Abstract
We study the grammatical production complexity of finite languages w.r.t. (i) different interpretations of approximation, i.e., equivalence, cover, and scattered cover, and (ii) whether the underlying grammar generates a finite or infinite language. In case the generated language is infinite, the intersection with all words up to a certain length has to be considered in order to obtain the finite language under consideration. In this way, we obtain six different measures for regular, linear context-free, and context-free grammars. We compare these measures according to the taxonomy introduced in [J. Dassow, Gh. Păun: Regulated Rewriting in Formal Language Theory, 1989] with each other by fixing the grammar type and varying the complexity measure and the other way around, that is, by fixing the complexity measure and varying the grammar type. In both of these cases, we develop an almost complete picture, which gives new and interesting insights into the old topic of grammatical production complexity.
This research was completed while the author was on leave at the Institut für Informatik, Universität Giessen, Arndtstr. 2, 35392 Giessen, Germany, in 2017 and is supported by the Vienna Science Fund (WWTF) project VRG12-004.
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- 1.
Observe that it is common in the literature to refer to the \(\mathsf {DFA}\) and the \(\mathsf {CFA}\) state complexity as \(\mathsf {sc}\) and \(\mathsf {csc}\), respectively. We adapted the notation in order to be consistent with the notation used throughout this paper.
- 2.
A regular grammar \(G = (N,\varSigma ,P,S)\) is in 2-normal form if all productions in P are of the form \(A\rightarrow a\) and \(A\rightarrow aB\), where \(A,B\in N\) and \(a\in \varSigma \cup \{\varepsilon \}\).
- 3.
A context-free grammar \(G = (N,\varSigma ,P,S)\) is in 2-Greibach normal form if all productions in P are of the form \(A\rightarrow a\), \(A\rightarrow aB\), \(A\rightarrow aBC\), or \(S \rightarrow \varepsilon \), where \(A\in N\), \(a\in \varSigma \), and \(B,C\in N\setminus \{S\}\). The transformation increases the number of productions by at most a polynomial of fourth degree [1].
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Holzer, M., Wolfsteiner, S. (2018). On the Grammatical 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_13
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DOI: https://doi.org/10.1007/978-3-319-94631-3_13
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