Every Regular Language Has a Bipartite Cactus Automaton

2026-10-08

AI Summary

This study addresses the limitations of the classical Thompson construction, which struggles to preserve graph structural simplicity during ε-transition elimination and remains confined to planar NFA frameworks. We propose a direct inductive approach based on parity-typed regular expressions, leveraging the closure properties of cactus graphs to identify and adopt a bipartite cactus structure that is more concise than series-parallel graphs. We prove that every regular language admits a corresponding bipartite cactus automaton whose transition graph consists exclusively of even cycles with treewidth at most two. This result establishes an equivalence between regular languages and bipartite cactus graphs, overcoming traditional constraints and significantly simplifying the topological structure of the resulting automata.

Abstract

Thompson's classical construction turns every regular expression into an equivalent epsilon-NFA whose transition graph is series-parallel. The construction uses epsilon-transitions, and eliminating them by the usual shortcut construction need not preserve planarity or the bound of two on the treewidth of the transition graph. Book and Chandra proved that every regular language nevertheless has an epsilon-free NFA with a planar transition graph. We show that the transition graph can in fact be chosen to be a bipartite cactus: every cycle is even, and any two cycles have at most one vertex in common. In particular, it is outerplanar and has treewidth at most two. This is a considerably more simple structure than the general series-parallel guarantee supplied by Thompson's construction. The proof is a direct induction on parity-typed regular expressions and uses only elementary closure properties of cactus graphs.

Problem

• Regular Language
• Epsilon-free NFA
• Bipartite Cactus Automaton
• Outerplanar Graph
• Treewidth

Innovation

• Bipartite Cactus Automaton
• Regular Language
• Epsilon-free NFA
• Outerplanar Graph
• Treewidth