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Distributed Markov Chains P S Thiagarajan School of Computing, National University of Singapore Joint work with Madhavan Mukund, Sumit K Jha and Ratul Saha

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Probabilistic dynamical systems Rich variety and theories of probabilistic dynamical systems – Markov chains, Markov Decision Processes (MDPs), Dynamic Bayesian networks Many applications Size of the model is a bottleneck – Can we exploit concurrency theory? We explore this in the setting of Markov chains.

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Our proposal A set of interacting sequential systems. – Synchronize on common actions. a a

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Our proposal A set of interacting sequential systems. – Synchronize on common actions. a

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Our proposal A set of interacting sequential systems. – Synchronize on common actions. a

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Our proposal A set of interacting sequential systems. – Synchronize on common actions. – This leads a joint probabilistic move by the participating agents. a, 0.8 a, 0.2

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Our proposal A set of interacting sequential systems. – Synchronize on common actions. – This leads a joint probabilistic move by the participating agents. a, 0.8 a, 0.2

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Our proposal A set of interacting sequential systems. – Synchronize on common actions. – This leads a joint probabilistic move by the participating agents. a, 0.8 a, 0.2

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Our proposal A set of interacting sequential systems. – Synchronize on common actions. – This leads a joint probabilistic move by the participating agents. a, 0.8 a, 0.2

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Our proposal A set of interacting sequential systems. – Synchronize on common actions. – This leads a joint probabilistic move by the participating agents. – More than two agents can take part in a synchronization. – More than two probabilistic outcomes possible. – There can also be just one agent taking part in a synchronization. Viewed as an internal probabilistic move (like in a Markov chain) by the agent.

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Our proposal This type of a system has been explored by Pighizzini et.al (“Probabilistic asynchronous automata”; 1996) – Language-theoretic study. Our key idea: – impose a “determinacy of communications” restriction. – Study formal verification problems using partial order based methods. We study here just one simple verification method.

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Some notations

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{a} Determinacy of communications. s s’ s’’ i {a}

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Determinacy of communications. s s’ s’’ i j

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{a} Determinacy of communications. s s’ s’’ i j loc(a) = {i, j} (s, s’), (s, s’’) en a a a a

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{a} Not allowed! s s’ i j s’’ k act(s) will have more than one action.

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Some notations

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Example – Two players each toss a fair coin – If the outcome is the same, they toss again – If the outcomes are different, the one who tosses Heads wins

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Example Two component DMC

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Interleaved semantics. Coin tosses are local actions, deciding a winner is synchronized action

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Goal We wish to analyze the behavior of a DMC in terms of its interleaved semantics. Follow the Markov chain route. – Construct the path space. The set of infinite paths from the initial state. Basic cylinder: a set of infinite paths with a common finite prefix. Close under countable unions and complements.

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The transition system view /5 3/ /5 3/ B Pr(B) = 1 2/5 1 1 = 2/5 B – The set of all paths that have the prefix

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Concurrency Events can occur independent of each other. Interleaved runs can be (concurrency) equivalent. We use Mazurkiewicz trace theory to group together equivalent runs: trace paths. Infinite trace paths do not suffice. We work with maximal infinite trace paths.

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(in 1, in 2 ) (T 1, in 2 )(in 1, H 2 ) (in 1, T 2 ) (H 1, in 2 ) t1, 0.5 t2, 0.5 h1, 0.5 h2, 0.5 (H 1, H 2 )(T 1, H 2 )(H 1, T 2 )(T 1, T 2 ) W1, L2 w1 l2 w1 L1, W2

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The trace space A basic trace cylinder is the one generated by a finite trace Construct the -algebra by closing under countable unions and complements. We must construct a probability measure over this -algebra. For a basic trace cylinder we want its probability to be the product of the probabilities of all the events in the trace.

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(in 1, in 2 ) (T 1, in 2 )(in 1, H 2 ) (in 1, T 2 ) (H 1, in 2 ) t1, 0.5 t2, 0.5 h1, 0.5 h2, 0.5 (H 1, H 2 )(T 1, H 2 )(H 1, T 2 )(T 1, T 2 ) W1, L2 w1 l2 w1 L1, W2 B Pr(B) = 0.5 0.5 = 0.25

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The probability measure over the trace space. But proving that this extends to a unique probability measure over the whole -algebra is hard. To solve this problem : – Define a Markov chain semantics for a DMC. – Construct a bijection between the maximal traces of the interleaved semantics and the infinite paths of the Markov chain semantics. Using Foata normal form – Transport the probability measure over the path space to the trace space.

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The Markov chain semantics.

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Markov chain semantics

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Probabilistic Product Bounded LTL

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PBLTL over interleaved runs

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Statistical model checking…

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SPRT based model checking

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Case study

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Case study…

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Distributed leader election protocol [Itai-Rodeh]

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Case study

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Case study… Dining Philosophers Problem

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Other examples Other PRISM case studies of randomized distributed algorithms – consensus protocols, gossip protocols… – Need to “translate" shared variables using a protocol Probabilistic choices in typical randomized protocols are local DMC model allows communication to influence probabilistic choices – We have not exploited this yet! – Not represented in standard PRISM benchmarks

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Summary and future work The interplay between concurrency and probabilistic dynamics is subtle and challenging. But concurrency theory may offer new tools for factorizing stochastic dynamics. – Earlier work on probabilistic event structures [Katoen et al, Abbes et al, Varacca et al] also attempt to impose probabilities on concurrent structures. – Our work shows that formal verification as the goal offers valuable guidelines Need to develop other model checking methods for DMCs. – Finite unfoldings – Stubborn sets for PCTL like specifications.

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