Prolegomena to Dynamic Logic for Belief Revision

We contrast Bonanno's ‘Belief Revision in a Temporal Framework’ (Bonanno, 2008) with preference change and belief revision from the perspective of dynamic epistemic logic (DEL). For that, we extend the logic of communication and change of van Benthem et al. (2006b) with relational substitutions (van Benthem, 2004) for preference change, and show that this does not alter its properties. Next we move to a more constrained context where belief and knowledge can be defined from preferences (Grove, 1988; Board, 2002; Baltag and Smets, 2006, 2008b), prove completeness of a very expressive logic of belief revision, and define a mechanism for updating belief revision models using a combination of action priority update (Baltag and Smets, 2008b) and preference substitution (van Benthem, 2004).

AGM’s belief revision is one of the main paradigms in the study of belief change operations. Recently, several logics for belief and information change have been proposed in the literature, which were used to encode belief change operations in a rich and expressive framework. While the connection of AGM-like operations and their encoding in dynamic doxastic logics have been studied before, by the exceptional work of Segerberg, most work on the area of Dynamic Epistemic Logics (DEL) have not attempted to characterize belief change operators by means of their logical properties. This work investigates how Dynamic Preference Logic, a logic in the DEL family, can be used to characterise properties of dynamic belief change operators, focusing on well-known postulates of iterated belief change.

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.

This contribution is a gentle introduction to so-called dynamic epistemic logics, that can describe how agents change their knowledge and beliefs. We start with a concise introduction to epistemic logic, through the example of one, two and finally three players holding cards; and, mainly for the purpose of motivating the dynamics, we also very summarily introduce the concepts of general and common knowledge. We then pay ample attention to the logic of public announcements, wherein agents change their knowledge as the result of, indeed, public announcements. One crucial topic in that setting is that of unsuccessful updates: formulas that become false when announced. The Moore-sentences that were already extensively discussed at the conception of epistemic logic in [15] give rise to such unsuccessful updates. After that, we present a few examples of more complex epistemic updates. Our closing observations are on recent developments that link the ‘standard’ topic of (theory) belief revision [1] to the dynamic epistemic logics introduced here.

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.

Dynamic epistemic logic (\(\mathbf {DEL}\)) is known as a large family of logics that extend standard epistemic logic with dynamic operators. Such dynamic operators can be regarded as epistemic actions over Kripke semantics (or its variant). Therefore, \(\mathbf {DEL}\) is often used to model changes of agents’ knowledge, belief or preference over Kripke semantics in terms of dynamic operators in many literatures. As a variant of \(\mathbf {DEL}\), (van Benthem and Liu, J Appl Non-Classical Logics 17(2):157–18 2007; Liu, Reasoning about preference dynamics, Springer Science & Business Media, Berlin 2011) proposed dynamic logic of relation changers (\(\mathbf {DLRC}\)). They provided a general framework to capture many dynamic operators in terms of relation changing operation written in programs of propositional dynamic logic, and they also provided a sound and complete Hilbert-style axiomatization for \(\mathbf {DLRC}\). While \(\mathbf {DLRC}\) can cover many dynamic operators in a uniform manner, proof theory for \(\mathbf {DLRC}\) is not well-studied except the Hilbert-style axiomatization. Therefore, we propose a cut-free labelled sequent calculus for \(\mathbf {DLRC}\). We show that our sequent calculus is equipollent with the Hilbert-style axiomatization.

This paper discusses the possibility of modelling inductive inference (Gold 1967) in dynamic epistemic logic (see e.g. van Ditmarsch et al. 2007). The general purpose is to propose a semantic basis for designing a modal logic for learning in the limit. First, we analyze a variety of epistemological notions involved in identification in the limit and match it with traditional epistemic and doxastic logic approaches. Then, we provide a comparison of learning by erasing (Lange et al. 1996) and iterated epistemic update (Baltag and Moss 2004) as analyzed in dynamic epistemic logic. We show that finite identification can be modelled in dynamic epistemic logic, and that the elimination process of learning by erasing can be seen as iterated belief-revision modelled in dynamic doxastic logic. Finally, we propose viewing hypothesis spaces as temporal frames and discuss possible advantages of that perspective.

