*Semantics Club 02 03 2001*

# Mathematical Modelling of Real Time Processes

### Abstract

In this talk, we will present a mathematical model of real time
process algebras like Temporal CCS [MT90].

The two main notions will be a linearly ordered monoid M, representing
the domain of time, and actions of that monoid. The monoid actions
will serve to describe two different features: a total (left) action
denoting a delaying operation on processes, and a partial (right)
action denoting the passage of time, related in a suitable way.

Technically, the main ingredient is a "real time comonad" on the
category of M-actions whose coalgebras will not only carry an M-action
structure but also an additional *partial* M-action structure.
Therefore we will call those structures "M-biactions".

We will then generalise the framework of [TP97], where it was shown
that a set of GSOS rules corresponds to a natural transformation of a
certain type parametrised in functorial notions of syntax and
behaviour on the category Set, in order to accdomodate the view of
transition systems as coalgebras for comonads, rather than just
coalgebras for a behaviour endofunctor.

This will eventually lead to both a format for and side conditions on
operational rules which will ensure that the transition system on
processes as defined by the rules will be indeed an M-biaction
structure.

[MT90] F. Moller, C. Tofts, A Temporal Calculus of Communicating
Systems, CONCUR'90

[TP97] D. Turi, G. Plotkin, Towards a Mathematical Operational
Semantics, LICS'97