188 research outputs found
Deciding Probabilistic Bisimilarity Distance One for Probabilistic Automata
Probabilistic bisimilarity, due to Segala and Lynch, is an equivalence relation that captures which states of a probabilistic automaton behave exactly the same. Deng, Chothia, Palamidessi and Pang proposed a robust quantitative generalization of probabilistic bisimilarity. Their probabilistic bisimilarity distances of states of a probabilistic automaton capture the similarity of their behaviour. The smaller the distance, the more alike the states behave. In particular, states are probabilistic bisimilar if and only if their distance is zero.
Although the complexity of computing probabilistic bisimilarity distances for probabilistic automata has already been studied and shown to be in NP cap coNP and PPAD, we are not aware of any practical algorithm to compute those distances. In this paper we provide several key results towards algorithms to compute probabilistic bisimilarity distances for probabilistic automata. In particular, we present a polynomial time algorithm that decides distance one. Furthermore, we give an alternative characterization of the probabilistic bisimilarity distances as a basis for a policy iteration algorithm
Computing Probabilistic Bisimilarity Distances for Probabilistic Automata
The probabilistic bisimilarity distance of Deng et al. has been proposed as a
robust quantitative generalization of Segala and Lynch's probabilistic
bisimilarity for probabilistic automata. In this paper, we present a
characterization of the bisimilarity distance as the solution of a simple
stochastic game. The characterization gives us an algorithm to compute the
distances by applying Condon's simple policy iteration on these games. The
correctness of Condon's approach, however, relies on the assumption that the
games are stopping. Our games may be non-stopping in general, yet we are able
to prove termination for this extended class of games. Already other algorithms
have been proposed in the literature to compute these distances, with
complexity in and \textbf{PPAD}. Despite the
theoretical relevance, these algorithms are inefficient in practice. To the
best of our knowledge, our algorithm is the first practical solution.
The characterization of the probabilistic bisimilarity distance mentioned
above crucially uses a dual presentation of the Hausdorff distance due to
M\'emoli. As an additional contribution, in this paper we show that M\'emoli's
result can be used also to prove that the bisimilarity distance bounds the
difference in the maximal (or minimal) probability of two states to satisfying
arbitrary -regular properties, expressed, eg., as LTL formulas
Cost Preserving Bisimulations for Probabilistic Automata
Probabilistic automata constitute a versatile and elegant model for
concurrent probabilistic systems. They are equipped with a compositional theory
supporting abstraction, enabled by weak probabilistic bisimulation serving as
the reference notion for summarising the effect of abstraction. This paper
considers probabilistic automata augmented with costs. It extends the notions
of weak transitions in probabilistic automata in such a way that the costs
incurred along a weak transition are captured. This gives rise to
cost-preserving and cost-bounding variations of weak probabilistic
bisimilarity, for which we establish compositionality properties with respect
to parallel composition. Furthermore, polynomial-time decision algorithms are
proposed, that can be effectively used to compute reward-bounding abstractions
of Markov decision processes in a compositional manner
Computing Probabilistic Bisimilarity Distances for Probabilistic Automata
The probabilistic bisimilarity distance of Deng et al. has been proposed as a robust quantitative generalization of Segala and Lynch’s probabilistic bisimilarity for probabilistic automata. In this paper, we present a novel characterization of the bisimilarity distance as the solution of a simple stochastic game. The characterization gives us an algorithm to compute the distances by applying Condon’s simple policy iteration on these games. The correctness of Condon’s approach, however, relies on the assumption that the games are stopping. Our games may be non-stopping in general, yet we are able to prove termination for this extended class of games. Already other algorithms have been proposed in the literature to compute these distances, with complexity in UP ∩ coUP and PPAD. Despite the theoretical relevance, these algorithms are inefficient in practice. To the best of our knowledge, our algorithm is the first practical solution. In the proofs of all the above-mentioned results, an alternative presentation of the Hausdorff distance due to Mémoli plays a central rôle
Expected-Delay-Summing Weak Bisimilarity for Markov Automata
A new weak bisimulation semantics is defined for Markov automata that, in
addition to abstracting from internal actions, sums up the expected values of
consecutive exponentially distributed delays possibly intertwined with internal
actions. The resulting equivalence is shown to be a congruence with respect to
parallel composition for Markov automata. Moreover, it turns out to be
comparable with weak bisimilarity for timed labeled transition systems, thus
constituting a step towards reconciling the semantics for stochastic time and
deterministic time.Comment: In Proceedings QAPL 2015, arXiv:1509.0816
Computing Probabilistic Bisimilarity Distances
Behavioural equivalences like probabilistic bisimilarity rely on the transition probabilities and, as a result, are sensitive to minuscule changes of those probabilities. Such behavioural equivalences are not robust, as first observed by Giacalone, Jou and Smolka. Probabilistic bisimilarity distances, a robust quantitative generalization of probabilistic bisimilarity, capture the similarity of the behaviour of states of a probabilistic model. The smaller the distance, the more alike the states behave. In particular, states are probabilistic bisimilar if and only if the distance between them is zero. In this dissertation, we focus on algorithms to compute probabilistic bisimilarity distances for two probabilistic models: labelled Markov chains and probabilistic automata.
