Algorithms to Compute Probabilistic Bisimilarity Distances for Labelled Markov Chains

In the late nineties, Desharnais, Gupta, Jagadeesan and Panangaden presented probabilistic bisimilarity distances on the states of a labelled Markov chain. This provided a quantitative generalisation of probabilistic bisimilarity introduced by Larsen and Skou a decade earlier. In the last decade, several algorithms to approximate and compute these probabilistic bisimilarity distances have been put forward. In this paper, we correct, improve and generalise some of these algorithms. Furthermore, we compare their performance experimentally

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

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 $\textbf{UP} \cap \textbf{coUP}$ 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 $\omega$-regular properties, expressed, eg., as LTL formulas

Computing Probabilistic Bisimilarity Distances via Policy Iteration

A transformation mapping a labelled Markov chain to a simple stochastic game is presented. In the resulting simple stochastic game, each vertex corresponds to a pair of states of the labelled Markov chain. The value of a vertex of the simple stochastic game is shown to be equal to the probabilistic bisimilarity distance, a notion due to Desharnais, Gupta, Jagadeesan and Panangaden, of the corresponding pair of states of the labelled Markov chain. Bacci, Bacci, Larsen and Mardare introduced an algorithm to compute the probabilistic bisimilarity distances for a labelled Markov chain. A modification of a basic version of their algorithm for a labelled Markov chain is shown to be the policy iteration algorithm applied to the corresponding simple stochastic game. Furthermore, it is shown that this algorithm takes exponential time in the worst case

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

Asymmetric distances for approximate differential privacy

Differential privacy is a widely studied notion of privacy for various models of computation, based on measuring differences between probability distributions. We consider (epsilon,delta)-differential privacy in the setting of labelled Markov chains. For a given epsilon, the parameter delta can be captured by a variant of the total variation distance, which we call lv_{alpha} (where alpha = e^{epsilon}). First we study lv_{alpha} directly, showing that it cannot be computed exactly. However, the associated approximation problem turns out to be in PSPACE and #P-hard. Next we introduce a new bisimilarity distance for bounding lv_{alpha} from above, which provides a tighter bound than previously known distances while remaining computable with the same complexity (polynomial time with an NP oracle). We also propose an alternative bound that can be computed in polynomial time. Finally, we illustrate the distances on case studies

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

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