Additive Cellular Automata

  • Burton VoorheesEmail author
Reference work entry
Part of the Encyclopedia of Complexity and Systems Science Series book series (ECSSS)


Additive cellular automata

An additive cellular automaton is a cellular automaton whose update rule satisfies the condition that its action on the sum of two states is equal to the sum of its actions on the two states separately.

Alphabet of a cellular automaton

The alphabet of a cellular automaton is the set of symbols or values that can appear in each cell. The alphabet contains a distinguished symbol called the null or quiescent symbol, usually indicated by 0, which satisfies the condition of an additive identity: 0 + x = x.

Basins of attraction

The basins of attraction of a cellular automaton are the equivalences classes of cyclic states together with their associated transient states, with two states being equivalent if they lie on the same cycle of the update rule.

Cellular automata rule

The rule, or update rule of a cellular automaton describes how any given state is transformed into its successor state. The update rule of a cellular automaton is described by a rule...


  1. This article provides a brief survey of some of the significant theoretical results on additive cellular automata, together with references to applications of both cellular automata in general and additive cellular automata in particular. For historical information, references von Neumann (1963), von Neumann and Burk (1966), Arbib (1966), Codd (1968), Sarkar (2000) are recommended. References Toffoli and Margolis (1987), Duff and Preston (1984), Chopard and Droz (1998), Lindenmayer and Rozenberg (1976) provide a general background in the use of cellular automata in modeling, as well as a number of examples. Specific exemplary cases of applications are found in Mackay (1976), Langer (1980), Lin and Goldenfeld (1990), Greenberg and Hastings (1978), Greenberg et al. (1978), Madore and Freedman (1983), Adamatzky et al. (2005), Oono and Kohmoto (1985), Falk (1986), Canning and Droz (1991), Vitanni (1973), Young (1984), Dutching and Vogelsaenger (1985), Moreira and Deutsch (2002), Sieburg et al. (1991), Santos and Continho (2001), Beauchemin et al. (2005), Burks and Farmer (1984), Moore and Hahn (2002), Gerola and Seiden (1978), Flache and Hegselmann (1998), Chen et al. (1990), Drossel and Schwabl (1992), Smith (1972), Sommerhalder and van Westrhenen (1983), Ibarra et al. (1985), Morita and Ueno (1994), Jen (1986a, b, 1988b, 1989), Raghavan (1993), Chattopadhyay et al. (2000), Rosenfeld (1979), Sternberg (1980), Hopcroft and Ullman (1972), Cole (1969), Benjamin and Johnson (1997), Carter (1984), Hillis (1984), Manning (1977), Atrubin (1965), Nishio (1981), Fischer (1965), Pries et al. (1986), Chaudhuri et al. (1997), Bardell and McAnney (1986), Hortensius et al. (1989, 1990), Tsalides et al. (1991), Matsumoto (1998), Tomassini et al. (2000), Das and Chaudhuri (1989, 1993), Serra (1990), Tziones et al. (1994), Mrugalski et al. (2000), Sikdar et al. (2002), Serra et al. (1990), Dasgupta et al. (2001), Chowdhury et al. (1994, 1995a, b), Nandi et al. (1994), Cattell and Muzio (1996), Cattell et al. (1999). Chaudhuri et al. (1997), which deals extensively with the use of additive cellular automata in computing applications and VLSI chip design, is of particular value. Willson (1984a, b, 1987a, b, 1992), Peitgen and Richter (1986), Culik and Dube (1989), Voorhees (1988), von Haeseler et al. (1992a, b, 1993, 1995, 2001a, b), Allouche et al. (1996), Barbé et al. (1995, 2003), Nagler and Claussen (2005), Takahashi (1990, 1992) provide a good background in the relation between cellular automata and fractal patterns. Voorhees (1993, 1994, 1996, 2008), Martin et al. (1984), Guan and He (1986), Das et al. (1992), Tadaki (1994), Davis (1979), Culik (1987), Toffoli and Margolis (1990), Kari (1990), Morita (1994), Moraal (2000), Lind (1984), Elspas (1959), Stevens et al. (1993, 1999), Thomas et al. (2006), Sutner (1988a, b, 1991, 2000, 2001), Lidl and Pilz (1984), Bulitko et al. (2006) deal with the theoretical analysis of additive cellular automata. A good survey of work in cellular automata up to the mid-1990s is Illichinsky (2001). An important computational survey of cellular automata dynamics is given in Wuensche and Lesser (1992).Google Scholar

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  33. Websites

    1. Gives many links to other sites on cellular automata
    2. Provides access to a number of worthwhile unpublished papers and a number of useful references
    3. An excellent site; it provides access to the Discrete Dynamics Lab program, a valuable asset in work on cellular automata and random Boolean networks
    4. Provides reviews of theoretical aspects of cellular automata

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Center for ScienceAthabasca UniversityAthabascaCanada

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