Counter Machines and Distributed Automata: A Story about Exchanging Space and Time

TL;DR

This paper establishes an equivalence between two classes of counter machines and a class of distributed automata, revealing deep connections between space and time complexity in automata theory.

Contribution

It proves the equivalence of counter machines with different features and a class of distributed automata operating on path graphs, linking space and time computational models.

Findings

Counter machines with copy and copyless sum features are equivalent.

Distributed automata on path graphs are equivalent to certain counter machine classes.

The models are related to linear-time one-way cellular automata.

Abstract

We prove the equivalence of two classes of counter machines and one class of distributed automata. Our counter machines operate on finite words, which they read from left to right while incrementing or decrementing a fixed number of counters. The two classes differ in the extra features they offer: one allows to copy counter values, whereas the other allows to compute copyless sums of counters. Our distributed automata, on the other hand, operate on directed path graphs that represent words. All nodes of a path synchronously execute the same finite-state machine, whose state diagram must be acyclic except for self-loops, and each node receives as input the state of its direct predecessor. These devices form a subclass of linear-time one-way cellular automata.