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Cover Complexity of Finite Languages

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Descriptional Complexity of Formal Systems (DCFS 2018)

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Abstract

We consider the notion of cover complexity of finite languages on three different levels of abstraction. For arbitrary cover complexity measures, we give a characterisation of the situations in which they collapse to a bounded complexity measure. Moreover, we show for a restricted class of context-free grammars that its grammatical cover complexity measure w.r.t. a finite language L is unbounded and that the cover complexity of L can be computed from the exact complexities of a finite number of covers \(L' \supseteq L\). We also investigate upper and lower bounds on the grammatical cover complexity of the language operations intersection, union, and concatenation on finite languages for several different types of context-free grammars.

Supported by the Vienna Science Fund (WWTF) project VRG12-004 and the Austrian Science Fund (FWF) project P25160.

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Notes

  1. 1.

    A context-free grammar \(G = (N,\varSigma ,P,S)\) is said to be in Chomsky normal form if all productions are of the form \(A \rightarrow BC\), \(A \rightarrow a\), or \(A \rightarrow \varepsilon \), where \(A,B,C \in N\) and \(a \in \varSigma \).

References

  1. Alspach, B., Eades, P., Rose, G.: A lower-bound for the number of productions required for a certain class of languages. Discrete Appl. Math. 6(2), 109–115 (1983)

    Article  MathSciNet  Google Scholar 

  2. Bucher, W.: A note on a problem in the theory of grammatical complexity. Theor. Comput. Sci. 14, 337–344 (1981)

    Article  MathSciNet  Google Scholar 

  3. Bucher, W., Maurer, H.A., Culik II, K.: Context-free complexity of finite languages. Theor. Comput. Sci. 28, 277–285 (1984)

    Article  MathSciNet  Google Scholar 

  4. Bucher, W., Maurer, H.A., Culik II, K., Wotschke, D.: Concise description of finite languages. Theor. Comput. Sci. 14, 227–246 (1981)

    Article  MathSciNet  Google Scholar 

  5. Câmpeanu, C., Sântean, N., Yu, S.: Minimal cover-automata for finite languages. In: Champarnaud, J.-M., Ziadi, D., Maurel, D. (eds.) WIA 1998. LNCS, vol. 1660, pp. 43–56. Springer, Heidelberg (1999). https://doi.org/10.1007/3-540-48057-9_4

    Chapter  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Černý, A.: Complexity and minimality of context-free grammars and languages. In: Gruska, J. (ed.) MFCS 1977. LNCS, vol. 53, pp. 263–271. Springer, Heidelberg (1977). https://doi.org/10.1007/3-540-08353-7_144

    Chapter  Google Scholar 

  8. Dassow, J.: Descriptional complexity and operations – two non-classical cases. In: Pighizzini, G., Câmpeanu, C. (eds.) DCFS 2017. LNCS, vol. 10316, pp. 33–44. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-60252-3_3

    Chapter  MATH  Google Scholar 

  9. Dassow, J., Harbich, R.: Production complexity of some operations on context-free languages. In: Kutrib, M., Moreira, N., Reis, R. (eds.) DCFS 2012. LNCS, vol. 7386, pp. 141–154. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-31623-4_11

    Chapter  MATH  Google Scholar 

  10. Eberhard, S., Hetzl, S.: Compressibility of finite languages by grammars. In: Shallit, J., Okhotin, A. (eds.) DCFS 2015. LNCS, vol. 9118, pp. 93–104. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-19225-3_8

    Chapter  MATH  Google Scholar 

  11. Eberhard, S., Hetzl, S.: On the compressibility of finite languages and formal proofs. Inf. Comput. 259, 191–213 (2018)

    Article  MathSciNet  Google Scholar 

  12. Gruska, J.: On a classification of context-free languages. Kybernetika 3(1), 22–29 (1967)

    MathSciNet  MATH  Google Scholar 

  13. Gruska, J.: Some classifications of context-free languages. Inf. Control 14(2), 152–179 (1969)

    Article  MathSciNet  Google Scholar 

  14. Gruska, J.: Complexity and unambiguity of context-free grammars and languages. Inf. Control 18(5), 502–519 (1971)

    Article  MathSciNet  Google Scholar 

  15. Gruska, J.: On the size of context-free grammars. Kybernetika 8(3), 213–218 (1972)

    MathSciNet  MATH  Google Scholar 

  16. 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). https://doi.org/10.1007/978-3-642-28332-1_26

    Chapter  MATH  Google Scholar 

  17. 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 (LNAI), vol. 8562, pp. 240–254. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-08587-6_17

    Chapter  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  19. Hetzl, S., Leitsch, A., Weller, D.: Towards algorithmic cut-introduction. In: Bjørner, N., Voronkov, A. (eds.) LPAR 2012. LNCS, vol. 7180, pp. 228–242. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-28717-6_19

    Chapter  Google Scholar 

  20. Pudlák, P.: Twelve problems in proof complexity. In: Hirsch, E.A., Razborov, A.A., Semenov, A., Slissenko, A. (eds.) CSR 2008. LNCS, vol. 5010, pp. 13–27. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-79709-8_4

    Chapter  Google Scholar 

  21. Tuza, Z.: On the context-free production complexity of finite languages. Discrete Appl. Math. 18(3), 293–304 (1987)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to thank Markus Holzer and the anonymous reviewers for several useful comments and suggestions concerning the results in this paper.

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Authors and Affiliations

  1. Institute of Discrete Mathematics and Geometry, TU Wien, Wiedner Hauptstraße 8–10, 1040, Wien, Austria

    Stefan Hetzl & Simon Wolfsteiner

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  1. Stefan Hetzl
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  2. Simon Wolfsteiner
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Correspondence to Simon Wolfsteiner .

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Editors and Affiliations

  1. Department of Mathematics and Computing Science, Saint Mary’s University, Halifax, Nova Scotia, Canada

    Stavros Konstantinidis

  2. Dipartimento di Informatica e Comunicazi, Universita degli Studi di Milano, Milan, Italy

    Giovanni Pighizzini

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Hetzl, S., Wolfsteiner, S. (2018). Cover Complexity of Finite Languages. In: Konstantinidis, S., Pighizzini, G. (eds) Descriptional Complexity of Formal Systems. DCFS 2018. Lecture Notes in Computer Science(), vol 10952. Springer, Cham. https://doi.org/10.1007/978-3-319-94631-3_12

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  • DOI: https://doi.org/10.1007/978-3-319-94631-3_12

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  • Online ISBN: 978-3-319-94631-3

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