Abstract
In multiparty session types, interconnection networks identify which roles in a session engage in communication (i.e., two roles are connected if they exchange a message). In session-based interpretations of linear logic the analogue notion corresponds to determining which processes are composed, or cut, using compatible channels typed by linear propositions. In this work, we show that well-formed interactions represented in a session-based interpretation of classical linear logic (CLL) form strictly less-expressive interconnection networks than those of a multiparty session calculus. To achieve this result, we introduce a new compositional synthesis property dubbed partial multiparty compatibility (PMC), enabling us to build a global type denoting the interactions obtained by iterated composition of well-typed CLL threads. We then show that CLL composition induces PMC global types without circular interconnections between three (or more) participants. PMC is then used to define a new CLL composition rule that can form circular interconnections but preserves the deadlock-freedom of CLL.
Highlights
The discovery of linear logic [28] and the early studies of its connections with concurrent processes [1, 2, 4] can be seen as the origin of a Curry-Howard correspondence for linear logic with typed interactive behaviours, which have led to the developments connecting linear logic and session types [8]
To reason about the induced connection topologies, we build on the synthesis approaches to multiparty sessions [36, 37] that invert the projection-based proposals: Instead of starting from a global type and producing the certifiably deadlock-free local communication specifications, the works based on synthesis take a collection of local specifications and study the conditions, dubbed multiparty compatibility, under which the local views form a deadlock-free global interaction. Our work extends these approaches by introducing a compositional, or partial, notion of multiparty compatibility (PMC), which allows us to synthesise a global type that represents the interactions obtained by iterated composition of Classical Linear Logic (CLL) processes, providing the necessary tools to precisely study CLL connection topologies
We develop a compositional synthesis property, partial multiparty compatibility (PMC) (Section 4), which we use to show that the interconnectability of CLL is strictly less expressive than that of a single multiparty session in multiparty session types (MP) (Section 5);
Summary
The discovery of linear logic [28] and the early studies of its connections with concurrent processes [1, 2, 4] can be seen as the origin of a Curry-Howard correspondence for linear logic with typed interactive behaviours, which have led to the developments connecting linear logic and (binary) session types [8]. This section introduces the two calculi used in our work: The binary session calculus CLL typed using the session type interpretation of classical linear logic [9, 60]; and the multiparty session calculus MP [18, 31] In both settings, the notion of a session consists of a (predetermined) sequence of interactions on a given communication channel. The reduction semantics for MP processes is given below (omitting closure under structural congruence) They are fundamentally identical to the reduction rules of CLL but require not just the session channel to match and the role assignment to be consistent: s[p][q] s [p ] ; P | s[q][p](x ); Q →− P | Q {s [p ]/x }.
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