Chapter 12: The Principle of Computational Equivalence

Section 8: Undecidability and Intractability

P completeness

If one allows arbitrary initial conditions in a cellular automaton with nearest-neighbor rules, then to compute the color of a particular cell after t steps in general requires specifying as input the colors of all 2t+1 initial cells up to distance t away (see page 960). And if one always does computations using systems that have only nearest-neighbor rules then just combining 2t+1 bits of information can take up to t steps—even if the bits are combined in a way that is not computationally irreducible. So to avoid this one can consider systems that are more like circuits in which any element can get data from any other. And given t elements operating in parallel one can consider the class NC studied by Nicholas Pippenger in 1978 of computations that can be done in a number of steps that is at most some power of Log[t]. Among such computations are Plus, Times, Divide, Det and LinearSolve for integers, as well as determining outcomes in additive cellular automata (see page 609). But I strongly suspect that computational irreducibility prevents outcomes in systems like rule 30 and rule 110 from being found by computations that are in NC—implying in effect that allowing arbitrary connections does not help much in computing the evolution of such systems. There is no way yet known to establish this for certain, but just as with NP and P one can consider showing that a computation is P-complete with respect to transformations in NC. It turns out that finding the outcome of evolution in any standard universal Turing machine or cellular automaton is P-complete in this sense, since the process of emulating any such system by any other one is in NC. Results from the mid-1970s established that finding the output from an arbitrary circuit with And or Or gates is P-complete, and this has made it possible to show that finding the outcome of evolution in various systems not yet known to be universal is P-complete. A notable example due to Cristopher Moore from 1996 is the 3D majority cellular automaton with rule UnitStep[a+AxesTotal[a, 3]-4] (see page 927); another example is the Ising model cellular automaton from page 982.

From Stephen Wolfram: A New Kind of Science [citation]