Alexandru Baltag

The logic of public announcements, common knowledge, and private suspicions

This paper presents a logical system in which various group-level epistemic actions are incorporated into the object language. That is, we consider the standard modeling of knowledge among a set of agents by multi-modal Kripke structures. One might want to consider actions that take place, such as announcements to groups privately, announcements with suspicious outsiders, etc. In our system, such actions correspond to additional modalities in the object language. That is, we do not add machinery on top of models (as in Fagin et al [1], but we reify aspects of the machinery in the logical language. n nThe first case of the kind of logics we consider appears in Gerbrandy and Groeneveld [3]. They introduced a language in which one can faithfully represent all of the reasoning in examples such as the Muddy Children scenario. In that paper we find operators for updating worlds via announcements to groups of agents who are isolated from all others. We advance this by considering many more actions; and by using a more general semantics. n nOur logic contains the infinitary operators used in the standard modeling of common knowledge. We present a sound and complete logical system for the logic, and we study its expressive power.

Logics for Epistemic Programs

We construct logical languages which allow one to represent a variety of possible types of changes affecting the information states of agents in a multi-agent setting. We formalize these changes by defining a notion of epistemic program. The languages are two-sorted sets that contain not only sentences but also actions or programs. This is as in dynamic logic, and indeed our languages are not significantly more complicated than dynamic logics. But the semantics is more complicated. In general, the semantics of an epistemic program is what we call aprogram model. This is a Kripke model of ‘actions’,representing the agents uncertainty about the current action in a similar way that Kripke models of ‘states’ are commonly used in epistemic logic to represent the agents uncertainty about the current state of the system. Program models induce changes affecting agents information, which we represent as changes of the state model, called epistemic updates. Formally, an update consists of two operations: the first is called the update map, and it takes every state model to another state model, called the updated model; the second gives, for each input state model, a transition relation between the states of that model and the states of the updated model.Each variety of epistemic actions, such as public announcements or completely private announcements to groups, gives what we call an action signature, and then each family of action signatures gives a logical language. The construction of these languages is the main topic of this paper. We also mention the systems that capture the valid sentences of our logics. But we defer to a separate paper the completeness proof.The basic operation used in the semantics is called the update product. A version of this was introduced in Baltag et al. (1998), and the presentation here improves on the earlier one. The update product is used to obtain from any program model the corresponding epistemic update, thus allowing us to compute changes of information or belief. This point is of interest independently of our logical languages. We illustrate the update product and our logical languages with many examples throughout the paper.

A qualitative theory of dynamic interactive belief revision

We present a logical setting that incorporates a belief-revision mechanism within Dynamic-Epistemic logic. As the “static” basis for belief revision, we use epistemic plausibility models, together with a modal language based on two epistemic operators: a “knowledge” modality K (the standard S5, fully introspective, notion), and a “safe belief” modality □ (“weak”, non-negatively-introspective, notion, capturing a version of Lehrer’s “indefeasible knowledge”). To deal with “dynamic” belief revision, we introduce action plausibility models, representing various types of “doxastic events”. Action models “act” on state models via a modified update product operation: the “Action-Priority” Update. This is the natural dynamic generalization of AGM revision, giving priority to the incoming information (i.e., to “actions”) over prior beliefs. We completely axiomatize this logic, and show how our update mechanism can “simulate”, in a uniform manner, many different belief-revision policies.

Conditional Doxastic Models: A Qualitative Approach to Dynamic Belief Revision

In this paper, we present a semantical approach to multi-agent belief revision and belief update. For this, we introduce relational structures called conditional doxastic models (CDMs, for short). We show this setting to be equivalent to an epistemic version of the classical AGM Belief Revision theory. We present a logic of conditional beliefs that is complete w.r.t. CDMs. Moving then to belief updates (sometimes called dynamic belief revision) induced by epistemic actions, we consider two particular cases: public announcements and private announcements to subgroups of agents. We show how the standard semantics for these types of updates can be appropriately modified in order to apply it to CDMs, thus incorporating belief revision into our notion of update. We provide a complete axiomatization of the corresponding dynamic doxastic logics. As an application, we solve a cheating version of the Muddy Children Puzzle.

‘KNOWABLE’ AS ‘KNOWN AFTER AN ANNOUNCEMENT’

Public announcement logic is an extension of multi-agent epistemic logic with dynamic operators to model the informational consequences of announcements to the entire group of agents. We propose an extension of public announcement logic with a dynamic modal operator that expresses what is true after any announcement: ♦ϕ expresses that there is a truthful announcement ψ after which ϕ is true. This logic gives a perspective on Fitch’s knowability issues: for which formulas ϕ does it hold that ϕ → ♦Kϕ? We give various semantic results, and we show completeness for a Hilbert-style axiomatization of this logic. There is a natural generalization to a logic for arbitrary events.

