Abstract
Families of DFAs (FDFAs) have recently been introduced as a new representation of \(\omega \)-regular languages. They target ultimately periodic words, with acceptors revolving around accepting some representation \(u\cdot v^\omega \). Three canonical FDFAs have been suggested, called periodic, syntactic, and recurrent. We propose a fourth one, limit FDFAs, which can be exponentially coarser than periodic FDFAs and are more succinct than syntactic FDFAs, while they are incomparable (and dual to) recurrent FDFAs. We show that limit FDFAs can be easily used to check not only whether \(\omega \)-languages are regular, but also whether they are accepted by deterministic Büchi automata. We also show that canonical forms can be left behind in applications: the limit and recurrent FDFAs can complement each other nicely, and it may be a good way forward to use a combination of both. Using this observation as a starting point, we explore making more efficient use of Myhill-Nerode’s right congruences in aggressively increasing the number of don’t-care cases in order to obtain smaller progress automata. In pursuit of this goal, we gain succinctness, but pay a high price by losing constructiveness.
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Notes
- 1.
- 2.
This enables to learn L via learning the regular language \(L_{\$}\) [9].
- 3.
Minimizing DFAs with don’t care words is NP-complete [20].
- 4.
Defining directly a progress RC \(\approx ^u\) that recognizes \(V_u\) is hard since \(V_u\) is quantified over all v-extensions.
- 5.
In the language \(L = a^{\omega } + ab^{\omega }\) from the example of Fig. 1, for example, we have \(a \approx ^{ab}_N \epsilon \) and \(a \approx ^{ab}_N b\), but \(b \not \approx ^{ab}_N \epsilon \).
- 6.
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Acknowledgements
We thank the anonymous reviewers for their valuable feedback. This work has been supported by the EPSRC through grants EP/X021513/1 and EP/X017796/1.
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Li, Y., Schewe, S., Tang, Q. (2023). A Novel Family of Finite Automata for Recognizing and Learning \(\omega \)-Regular Languages. In: André, É., Sun, J. (eds) Automated Technology for Verification and Analysis. ATVA 2023. Lecture Notes in Computer Science, vol 14215. Springer, Cham. https://doi.org/10.1007/978-3-031-45329-8_3
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