Skip to main content
SpringerLink
Log in
Book cover

International Conference on Developments in Language Theory

DLT 2017: Developments in Language Theory pp 308–319Cite as

Limited Automata and Unary Languages

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10396))

Abstract

Limited automata are one-tape Turing machines that are allowed to rewrite the content of any tape cell only in the first d visits, for a fixed constant d. When \(d=1\) these models characterize regular languages. An exponential gap between the size of limited automata accepting unary languages and the size of equivalent finite automata is proved. Since a similar gap was already known from unary context-free grammars to finite automata, also the conversion of such grammars into limited automata is investigated. It is proved that from each unary context-free grammar it is possible to obtain an equivalent 1-limited automaton whose description has a size which is polynomial in the size of the grammar. Furthermore, despite the exponential gap between the sizes of limited automata and of equivalent unary finite automata, there are unary regular languages for which d-limited automata cannot be significantly smaller than equivalent finite automata, for any arbitrarily large d.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    A language L is local if there exist sets \(\mathcal {A}\subseteq \varSigma \times \varSigma \), \(\mathcal {I}\subseteq \varSigma \), and \(\mathcal {F}\subseteq \varSigma \) such that \(w \in L\) if and only if all factors of length 2 in w belong to \(\mathcal {A}\) and the first and the last symbols of w belong to \(\mathcal {I}\) and \(\mathcal {F}\), respectively [6].

References

  1. Ginsburg, S., Rice, H.G.: Two families of languages related to ALGOL. J. ACM 9(3), 350–371 (1962). http://doi.acm.org/10.1145/321127.321132

    Article  MathSciNet  MATH  Google Scholar 

  2. Hibbard, T.N.: A generalization of context-free determinism. Inf. Control 11(1/2), 196–238 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  3. Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages and Computation. Addison-Wesley, Reading (1979)

    MATH  Google Scholar 

  4. Kutrib, M., Pighizzini, G., Wendlandt, M.: Descriptional complexity of limited automata. Inf. Comput. (to appear)

    Google Scholar 

  5. Kutrib, M., Wendlandt, M.: On simulation cost of unary limited automata. In: Shallit, J., Okhotin, A. (eds.) DCFS 2015. LNCS, vol. 9118, pp. 153–164. Springer, Cham (2015). doi:10.1007/978-3-319-19225-3_13

    Chapter  Google Scholar 

  6. McNaughton, R., Papert, S.A.: Counter-Free Automata. M.I.T. Research Monograph, vol. 65. The MIT Press, Cambridge (1971)

    MATH  Google Scholar 

  7. Mereghetti, C., Pighizzini, G.: Two-way automata simulations and unary languages. J. Autom. Lang. Comb. 5(3), 287–300 (2000)

    MathSciNet  MATH  Google Scholar 

  8. Okhotin, A.: Non-erasing variants of the Chomsky–Schützenberger theorem. In: Yen, H.-C., Ibarra, O.H. (eds.) DLT 2012. LNCS, vol. 7410, pp. 121–129. Springer, Heidelberg (2012). doi:10.1007/978-3-642-31653-1_12

    Chapter  Google Scholar 

  9. Pighizzini, G., Shallit, J., Wang, M.: Unary context-free grammars and pushdown automata, descriptional complexity and auxiliary space lower bounds. J. Comput. Syst. Sci. 65(2), 393–414 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Pighizzini, G.: Strongly limited automata. Fundam. Inform. 148(3–4), 369–392 (2016). http://dx.doi.org/10.3233/FI-2016-1439

    MathSciNet  Google Scholar 

  11. Pighizzini, G., Pisoni, A.: Limited automata and regular languages. Int. J. Found. Comput. Sci. 25(7), 897–916 (2014). http://dx.doi.org/10.1142/S0129054114400140

    Article  MathSciNet  MATH  Google Scholar 

  12. Pighizzini, G., Pisoni, A.: Limited automata and context-free languages. Fundam. Inf. 136(1–2), 157–176 (2015). http://dx.doi.org/10.3233/FI-2015-1148

    MathSciNet  MATH  Google Scholar 

  13. Sloane, N.J.A.: The on-line encyclopedia of integer sequences. http://oeis.org/A007814

  14. Wagner, K.W., Wechsung, G.: Computational Complexity. D. Reidel Publishing Company, Dordrecht (1986)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Informatica, Università degli Studi di Milano, Milan, Italy

    Giovanni Pighizzini & Luca Prigioniero

Authors
  1. Giovanni Pighizzini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Luca Prigioniero
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Giovanni Pighizzini .

Editor information

Editors and Affiliations

  1. Université de Liège, Liège, Belgium

    Émilie Charlier

  2. Université de Liège, Liège, Belgium

    Julien Leroy

  3. Université de Liège, Liège, Belgium

    Michel Rigo

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Pighizzini, G., Prigioniero, L. (2017). Limited Automata and Unary Languages. 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_23

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-62809-7_23

  • 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)

Publish with us

Policies and ethics

Search

Navigation