Merging Frameworks for Interaction: DEL and ETL
Johan van Benthem, Jelle Gerbrandy, Eric Pacuit

Abstract:
Many logical systems today describe behaviour of intelligent
interacting agents over time. Frameworks include Interpreted Systems
(IS, Fagin et al.), Epistemic-Temporal Logic (ETL, Parikh &
Ramanujam), STIT (Belnap et al.), Process Algebra and Game Semantics
(Abramsky).  This proliferation is an asset, as different modeling
tools can be fine-tuned to specific applications.  But it may also be
an obstacle, when barriers between paradigms and schools go up.

This paper takes a closer look at one particular interface, between
two systems that both address the dynamics of knowledge and
information flow in multi-agent systems. One is IS/ETL (IS and ETL are
basically the same up to model transformations), which uses linear or
branching time models with added epistemic structure induced by
agents' different capabilities for observing events. These models
provide a Grand Stage where histories of some process unfold
constrained by a protocol, and a matching epistemic-temporal language
describes what happens. The other framework is Dynamic Epistemic Logic
(DEL) which describes interactive processes in terms of epistemic
event models which may occur inside modalities of the
language. Temporal evolution is then computed from some initial
epistemic model through a process of successive 'product updates'.  It
has long been unclear how to best compare IS/ETL and DEL. Other papers
have investigated various aspects, but in this paper, we strengthen
the interface to a considerable extent.

We first show how to transform DEL protocols into classes of ETL
models, leading to a simple language translation from dynamic
modalities to temporal operators. Next, we prove a new representation
theorem characterizing the largest class of ETL models corresponding
to DEL protocols in terms of notions of Perfect Recall, No Miracles,
and Bisimulation Invariance. These describe the sort of idealized
agent presupposed in standard DEL. Next, we consider further
assumptions on agents, and introduce a new technique of modal
correspondence analysis relating special properties of DEL protocols
to corresponding ETL-style properties. Finally, we how the DEL ETL
analogy suggests new issues of completeness. Our new contribution is
an axiomatization for the dynamic logic of public announcements
constrained by protocols, which has been an open problem for some
years, as it does not fit the usual 'reduction axiom' format of DEL.

Once again, we are not reducing one framework to another. We show
rather how ETL and DEL lead to interesting new issues when merged as
accounts of intelligent agents.

Keywords: Dynamic Epistemic Logic;  Epistemic Temporal Logic