The paper establishes a Myhill-Nerode type theorem for higher-dimensional automata (HDAs), stating that a language is regular if and only if it has finite prefix quotient. HDAs extend standard automata with additional structure to distinguish between interleavings and concurrency.
The key highlights and insights are:
The authors introduce a stronger equivalence relation "strong equivalence" (≈L) on ipomsets, which takes concurrency of events into account, in contrast to the standard "weak equivalence" (∼L) used in the classical Myhill-Nerode theorem.
They construct an HDA MN(L) whose cells are ≈L-equivalence classes of ipomsets, and show that MN(L) recognizes the language L. If the prefix quotient of L is finite, then the essential part of MN(L) is also finite.
The authors show that there exist regular languages that cannot be recognized by deterministic HDAs, in contrast to the classical Myhill-Nerode theorem for finite automata. They develop a language-internal characterization of deterministic languages.
The paper also develops analogues of the Myhill-Nerode construction and of determinism for HDAs with interfaces (iHDAs), which allow for more principled handling of non-accessible parts.
Overall, the paper extends the classical Myhill-Nerode theorem to the higher-dimensional setting of HDAs, highlighting the additional challenges and insights that arise from the richer structure of HDAs compared to standard automata.
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by Uli ... às arxiv.org 04-22-2024
https://arxiv.org/pdf/2210.08298.pdfPerguntas Mais Profundas