Restarting automata with restricted utilization of auxiliary symbols

The restarting automaton is a restricted model of computation that was introduced by Jancar et al. to model the so-called analysis by reduction, which is a technique used in linguistics to analyze sentences of natural languages. The most general models of restarting automata make use of auxiliary symbols in their rewrite operations, although this ability does not directly correspond to any aspect of the analysis by reduction. Here we put restrictions on the way in which restarting automata use auxiliary symbols, and we investigate the influence of these restrictions on their expressive power. In fact, we consider two types of restrictions. First, we consider the number of auxiliary symbols in the tape alphabet of a restarting automaton as a measure of its descriptional complexity. Secondly, we consider the number of occurrences of auxiliary symbols on the tape as a dynamic complexity measure. We establish some lower and upper bounds with respect to these complexity measures concerning the ability of restarting automata to recognize the (deterministic) context-free languages and some of their subclasses.

On the Gap-Complexity of Simple RL-Automata

Analysis by reduction is a method used in linguistics for checking the correctness of sentences of natural languages. This method is modelled by restarting automata. All types of restarting automata considered in the literature up to now accept at least the deterministic context-free languages. Here we introduce and study a new type of restarting automaton, the so-called t-RL-automaton, which is an RL-automaton that is rather restricted in that it has a window of size one only, and that it works under a minimal acceptance condition. On the other hand, it is allowed to perform up to t rewrite (that is, delete) steps per cycle. Here we study the gap-complexity of these automata. The membership problem for a language that is accepted by a t-RL-automaton with a bounded number of gaps can be solved in polynomial time. On the other hand, t-RL-automata with an unbounded number of gaps accept NP-complete languages.

On the Descriptional Complexity of Simple RL-Automata

Analysis by reduction is a method used in linguistics for checking the correctness of sentences of natural languages. This method is modelled by restarting automata. Here we study a new type of restarting automaton, the so-called t-sRL-automaton, which is an RL-automaton that is rather restricted in that it has a window of size 1 only, and that it works under a minimal acceptance condition. On the other hand, it is allowed to perform up to t rewrite (that is, delete) steps per cycle. We focus on the descriptional complexity of these automata, establishing two complexity measures that are both based on the description of t-sRL-automata in terms of so-called meta-instructions. We present some hierarchy results as well as a non-recursive trade-off between deterministic 2-sRL-automata and finite-state acceptors.

The Degree of Word-Expansion of Lexicalized RRWW-Automata

Restarting automata can be seen as analytical variants of classical automata as well as of regulated rewriting systems. We study a measure for the degree of nondeterminism of (context-free) languages in terms of deterministic restarting automata that are (strongly) lexicalized. This measure is based on the number of auxiliary symbols (categories) used for recognizing a language as the projection of its characteristic language onto its input alphabet. This type of recognition is typical for analysis by reduction, a method used in linguistics for the creation and verification of formal descriptions of natural languages. Our main results establish a hierarchy of classes of context-free languages and two hierarchies of classes of non-context-free languages that are based on the expansion factor of a language.