## Section: Scientific Foundations

### Infinite systems

Participants : Catuscia Palamidessi, Frank Valencia.

This research is carried over in cooperation with Biorn Victor (Uppsala University), Vijay Saraswat (IBM, USA), and Stefan Dantchev (University of Durham, UK)

- identification
Constraints and process calculi approaches for proving properties of infinite-state systems

Verifying infinite systems is a particularly challenging and a relatively new area. Practical applications of this are still at a preliminary stage.

#### Constraints approach

Constraint-based verification [61] , [55] has shown to be promising approach for infinite systems since a constraint formula is a natural symbolic representation of an infinite state set.

*Open Constraint Satisfaction Problems* have been recently introduced
for specifying and solving constraints problems in highly distributed
networks. In such a context typically there is no bound on the number
of devices/resources that can be part of a given network. Algorithms
for this kind of problems and their applications have been considered
in [40] , [43] , [60] . Nevertheless
little attention has been paid to the computational limits of these
problems. I.e., studies establishing, for interesting classes of these
problems are actually computationally solvable. This is certainly
an issue when you allow unbounded number of resources as it is the
case in DMS.

#### Process calculi approach

The study of expressive power of different forms of specifying infinite-behavior in Process Calculi is a recent line of research bringing understanding for infinite behavior of concurrent systems in terms of decidability.

Our work in [79] (see also [6] , [89] ), to our knowledge the first of this kind, deepened the understanding of process calculi for concurrent constraint programming by establishing an expressive power hierarchy of several temporal ccp languages which were proposed in the literature by other authors. These calculi, differ in their way of defining infinite behavior (i.e., replication or recursion) and the scope of variables (i.e., static or dynamic scope). In particular, it is shown that (1) recursive procedures with parameters can be encoded into parameterless recursive procedures with dynamic scoping, and vice-versa; (2) replication can be encoded into parameterless recursive procedures with static scoping, and vice-versa; (3) the calculi from (1) are strictly more expressive than the calculi from (2). Moreover, it is shown that the behavioral equivalence for these calculi is undecidable for those from (1), but decidable for those from (2). Interestingly, the undecidability result holds even if the variables in the corresponding languages take values from a fixed finite domain whilst the decidability holds for arbitrary domains. The works [45] , [46] , [47] present similar results in the context of the calculus for communicating systems (CCS).

Both the expressive power hierarchy and decidability/undecidability results give theoretical distinctions among different ways of expressing infinite behavior. The above work, however, pay little attention to the existence efficient algorithms for the corresponding decidability questions or the existence of semi-decision procedures for the undecidable cases. These issues are fundamental if we wish to verify infinite-state process specifications, and hence we shall address it in this project.