Deterministic pushdown automata and unary languages

TL;DR

This paper explores the complexity of simulating unary deterministic pushdown automata with finite automata, establishing exponential bounds and tightness, and analyzing the succinctness of automata and grammars for unary languages.

Contribution

It provides tight bounds on the simulation cost of unary deterministic pushdown automata by finite automata and compares their succinctness to context-free grammars.

Findings

Simulation of unary DPDA by DFA is exponential and tight.

Two-way nondeterministic automata cannot reduce simulation cost.

Unary languages allow more succinct context-free grammars.

Abstract

The simulation of deterministic pushdown automata defined over a one-letter alphabet by finite state automata is investigated from a descriptional complexity point of view. We show that each unary deterministic pushdown automaton of size s can be simulated by a deterministic finite automaton with a number of states that is exponential in s. We prove that this simulation is tight. Furthermore, its cost cannot be reduced even if it is performed by a two-way nondeterministic automaton. We also prove that there are unary languages for which deterministic pushdown automata cannot be exponentially more succinct than finite automata. In order to state this result, we investigate the conversion of deterministic pushdown automata into context-free grammars. We prove that in the unary case the number of variables in the resulting grammar is strictly smaller than the number of variables needed in the case of nonunary alphabets.