---
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 *adding blind counters* and *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. 
author:
- Georg Zetzsche
bibliography:
- 'bibliography.bib'
title: Computing downward closures for stacked counter automata
---

Introduction
============

Preliminaries {#sec:preliminaries}
=============

A hierarchy of language classes {#sec:hierarchy}
===============================

This section introduces a hierarchy of language classes that divides the class of languages accepted by stacked counter automata into levels.  This will allow us to apply recursion with respect to these levels. 

Parikh annotations {#sec:pa}
==================

Computing downward closures {#sec:dclosure}
===========================

Strictness of the hierarchy {#sec:strictness}
===========================