In the late nineties, Desharnais, Gupta, Jagadeesan and Panangaden defined probabilistic bisimilarity distances on the states of a labelled Markov chain. This provided a quantitative generalization of probabilistic bisimilarity, which was introduced by Larsen and Skou a decade earlier. Several algorithms to approximate and compute these probabilistic bisimilarity distances have been put forward. In this dissertation, we correct and generalize some of these policy iteration algorithms. Moreover, we develop several new algorithms which have better performance in practice and can handle much larger systems.
Similarly, Deng, Chothia, Palamidessi and Pang presented probabilistic bisimilarity distances on the states of a probabilistic automaton. This provided a robust quantitative generalization of probabilistic bisimilarity introduced by Segala and Lynch. Although the complexity of computing probabilistic bisimilarity distances for probabilistic automata has already been studied and shown to be in NP coNP and PPAD, we are not aware of any practical algorithms to compute those distances. In this dissertation, we provide several key results that may prove to be useful for the development of algorithms to compute probabilistic bisimilarity distances for probabilistic automata. In particular, we present a polynomial time algorithm that decides distance one. Furthermore, we give an alternative characterization of the probabilistic bisimilarity distances as a basis for a policy iteration algorithm
When equivalence and bisimulation join forces in probabilistic automata
Probabilistic automata were introduced by Rabin in 1963 as language acceptors. Two automata are equivalent if and only if they accept each word with the same probability. On the other side, in the process algebra community, probabilistic automata were re-proposed by Segala in 1995 which are more general than Rabin's automata. Bisimulations have been proposed for Segala's automata to characterize the equivalence between them. So far the two notions of equivalences and their characteristics have been studied most independently. In this paper, we consider Segala's automata, and propose a novel notion of distribution-based bisimulation by joining the existing equivalence and bisimilarities. Our bisimulation bridges the two closely related concepts in the community, and provides a uniform way of studying their characteristics. We demonstrate the utility of our definition by studying distribution-based bisimulation metrics, which gives rise to a robust notion of equivalence for Rabin's automata. © 2014 Springer International Publishing Switzerland
Comparing Labelled Markov Decision Processes
A labelled Markov decision process is a labelled Markov chain with nondeterminism, i.e., together with a strategy a labelled MDP induces a labelled Markov chain. The model is related to interval Markov chains. Motivated by applications of equivalence checking for the verification of anonymity, we study the algorithmic comparison of two labelled MDPs, in particular, whether there exist strategies such that the MDPs become equivalent/inequivalent, both in terms of trace equivalence and in terms of probabilistic bisimilarity. We provide the first polynomial-time algorithms for computing memoryless strategies to make the two labelled MDPs inequivalent if such strategies exist. We also study the computational complexity of qualitative problems about making the total variation distance and the probabilistic bisimilarity distance less than one or equal to one
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