On the complexity of asynchronous freezing cellular automata

Eric Goles, Diego Maldonado, Pedro Montealegre, Martín Ríos-Wilson

Research output: Contribution to journalArticlepeer-review


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. We define the problem ASYNCUNSTABILITY, which consists in deciding there exists a sequential updating scheme that changes the state of a given cell. 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). 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.

Original languageEnglish
Article number104764
JournalInformation and Computation
StatePublished - Dec 2021
Externally publishedYes


  • Cellular automata
  • Computational complexity
  • Freezing dynamics


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