, There exists an intrinsically universal for 2D cellular automaton with von Neumann neighborhood and 2 states

, We calculate V +1 as if rule 0 were quiescent. If in V +1 there is a cell with all its neighbors inactive, so u is not stable, otherwise the algorithm for the rule with 0 quiescent will give the correct answer, because there are no cells with all its neighbors inactive in V +1 nor in B +1 , because all its cells have at least one active cell, The following algorithm is able to solve AsyncStability using the algorithms for rules with 0 as a quiescent state

, Compute the V +1 = {v ? Z 2 : x v = 0 ? |x N (v) | 1 + 1 ? I F }. 2: for all v ? V +1 do in parallel 3: if |x N (v) | 1 for 7: To solve AsyncStability for the rule same rule, Algorithm 15 AsyncStability solving 01, 012 and 013 in the triangular grid and 01, 012, 013, 014, 0123, 0134 and 0124 in the square grid

, Steps 3-6 are computed in time O(log N ) using O(N ) processors, one per cell to compute the sum of its neighbors

, It is NP because we can verify if a given updating scheme activates a cell in a polynomial time for any FTACA. Note that if we consider the rule 2 in a tri-dimensional space, we can build the logic gates AND and OR and duplicate signals over a plane simulating a two dimensional FTCA 2. This work also in an asynchronous way. Instead of using an XOR gate for wire crossings, we can simply use the third dimension to pass one wire over the other, similar to Figure 4.6. We will build an abstract system similar to CNF formulas, but considering three state instead of two

, 0), (0, 0)}. To give an intuition, We consider the following states T = {(0, 1)

D. Maldonado, A. Moreira, and A. Gajardo, Universal time-symmetric number-conserving cellular automaton, Cellular Automata and Discrete Complex Systems -21st IFIP WG 1.5 International Workshop, AUTOMATA 2015, vol.9099, pp.155-168, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01442471

F. Becker, D. Maldonado, N. Ollinger, and G. Theyssier, Universality in freezing cellular automata, Sailing Routes in the World of Computation14th Conference on Computability in Europe, pp.50-59, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01772500

E. Goles, D. Maldonado, P. Montealegre, and N. Ollinger, On the computational complexity of the freezing non-strict majority automata, Cellular Automata and Discrete Complex Systems -23rd IFIP WG 1.5 International Workshop, AUTOMATA 2017, pp.109-119, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01656355

E. Goles, D. Maldonado, P. Montealegre-barba, and N. Ollinger, Fast-parallel algorithms for freezing totalistic asynchronous cellular automata, Cellular Automata -13th International Conference on Cellular Automata for Research and Industry, pp.406-415, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01985010

M. Gardner, The fantastic combinations of John Conway's new solitaire game "life, Scientific American, vol.223, pp.120-123, 1970.

J. Hardy, Y. Pomeau, and O. De-pazzis, Time evolution of a two-dimensional classical lattice system, Phys. Rev. Lett, vol.31, pp.276-279, 1973.

J. Hardy, O. De-pazzis, and Y. Pomeau, Molecular dynamics of a classical lattice gas: Transport properties and time correlation functions, Phys. Rev. A, vol.13, pp.1949-1961, 1976.

R. Landauer, Irreversibility and heat generation in the computing process, IBM Journal of Research and Development, vol.5, issue.3, pp.183-191, 1961.

A. Church, A Set of Postulates for the Foundation of Logic

E. L. Post, Formal reductions of the general combinatorial decision problem, Am. J. Math, vol.65, pp.197-215, 1943.

M. L. Minsky, Computation: Finite and Infinite Machines, 1967.

A. M. Turing, On computable numbers, with an application to the Entscheidungsproblem. Proceedings of the, vol.2, pp.230-265, 1936.

J. V. Neumann, Theory of Self-Reproducing Automata, 1966.

A. R. Smith, Simple computation-universal cellular spaces and self-reproduction, 9th Annual Symposium on Switching and Automata Theory (swat 1968)(FOCS), vol.00, pp.269-277, 1968.

K. Lindgren and M. G. Nordahl, Universal computation in simple one-dimensional cellular automata, Complex Systems, vol.4, issue.3, 1990.

