## Section: New Results

### Distributed programming

Participant : Bernard Serpette [correspondant] .

#### Logical behavioural semantics of Esterel

We have formalised, with the Coq system, the logical behavioural semantics of Esterel as described in Gérard Berry's book. In order to define the properties of reactivity and determinism, we have defined a new semantics using contexts with a proven correspondence between the two semantics.

The specification and the proofs of the correspondence take 3500 lines of Coq.

#### Abstract distributed machine

We have experimented an abstract machine composed of distributed nodes. Each node has exactly two named links to other nodes and an instruction able to modify one link of a reachable node. This instruction is executed when a token is received, once the instruction is achieved the token is transmitted to another reachable node.

This abstract machine is turing complete. The $\lambda $-calculus and the $\pi $-calculus can be compiled to the instruction set of this machine.

The execution of one individual node may involve paths of arbitrary length, for example, when compiling the $\lambda $-calculus or the $\pi $-calculus, the path length for accessing a variable is proportional to it's de Bruijn index and therefore is not bounded. Given a machine with instructions of unbounded paths, we can build an equivalent machine where all the paths are bounded by two: a node is only able to access it's own links and the links of its neighbour. Moreover, this transformation uses only 6 different instructions.