Computing downward closures for stacked counter automata

TL;DR

This paper proves that the downward closure of languages accepted by stacked counter automata, which generalize pushdown and blind counter automata, is computable using novel Parikh annotations, advancing understanding of language hierarchies.

Contribution

It introduces Parikh annotations to compute downward closures for stacked counter automata, a class extending context-free languages with new closure properties.

Findings

Downward closure is computable for stacked counter automata.

Hierarchy of languages based on these automata is strict at every level.

Parikh annotations are a key tool for these computations.

Abstract

The downward closure of a language $L$ of words is the set of all (not necessarily contiguous) subwords of members of $L$. It is well known that the downward closure of any language is regular. Although the downward closure seems to be a promising abstraction, there are only few language classes for which an automaton for the downward closure is known to be computable. It is shown here that for stacked counter automata, the downward closure is computable. Stacked counter automata are finite automata with a storage mechanism obtained by \emph{adding blind counters} and \emph{building stacks}. Hence, they generalize pushdown and blind counter automata. The class of languages accepted by these automata are precisely those in the hierarchy obtained from the context-free languages by alternating two closure operators: imposing semilinear constraints and taking the algebraic extension. The main tool for computing downward closures is the new concept of Parikh annotations. As a second application of Parikh annotations, it is shown that the hierarchy above is strict at every level.