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Section: New Results

CCSL as a Logical Clock Calculus Algebra: expressiveness and decidability results

Participants : Robert de Simone, Julien Deantoni, Frédéric Mallet, Qingguo Xu.

CCSL is a language dedicated to the expression of time constraints, based on so-called logical clocks. Its declarative nature is akin to the Lustre or (even closer to) the Signal language, but without values (to clock/event occurrences) and with both synchronous and asynchronous constraints. Solving a set of CCSL constraints amounts to the production of a feasible schedule of the system. While the TimeSquare tool may attempt to generate such a schedule trace by insightful simulation, it is not guaranteed to be complete in its search.So the issue of expressiveness and decidability was left open to this day.

Still, in previous years, we had established the CCSL constraints could be translated into parallel products (extended, transition-labelled) Büchi machines, but some of these machines had to contain integer shift counters, and were thus not fully FSMs. Our (misled) conjecture that CCSL had semilinear, Presburger-arithmetic power was defeated by a new translation expressing (unitary then general) Petri Nets and Vector Addition Systems into CCSL by encoding. The new conjecture that CCSL was then as powerful as Petri Nets was again defeated by a construction interpreting the features of inhibitor arcs in CCSL. As such inhibitor arcs extend the expressive power of Petri Nets to become universal (Turing-complete), CCSL enjoys the same universal property (which makes it unfortunately impossible to solve automatically in general).

Despite this negative result we could show that, under natural restrictions such as the assumption that "input" clocks have bounded jitter around a mean rate, and even if those bounds are not exactly known (but may be used as a parameter), then expressiveness remains in the semi-linear, Presburger-arithmetic range.

As a side-effect of this work we provided the translation of CCSL constraints into Büchi components by using a well-defined fragment of the Esterel syntax to express the Buchi automata.

Preliminary results are exposed in a research report. A much more ambitious article is in preparation.

As part of Professor Xu sabbatical in Aoste, we also considered the topic of machine-assisted proof of schedulability using theorem-provers (in our case PVS) [54] ).