Formalizing Propagation of Priorities in Reo, Using Eight Colors

Abstract

Reo is a language for programming of coordination protocols among concurrent processes. Central to Reo are connectors: programmable synchronization/communication mediums used by processes to exchange data. Every connector runs at a clock; at every tick, it enacts an enabled synchronization/communication among processes.

Connectors may prioritize certain synchronizations/communications over others. “Passive” connectors use their priorities only at clock ticks, to decide which enabled synchronization/communication to enact. “Active” connectors, in contrast, use their priorities also between clock ticks, to influence which synchronizations/communications become enabled; they are said to “propagate their priorities”.

This paper addresses the problem of formalizing propagation of priorities in Reo. Specifically, this paper presents a new instantiation of the connector coloring framework, using eight colors. The resulting formalization of propagation of priorities is evaluated by proving several desirable behavioral equalities.

A Definitions

Definition 1

(Structure). $\mathbb {V}$ is the set of all vertices. $\mathbb {T}$ is the set of all types. The structure of a connector is a tuple $g = (V, E)$, where $V \subseteq \mathbb {V}$ and $E \subseteq (2^{V} \times \mathbb {T}\times 2^{V}) \setminus \{(\emptyset , t, \emptyset )\mid t \in \mathbb {T}\}$. $\mathbb {G}$ is the set of all structures.

Definition 2

(Structural composition). $\mathsf {S}, \mathsf {T}: 2^{(2^{\mathbb {V}} \times \mathbb {T}\times 2^{\mathbb {V}})} \rightarrow 2^{\mathbb {V}}$ are the functions defined by the following equations:

$$\begin{aligned} \mathsf {S}(E) = \textstyle \bigcup \{V\mid (V, t, V') \in E\} \setminus \bigcup \{V'\mid (V, t, V') \in E\} \\ \mathsf {T}(E) = \textstyle \bigcup \{V'\mid (V, t, V') \in E\} \setminus \bigcup \{V\mid (V, t, V') \in E\} \end{aligned}$$

${\bowtie }: \mathbb {G}\times \mathbb {G}\rightarrow \mathbb {G}$ is the partial operation defined by the following equation:

$$\begin{aligned}&(V_1, E_1) \mathbin {{\bowtie }}(V_2, E_2) = {\left\{ \begin{array}{ll} (V_1 \cup V_2, E_1 \cup E_2) &{} \mathrm{if}\!\!:\; \mathsf {S}(E_1) \cap \mathsf {T}(E_2) = \mathsf {S}(E_2) \cap \mathsf {T}(E_1) \\ \bot &{} \mathrm{otherwise} \end{array}\right. } \end{aligned}$$

Definition 3

(Behavior). $\mathbb {C}$ is the set of all colors. A coloring $\gamma$ over V is a function from V to $\mathbb {C}$. $\mathbb {C}\textsc {ol}(V) = V \rightarrow \mathbb {C}$ is the set of all colorings over V. The behavior of a connector $(V, E)$ is a set $\varGamma \subseteq \mathbb {C}\textsc {ol}(V)$ of colorings.

Definition 4

(Behavioral composition).

${\bowtie }: (\mathbb {C}\textsc {ol}(V_1) \times \mathbb {C}\textsc {ol}(V_2) \rightharpoonup \mathbb {C}\textsc {ol}(V_1 \cup V_2)) \cup (2^{\mathbb {C}\textsc {ol}(V_1)} \times 2^{\mathbb {C}\textsc {ol}(V_2)} \rightharpoonup 2^{\mathbb {C}\textsc {ol}(V_1 \cup V_2)})$ is the partial function defined by the following equations:

$$\begin{aligned} \gamma _1 \mathbin {{\bowtie }}\gamma _2&= {\left\{ \begin{array}{ll} \gamma _1 \cup \gamma _2 &{} \mathrm{if}\!\!:\; \gamma _1(p) = \gamma _2(p) \,\,\mathrm{for}\text {-}\mathrm{all} \,\,p \in dom (\gamma _1) \cap dom (\gamma _2) \\ \bot &{} \mathrm{otherwise} \end{array}\right. } \\ \varGamma _1 \mathbin {{\bowtie }}\varGamma _2&= \{\gamma _1 \mathbin {{\bowtie }}\gamma _2\mid \gamma _1 \in \varGamma _1 \,\,\mathrm{and } \,\, \gamma _2 \in \varGamma _2 \,\,\mathrm{and } \,\, \gamma _1 \mathbin {{\bowtie }}\gamma _2 \in dom (\mathbin {{\bowtie }})\} \end{aligned}$$

Definition 5

(Denotation). With $\mathcal {T}: \mathbb {T}\rightarrow (2^{\mathbb {V}} \times 2^{\mathbb {V}}) \rightarrow \bigcup \{2^{\mathbb {C}\textsc {ol}(V)}\mid V \subseteq \mathbb {V}\}$, is the function defined by the following equation:

Theorem 1