Pi-nets: interaction nets for pi-calculus
نویسندگان
چکیده
π-calculus is a framework that aims to describe concurrent calculations through a formal definition of processes. Originally, π-calculus is defined by a formal language and a set of reduction rules, much in the spirit of λ-calculus. Our aim is to provide a graphical representation of π-calculus using multi-wired interaction nets, in the spirit this time of proof-nets of linear logic. 1 π-calculus (without sums and promotion) A typical system in π-calculus are multiple processes that can compute concurrently. Processes can interact with one another by sending and receiving information threw a given set of channels. Many processes can “listen” on the same channel, but only one pair can communicate at a time on it. Therefore, the calculus is completely indeterministic in a strong way (e.g. not confluent), the choice of the pair being completely random. More than simple parallelism, π-calculus describes mobile processes. The content of the messages are channel names that the recipient process can then use to communicate with other processes that have access to it, thus changing the “wiring” of the whole process-net. Practically, a process is a sequence of three types of actions – sending, receiving and an invisible internal action, that can be combined by an operation of concurrency, putting two processes in parallel, the creation of new names, replication and summation, representing a definite indeterministic choice between two processes. In this paper, we focus on a part of π-calculus without replication and summation. We begin by defining the calculus formally. The language of π-calculus LetN be a numerable set of names ranged over by lower case Latin letters. Names represent communication channels, and are also the values being transmitted in communications. Definition 1.1 (Prefix). Actions are expressed by prefixes, of which they are three kinds: π ::= x̄y | (x)y | τ The first variable (here x in both cases) is called the name of the prefix, the second one is called its co-name. Informally, x̄y corresponds to the fact that the process can send the name y through the channel x; x(z) means the process can receive a name on the channel x (the role
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