# Complexity-Theoretic Aspects of Expanding Cellular Automata

## TL;DR

This paper explores the computational complexity of expanding cellular automata (XCA), showing their decision problem class aligns with NP-related classes and providing various characterizations and classifications.

## Contribution

It introduces a complexity-theoretic analysis of XCA, establishing their decision problem class as equivalent to polynomial-time NP-truth-table reducibility and offering multiple characterizations.

## Key findings

- XCA decision problems are in ${ ext{≤}_{tt}^p}( ext{NP})$
- XCAs with multiple accept/reject states are polynomial-time equivalent to original XCAs
- Acceptance condition variants are classified within ${ ext{≤}_{tt}^p}( ext{NP})$ and P

## Abstract

The expanding cellular automata (XCA) variant of cellular automata is investigated and characterized from a complexity-theoretical standpoint. An XCA is a one-dimensional cellular automaton which can dynamically create new cells between existing ones. The respective polynomial-time complexity class is shown to coincide with ${\le_{tt}^p}(\mathsf{NP})$, that is, the class of decision problems polynomial-time truth-table reducible to problems in $\mathsf{NP}$. An alternative characterization based on a variant of non-deterministic Turing machines is also given. In addition, corollaries on select XCA variants are proven: XCAs with multiple accept and reject states are shown to be polynomial-time equivalent to the original XCA model. Finally, XCAs with alternative acceptance conditions are considered and classified in terms of ${\le_{tt}^p}(\mathsf{NP})$ and the Turing machine polynomial-time class $\mathsf{P}$.