In this paper we compare standard ways to perform belief change with attempts to model such change with dynamic modal operators. We address both belief expansion, belief contraction, belief revision, and update. Public announcement logic is an implementation of a belief expansion operator in a dynamic epistemic setting. The postulates of success and minimal change cannot be satisfied in their original AGM formulation. Dynamic doxastic logic provides various implementations of a belief revision operator in a dynamic epistemic setting. We provide an example of the application of such a dynamic doxastic semantics, which can also be seen as a realization of a proposal by Spohn. The purpose of this paper is to provide an overview of different approaches to dynamic logics for belief change, including some of our own contributions to the area, in a way that is accessible to an interdisciplinary audience. It does not contain new technical results not published elsewhere. However, the comparison of different approaches provides, we hope, new insights not seen before. A final version of this paper may include new technical results.

20 Modal logic for games and information

Epistemic Logic and Information Update

In this paper we show how to formalise false-belief tasks like the Sally-Anne task and the second-order chocolate task in Dynamic Epistemic Logic (DEL). False-belief tasks are used to test the strength of the Theory of Mind (ToM) of humans, that is, a human’s ability to attribute mental states to other agents. Having a ToM is known to be essential to human social intelligence, and hence likely to be essential to social intelligence of artificial agents as well. It is therefore important to find ways of implementing a ToM in artificial agents, and to show that such agents can then solve false-belief tasks. In this paper, the approach is to use DEL as a formal framework for representing ToM, and use reasoning in DEL to solve false-belief tasks. In addition to formalising several false-belief tasks in DEL, the paper introduces some extensions of DEL itself: edge-conditioned event models and observability propositions. These extensions are introduced to provide better formalisations of the false-belief tasks, but expected to have independent future interest.

In this paper we attempt to shed light on the concept of an agent’s knowledge after a non-deterministic action is executed. We start by making a comparison between notions of non-deterministic choice, and between notions of sequential composition, of settings with dynamic and/or epistemic character; namely Propositional Dynamic Logic (PDL), Dynamic Epistemic Logic (DEL), and the more recent logic of Semi-Public Environments (SPE). These logics represent two different approaches for defining the aforementioned actions, and in order to provide unified frameworks that encompass both, we define the logics DELVO (DEL+Vision+Ontic change) and PDLVE (PDL+Vision+Epistemic operators). DELVO is given a sound and complete axiomatisation.

The Sum-and-Product riddle was first published in the reference H. Freudenthal (1969, Nieuw Archief voor Wiskunde 3, 152) [6]. 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. We analyse one of the announcements towards the solution of the riddle as a so-called unsuccessful update: a formula that becomes 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 and the complexity is experimentally investigated for several representations and parameter settings.

The paper analyzes dynamic epistemic logic from a topological perspective. The main contribution consists of a framework in which dynamic epistemic logic satisfies the requirements for being a topological dynamical system thus interfacing discrete dynamic logics with continuous mappings of dynamical systems. The setting is based on a notion of logical convergence, demonstratively equivalent with convergence in Stone topology. Presented is a flexible, parametrized family of metrics inducing the latter, used as an analytical aid. We show maps induced by action model transformations continuous with respect to the Stone topology and present results on the recurrent behavior of said maps.

This paper surveys the interface between the two major logical trends that describe agents' intelligent interaction over time: dynamic epistemic logic (DEL) and epistemic temporal logic (ETL). The initial attempt to "merge" DEL and ETL was made in van Benthem et al. (Merging frameworks for interaction: DEL and ETL, 2007) and followed up by van Benthem et al. (J Phil Logic 38(5):491---526, 2009) and Hoshi (Epistemic dynamics and protocol information. Ph.D. thesis, Stanford University Stanford, 2009a). The merged framework provides a systematic comparison between these two logical systems and studies new logics of intelligent interaction. This paper presents the main results and the recent developments at the interface between DEL and ETL.