Simulation and Intrinsic Universality Among Reversible Cellular Automata, the Partition Cellular Automata Leverage

Abstract

This chapter presents the use of Partitioned Cellular Automata—introduced by Morita and colleagues— as the tool to tackle simulation and intrinsic universality in the context of Reversible Cellular Automata. Cellular automata (CA) are mappings over infinite lattices such that all cells are updated synchronously according to the states around each one and a common local function. A CA is reversible if its global function is invertible and its inverse can also be expressed as a CA. Kari proved in 1989 that invertibility is not decidable (for CA of dimension at least 2) and is thus hard to manipulate. Partitioned Cellular Automata (PCA) were introduced as an easy way to handle reversibility by partitioning the states of cells according to the neighborhood. Another approach by Margolus led to the definition of Block CA (BCA) where blocks of cells are updated independently. Both models allow easy check and design for reversibility. After proving that CA, BCA and PCA can simulate each other, it is proven that the reversible sub-classes can also simulate each other contradicting the intuition based on decidability results. In particular, it is proven that any d-dimensional reversible CA (d-R-CA) can be expressed as a BCA with \(d+1\) partitions. This proves a 1990 conjecture by Toffoli and Margolus (Physica D 45) improved and partially proved by Kari in 1996 (Mathematical System Theory 29). With the use of signals and reversible programming, a 1-R-CA that is intrinsically universal —able to simulate any 1-R-CA— is built. Finally, with a peculiar definition of simulation, it is proven that any CA (reversible or not) can be simulated by a reversible one. All these results extend to any dimension.