On restarting automata with auxiliary symbols and small window size

František Mráz,Friedrich Otto

doi:10.1051/ita/2021003

František Mráz, Friedrich Otto

Open Access

https://doi.org/10.1051/ita/2021003

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Abstract

Here we show that for monotone RWW- (and RRWW-) automata, window size two is sufficient, both in the nondeterministic as well as in the deterministic case. For the former case, this is done by proving that each context-free language is already accepted by a monotone RWW-automaton of window size two. In the deterministic case, we first prove that each deterministic pushdown automaton can be simulated by a deterministic monotone RWW-automaton of window size three, and then we present a construction that transforms a deterministic monotone RWW-automaton of window size three into an equivalent automaton of the same type that has window size two. Furthermore, we study the expressive power of shrinking RWW- and RRWW-automata the window size of which is just one or two. We show that for shrinking RRWW-automata that are nondeterministic, window size one suffices, while for nondeterministic shrinking RWW-automata, we already need window size two to accept all growing context-sensitive languages. In the deterministic case, shrinking RWW- and RRWW-automata of window size one accept only regular languages, while those of window size two characterize the Church-Rosser languages.

Highlights

The restarting automaton was introduced in [6] as a formal model for the linguistic technique of ‘analysis by reduction’
We have seen above that deterministic RWW- and RRWW-automata of window size two are necessarily monotone, which implies that they yield a characterization for the class DCFL of deterministic context-free languages
We have studied the expressive power of restarting automata and shrinking restarting automata of small window size

Summary

Introduction

The restarting automaton was introduced in [6] as a formal model for the linguistic technique of ‘analysis by reduction’. We first present a construction that turns a deterministic pushdown automaton (PDA) into a deterministic monotone RWW-automaton of window size three that accepts the same language. Based on the window size we have just two classes of languages that are accepted by monotone RWW-automata, both in the nondeterministic as well as in the deterministic case. Window size nine suffices to again obtain all language accepted by finite-change automata It remains open whether window size nine is the smallest possible, that is, whether nondeterministic shrinking RWW-automata of window size eight are really less expressive than those of window size nine. It turns out that deterministic shrinking RWW- and RRWW-automata of window size one just accept the regular languages, while with window size two, these automata characterize the Church-Rosser languages.

Definitions and notation

On nondeterministic monotone RWW-automata

On deterministic RWW-automata

On shrinking restarting automata

On deterministic shrinking restarting automata

Conclusion