Abstract
Using a novel rewriting problem, we show that several natural decision problems about finite automata are undecidable (i.e., recursively unsolvable). In contrast, we also prove three related problems are decidable. We apply one result to prove the undecidability of a related problem about k-automatic sets of rational numbers.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Alur, R., Deshmukh, J.V.: Nondeterministic streaming string transducers. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011. LNCS, vol. 6756, pp. 1–20. Springer, Heidelberg (2011). doi:10.1007/978-3-642-22012-8_1
Bar-Hillel, Y., Perles, M., Shamir, E.: On formal properties of simple phrase structure grammars. Z. Phonetik. Sprachwiss. Kommuniationsforsch. 14, 143–172 (1961)
Book, R.V., Otto, F.: String-Rewriting Systems. Springer, New York (1993). doi:10.1007/978-1-4613-9771-7
Cobham, A.: Uniform tag sequences. Math. Syst. Theor. 6, 164–192 (1972)
Engelfriet, J., Rozenberg, G.: Fixed point languages, equality languages, and representation of recursively enumerable languages. J. Assoc. Comput. Mach. 27, 499–518 (1980)
Ginsburg, S., Rose, G.F.: Some recursively unsolvable problems in ALGOL-like languages. J. Assoc. Comput. Mach. 10, 29–47 (1963)
Hoogeboom, H.J.: Are there undecidable properties of non-turing-complete automata? Posting on stackexchange, 20 October 2012. http://cs.stackexchange.com/questions/1697/are-there-undecidable-properties-of-non-turing-complete-automata
Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, Reading (1979)
Lyndon, R.C., Schützenberger, M.P.: The equation \(a^M = b^N c^P\) in a free group. Mich. Math. J. 9, 289–298 (1962)
Post, E.: Absolutely unsolvable problems and relatively undecidable propositions: account of an anticipation. In: Davis, M. (ed.) The Undecidable, pp. 338–433. Raven Press, Hewlett (1965)
Rowland, E., Shallit, J.: Automatic sets of rational numbers. Int. J. Found. Comput. Sci. 26, 343–365 (2015)
Rozenberg, G., Salomaa, A.: Cornerstones of Undecidability. Prentice-Hall, New York (1994)
Schaeffer, L., Shallit, J.: The critical exponent is computable for automatic sequences. Int. J. Found. Comput. Sci. 23, 1611–1626 (2012)
Shallit, J.: A Second Course in Formal Languages and Automata Theory. Cambridge University Press, Cambridge (2009)
Shallit, J.O.: Numeration systems, linear recurrences, and regular sets. Inf. Comput. 113, 331–347 (1994)
Turing, A.M.: On computable numbers, with an application to the Entscheidungsproblem. Proc. Lond. Math. Soc. 42, 230–265 (1936)
Acknowledgments
We thank Hendrik Jan Hoogeboom and the referees for their helpful comments.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Endrullis, J., Shallit, J., Smith, T. (2017). Undecidability and Finite Automata. In: Charlier, É., Leroy, J., Rigo, M. (eds) Developments in Language Theory. DLT 2017. Lecture Notes in Computer Science(), vol 10396. Springer, Cham. https://doi.org/10.1007/978-3-319-62809-7_11
Download citation
DOI: https://doi.org/10.1007/978-3-319-62809-7_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-62808-0
Online ISBN: 978-3-319-62809-7
eBook Packages: Computer ScienceComputer Science (R0)