Skip to main content
SpringerLink
Log in

Two-Way Automata and One-Tape Machines

Read Only Versus Linear Time

  • Conference paper
  • First Online:
Developments in Language Theory (DLT 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11088))

Included in the following conference series:

Abstract

It is well-known that one-tape Turing machines working in linear time are no more powerful than finite automata, namely they recognize exactly the class of regular languages. We study the costs, in terms of description sizes, of the conversion of nondeterministic finite automata into equivalent linear-time one-tape deterministic machines. We prove a polynomial blowup from two-way nondeterministic finite automata into equivalent weight-reducing one-tape deterministic machines that work in linear time. The blowup remains polynomial if the tape in the resulting machines is restricted to the portion which initially contains the input. However, in this case the machines resulting from our construction are not weight reducing, unless the input alphabet is unary.

D. Průša was supported by the Czech Science Foundation, grant 16-05872S.

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

Access this chapter

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

Institutional subscriptions

Notes

  1. 1.

    Actually, the model considered by Hennie was deterministic. Several extensions of this result, including that to the nondeterministic case and greater time lower bounds for nonregular-language recognition, have been stated in literature [4, 11, 12, 20, 21].

  2. 2.

    For the sake of completeness, we mention that it is decidable whether or not a machine makes at most \(cm+d\) steps on input of length m, for any fixed \(c,d>0\) [1].

References

  1. Gajser, D.: Verifying time complexity of turing machines. Theor. Comput. Sci. 600, 86–97 (2015)

    Article  MathSciNet  Google Scholar 

  2. Geffert, V., Guillon, B., Pighizzini, G.: Two-way automata making choices only at the endmarkers. Inf. Comp. 239, 71–86 (2014)

    Article  MathSciNet  Google Scholar 

  3. Geffert, V., Mereghetti, C., Pighizzini, G.: Converting two-way nondeterministic unary automata into simpler automata. Theor. Comput. Sci. 295(1–3), 189–203 (2003)

    Article  MathSciNet  Google Scholar 

  4. Hartmanis, J.: Computational complexity of one-tape Turing machine computations. J. ACM 15(2), 325–339 (1968)

    Article  MathSciNet  Google Scholar 

  5. Hennie, F.C.: One-tape, off-line Turing machine computations. Inf. Control 8(6), 553–578 (1965)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  7. Hromkovič, J., Schnitger, G.: Nondeterminism versus determinism for two-way finite automata: generalizations of Sipser’s separation. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 439–451. Springer, Heidelberg (2003). https://doi.org/10.1007/3-540-45061-0_36

    Chapter  MATH  Google Scholar 

  8. Kapoutsis, C.A.: Nondeterminism is essential in small two-way finite automata with few reversals. Inf. Comput. 222, 208–227 (2013)

    Article  MathSciNet  Google Scholar 

  9. Kapoutsis, C.A.: Two-way automata versus logarithmic space. Theory Comput. Syst. 55(2), 421–447 (2014)

    Article  MathSciNet  Google Scholar 

  10. Kuroda, S.: Classes of languages and linear-bounded automata. Inf. Control 7(2), 207–223 (1964)

    Article  MathSciNet  Google Scholar 

  11. Michel, P.: An NP-complete language accepted in linear time by a one-tape Turing machine. Theor. Comput. Sci. 85(1), 205–212 (1991)

    Article  MathSciNet  Google Scholar 

  12. Pighizzini, G.: Nondeterministic one-tape off-line Turing machines. J. Autom. Lang. Comb. 14(1), 107–124 (2009)

    MathSciNet  MATH  Google Scholar 

  13. Pighizzini, G.: Two-way finite automata: old and recent results. Fundam. Inform. 126(2–3), 225–246 (2013)

    MathSciNet  MATH  Google Scholar 

  14. Průša, D.: Weight-reducing Hennie machines and their descriptional complexity. In: Dediu, A.-H., Martín-Vide, C., Sierra-Rodríguez, J.-L., Truthe, B. (eds.) LATA 2014. LNCS, vol. 8370, pp. 553–564. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-04921-2_45

    Chapter  MATH  Google Scholar 

  15. Rabin, M.O., Scott, D.S.: Finite automata and their decision problems. IBM J. Res. Dev. 3(2), 114–125 (1959)

    Article  MathSciNet  Google Scholar 

  16. Sakoda, W.J., Sipser, M.: Nondeterminism and the size of two way finite automata. In: STOC 1978, pp. 275–286 (1978)

    Google Scholar 

  17. Savitch, W.J.: Relationships between nondeterministic and deterministic tape complexities. J. Comput. Syst. Sci. 4(2), 177–192 (1970)

    Article  MathSciNet  Google Scholar 

  18. Shepherdson, J.C.: The reduction of two-way automata to one-way automata. IBM J. Res. Dev. 3(2), 198–200 (1959)

    Article  MathSciNet  Google Scholar 

  19. Sipser, M.: Lower bounds on the size of sweeping automata. J. Comput. Syst. Sci. 21(2), 195–202 (1980)

    Article  MathSciNet  Google Scholar 

  20. Tadaki, K., Yamakami, T., Lin, J.C.H.: Theory of one-tape linear-time Turing machines. Theor. Comput. Sci. 411(1), 22–43 (2010)

    Article  MathSciNet  Google Scholar 

  21. Trakhtenbrot, B.A.: Turing machine computations with logarithmic delay. Algebra I Log. 3, 33–48 (1964). (in Russian)

    Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Bruno Guillon, Giovanni Pighizzini & Luca Prigioniero

  2. Faculty of Electrical Engineering, Czech Technical University, Prague, Czech Republic

    Daniel Průša

Authors
  1. Bruno Guillon
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Giovanni Pighizzini
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Luca Prigioniero
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Daniel Průša
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Luca Prigioniero .

Editor information

Editors and Affiliations

  1. National Archives of Japan, Tokyo, Japan

    Mizuho Hoshi

  2. The University of Electro-Communications, Tokyo, Japan

    Shinnosuke Seki

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Guillon, B., Pighizzini, G., Prigioniero, L., Průša, D. (2018). Two-Way Automata and One-Tape Machines. In: Hoshi, M., Seki, S. (eds) Developments in Language Theory. DLT 2018. Lecture Notes in Computer Science(), vol 11088. Springer, Cham. https://doi.org/10.1007/978-3-319-98654-8_30

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-98654-8_30

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-98653-1

  • Online ISBN: 978-3-319-98654-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation