How to turn a second-order cellular automaton into a lattice gas: a new inversion scheme

  • Authors:

  • Tommaso Toffoli;Silvio Capobianco;Patrizia Mentrasti

  • Affiliations:

  • Electrical and Computer Engineering, Boston University, 8 Saint Mary's St., Boston, MA;Dip. di Matematica, Università di Roma "La Sapienza";Dip. di Matematica, Università di Roma "La Sapienza"

  • Venue:
  • Theoretical Computer Science - Special issue: Theoretical aspects of cellular automata
  • Year:
  • 2004

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Abstract

In second-order cellular automata, the signals made available to the local transition function include, besides the current state of a site's neighbors, also the previous state of the site itself. Some similarities had been noted, especially in physics simulations, between certain second-order cellular automata and certain lattice gases.Here we show how to construct, for any second-order cellular automaton, a lattice gas having isomorphic functional behavior. (Paradoxically, this isomorphism of function is achieved by compromising on the isomorphism of structure: namely, the group of translation symmetries of the resulting lattice gas is coarser than that of the original cellular automaton.) The advantage of our construction is that, while invertibility in cellular automata is not directly deducible from the local map (indeed, it is in general undecidable), in the second-order case the invertibility of the cellular automaton goes hand-in-hand with that of the corresponding lattice gas--and for lattices gases invertibility is trivially decidable on the basis of the local structure.From a physical viewpoint, our construction illustrates a trade-off between two ways of achieving a "force" of a given range. In a cellular automaton, multiple copies of a signal are made and distributed in parallel to several sites, explicitly providing the desired fanout width. In a lattice gas, with our construction, one can let a single signal serially service a number of sites along an appropriate spacetime route. The latter method is better suited to a nondissipative implementation of the dynamics.