Expressiveness modulo Bisimilarity of Regular Expressions with Parallel Composition (Extended Abstract)

The languages accepted by finite automata are precisely the languages denoted by regular expressions. In contrast, finite automata may exhibit behaviours that cannot be described by regular expressions up to bisimilarity. In this paper, we consider extensions of the theory of regular expressions with various forms of parallel composition and study the effect on expressiveness. First we prove that adding pure interleaving to the theory of regular expressions strictly increases its expressiveness up to bisimilarity. Then, we prove that replacing the operation for pure interleaving by ACP-style parallel composition gives a further increase in expressiveness. Finally, we prove that the theory of regular expressions with ACP-style parallel composition and encapsulation is expressive enough to express all finite automata up to bisimilarity. Our results extend the expressiveness results obtained by Bergstra, Bethke and Ponse for process algebras with (the binary variant of) Kleene’s star operation.

💡 Research Summary

The paper investigates how the expressive power of regular expressions changes when various forms of parallel composition are added, with the equivalence notion being bisimilarity rather than ordinary language equality. It proceeds in three main steps, each establishing a strict increase in expressive power, and culminates in a result showing that a suitably enriched regular‑expression language can capture the behaviour of any finite automaton up to bisimilarity.

1. From Sequential Regular Expressions to Interleaving.
Regular expressions (RE) traditionally consist of sequential composition, choice, and Kleene star, and they are known to denote exactly the languages accepted by finite automata. However, when we view processes as labelled transition systems and compare them using bisimilarity, many behaviours of finite automata cannot be expressed by plain RE. The authors introduce a pure interleaving operator “‖” that, given two sub‑expressions, produces all possible interleavings of their actions. The resulting language, denoted RE‖, is shown to be strictly more expressive than RE with respect to bisimilarity. The proof uses a simple counter‑example: the process that can perform either the sequence a·b or b·a cannot be represented by any RE, because any RE that generates both traces would also generate undesirable interleavings. By contrast, the expression a‖b directly yields exactly the two required interleavings, establishing RE‖ ⊃ RE.

2. Adding ACP‑style Parallel Composition.
The second extension replaces the pure interleaving operator with the ACP (Algebra of Communicating Processes) parallel composition, which not only interleaves actions but also synchronises on a set of shared actions. This operator is denoted “|” together with a communication function. The authors define a new calculus RE‖_ACP and prove that it is strictly more expressive than RE‖. The key observation is that some behaviours require simultaneous execution of matching actions, something that pure interleaving cannot enforce. For instance, a process that must perform a and b concurrently before a third action c can be expressed as (a | b)·c in ACP, but any RE‖ expression would either miss the required synchronisation or introduce extra interleavings. By constructing such synchronisation‑dependent examples, the paper demonstrates a proper hierarchy: RE ⊂ RE‖ ⊂ RE‖_ACP.

3. Encapsulation and Full Expressiveness.
The final step augments ACP parallel composition with an encapsulation (or restriction) operator ∂_H, which hides or blocks a chosen set H of actions. This operator allows internal communications to be concealed from the external observer, aligning the observable behaviour of the expression with that of a target automaton. The authors prove that the combined calculus RE‖_ACP,encap can represent every finite automaton up to bisimilarity. The construction proceeds as follows: each state s of the automaton is mapped to a unique atomic process p_s; each transition s ─a→ t becomes the sequential term p_s·a·p_t; all such terms are combined using the ACP parallel operator to form a global system; finally, encapsulation hides the internal p_s actions, leaving only the original alphabet visible. The resulting regular‑expression term is bisimilar to the original automaton, establishing expressive completeness.

Relation to Prior Work.
Bergstra, Bethke and Ponse previously studied the expressive power of process algebras that extend Kleene’s star with binary parallel composition. This paper extends their line of inquiry by focusing on the regular‑expression syntax itself and by using bisimilarity as the comparison criterion. The hierarchy RE ⊂ RE‖ ⊂ RE‖_ACP ⊂ RE‖_ACP,encap mirrors the hierarchy found in process algebras, but the results are obtained directly for regular‑expression languages, offering a more fine‑grained view of how each additional operator contributes to expressive power.

Implications and Future Directions.
The findings have several practical implications. First, they show that any tool or language based on regular expressions can be made capable of modelling concurrent systems if it incorporates parallel composition and restriction operators. Second, because bisimilarity is a robust equivalence used in model checking, the results open the door to automated verification techniques that operate directly on enriched regular‑expression specifications. Finally, the paper suggests several avenues for further research: extending the results to infinite‑state systems, developing efficient algorithms for translating between finite automata and the enriched regular‑expression terms, and exploring how these ideas can be integrated into real‑world programming languages and verification frameworks.