M. Cook, Universality in elementary cellular automata, Complex Systems, vol.15, issue.1, pp.1-40, 2004.

J. Albert and K. C. Ii, A simple universal cellular automaton and its one-way and totalistic version, Complex Systems, vol.1, issue.1, 1987.

B. Durand and Z. Róka, The Game of Life: Universality Revisited, pp.51-74, 1999.

N. Ollinger and G. Richard, Four states are enough! Theor, Comput. Sci, vol.412, issue.1-2, pp.22-32, 2011.

A. Moreira, Universality and decidability of number-conserving cellular automata, Algorithms in Quantum Information Prcoessing, vol.292, pp.711-721, 2003.

J. Durand-lose, Intrinsic universality of a 1-dimensional reversible cellular automaton, STACS, vol.1200, pp.439-450, 1997.
URL : https://hal.archives-ouvertes.fr/hal-01559642

K. Morita, Universality of one-dimensional reversible and number-conserving cellular automata, EPTCS, vol.90, pp.142-150, 2012.

A. Gajardo, J. Kari, and A. Moreira, On time-symmetry in cellular automata, J. Comput. System Sci, vol.78, issue.4, pp.1115-1126, 2012.

M. Delorme, J. Mazoyer, N. Ollinger, and G. Theyssier, Bulking I: an abstract theory of bulking, Theoret. Comput. Sci, vol.412, issue.30, pp.3866-3880, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00980376

M. Delorme, J. Mazoyer, N. Ollinger, and G. Theyssier, Bulking II: classifications of cellular automata, Theoret. Comput. Sci, vol.412, issue.30, pp.3881-3095, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00980377

J. Hartmanis and R. E. Stearns, On the computational complexity of algorithms, vol.117, pp.285-285, 1965.

J. Edmonds, Paths, trees, and flowers, Canad. J. Math, vol.17, pp.449-467, 1965.

S. A. Cook, The complexity of theorem-proving procedures, Proceedings of the Third Annual ACM Symposium on Theory of Computing, STOC '71, pp.151-158, 1971.

L. A. Levin, Universal sequential search problems. Problems of Information Transmission, vol.9, pp.265-266, 1973.

D. Griffeath and C. Moore, Life without death is P-Complete. Working papers, 1997.

C. Moore, Majority-vote cellular automata, ising dynamics, and p-completeness. Working papers, 1996.

T. Neary and D. Woods, P-completeness of cellular automaton rule 110, Automata, Languages and Programming, pp.132-143, 2006.

E. Goles, N. Ollinger, and G. Theyssier, Introducing Freezing Cellular Automata, Cellular Automata and Discrete Complex Systems, 21st International Workshop (AUTOMATA 2015, vol.24, pp.65-73, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01294144

J. Chalupa, P. L. Leath, and G. R. Reich, Bootstrap percolation on a bethe lattice, Journal of Physics C: Solid State Physics, vol.12, issue.1, p.31, 1979.

T. Ghisu, B. Arca, G. Pellizzaro, and P. Duce, An improved cellular automata for wildfire spread, International Conference On Computational Science, vol.51, pp.2287-2296, 2015.

M. Fuentes and M. Kuperman, Cellular automata and epidemiological models with spatial dependence, Physica A: Statistical Mechanics and its Applications, vol.267, issue.3, pp.471-486, 1999.

M. J. Patitz, An introduction to tile-based self-assembly, Unconventional Computation and Natural Computation, pp.34-62, 2012.

P. Bak, K. Chen, and C. Tang, A forest-fire model and some thoughts on turbulence, Physics Letters A, vol.147, issue.5-6, pp.297-300, 1990.

E. Goles, P. Montealegre-barba, and I. Todinca, The complexity of the bootstraping percolation and other problems, Theoretical Computer Science, vol.504, pp.73-82, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00914603

S. Wolfram, Statistical mechanics of cellular automata, Rev. Mod. Phys, vol.55, issue.3, pp.601-644, 1983.

G. Hedlund, Endomorphisms and automorphisms of the shift dynamical system, Math. Syst Theory, vol.3, issue.4, pp.320-375, 1969.

F. Robert, Discrete iterations: a metric study. Springer series in computational mathematics, 1986.

M. Sipser, Introduction to the Theory of Computation, Cengage Learning, 2012.

J. Jájá, An Introduction to Parallel Algorithms, 1992.

R. Greenlaw, H. Hoover, and W. Ruzzo, Limits to Parallel Computation: P-completeness Theory, 1995.

J. Jájá and J. Simon, Parallel algorithms in graph theory: Planarity testing, SIAM J. Comput, vol.11, issue.2, pp.314-328, 1982.