A Logic for Suspicious Players: Epistemic Actions and Belief–Updates in Games

In this paper, we introduce a notion of ``epistemic action to describe changes in the information states of the players in a game. For this, we use ideas that we have developed in our previous papers [BMS], [BMS2] and [B], enriching them to cover, not just {it purely epistemic} actions, but also {it fact-changing actions} (``real moves, e.g. choosing a card, exchanging cards etc.) and {it nondeterministic actions} and {it strategies} (conditional actions having knowledge tests as conditions). We consider natural {it operations with epistemic actions} and we use them to describe significant aspects of the interaction between beliefs and actions in a game. For this, we use a logic that combines in a specific way a multi-agent epistemic logic with a dynamic logic of ``epistemic actions. We give (without proof) {it a complete and decidable proof system} for this logic. As an application, we analyze a specific example of a dialogue game (a version of the Muddy Children Puzzle, in which some of the children can ``cheat by engaging in secret communication moves, while others may be punished for their credulity). We also present a sketch of a ``rule-based approach to games with imperfect information (allowing ``sneaky possibilities, such as: cheating, being deceived and suspecting the others to be cheating).

Keep ‘hoping’ for rationality: a solution to the backward induction paradox

We formalise a notion of dynamic rationality in terms of a logic of conditional beliefs on (doxastic) plausibility models. Similarly to other epistemic statements (e.g. negations of Moore sentences and of Muddy Children announcements), dynamic rationality changes its meaning after every act of learning, and it may become true after players learn it is false. Applying this to extensive games, we “simulate” the play of a game as a succession of dynamic updates of the original plausibility model: the epistemic situation when a given node is reached can be thought of as the result of a joint act of learning (via public announcements) that the node is reached. We then use the notion of “stable belief”, i.e. belief that is preserved during the play of the game, in order to give an epistemic condition for backward induction: rationality and common knowledge of stable belief in rationality. This condition is weaker than Aumann’s and compatible with the implicit assumptions (the “epistemic openness of the future”) underlying Stalnaker’s criticism of Aumann’s proof. The “dynamic” nature of our concept of rationality explains why our condition avoids the apparent circularity of the “backward induction paradox”: it is consistent to (continue to) believe in a player’s rationality after updating with his irrationality.

EPISTEMIC LOGIC AND INFORMATION UPDATE

Epistemic logic investigates what agents know or believe about certain factual descriptions of the world, and about each other. It builds on a model of what information is (statically) available in a given system, and isolates general principles concerning knowledge and belief. The information in a system may well change as a result of various changes: events from the outside, observations by the agents, communication between the agents, etc. This requires information updates. These have been investigated in computer science via interpreted systems ; in philosophy and in artificial intelligence their study leads to the area of belief revision. A more recent development is called dynamic epistemic logic. Dynamic epistemic logic is an extension of epistemic logic with dynamic modal operators for belief change (i.e., information update). It is the focus of our contribution, but its relation to other ways to model dynamics will also be discussed in some detail.

Epistemic Actions as Resources

We provide algebraic semantics together with a sound and complete sequent calculus for information update due to epistemic actions. This semantics is flexible enough to accommodate incomplete as well as wrong information e.g. due to secrecy and deceit, as well as nested knowledge. We give a purely algebraic treatment of the muddy children puzzle, which moreover extends to situations where the children are allowed to lie and cheat. Epistemic actions, that is, information exchanges between agents A,B, . . . ∈ A, are modeled as elements of a quantale. The quantale (Q, ∨ , •) acts on an underlyingQ-right module (M, ∨ ) of epistemic propositions and facts. The epistemic content is encoded by appearance maps, one pair f A : M → M and f Q A : Q → Q of (lax) morphisms for each agent A ∈ A, which preserve the module and quantale structure respectively. By adjunction, they give rise to epistemic modalities [12], capturing the agents’ knowledge on propositions and actions. The module action is epistemic update and gives rise to dynamic modalities [21]— cf.weakest precondition. This model subsumes the crucial fragment of Baltag, Moss and Solecki’s [6] dynamic epistemic logic, abstracting it in a constructive fashion while introducing resource-sensitive structure on the epistemic actions.

What can we achieve by arbitrary announcements?: A dynamic take on Fitch's knowability

Public announcement logic is an extension of multi-agent epistemic logic with dynamic operators to model the informational consequences of announcements to the entire group of agents. We propose an extension of public announcement logic with a dynamic modal operator that expresses what is true after any announcement: □ϕ expresses that ϕ is true after an arbitrary announcement ψ. As this includes the trivial announcement ⊤, one might as well say that □ϕ expresses what remains true after any announcement: it therefore corresponds to truth persistence after (definable) relativisation. The dual operation ⋄ϕ expresses that there is an announcement after which ϕ. This gives a perspective on Fitchs knowability issues: for which formulas ϕ does it hold that ϕ → ⋄Kϕ? We give various semantic results, and we show completeness for a Hilbert-style axiomatisation of this logic.