State Grammars with Stores

TL;DR

This paper explores the computational power of state grammars with stores, showing how adding reversal-bounded counters affects their generative capacity and their relation to machine models, with implications for complexity and language classes.

Contribution

It introduces and analyzes state grammars with stores, demonstrating their equivalence to certain machine models and examining the impact of reversal-bounded counters on their generative power.

Findings

Adding reversal-bounded counters does not increase capacity with standard derivation.

State grammars with counters under leftmost derivations are weaker than those with standard derivation.

The complexity of the emptiness problem for these grammars is studied.

Abstract

State grammars are context-free grammars where the productions have states associated with them, and a production can only be applied to a nonterminal if the current state matches the state in the production. Once states are added to grammars, it is natural to add various stores, similar to machine models. With such extensions, productions can only be applied if both the state and the value read from each store matches between the current sentential form and the production. Here, generative capacity results are presented for different derivation modes, with and without additional stores. In particular, with the standard derivation relation, it is shown that adding reversal-bounded counters does not increase the capacity, and states are enough. Also, state grammars with reversal-bounded counters that operate using leftmost derivations are shown to coincide with languages accepted by one-way machines with a pushdown and reversal-bounded counters, and these are surprisingly shown to be strictly weaker than state grammars with the standard derivation relation (and no counters). The complexity of the emptiness problem involving state grammars with reversal-bounded counters is also studied.