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Modal Stochastic Games

Abstraction-Refinement of Probabilistic Automata

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Models, Algorithms, Logics and Tools

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10460))


This paper presents an abstraction-refinement framework for Segala’s probabilistic automata (PA), a slight variant of Markov decision processes. We use Condon and Ladner’s two-player probabilistic game automata extended with possible and required transitions—as in Larsen and Thomsen’s modal transition systems—as abstract models. The key idea is to refine player-one and player-two states separately resulting in a nested abstract-refine loop. We show the adequacy of this approach for obtaining tight bounds on extremal reachability probabilities.

This work has been partially funded by the Excellence Initiative of the German federal and state government and the CDZ project CAP (GZ 1023).

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  1. 1.

    As this paper does not cover parallel composition all PGAs are closed. For modeling PGAs in a compositonal manner though, the distinction between internal and other actions is important, see [7].

  2. 2.

    For example, let \(x=\max \) in \( \mathrm {Pr}^x(T^{\prime })\) then \(\mathbf {1}=\max \) and \(\mathbf {2}=\min \) (player-one maximizes whereas the player-two minimizes the probability) or vice versa.

  3. 3.

    This may converge slower than allowing for coarser splittings (as in [5]), but yields smaller state spaces.


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This work is strongly inspired by and heavily builds upon the work of Kim G. Larsen. The idea of using possible (may) and required (must) transitions goes back to his seminal work with Thomsen [28]. Simulation and refinement relations for probabilistic models originated in his work with Jonsson [16]. Kim developed one of the first, if not the very first, abstraction-refinement technique for MDPs [19]. His work on constraint Markov chains [29] provided the basis for our joint work on abstract PAs [4]. The uncertainty of the non-deterministic choices in APA is modeled by modal transitions while uncertainty of the stochastic behavior is expressed—as in constraint Markov chains—by (underspecified) stochastic constraints. Besides the influence of all these work, Kim has always been extremely inspiring. This started in 1996 at the conference FTRTFT in Uppsala, when he stimulated us to use Uppaal—at those days in its very early stage of development [30]—to take up the challenge of modeling and verifying Philips’ bounded retransmission protocol [31]. This relationship has continued over the years and has led to several joint EU projects. It has been a great pleasure and enormous honor to work with Kim. This paper is a salute to his 60th birthday.

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Katoen, JP., Sher, F. (2017). Modal Stochastic Games. In: Aceto, L., Bacci, G., Bacci, G., Ingólfsdóttir, A., Legay, A., Mardare, R. (eds) Models, Algorithms, Logics and Tools. Lecture Notes in Computer Science(), vol 10460. Springer, Cham.

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