Skip to main content
SpringerLink
Log in

Compressibility of Finite Languages by Grammars

  • Conference paper
  • First Online:
Book cover Descriptional Complexity of Formal Systems (DCFS 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9118))

Included in the following conference series:

Abstract

We consider the problem of simultaneously compressing a finite set of words by a single grammar. The central result of this paper is the construction of an incompressible sequence of finite word languages. This result is then shown to transfer to tree languages and (via a previously established connection between proof theory and formal language theory) also to formal proofs in first-order predicate logic.

Research supported by the Vienna Science Fund (WWTF) project VRG12-04.

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

References

  1. Buss, S.: An Introduction to proof theory. In: Buss, S. (ed.) The Handbook of Proof Theory, pp. 2–78. Elsevier, Amsterdam (1998)

    Google Scholar 

  2. Buss, S.R.: On Herbrand’s theorem. In: Leivant, D. (ed.) LCC’94. LNCS, vol. 960, pp. 195–209. Springer, Heidelberg (1995)

    Chapter  Google Scholar 

  3. Câmpeanu, C., Sântean, N., Yu, S.: Minimal cover-automata for finite languages. In: Champarnaud, J.-M., Maurel, D., Ziadi, D. (eds.) WIA 1998. LNCS, vol. 1660, pp. 43–56. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  4. Câmpeanu, C., Santean, N., Yu, S.: Minimal cover-automata for finite languages. Theoret. Comput. Sci. 267(1–2), 3–16 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  5. Charikar, M., Lehman, E., Liu, D., Panigrahy, R., Prabhakaran, M., Sahai, A., Shelat, A.: The smallest grammar problem. IEEE Trans. Inf. Theory 51(7), 2554–2576 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  6. Herbrand, J.: Recherches sur la théorie de la démonstration. Ph.D. thesis, Université de Paris (1930)

    Google Scholar 

  7. Hetzl, S.: Applying tree languages in proof theory. In: Dediu, A.-H., Martín-Vide, C. (eds.) LATA 2012. LNCS, vol. 7183, pp. 301–312. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  8. Hetzl, S., Leitsch, A., Reis, G., Tapolczai, J., Weller, D.: Introducing quantified cuts in logic with equality. In: Demri, S., Kapur, D., Weidenbach, C. (eds.) IJCAR 2014. LNCS, vol. 8562, pp. 240–254. Springer, Heidelberg (2014)

    Google Scholar 

  9. Hetzl, S., Leitsch, A., Reis, G., Weller, D.: Algorithmic introduction of quantified cuts. Theoret. Comput. Sci. 549, 1–16 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  10. Hetzl, S., Leitsch, A., Weller, D.: Towards algorithmic cut-introduction. In: Bjørner, N., Voronkov, A. (eds.) LPAR-18 2012. LNCS, vol. 7180, pp. 228–242. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

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

    MATH  Google Scholar 

  12. Jacquemard, F., Klay, F., Vacher, C.: Rigid tree automata. In: Dediu, A.H., Ionescu, A.M., Martín-Vide, C. (eds.) LATA 2009. LNCS, vol. 5457, pp. 446–457. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  13. Jacquemard, F., Klay, F., Vacher, C.: Rigid tree automata and applications. Inf. Comput. 209, 486–512 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  14. Jeż, A.: Approximation of grammar-based compression via recompression. In: Fischer, J., Sanders, P. (eds.) CPM 2013. LNCS, vol. 7922, pp. 165–176. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  15. Jeż, A.: A really simple approximation of smallest grammar. In: Kulikov, A.S., Kuznetsov, S.O., Pevzner, P. (eds.) CPM 2014. LNCS, vol. 8486, pp. 182–191. Springer, Heidelberg (2014)

    Google Scholar 

  16. Kieffer, J.C., Yang, E.H.: Grammar-based codes: a new class of universal lossless source codes. IEEE Trans. Inf. Theory 46(3), 737–754 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  17. Larsson, N.J., Moffat, A.: Offline dictionary-based compression. In: Data Compression Conference (DCC 1999). pp. 296–305. IEEE Computer Society (1999)

    Google Scholar 

  18. Lohrey, M.: Algorithmics on SLP-compressed strings: a survey. Groups Complex. Cryptol. 4(2), 241–299 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  19. Nevill-Manning, C.G., Witten, I.H.: Identifying hierarchical structure in sequences: a linear-time algorithm. J. Artif. Intell. Res. 7, 67–82 (1997)

    MATH  Google Scholar 

  20. Orevkov, V.: Lower bounds for increasing complexity of derivations after cut elimination. Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta 88, 137–161 (1979)

    MATH  MathSciNet  Google Scholar 

  21. Pudlák, P.: The Lengths of proofs. In: Buss, S. (ed.) Handbook of Proof Theory, pp. 547–637. Elsevier, Amsterdam (1998)

    Chapter  Google Scholar 

  22. Pudlák, P.: Twelve problems in proof complexity. In: Hirsch, E.A., Razborov, A.A., Semenov, A., Slissenko, A. (eds.) Computer Science – Theory and Applications. LNCS, vol. 5010, pp. 13–27. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  23. Rytter, W.: Application of Lempel-Ziv factorization to the approximation of grammar-based compression. Theoret. Comput. Sci. 302(1–3), 211–222 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  24. Sakamoto, H.: A fully linear-time approximation algorithm for grammar-based compression. J. Discrete Algorithms 3(2–4), 416–430 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  25. Shallit, J., Breitbart, Y.: Automaticity I: properties of a measure of descriptional complexity. J. Comput. Syst. Sci. 53, 10–25 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  26. Shallit, J., Wang, M.W.: Automatic complexity of strings. J. Automata, Lang. Comb. 6(4), 537–554 (2001)

    MATH  MathSciNet  Google Scholar 

  27. Statman, R.: Lower bounds on Herbrand’s theorem. Proc. Am. Math. Soc. 75, 104–107 (1979)

    MATH  MathSciNet  Google Scholar 

  28. Storer, J.A., Szymanski, T.G.: The macro model for data compression (extended abstract). In: Proceedings of the Tenth Annual ACM Symposium on Theory of Computing (STOC ’78). pp. 30–39. ACM, New York (1978)

    Google Scholar 

Download references

Acknowledgments

The authors would like to thank Manfred Schmidt-Schauß for several helpful conversations about the topic of this paper, Werner Kuich for a number of remarks that improved the presentation of the results, and the anonymous reviewers for numerous important comments and suggestions.

Author information

Authors and Affiliations

  1. Institute of Discrete Mathematics and Geometry, Vienna University of Technology, Wiedner Hauptstraße 8-10, 1040, Wien, Austria

    Sebastian Eberhard & Stefan Hetzl

Authors
  1. Sebastian Eberhard
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stefan Hetzl
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stefan Hetzl .

Editor information

Editors and Affiliations

  1. School of Computer Science, University of Waterloo, Waterloo, Ontario, Canada

    Jeffrey Shallit

  2. Department of Mathematics, University of Turku, Turku, Finland

    Alexander Okhotin

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Eberhard, S., Hetzl, S. (2015). Compressibility of Finite Languages by Grammars. In: Shallit, J., Okhotin, A. (eds) Descriptional Complexity of Formal Systems. DCFS 2015. Lecture Notes in Computer Science(), vol 9118. Springer, Cham. https://doi.org/10.1007/978-3-319-19225-3_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-19225-3_8

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-19224-6

  • Online ISBN: 978-3-319-19225-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation