Given a formula of the propositional µ-calculus, we construct a tableau of the formula and define an infinite game of two players of which one wants to show that the formula is satisfiable, and the other seeks the opposite. The strategy for the first player can be further transformed into a model of the formula while the strategy for the second forms what we call a refutation of the formula. Using Martin's Determinacy Theorem, we prove that any formula has either a model or a refutation. This completeness result is a starting point for the completeness theorem for the µ-calculus to be presented elsewhere. However, we argue that refutations have some advantages of their own. They are generated by a natural system of sound logical rules and can be presented as regular trees of the size exponential in the size of a refuted formula. This last aspect completes the small model theorem for the µ-calculus established by Emerson and Jutla [3]. Thus, on a more practical side, refutations can be...

Abstract. People often need to reason about policy changes before they are adopted. For example, suppose a website manager knows that users want to enter her site without going through the welcome page. To decide whether or not to permit this, the wise manager will consider the consequences of modifying the policies (e.g., would this allow users to bypass advertisements and legal notices?). Similiarly, people often need to compare policy sets. For example, consider a person who wants to buy health insurance. Before choosing a provider, the customer will want to compare the different policies. In other words, the customer wants to reason about the effect of choosing one policy set over another. We introduce a logic, based on propositional dynamic logic, in which these tasks can be done. We give a sound and complete axiomatization for our logic, and also show that it is decidable. More precisely, the satisfiability problem is decidable in nondeterministic exponential time. 1

Logical approaches to reasoning about agents often rely on idealisations about belief ascription and logical omniscience which make it difficult to apply the results obtained to real agents. In this paper, we show how to ascribe beliefs and an ability to reason in an arbitrary decidable logic to an agent in a computationally grounded way. We characterise those cases in which the assumption that an agent is logically omniscient in a given logic is `harmless' in the sense that it does not lead to making incorrect predictions about the agent, and show that such an assumption is not harmless when our predictions have a temporal dimension: `now the agent believes p', and the agent requires time to derive the consequences of its beliefs. We present a family of logics for reasoning about the beliefs of an agent which is a perfect reasoner in an arbitrary decidable logic L but only derives the consequences of its beliefs after some delay #. We investigate two members of this family in detail, L# in which all the consequences are derived at the next tick of the clock, and L # # in which the agent adds at most one new belief to its set of beliefs at every tick of the clock, and show that these are sound, complete and decidable.

The propositional µ-calculus as introduced by Kozen in [12] is considered. In that paper

Abstract. The effects of public announcements, private communications, deceptive messages to groups, and so on, can all be captured by a general mechanism of updating multi-agent models with update action models [3], now in widespread use (see [10] for a textbook treatment). There is a natural extension of the definition of a bisimulation to action models. Surely enough, updating with bisimilar action models gives the same result (modulo bisimulation). But the converse turns out to be false: update models may have the same update effects without being bisimilar. We propose action emulation as a notion of structural equivalence more appropriate for action models, and generalizing standard bisimulation. It is proved that action emulation provides a full characterization of update effect, provided we confine attention to ‘smooth ’ action models. We also give a recipe for turning any action model into a smooth one with the same update effect. Together, this yields a simplification procedure for action models, and it gives designers of multi-agent systems a useful tool for comparing different ways of representing a particular communicative action. 1.

We give an optimal (exptime), sound and complete tableau-based algorithm for deciding satisfiability for propositional dynamic logic. Our main contribution is a sound method to track unfulfilled eventualities “on the fly” which allows us to detect “bad loops” sooner rather than in multiple subsequent passes. We achieve this by propagating and updating the “status” of nodes throughout the underlying graph as soon as is possible. We give sufficient details to enable an easy implementation by others. Preliminary experimental results from our unoptimised OCaml implementation indicate that our algorithm is feasible.

Abstract: In Kripke semantics for modal logic, “pos-sible world^ " and the possibility relation are both primitive notions. This has both technical and con-ceptual shortcomings. From a technical point of view, the mathematics associated with Kripke. se-mantics is often quite complicated. From a concep-tual point of view, it is not clear how to use Kripke structures to model know!edge and belief, where one wants a clearer understanding of the notions that are primitive in Kripke semantics. We introduce modal structures as models for modal logic. We use the idea of possible worlds, but by directly describing the “internal semantics ” of each possible world. It is much easier to study the standard logical questions, such as completeness, decidability, and compactness, ushg modal structures. Furthermore, modal struc-tures offer a much more intuitive approach to mod-elling knowledge and belief. 1.

Current dynamic epistemic logics often become cumbersome and opaque when common knowledge is added for groups of agents. Still, postconditions regarding common knowledge express the essence of what communication achieves. We present some methods that yield so-called reduction axioms for common knowledge. We investigate the expressive power of public announcement logic with relativized common knowledge, and present reduction axioms that give a detailed account of the dynamics of common knowledge in some major communication types.

We argue the need for a logic of knowledge revision, in which one can explicitly reason about how agents' actions change their states of knowledge. However, we also show a strong negative result for a `natural' modal logic defined on these systems, suggesting that more subtlety is called for. 1 Motivation The problem of modelling knowledge revision has puzzled philosophers for a very long time, and in recent years has interested computer scientists, particularly in the areas of artificial intelligence and distributed computing. The interaction between knowledge and action makes the problem quite nontrivial, and when there are many agents in the system, they can reason about each others' knowledge as well, complicating the picture further. The situation can be understood as follows : at any time, an agent is in a particular state of knowledge and each of its actions takes it to a new state of knowledge. The questions that are often asked of any model of knowledge revision are: 1. Ho...

This talk shows how propositional dynamic logic (PDL) can be interpreted as a logic for multi-agent belief revision. For that we revise and extend the logic of communication and change (LCC) of [9]. Like LCC, our logic uses PDL as a base epistemic language. Unlike LCC, we start out from agent plausibilities, add their converses, and build knowledge and belief operators from these with the PDL constructs. We extend the update mechanism of LCC to an update mechanism that handles belief change as relation substitution, and we show that the update part of this logic is more expressive than either that of LCC or that of epistemic/doxastic PDL with a belief change modality. Next, we show that the properties of knowledge and belief are preserved under any update, unlike in LCC. We prove completeness of the logic and give examples of its use. If there is time, we will also look at the preference interpretation of the system, and at preference change scenarios that can be modelled with it.