The theoretical foundation of functional programming is the Curry-Howard correspondence, also known as the propositions as types paradigm. Types in simply typed lambda calculus correspond to propositions in intuitionistic logic: function types correspond to logical implications, and product types correspond to logical conjunctions. Not only that, programs correspond to proofs and computation corresponds to a procedure of cut elimination or proof normalisation in which proofs are progressively simplified. The Curry-Howard view has proved to be robust and general and has been extended to varied and more powerful type systems and logics. In one of these extensions the language is a form of pi calculus and the logic is linear logic, with its propositions interpreted as session types. In this chapter we present this system and its key results.

This chapter introduces a variant of the pi calculus defined in Chapter 2. We drop choice and channel closing, retaining only message passing. To compensate, messages may now carry values other than channel endpoints: we introduce record and variant values, both of which are standard in functional languages. We further introduce new processes to eliminate the new values. We play the same game at the type level: from input/output, external/internal choice and the two end types, we retain only input/output types. In return we incorporate record and variant types, again standard from functional languages. Unlike the input and output types in all preceding chapters, those in this chapter have no continuation. These changes lead to a linear pi calculus with record and variant types. The interesting characteristic of this calculus is that it allows us to faithfully encode the pi calculus with session types, even though it has no specific support for session types. We present an encoding based on work by Dardha, Giachino and Sangiorgi.

The previous chapters present a linear world in which each channel endpoint is possessed by exactly one thread. Linearity allows precise control of interference and that can be useful. If two processes engage in a series of consecutive message exchanges then we can be sure that messages are not sent to or received from foreign processes. Although linearity is often convenient, it can also be detrimental in concurrent systems. Purely linear systems cannot describe e-commerce interactions where a server (or a pool of servers) is expected to serve a much larger number of clients. Linear systems also fail to describe an auction, where at the end, only one bidder is to get the item for sale. This chapter introduces a type discipline that provides for channels that can be shared and consequently for processes that compete for resources, thus creating race conditions and nondeterminism. We do this in such a way that we do not need to revisit process formation or process reduction, and process typing only requires mild changes. We define the language and type system as extensions of those in Chapter 3.