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Register a new userCompositional closure for Bayes Risk in probabilistic noninterference
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Larissa Meinicke
Publication date
2010
fields
Informatics Engineering
and research's language is
English
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We give a sequential model for noninterference security including probability (but not demonic choice), thus supporting reasoning about the likelihood that high-security values might be revealed by observations of low-security activity. Our novel methodological contribution is the definition of a refinement order and its use to compare security measures between specifications and (their supposed) implementations. This contrasts with the more common practice of evaluating the security of individual programs in isolation. The appropriateness of our model and order is supported by our showing that our refinement order is the greatest compositional relation --the compositional closure-- with respect to our semantics and an elementary order based on Bayes Risk --- a security measure already in widespread use. We also relate refinement to other measures such as Shannon Entropy. By applying the approach to a non-trivial example, the anonymous-majority Three-Judges protocol, we demonstrate by example that correctness arguments can be simplified by the sort of layered developments --through levels of increasing detail-- that are allowed and encouraged by compositional semantics.
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The probabilistic bisimilarity distance of Deng et al. has been proposed as a robust quantitative generalization of Segala and Lynchs 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 Condons simple policy iteration on these games. The correctness of Condons 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 Memoli. As an additional contribution, in this paper we show that Memolis 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.
In [1], we introduced the weakly synchronizing languages for probabilistic automata. In this report, we show that the emptiness problem of weakly synchronizing languages for probabilistic automata is undecidable. This implies that the decidability result of [1-3] for the emptiness problem of weakly synchronizing language is incorrect.
We characterize the class of nondeterministic ${omega}$-automata that can be used for the analysis of finite Markov decision processes (MDPs). We call these automata `good-for-MDPs (GFM). We show that GFM automata are closed under classic simulation as well as under more powerful simulation relations that leverage properties of optimal control strategies for MDPs. This closure enables us to exploit state-space reduction techniques, such as those based on direct and delayed simulation, that guarantee simulation equivalence. We demonstrate the promise of GFM automata by defining a new class of automata with favorable properties - they are Buchi automata with low branching degree obtained through a simple construction - and show that going beyond limit-deterministic automata may significantly benefit reinforcement learning.
This short note aims at proving that the isolation problem is undecidable for probabilistic automata with only one probabilistic transition. This problem is known to be undecidable for general probabilistic automata, without restriction on the number of probabilistic transitions. In this note, we develop a simulation technique that allows to simulate any probabilistic automaton with one having only one probabilistic transition.
We answer two open questions by (Gruber, Holzer, Kutrib, 2009) on the state-complexity of representing sub- or superword closures of context-free grammars (CFGs): (1) We prove a (tight) upper bound of $2^{mathcal{O}(n)}$ on the size of nondeterministic finite automata (NFAs) representing the subword closure of a CFG of size $n$. (2) We present a family of CFGs for which the minimal deterministic finite automata representing their subword closure matches the upper-bound of $2^{2^{mathcal{O}(n)}}$ following from (1). Furthermore, we prove that the inequivalence problem for NFAs representing sub- or superword-closed languages is only NP-complete as opposed to PSPACE-complete for general NFAs. Finally, we extend our results into an approximation method to attack inequivalence problems for CFGs.
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