The notion of preference occurs across many areas, including the philosophy of action, decision theory, optimality theory, and game theory. In these settings, individual preferences between worlds or actions can be used to predict behavior by rational agents. In a more abstract sense, the notion of preference also

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abstract. We survey a number of decidablity and undecidablity results concerning epistemic temporal logic. The goal is to provide a general picture which will facilitate the ‘sharing of ideas ’ from a number of different areas concerned with modeling agents in interactive social situations. 1

The syntactic approach to epistemic logic avoids the logical omniscience problem by taking knowledge as primary rather than as defined in terms of possible worlds. In this study, we combine the syntactic approach with modal logic, using transition systems to model reasoning. We use two syntactic epistemic modalities: ‘knowing at least ’ a set of formulae and ‘knowing at most’ a set of formulae. We are particularly interested in models restricting the set of formulae known by an agent at a point in time to be finite. The resulting systems are investigated from the point of view of axiomatization and complexity. We show how these logics can be used to formalise non-omniscient agents who know some inference rules, and study their relationship to other systems of syntactic epistemic logics,

The Sum-and-Product riddle was first published in [Fre69]. We provide an overview on the history of the dissemination of this riddle through the academic and puzzle-math community. This includes some references to precursors of the riddle, that were previously (as far as we know) unknown. We then model the Sum-and-Product riddle in a modal logic called public announcement logic. This logic contains operators for knowledge, but also operators for the informational consequences of public announcements. The logic is interpreted on multi-agent Kripke models. The information in the riddle can be represented in the traditional way by number pairs, so that Sum knows their sum and Product their product, but also as an interpreted system, so that Sum and Product at least know their local state. We show that the different representations are isomorphic. We also provide characteristic formulas of the initial epistemic state of the riddle. Finally we analyze one of the announcements towards the solution of the riddle as a so-called unsuccessful update: a formula that become false because it is announced. The riddle is then implemented and its solution verified in the epistemic model checker DEMO. This can be done, we think, surprisingly elegantly. The results are compared with other work in epistemic model checking.

Diversity of agents is investigated in the context of standard epistemic logic, dynamic information update, and belief revision. We provide a systematic discussion of different sources of diversities, such as introspection ability, powers of observation, memory capacity, and revision policies. In each case, we show how this diversity can be encoded in a logical system allowing for individual variation among rational agents. We conclude by raising some general issues concerning this view of a logic as a system for encoding a society of diverse agents and their interaction. 1 Diversity Inside Logical Systems Logical systems seem to prescribe one norm for an “idealized agent”. Any discrepancies with actual human behavior are then irrelevant, since the logic is meant to be normative, not descriptive. But logical systems would not be of much appeal if they did not have a plausible link with reality. And this is not just a matter of confronting one ideal norm with one kind of practical behavior. The striking fact is that human and virtual agents are not all the same: actual reasoning takes place in societies of diverse agents. This diversity shows itself particularly clearly in epistemic logic. There have been long debates about the appropriateness of various basic axioms, and they have to do with agents ’ different powers. In particular,

This paper provides a sound and complete proof system for a language Le+Y that adds to Dynamic Epistemic Logic (DEL) a discrete previous-time operator as well as single symbol formulas that partially reveal the most recent event that occured. The completeness theorem is by filtration followed by model unravelling and other model transformations. Decidability follows from the completeness proof. The degree to which it is important to include the additional single symbol formulas is addressed in a discussion about the difficulties of the completeness for a language LY that only adds the previous-time operator to DEL. Discussion is also given regarding the completeness for a language obtained by removing common knowledge operators from Le+Y.

Justification Logic is the study of a family of logics used to reason about justified true belief. Dynamic Epistemic Logic is the study of a family of logics obtained by adding various kinds of communication to the language of multi-modal logic, yielding languages for reasoning about communication and true belief. This paper is a first-step in merging these two areas, in that it brings the most basic kind of communication studied in Dynamic Epistemic Logic—the public announcement—over to Justification Logic. This gives us a language for reasoning about public announcements and justified true belief. After giving an overview of Justification Logic, the paper introduces a notion of bisimulation for Justification Logic. Bisimulation allows us to study the affect on language expressivity when we add various kinds of communication to the language. Among a number of expressivity results, we show that adding public announcements to the language of Justification Logic strictly increases language expressivity. This stands in contrast to the Plaza-Gerbrandy Theorem, which states that adding public announcements to multi-modal logic does not increase language expressivity. This leads us to extend the language of Justification Logic in order to provide a Plaza-Gerbrandy analog of multi-modal logic that we can use to reason about justified true belief. 1

Abstract. We model the well-known Sum-and-Product problem in a modal logic, and verify its solution in a model checker. The modal logic is public announcement logic. This logic contains operators for knowledge, but also for the informational consequences of public announcements. The logic is interpreted on multi-agent Kripke models. The information in the riddle can be represented in the traditional way by number pairs, so that Sum knows their sum and Product their product, but also as an interpreted system, so that Sum and Product at least know their local state. We show that the different representations are isomorphic. The riddle is then implemented and its solution verified in the epistemic model checker DEMO. This can be done, we think, surprisingly elegantly. It involves reformulations to facilitate the computation. 1

Pit is a multi-player card game that simulates the commodities trading market, and where actions consist of bidding and of swapping cards. We define a simplification of that game for which we present a detailed description of all dynamical game features. The description is in a standard language for dynamic epistemics. This formalization is then used to outline the game theory for a simplification of the Pit game. This uncovers some interesting equilibria.