Converting Nondeterministic Automata and Context-Free Grammars into Parikh Equivalent One-Way and Two-Way Deterministic Automata

TL;DR

This paper explores converting nondeterministic automata and context-free grammars into Parikh equivalent deterministic automata, establishing tight bounds on the state complexity for such conversions.

Contribution

It provides tight bounds on the state complexity of converting nondeterministic automata and grammars into Parikh equivalent deterministic automata, including special cases with polynomial bounds.

Findings

Deterministic automata with exponential and polynomial states exist for given nondeterministic automata.

For automata accepting words with at least two different letters, polynomial state deterministic automata are possible.

Bounds for context-free grammars are tight, with exponential states in terms of variables.

Abstract

We investigate the conversion of one-way nondeterministic finite automata and context-free grammars into Parikh equivalent one-way and two-way deterministic finite automata, from a descriptional complexity point of view. We prove that for each one-way nondeterministic automaton with $n$ states there exist Parikh equivalent one-way and two-way deterministic automata with $e^{O(\sqrt{n \ln n})}$ and $p(n)$ states, respectively, where $p(n)$ is a polynomial. Furthermore, these costs are tight. In contrast, if all the words accepted by the given automaton contain at least two different letters, then a Parikh equivalent one-way deterministic automaton with a polynomial number of states can be found. Concerning context-free grammars, we prove that for each grammar in Chomsky normal form with h variables there exist Parikh equivalent one-way and two-way deterministic automata with $2^{O(h^2)}$ and $2^{O(h)}$ states, respectively. Even these bounds are tight.