# Variations of Checking Stack Automata: Obtaining Unexpected Decidability Properties

## TL;DR

This paper explores various models of checking stack automata, revealing unexpected decidability properties and differences between deterministic and nondeterministic versions, including the first one-way model with certain decidability characteristics.

## Contribution

It introduces new variants of checking stack automata with unique decidability properties, including the first one-way model exhibiting specific decidability and undecidability behaviors.

## Key findings

- Deterministic checking stack automata have decidable membership but undecidable emptiness.
- Nondeterministic models can have both undecidable membership and emptiness.
- Two-way deterministic machines with multiple stacks and counters have decidable membership.

## Abstract

We introduce a model of one-way language acceptors (a variant of a checking stack automaton) and show the following decidability properties: (1) The deterministic version has a decidable membership problem but has an undecidable emptiness problem. (2) The nondeterministic version has an undecidable membership problem and emptiness problem. There are many models of accepting devices for which there is no difference with these problems between deterministic and nondeterministic versions, and the same holds for the emptiness problem. As far as we know, the model we introduce above is the first one-way model to exhibit properties (1) and (2). We define another family of one-way acceptors where the nondeterministic version has an undecidable emptiness problem, but the deterministic version has a decidable emptiness problem. We also know of no other model with this property in the literature. We also investigate decidability properties of other variations of checking stack automata (e.g., allowing multiple stacks, two-way input, etc.). Surprisingly, two-way deterministic machines with multiple checking stacks and multiple reversal-bounded counters are shown to have a decidable membership problem, a very general model with this property.