## 5 Some perspectives

There are numerous perspectives in static analysis of concurrent programs,
computability and complexity issues in fault-tolerant distributed systems as
well as in concurrent database theory, as I have been trying to explain in
the previous sections. The aim here is not to list the possible research that
could be carried on in these directions (good references for this are
[Fajstrup et al., 1999] and [Herlihy and Rajsbaum, 1999]), but to look at other possible use of these
techniques. For instance, Squier's theorem in rewriting systems theory, which
gives a necessary condition for the existence of a presentation of a given
monoid by a finite
canonical rewriting system in terms of its homology (must be of finite dimension),
seems very much related to the techniques presented above. It is definitely a
computability result, as we have in fault-tolerant distributed systems theory,
but for something which looks sequential (rewriting). As hinted in [Goubault, 1995a],
this can be understood as a problem of concurrency theory in that the study of
the confluence of rewriting systems is related to parallel reduction techniques
(as in [Lévy, 1978] for instance). The resolutions used in most of the
proofs of this theorem, like in [Kobayashi, 1990], [Groves, 1991],
[Anick, 1986], [Farkas, 1992] and [Lafont and Prouté, 1990] are very much like
a Knuth-Bendix completion procedure, where higher-dimensional objects are filling
in possible defects of local confluence. This looks like building higher-dimensional
transitions implementing the parallel (confluent) reductions (see in particular
[Groves, 1991] where the resolution is a cubical complex and
in dimension one it is generated by
the transition system coming from the reduction relation). Some other proof
techniques use something which is very much like some kind of directed homotopy,
as in
[Squier et al., 1994] for instance. Other interesting relations should be
studied concerning ``higher-dimensional'' word problems, as in
[Burroni, 1991].
A hope is that geometry can also give some insight in logics, especially modal
logics as in [Goubault-Larrecq and Goubault, 1999]. Finally, there is some intuition from
theoretical physics that
seems relevant to semantics, in particular concerning time and dynamical systems.
Some of the concepts of M. Raussen's and L. Fasjtrup's articles in this issue are
based on similar notions as in [Penrose, 1972]: some areas of physics (not
classical mechanics though) have to consider time as non-reversible, hence have
to construct some kind of directed topology.