Abstract
Although regular expressions do not correspond univocally to regular languages, it is still worthwhile to study their properties and algorithms. For the average case analysis one often relies on the uniform random generation using a specific grammar for regular expressions, that can represent regular languages with more or less redundancy. Generators that are uniform on the set of expressions are not necessarily uniform on the set of regular languages. Nevertheless, it is not straightforward that asymptotic estimates obtained by considering the whole set of regular expressions are different from those obtained using a more refined set that avoids some large class of equivalent expressions. In this paper we study a set of expressions that avoid a given absorbing pattern. It is shown that, although this set is significantly smaller than the standard one, the asymptotic average estimates for the size of the Glushkov automaton for these expressions does not differ from the standard case.
This work was partially supported by CMUP, through FCT - Fundação para a Ciência e a Tecnologia, I.P., under the project with reference UIDB/00144/2020.
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Notes
- 1.
It is actually true that \(|\varDelta _k(z)-p_k(z)| < |p_k(z)|\) for all \(|z| = \frac{1}{\sqrt{8+8k}}\) and \(k\ge 2\).
- 2.
These polynomials are quite large, e.g. \(q_k\) has 437 monomials and degree \(10+28k\).
References
Broda, S., Machiavelo, A., Moreira, N., Reis, R.: On the average size of Glushkov and partial derivative automata. Int. J. Found. Comput. Sci. 23(5), 969–984 (2012)
Broda, S., Machiavelo, A., Moreira, N., Reis, R.: On average behaviour of regular expressions in strong star normal form. Int. J. Found. Comput. Sci. 30(6–7), 899–920 (2019)
Broda, S., Machiavelo, A., Moreira, N., Reis, R.: Analytic combinatorics and descriptional complexity of regular languages on average. ACM SIGACT News 51(1), 38–56 (2020)
Buchberger, B.: Gröbner bases: a short introduction for systems theorists. In: Moreno-Díaz, R., Buchberger, B., Luis Freire, J. (eds.) EUROCAST 2001. LNCS, vol. 2178, pp. 1–19. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-45654-6_1
Flajolet, P., Sedgewick, R.: Analytic Combinatorics. CUP (2008)
Glushkov, V.M.: The abstract theory of automata. Russ. Math. Surv. 16(5), 1–53 (1961)
Koechlin, F., Nicaud, C., Rotondo, P.: Uniform random expressions lack expressivity. In: Rossmanith, P., Heggernes, P., Katoen, J. (eds.) 44th MFCS 2019, vol. 138, pp. 51:1–51:14. LIPIcs (2019)
Koechlin, F., Nicaud, C., Rotondo, P.: On the degeneracy of random expressions specified by systems of combinatorial equations. In: Jonoska, N., Savchuk, D. (eds.) DLT 2020. LNCS, vol. 12086, pp. 164–177. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-48516-0_13
Nicaud, C.: On the average size of Glushkov’s automata. In: Dediu, A.H., Ionescu, A.M., Martín-Vide, C. (eds.) LATA 2009. LNCS, vol. 5457, pp. 626–637. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-00982-2_53
Nicaud, C.: Random deterministic automata. In: Csuhaj-Varjú, E., Dietzfelbinger, M., Ésik, Z. (eds.) MFCS 2014. LNCS, vol. 8634, pp. 5–23. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-44522-8_2
Simon, B.: Basic Complex Analysis, vol. 2A. American Mathematical Society, Providence (2015)
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Broda, S., Machiavelo, A., Moreira, N., Reis, R. (2021). On the Uniform Distribution of Regular Expressions. In: Han, YS., Ko, SK. (eds) Descriptional Complexity of Formal Systems. DCFS 2021. Lecture Notes in Computer Science(), vol 13037. Springer, Cham. https://doi.org/10.1007/978-3-030-93489-7_2
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