Project-Team:ESPRESSO

Inria | Raweb 2013 | Presentation of the Project-Team ESPRESSO


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Section: Research Program

Compilation of Signal

Sequential code generation starting from a Signal specification starts with an analysis of its implicit synchronization and scheduling relations. This analysis yields the control and dataflow graphs that define the class of sequentially executable specifications and allow to generate code.

Synchronization and scheduling specifications

In Signal, the clock $\hat{} x$ of a signal $x$ denotes the set of instants at which the signal $x$ is present. It is represented by a signal that is true when $x$ is present and that is absent otherwise. Clock expressions represent control. The clock $𝚠𝚑𝚎𝚗 x$ (resp. $𝚠𝚑𝚎𝚗 𝚗𝚘𝚝 x$ ) represents the time tags at which a boolean signal $x$ is present and true (resp. false).

The empty clock is written $\hat{} 0$ and clock expressions $e$ combined using conjunction, disjunction and symmetric difference. Clock equations $E$ are Signal processes: the equation $e \hat{} = e^{'}$ synchronizes the clocks $e$ and $e^{'}$ while $e \hat{} < e^{'}$ specifies the containment of $e$ in $e^{'}$ . Explicit scheduling relations $x \to y 𝚠𝚑𝚎𝚗 e$ allow to schedule the calculation of signals (e.g. $x$ after $y$ at the clock $e$ ).

\begin{matrix} e & : : = & \hat{} x | 𝚠𝚑𝚎𝚗 x | 𝚗𝚘𝚝 x | e \hat{} + e^{'} | e \hat{} - e^{'} | e \hat{} * e^{'} | \hat{} 0 & (clock expression) \\ E & : : = & () | e \hat{} = e^{'} | e \hat{} < e^{'} | x \to y 𝚠𝚑𝚎𝚗 e | E | E^{'} | E / x & (clock relations) \end{matrix}

Synchronization and scheduling analysis

A Signal process $P$ corresponds to a system of clock and scheduling relations $E$ that denotes its timing structure. It can be defined by induction on the structure of $P$ using the inference system $P : E$ of Figure 5 .

x := y$ init v : ^x ^= ^y

x := y when z : ^x ^= ^y when z | y -> x when ^x

x := y default z : ^x ^= ^y default ^z | y -> x when ^y | z -> x when ^z ^- ^y

Figure 5. Clock inference system

Hierarchization

The clock and scheduling relations $E$ of a process $P$ define the control flow and dataflow graphs that hold all necessary information to compile a Signal specification upon satisfaction of the property of endochrony. A process is said endochronous iff, given a set of input signals and flow-equivalent input behaviors, it has the capability to reconstruct a unique synchronous behavior up to clock-equivalence: the input and output signals are ordered in clock-equivalent ways.

Figure 6. Hierarchization of clocks

To determine the order $x ⪯ y$ in which signals are processed during the period of a reaction, clock relations $E$ play an essential role. The process of determining this order is called hierarchization and consists of an insertion algorithm which hooks elementary control flow graphs (in the form of if-then-else structures) one to the others. Figure 6 , right, let h3 be a clock computed using h1 and h2. Let h be the head of a tree from which h1 and h2 are computed (an if-then-else), h3 is computed after h1 and h2 and placed under h [27] .

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