E. R. Banks, Universality in cellular automata, SWAT (FOCS), pp.194-215, 1970.

S. Wolfram, Universality and complexity in cellular automata, Physica D, vol.10, pp.1-35, 1984.

E. R. Banks, Information processing and transmission in cellular automata, 1971.

N. Ollinger and G. Richard, A particular universal cellular automaton, Proceedings International Workshop on The Complexity of Simple Programs, pp.205-214, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00095821

A. Maruoka and M. Kimura, Condition for injectivity of global maps for tessellation automata, Information and Control, vol.32, issue.2, pp.158-162, 1976.

K. C. Ii, J. Pachl, and S. Yu, On the limit sets of cellular automata, SIAM Journal on Computing, vol.18, issue.4, pp.831-842, 1989.

K. Sutner, Model checking one-dimensional cellular automata, J. Cellular Automata, vol.4, pp.213-224, 2009.

J. Kari, The nilpotency problem of one-dimensional cellular automata, SIAM J. Comput, vol.21, issue.3, pp.571-586, 1992.

S. Kirkpatrick, W. W. Wilcke, R. B. Garner, and H. Huels, Percolation in dense storage arrays, Physica A: Statistical Mechanics and its Applications, vol.314, issue.1, pp.220-229, 2002.

H. Amini, Bootstrap percolation in living neural networks, Journal of Statistical Physics, vol.141, issue.3, pp.459-475, 2010.

I. Karafyllidis and A. Thanailakis, A model for predicting forest fire spreading using cellular automata, Ecological Modelling, vol.99, issue.1, pp.87-97, 1997.

J. H. Holland, A universal computer capable of executing an arbitrary number of subprograms simultaneously, Essays on Cellular Automata, pp.264-276, 1970.

J. W. Thatcher, Universality in the von neumman cellular model, Essays on Cellular Automata, pp.132-186, 1970.

S. Ulam and R. Schrandt, On recursively defined geometric objects and patterns of growth, 1967.

J. Gravner and D. Griffeath, Cellular automaton growth on z2: Theorems, examples, and problems, Advances in Applied Mathematics, vol.21, issue.2, pp.241-304, 1998.

B. Bollobás, P. Smith, and A. Uzzell, Monotone cellular automata in a random environment, Combinatorics, Probability and Computing, vol.24, issue.4, pp.687-722, 2015.

D. Doty, J. H. Lutz, M. J. Patitz, R. T. Schweller, S. M. Summers et al., The tile assembly model is intrinsically universal, FOCS 2012 Proceedings, pp.302-310, 2012.

P. Meunier, M. J. Patitz, S. M. Summers, G. Theyssier, A. Winslow et al., Intrinsic universality in tile self-assembly requires cooperation, SODA 2014 Proceedings, pp.752-771, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00943802

R. Vollmar, On cellular automata with a finite number of state changes, Parallel Processes and Related Automata, vol.3, pp.181-191, 1981.

D. A. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, 1995.

N. Ollinger, Universalities in cellular automata, Handbook of Natural Computing, pp.189-229, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00980362

L. M. Goldschlager, The monotone and planar circuit value problems are log space complete for, P. SIGACT News, vol.9, issue.2, pp.25-29, 1977.

E. Goles, P. Montealegre, K. Perrot, and G. Theyssier, On the complexity of two-dimensional signed majority cellular automata, J. Comput. Syst. Sci, vol.91, pp.1-32, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01472161

, A New Kind of Science. Wolfram Media Inc, 2002.

J. G. Zabolitzky, Critical properties of rule 22 elementary cellular automata, Journal of Statistical Physics, vol.50, issue.5, pp.1255-1262, 1988.

P. Grassberger, Long-range effects in an elementary cellular automaton, Journal of Statistical Physics, vol.45, issue.1, pp.27-39, 1986.

J. Zhisong and W. Yi, Complexity of limit language of the elementary cellular automaton of rule 22, Applied Mathematics-A Journal of Chinese Universities, vol.20, issue.3, pp.268-276, 2005.