that the play (a timed word) is accepted by the timed automaton with probability one. Player II aims at the opposite. We prove that whenever player I has a winning strategy, then she also has a strategy that can be specified by a timed automaton. The strategy automaton reads the history of a play, and the decisions taken by the strategy depend only on the region of the resulting configuration. We also give an exponential-time algorithm which computes a winning timed automaton strategy if it exists.

Continuous-Time Stochastic Games with Time-Bounded Reachability

by Tomáš Brázdil, Vojtěch Forejt, Jan Krčál, Jan Křetínský, Antonín Kučera, A full version of the paper presented at FST&TCS 2009. October 2009, 46 pages.

FIMU-RS-2009-09. Available as Postscript, PDF.

Abstract:

We study continuous-time stochastic games with time-bounded reachability objectives. We show that each vertex in such a game has a value (i.e., an equilibrium probability), and we classify the conditions under which optimal strategies exist. Finally, we show how to compute optimal strategies in finite uniform games, and how to compute e-optimal strategies in finitely-branching games with bounded rates (for finite games, we provide detailed complexity estimations).

Checking Thorough Refinement on Modal Transition Systems Is EXPTIME-Complete

by Nikola Beneš, Jan Křetínský, Kim Guldstrand Larsen, Jiří Srba, A full version of the paper presented at conference ICTAC 2009. July 2009, 28 pages.

FIMU-RS-2009-03. Available as Postscript, PDF.

Abstract:

Modal transition systems (MTS), a specification formalism introduced more than 20 years ago, has recently received a considerable attention in several different areas. Many of the fundamental questions related to MTSs have already been answered. However, the problem of the exact computational complexity of thorough refinement checking between two finite MTSs remained unsolved.

We settle down this question by showing EXPTIME-completeness of thorough refinement checking on finite MTSs. The upper-bound result relies on a novel algorithm running in single exponential time providing a direct goal-oriented way to decide thorough refinement. If the right-hand side MTS is moreover deterministic, or has a fixed size, the running time of the algorithm becomes polynomial. The lower-bound proof is achieved by reduction from the acceptance problem of alternating linear bounded automata and the problem remains EXPTIME-hard even if the left-hand side MTS is fixed.

The Satisfiability Problem for Probabilistic CTL

by Tomáš Brázdil, Vojtěch Forejt, Jan Křetínský, Antonín Kučera, A full version of the paper presented at LICS 2008. June 2008, 34 pages.

FIMU-RS-2008-03. Available as Postscript, PDF.

Abstract:

We study the satisfiability problem for qualitative PCTL (Probabilistic Computation Tree Logic), which is obtained from "ordinary" CTL by replacing the EX, AX, EU, and AU operators with their qualitative counterparts X>0, X=1, U>0, and U=1, respectively. As opposed to CTL, qualitative PCTL does not have a small model property, and there are even qualitative PCTL formulae which have only infinite-state models. Nevertheless, we show that the satisfiability problem for qualitative PCTL is EXPTIME-complete and we give an exponential-time algorithm which for a given formula computes a finite description of a model (if it exists), or answers "not satisfiable" (otherwise). We also consider the finite satisfiability problem and provide analogous results. That is, we show that the finite satisfiability problem for qualitative PCTL is EXPTIME-complete, and every finite satisfiable formula has a model of an exponential size which can effectively be constructed in exponential time. Finally, we give some results about the quantitative PCTL, where the numerical bounds in probability constraints can be arbitrary rationals between 0 and 1. We prove that the problem whether a given quantitative PCTL formula has a model of the branching degree at most k, where k > 1 is an arbitrary but fixed constant, is highly undecidable. We also show that every satisfiable formula F has a model with branching degree at most |F| + 2. However, this does not yet imply the undecidability of the satisfiability problem for quantitative PCTL, and we in fact conjecture the opposite.

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