Abstract
In an interactive multi-agent system, agents
communicate with each other and this communication
changes their information state. In order to reason
about information updates in such settings, one has to
take into account the dynamic as well as the epistemic
aspect of communication. The traditional approaches
only consider the epistemic aspect and dismiss the
dynamic one. Recent development of Dynamic Epistemic
Logic integrates both in a {\sf PDL} style logic with
kripke semantics.
OUTLINE: I shall first show how this combinatorial
setting can be generalized to an order theoretic
structure that has a non-boolean resource-sensitive
nature and considers communication actions as
fundamental operations of an algebra rather than
concrete constructions on a Kripke model. The algebra
consists of an \em epistemic system \em
$(M,Q,\{f_A\}_{A \in {\cal A}})$, which is a
quantale-module pair $(M,Q)$ endowed with a family
of ``appearance maps" for each agent $f_A = (f_A^M:M
\to M, f_A^Q:Q \to Q)$. The right Galois adjoint to
the appearance map gives rise to the ``box" modality
of epistemic logic (expressing knowledge or belief of
some agent).
I shall then briefly present a sound and complete
Lambek - style sequent calculus based on the algebra
that enables us to reason about communications and
their effects in a semi-automatic way. Interesting
examples will be presented including a cheating
version of the muddy children puzzle and a security
protocol!
Finally and most importantly, I shall show how moving
to a sup-enriched categorical setting provides us a
super elegant structure and allows each agent to have
his own update schema.