Reversible sessions with flexible choices

Abstract

We propose a calculus for concurrent reversible multiparty sessions, equipped with a flexible choice operator allowing for different sets of participants in each branch. This operator is inspired by the notion of connecting action recently introduced by Hu and Yoshida to describe protocols with optional participants. We argue that this choice operator allows for a natural description of typical communication protocols. Our calculus also supports a compact representation of the history of processes and types, which facilitates the definition of rollback. Moreover, it implements a fine-tuned strategy for backward computation. We present a session type system for the calculus and show that it enforces the expected properties of session fidelity, forward progress and backward progress.

A: Reductions of networks and systems

It is easy to define a bijection between networks and initial systems:

$$\begin{aligned} {\mathcal {S}}(\mathbb {N})=\mathbb {N}\between \langle -,\mathbb {P},\emptyset \rangle \qquad \qquad {\mathcal {N}}(\mathbb {N}\between \langle -,\mathbb {P},\emptyset \rangle )=\mathbb {N}\end{aligned}$$

where $\mathbb {P}$ is the set of participants in $\mathbb {N}$.

We establish now an operational correspondence between networks and their associated systems. More precisely we prove the following:

• Communication preservation and reflection: every network communication is simulated by a communication in the associated system and vice-versa;

• Rollback preservation: every network rollback is simulated by a sequence of backward moves in the associated system, initiating with a starting backward move and terminating with an ending backward move;

• Rollback reflection: every starting backward move in a system can be extended to a complete sequence of backward moves such that the resulting system is the image of a rollback in the source network.

In the following theorem we denote transitive closure of and .

Theorem 5

• 1. If , then .

• 2. If , then .

• 3. If , then .

• 4. If , then and .

Proof

(1) and (2). The result is immediate since Rule [Com] of Fig. 1 and Rule [ComS] of Fig. 2 have the same antecedents.

(3). If , then the applied rule is

where

$$\begin{aligned} \begin{array}{c}\mathbb {N}=\textsf {p} \llbracket \,P\,\rrbracket \mathrel {\Vert }\varPi _{1\le i\le n}\textsf {p} _i\llbracket \,P_i\,\rrbracket \quad \text { and }\quad \mathbb {N}'=\textsf {p} \llbracket \,P'\,\rrbracket \mathrel {\Vert }\varPi _{1\le i\le m}\textsf {p} _i\llbracket \,P'_i\,\rrbracket \mathrel {\Vert }\varPi _{m+1\le i\le n}\textsf {p} _i\llbracket \,P_i\,\rrbracket \end{array} \end{aligned}$$

Then ${\mathcal {S}}(\mathbb {N})=\mathbb {N}\between \sigma$ and ${\mathcal {S}}(\mathbb {N}')=\mathbb {N}'\between \sigma$ with $\sigma =\langle -,\{\textsf {p} \}\cup \{\textsf {p} _i\mid 1\le i \le n\},\emptyset \rangle$. We get

(4). If , then Rule [BackS] has been applied, so

Since $\mathbb {P}_2$ can be split in two subsets of participants, according to whether the associated processes satisfy the premise of Rule [BackY] or of Rule [BackN] in Fig. 2, we may assume without loss of generality that

$$\begin{aligned} \begin{array}{c} \mathbb {N}_1=\varPi _{1\le i\le m}\textsf {p} _i\llbracket \,P_i\,\rrbracket \mathrel {\Vert }\varPi _{m+1\le i\le n}\textsf {p} _i\llbracket \,P_i\,\rrbracket \end{array} \end{aligned}$$

where for all i, $1\le i\le n$, if $i\le m$ and otherwise. With a sequence of reductions as in the proof of point (3), we then obtain for some network $\mathbb {N}''$:

Since $P$ and all the $P_i$ for $1\le i\le n$ satisfy the premises of Rule [Back] in Fig. 1, we may apply this rule to $\mathbb {N}$ to conclude , as required. $\square$