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Distributed automata and logic

Abstract : Distributed automata are finite-state machines that operate on finitedirected graphs. Acting as synchronous distributed algorithms, they use their input graph as a network in which identical processors communicate for a possibly infinite number of synchronous rounds. For the local variant of those automata, where the number of rounds is bounded by a constant, Hella et al. (2012, 2015) have established a logical characterization in terms of basic modal logic. In this thesis, we provide similar logical characterizations for two more expressive classes of distributed automata.The first class extends local automata with a global acceptance condition and the ability to alternate between non deterministic and parallel computations. We show that it is equivalent to monadic second-order logic on graphs. By restricting transitions to be non deterministic or deterministic, we also obtain two strictly weaker variants for which the emptiness problem is decidable.Our second class transfers the standard notion of asynchronous algorithm to the setting of non local distributed automata. There sulting machines are shown to be equivalent to a small fragment of least fixpoint logic, and more specifically, to a restricted variantof the modal μ -calculus that allows least fixpoints but forbids greatest fixpoints. Exploiting the connection with logic, we additionally prove that the expressive power of those asynchronous automata is independent of whether or not messages can be lost.We then investigate the decidability of the emptiness problem forseveral classes of nonlocal automata. We show that the problem isundecidable in general, by simulating a Turing machine with adistributed automaton that exchanges the roles of space and time. Onthe other hand, the problem is found to be decidable in logspace for a class of forgetful automata, where the nodes see the messages received from their neighbors but cannot remember their own state. As a minor contribution, we also give new proofs of the strictness of several set quantifier alternation hierarchies that are based on modallogic.
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Submitted on : Thursday, December 13, 2018 - 1:05:05 AM
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Fabian Reiter. Distributed automata and logic. Technology for Human Learning. Université Sorbonne Paris Cité, 2017. English. ⟨NNT : 2017USPCC034⟩. ⟨tel-01953461v2⟩

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