Descriptional complexity of bounded context-free languages

TL;DR

This paper explores the descriptional complexity of finite-turn pushdown automata accepting bounded context-free languages, showing that certain trade-offs are recursive and providing algorithms with optimal bounds for converting and reducing turns.

Contribution

It introduces algorithms for converting finite-turn PDAs accepting bounded languages and establishes tight bounds, extending results from letter-bounded to word-bounded languages.

Findings

Non-recursive trade-off reduces to exponential for letter-bounded languages

Conversion algorithms achieve optimal bounds with tight lower bounds

Polynomial bounds for reducing the number of turns in finite-turn PDAs

Abstract

Finite-turn pushdown automata (PDA) are investigated concerning their descriptional complexity. It is known that they accept exactly the class of ultralinear context-free languages. Furthermore, the increase in size when converting arbitrary PDAs accepting ultralinear languages to finite-turn PDAs cannot be bounded by any recursive function. The latter phenomenon is known as non-recursive trade-off. In this paper, finite-turn PDAs accepting bounded languages are considered. First, letter-bounded languages are studied. We prove that in this case the non-recursive trade-off is reduced to a recursive trade-off, more precisely, to an exponential trade-off. A conversion algorithm is presented and the optimality of the construction is shown by proving tight lower bounds. Furthermore, the question of reducing the number of turns of a given finite-turn PDA is studied. Again, a conversion algorithm is provided which shows that in this case the trade-off is at most polynomial. Finally, the more general case of word-bounded languages is investigated. We show how the results obtained for letter-bounded languages can be extended to word-bounded languages.