🤖 AI Summary
The determinization of finite-state automata lacks a unified categorical semantics capturing its universal property across diverse automaton models.
Method: This paper reconstructs automaton determinization from the perspective of fibred category theory, introducing two novel determinization schemes: (i) canonical forward simulation via local adjunctions in SpanSet/Rel, and (ii) forward–backward simulation leveraging the multiset functor and relative adjunctions—provably preserving path structure strictly.
Contribution/Results: It establishes, for the first time, the universal property of determinization intrinsically within a fibred setting; systematically unifies Kleisli categories, the bicategories Rel and SpanSet, and forward–backward simulation theory; and yields a coherent categorical semantics for automaton determinization. The approach delivers two canonical constructions, with rigorous proofs of simulation preservation and path preservation, thereby enabling uniform modeling and reasoning about determinization across automaton variants.
📝 Abstract
Colcombet and Petric{s}an argued that automata may be usefully considered from a functorial perspective, introducing a general notion of"$mathcal{V}$-automaton"based on functors into $mathcal{V}$. This enables them to recover different standard notions of automata by choosing $mathcal{V}$ appropriately, and they further analyzed the determinization for extbf{Rel}-automata using the Kleisli adjunction between extbf{Set} and extbf{Rel}. In this paper, we revisit Colcombet and Petric{s}an's analysis from a fibrational perspective, building on Melli`es and Zeilberger's recent alternative but related definition of categorical automata as functors $p : mathcal{Q} o mathcal{C}$ satisfying the finitary fiber and unique lifting of factorizations property. In doing so, we improve the understanding of determinization in three regards: Firstly, we carefully describe the universal property of determinization in terms of forward-backward simulations. Secondly, we generalize the determinization procedure for extbf{Rel} automata using a local adjunction between extbf{SpanSet} and extbf{Rel}, which provides us with a canonical forward simulation. Finally we also propose an alterative determinization based on the multiset relative adjunction which retains paths, and we leverage this to provide a canonical forward-backward simulation.