Abstract
We compare pushdown automata (PDAs for short) against other representations. First, we show that there is a family of PDAs over a unary alphabet with \(n\) states and \(p \ge 2n + 4\) stack symbols that accepts one single long word for which every equivalent context-free grammar needs \(\varOmega (n^2(p-2n-4))\) variables. This family shows that the classical algorithm for converting a PDA into an equivalent context-free grammar is optimal even when the alphabet is unary. Moreover, we observe that language equivalence and Parikh equivalence, which ignores the ordering between symbols, coincide for this family. We conclude that, when assuming this weaker equivalence, the conversion algorithm is also optimal. Second, Parikh’s theorem motivates the comparison of PDAs against finite state automata. In particular, the same family of unary PDAs gives a lower bound on the number of states of every Parikh-equivalent finite state automaton. Finally, we look into the case of unary deterministic PDAs. We show a new construction converting a unary deterministic PDA into an equivalent context-free grammar that achieves best known bounds.
P. Ganty—has been supported by the Madrid Regional Government project S2013/ICE-2731, N-Greens Software - Next-GeneRation Energy-EfficieNt Secure Software, and the Spanish Ministry of Economy and Competitiveness project No. TIN2015-71819-P, RISCO - RIgorous analysis of Sophisticated COncurrent and distributed systems.
E. Gutiérrez—is partially supported by BES-2016-077136 grant from the Spanish Ministry of Economy, Industry and Competitiveness.
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Notes
- 1.
Their family has an alphabet of non-constant size.
- 2.
- 3.
But not necessarily at the same positions, e.g. \(ab \) and \(ba\) are Parikh-equivalent.
- 4.
- 5.
\( (w)_i \) is the \(i\)-th symbol of \(w\) if \(1\le i \le {\vert {w}\vert }\); else \( (w)_i=\varepsilon \). \({\vert {w}\vert }\) is the length of \(w\).
- 6.
When \(b=\varepsilon \) the move does not consume input.
- 7.
Note that if \(n \le Ck\) for some \(C>0\) then the \(n^3\) addend in \(\mathcal {O}(n^2k + n^3)\) becomes negligible compared to \(n^2k\), and the lower and upper bound coincide.
- 8.
The set of final states is given by \(F\subseteq Q\).
- 9.
As the blow up of our construction is \(\mathcal {O}(4^{n^2(k + 2n + 4)})\) for a lower bound of \(2^{n^2k}\), we say that it is close to optimal in the sense that \(2n^2(k + 2n + 4) \in \varTheta (n^2k)\), which holds when \(n\) is in linear relation with respect to \(k\) (see Remark 10).
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Acknowledgement
We thank Pedro Valero for pointing out the reference on smallest grammar problems [2]. We also thank the anonymous referees for their insightful comments and suggestions.
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Ganty, P., Gutiérrez, E. (2017). Parikh Image of Pushdown Automata. In: Klasing, R., Zeitoun, M. (eds) Fundamentals of Computation Theory. FCT 2017. Lecture Notes in Computer Science(), vol 10472. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55751-8_22
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