Abstract
The space of one-sided infinite words plays a crucial rôle in several parts of Theoretical Computer Science. Usually, it is convenient to regard this space as a metric space, the Cantor-space. It turned out that for several purposes topologies other than the one of the Cantor-space are useful, e.g. for studying fragments of first-order logic over infinite words or for a topological characterisation of random infinite words.
Continuing the work of [14], here we consider two different refinements of the Cantor-space, given by measuring common factors, and common factors occurring infinitely often. In particular we investigate the relation of these topologies to the sets of infinite words definable by finite automata, that is, to regular \(\omega \)-languages.
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Notes
- 1.
Observe that the relation \(\sim _{P}\) defined by \(w \sim _{P} v\) iff \(P/w=P/v\) is the Nerode right congruence of P.
- 2.
It is convenient to choose \(r=|X|\). Then every ball of radius \(r^{-n}\) is partitioned into exactly r balls of radius \(r^{-(n+1)}\).
- 3.
Observe that \(e\notin {\mathbf {pref}(\xi )}\,\mathsf {\Delta }\,{\mathbf {pref}(\eta )}\) and Eq. (1) imply \(\rho (\xi ,\eta ) = \inf \{r^{-|w|}: w\sqsubset \xi \wedge w\sqsubset \eta \}\).
- 4.
Those sequences are usually referred to as Cauchy sequences.
- 5.
They are also closed balls of radius \(r^{-(n+1)}\).
- 6.
A point \(\xi \) is referred to as isolated if \(\rho '(\xi , \eta )\ge \epsilon _{\xi }\) for all \(\eta \ne \xi \). Here the distance \(\epsilon _{\xi }>0\) may depend on \(\xi \).
- 7.
In particular, they satisfy \(F^{(\tau )}_{\gamma }/w=F^{(\tau )}_{\gamma } \) for all \(w\in X^{*}\).
References
Büchi, J.R.: On a decision method in restricted second order arithmetic. In: Proceedings of the 1960 International Congress for Logic, pp. 1–11. Stanford Univ. Press, Stanford (1962)
Calude, C.S., Marcus, S., Staiger, L.: A topological characterization of random sequences. Inform. Process. Lett. 88, 245–250 (2003)
Calude, C.S., Jürgensen, H., Staiger, L.: Topology on words. Theoret. Comput. Sci. 410, 2323–2335 (2009)
Diekert, V., Kufleitner, M.: Fragments of first-order logic over infinite words. In: Albers, S., Marion, J.-Y. (eds.) Proceedings of the STACS 2009, pp. 325–336. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2009)
Engelking, R.: General Topology. Państwowe wydawnictwo naukowe, Warszawa (1977)
Fernau, H., Staiger, L.: Iterated function systems and control languages. Inform. Comput. 168, 125–143 (2001)
Hoffmann, S.: Metriken zur Verfeinerung des Cantor-Raumes auf \(X^{\omega }\). Diploma thesis, Martin-Luther-Universität Halle-Wittenberg (2014)
Landweber, L.H.: Decision problems for \(\omega \)-automata. Math. Syst. Theory 3, 376–384 (1969)
Lothaire, M.: Algebraic Combinatorics on Words. Cambridge University Press, Cambridge (2002)
McNaughton, R.: Testing and generating infinite sequences by a finite automaton. Inform. Control 9, 521–530 (1966)
Perrin, D., Pin, J.-E.: Infinite Words. Elsevier, Amsterdam (2004)
Redziejowski, R.R.: Infinite word languages and continuous mappings. Theoret. Comput. Sci. 43, 59–79 (1986)
Rozenberg, G., Salomaa, A. (eds.): Handbook of Formal Languages. Springer, Berlin (1997)
Schwarz, S., Staiger, L.: Topologies refining the Cantor topology on \(X^{\omega }\). In: Calude, C.S., Sassone, V. (eds.) Theoretical Computer Science. IFIP, vol. 323, pp. 271–285. Springer, Berlin (2010)
Staiger, L.: Finite-state \(\omega \)-languages. J. Comput. Syst. Sci. 27, 434–448 (1983)
Staiger, L.: Sequential mappings of \(\omega \)-languages. ITA 21, 147–173 (1987)
Staiger, L.: Kolmogorov complexity and Hausdorff dimension. Inf. Comput. 103, 159–194 (1993)
Staiger, L.: \(\omega \)-languages. In: [13], vol. 3, pp. 339–387
Staiger, L.: Weighted finite automata and metrics in Cantor Space. J. Automata Lang. Comb. 8, 353–360 (2003)
Staiger, L.: Topologies for the set of disjunctive \(\omega \)-words. Acta Cybern. 17, 43–51 (2005)
Staiger, L.: Asymptotic subword complexity. In: Bordihn, H., Kutrib, M., Truthe, B. (eds.) Languages Alive. LNCS, vol. 7300, pp. 236–245. Springer, Heidelberg (2012)
Staiger, L., Wagner, K.: Automatentheoretische und automatenfreie Charakterisierungen topologischer Klassen regulärer Folgenmengen. Elektronische Informationsverarbeitung und Kybernetik 10, 379–392 (1974)
Thomas, W.: Automata on infinite objects. In: Van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, vol. B, pp. 133–191. Elsevier, Amsterdam (1990)
Thomas, W.: Languages, automata, and logic. In: [13], vol. 3, pp. 389–455
Trakhtenbrot, B.A.: Finite automata and monadic second order logic. Sibirsk. Mat. Ž. 3, 103–131 (1962). (Russian; English translation: AMS Transl. 59, 23–55, (1966))
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Hoffmann, S., Staiger, L. (2015). Subword Metrics for Infinite Words. In: Drewes, F. (eds) Implementation and Application of Automata. CIAA 2015. Lecture Notes in Computer Science(), vol 9223. Springer, Cham. https://doi.org/10.1007/978-3-319-22360-5_14
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