Theoretical Computer Science

# Does mathematical model for conccurent computations exist?

Turing machines can represent any computation. Can they also represent concurrent computations? Eg. multiple computations that can happen at the same time?

If yes, how are the concurrent computations represented and is it possible to convert them into turing machine?

And do they mathematically describe things like race condition and deadlocks?

## Solution

As already suggested above, process algebra or process calculus is the place to start. Quoting freely from the respective Wikipedia page,

History

In the first half of the 20th century, various formalisms were proposed to capture the informal concept of a computable function, with -recursive functions, Turing Machines and the lambda calculus possibly being the best-known examples today. The surprising fact that they are essentially equivalent, in the sense that they are all encodable into each other, supports the Church-Turing thesis. Another shared feature is more rarely commented on: they all are most readily understood as models of sequential computation. The subsequent consolidation of computer science required a more subtle formulation of the notion of computation, in particular explicit representations of concurrency and communication. Models of concurrency such as the process calculi, Petri nets in 1962, and the Actor model in 1973 emerged from this line of enquiry.

Research on process calculi began in earnest with Robin Milner's seminal work on the Calculus of Communicating Systems (CCS) during the period from 1973 to 1980. C.A.R. Hoare's Communicating Sequential Processes (CSP) first appeared in 1978, and was subsequently developed into a full-fledged process calculus during the early 1980s. There was much cross-fertilization of ideas between CCS and CSP as they developed. In 1982 Jan Bergstra and Jan Willem Klop began work on what came to be known as the Algebra of Communicating Processes (ACP), and introduced the term process algebra to describe their work.1 CCS, CSP, and ACP constitute the three major branches of the process calculi family: the majority of the other process calculi can trace their roots to one of these three calculi.