Bounded Languages Meet Cellular Automata with Sparse Communication

TL;DR

This paper explores how limiting communication in one-way cellular automata affects their computational power, showing that even with minimal communication, they can accept complex languages, and many decision problems become undecidable with increased communication.

Contribution

It introduces bounds on inter-cell communication in cellular automata and analyzes their impact on language acceptance and decidability, revealing new computational thresholds.

Findings

Devices with constant communication can accept non-semilinear bounded languages.

Logarithmic communication bounds lead to undecidability of several problems.

Linear communication bounds also result in undecidability.

Abstract

Cellular automata are one-dimensional arrays of interconnected interacting finite automata. We investigate one of the weakest classes, the real-time one-way cellular automata, and impose an additional restriction on their inter-cell communication by bounding the number of allowed uses of the links between cells. Moreover, we consider the devices as acceptors for bounded languages in order to explore the borderline at which non-trivial decidability problems of cellular automata classes become decidable. It is shown that even devices with drastically reduced communication, that is, each two neighboring cells may communicate only constantly often, accept bounded languages that are not semilinear. If the number of communications is at least logarithmic in the length of the input, several problems are undecidable. The same result is obtained for classes where the total number of communications during a computation is linearly bounded.