# Descriptive complexity for minimal time of cellular automata

## TL;DR

This paper uses descriptive complexity and logical characterizations to analyze the minimal real-time computational complexity of one-dimensional cellular automata, providing a logical framework for understanding their capabilities.

## Contribution

It establishes logical characterizations of real-time complexity classes for cellular automata using restrictions of existential second-order Horn logic, linking logic and parallel computation.

## Key findings

- Logical characterizations of real-time classes for cellular automata
- A normalization method transforming formulas into grid circuit mimics
- Insights into designing parallel programs via logical frameworks

## Abstract

Descriptive complexity may be useful to design programs in a natural declarative way. This is important for parallel computation models such as cellular automata, because designing parallel programs is considered difficult. Our paper establishes logical characterizations of the three classical complexity classes that model minimal time, called real-time, of one-dimensional cellular automata according to their canonical variants. Our logics are natural restrictions of the existential second-order Horn logic. They correspond to the three ways of deciding a language on a square grid circuit of side n according to the three canonical placements of an input word of length n on the grid. Our key tool is a normalization method that transforms a formula into an equivalent formula that literally mimics a grid circuit.