On the Complexity of Asynchronous Freezing Cellular Automata

TL;DR

This paper investigates the computational complexity of determining cell stability in asynchronous freezing cellular automata, revealing that most cases are efficiently solvable, with some exceptions being NP-complete.

Contribution

It characterizes the complexity of AsyncUnstability in FCA, especially for life-like freezing CA, and identifies cases solvable in NC and one NP-complete case.

Findings

AsyncUnstability is in NL for 1D FCA.

Most LFCA cases on grids are in NC.

One LFCA rule has NP-complete complexity.

Abstract

In this paper we study the family of freezing cellular automata (FCA) in the context of asynchronous updating schemes. A cellular automaton is called freezing if there exists an order of its states, and the transitions are only allowed to go from a lower to a higher state. A cellular automaton is asynchronous if at each time-step only one cell is updated. Given configuration, we say that a cell is unstable if there exists a sequential updating scheme that changes its state. In this context, we define the problem AsyncUnstability, which consists in deciding if a cell is unstable or not. In general AsyncUnstability is in NP, and we study in which cases we can solve the problem by a more efficient algorithm. We begin showing that AsyncUnstability is in NL for any one-dimensional FCA. Then we focus on the family of life-like freezing CA (LFCA), which is a family of two-dimensional two-state FCA that generalize the freezing version of the game of life, known as life without death. We study the complexity of AsyncUnstability for all LFCA in the triangular and square grids, showing that almost all of them can be solved in NC, except for one rule for which the problem is NP-complete.