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Register a new userOpen Problems in a Logic of Gossips
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Publication date
2019
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Informatics Engineering
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English
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Gossip protocols are programs used in a setting in which each agent holds a secret and the aim is to reach a situation in which all agents know all secrets. Such protocols rely on a point-to-point or group communication. Distributed epistemic gossip protocols use epistemic formulas in the component programs for the agents. The advantage of the use of epistemic logic is that the resulting protocols are very concise and amenable for a simple verification. Recently, we introduced a natural modal logic that allows one to express distributed epistemic gossip protocols and to reason about their correctness. We proved that the resulting protocols are implementable and that all aspects of their correctness, including termination, are decidable. To establish these results we showed that both the definition of semantics and of truth of the underlying logic are decidable. We also showed that the analogous results hold for an extension of this logic with the common knowledge operator. However, several, often deceptively simple, questions about this logic and the corresponding gossip protocols remain open. The purpose of this paper is to list and elucidate these questions and provide for them an appropriate background information in the form of partial of related results.
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Gossip protocols aim at arriving, by means of point-to-point or group communications, at a situation in which all the agents know each other secrets. Recently a number of authors studied distributed epistemic gossip protocols. These protocols use as guards formulas from a simple epistemic logic, which makes their analysis and verification substantially easier. We study here common knowledge in the context of such a logic. First, we analyze when it can be reduced to iterated knowledge. Then we show that the semantics and truth for formulas without nested common knowledge operator are decidable. This implies that implementability, partial correctness and termination of distributed epistemic gossip protocols that use non-nested common knowledge operator is decidable, as well. Given that common knowledge is equivalent to an infinite conjunction of nested knowledge, these results are non-trivial generalizations of the corresponding decidability results for the original epistemic logic, established in (Apt & Wojtczak, 2016). K. R. Apt & D. Wojtczak (2016): On Decidability of a Logic of Gossips. In Proc. of JELIA 2016, pp. 18-33, doi:10.1007/ 978-3-319-48758-8_2.
A shallow semantical embedding for public announcement logic with relativized common knowledge is presented. This embedding enables the first-time automation of this logic with off-the-shelf theorem provers for classical higher-order logic. It is demonstrated (i) how meta-theoretical studies can be automated this way, and (ii) how non-trivial reasoning in the target logic (public announcement logic), required e.g. to obtain a convincing encoding and automation of the wise men puzzle, can be realized. Key to the presented semantical embedding -- in contrast, e.g., to related work on the semantical embedding of normal modal logics -- is that evaluation domains are modeled explicitly and treated as additional parameter in the encodings of the constituents of the embedded target logic, while they were previously implicitly shared between meta logic and target logic.
Heifetz, Meier and Schipper (HMS) present a lattice model of awareness. The HMS model is syntax-free, which precludes the simple option to rely on formal language to induce lattices, and represents uncertainty and unawareness with one entangled construct, making it difficult to assess the properties of either. Here, we present a model based on a lattice of Kripke models, induced by atom subset inclusion, in which uncertainty and unawareness are separate. We show the models to be equivalent by defining transformations between them which preserve formula satisfaction, and obtain completeness through our and HMS results.
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In this note we consider the problem of introducing variables in temporal logic programs under the formalism of Temporal Equilibrium Logic (TEL), an extension of Answer Set Programming (ASP) for dealing with linear-time modal operators. To this aim, we provide a definition of a first-order version of TEL that shares the syntax of first-order Linear-time Temporal Logic (LTL) but has a different semantics, selecting some LTL models we call temporal stable models. Then, we consider a subclass of theories (called splittable temporal logic programs) that are close to usual logic programs but allowing a restricted use of temporal operators. In this setting, we provide a syntactic definition of safe variables that suffices to show the property of domain independence -- that is, addition of arbitrary elements in the universe does not vary the set of temporal stable models. Finally, we present a method for computing the derivable facts by constructing a non-temporal logic program with variables that is fed to a standard ASP grounder. The information provided by the grounder is then used to generate a subset of ground temporal rules which is equivalent to (and generally smaller than) the full program instantiation.
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