Choiceless Polynominal Time Computation and the Zero-One Law

This paper is a sequel to [2], a commentary on [7], and an abridged version of a planned paper that will contain complete proofs of all the results presented here. The BGS model of computation was defined in [2] with the intention of modeling computation with arbitrary finite relational structures as inputs, with essentially arbitrary data structures, with parallelism, but without arbitrary choices. In the absence of any resource bounds, the lack of arbitrary choices makes no difference, because an algorithm could take advantage of parallelism to produce all possible linear orderings of its input and then use each of these orderings to make whatever choices are needed. But if we require the total computation time (summed over all parallel subprocesses) to be polynomially bounded, then there isn’t time to construct all the linear orderings, and so the inability to make arbitrary choices really matters. In fact, it was shown that choiceless polynomial time C̃PTime, the complexity class defined by BGS programs subject to a polynomial time bound, does not contain the parity problem: Given a set, determine whether its cardinality is even. Several similar results were proved, all depending on symmetry considerations, i.e., on automorphisms of the input structure. Subsequently, Shelah [7] proved a zero-one law for C̃PTime properties of graphs. We shall state this law and discuss its proof later in this paper. For now, let us just mention a crucial difference from the earlier results in [2]: Almost all finite graphs have no non-trivial automorphisms, so symmetry considerations cannot be applied to them. Shelah’s proof therefore depends on a more subtle concept of partial symmetry, which we explain in Section 8 below. Finding the proof in (an early version of) [7] difficult to follow, we worked out a presentation of the argument for the main case, which we hope will be helpful for others interested in Shelah’s ideas. We also added some related results, indicating the need for certain aspects of the proof and clarifying some of the concepts involved in it. Unfortunately, this material is not yet fully written up.