Computation and Concurrency

We try to clarify the relationship between computation and concurrency. Base on the so-called pomsetc automata and step automata, we introduce communication and more operators, and establish the algebras modulo language equivalence and truly concurrent bisimilarities.

💡 Research Summary

The paper investigates the deep relationship between computation and concurrency by building a unified algebraic framework that incorporates both traditional process algebras and truly concurrent models. After a thorough review of interleaving concurrency (CCS, ACP) and true concurrency (event structures, Petri nets), the authors introduce pomset automata and step automata as the foundational semantic devices for representing concurrent behavior. On top of these devices they define a suite of communication operators—merge, encapsulation, silent step, abstraction, and parallel composition—extending the classic sequential and parallel operators. A central contribution is the generalized expansion law a ⊙ b = a·b + b·a + a∥b + a⊗b, which subsumes Milner’s original law and explicitly captures communication between parallel components. The paper systematically develops algebras modulo language equivalence and several truly concurrent bisimilarities (step, hp‑, hhp‑bisimulation), proving soundness, completeness, and a range of meta‑theoretical properties such as reduction, refinement, lifting, and decomposition. Exchange laws are established to manage interactions among operators, and a parallel star operator is introduced to model iterative parallelism. The authors then present detailed constructions that translate between regular expressions and both pomset and step automata, proving well‑nestedness to guarantee structural preservation during translation. Finally, a step‑based Turing machine is defined, and the Church‑Turing thesis is examined within this truly concurrent setting, demonstrating that the proposed model is computationally universal. Overall, the work offers a comprehensive, mathematically rigorous bridge between computation and concurrency, extending Kleene algebra with communication and parallelism, and laying groundwork for future formal verification and modeling tools that need to handle both aspects simultaneously.