Algorithms for computing preimages of cellular automata configurations

This paper investigates preimages (ancestors or past configurations) of specified configurations of one-dimensional cellular automata. Both counting and listing of preimages are discussed. The main graphical tools used are the de Bruijn diagram, and its extension the preimage network, which is created by concatenating de Bruijn diagrams. The counting of preimages is performed as multiplication of topological matrices of de Bruijn diagrams. Listing of preimages is described using two algorithms. The first algorithm traces paths in the preimage network and focuses on local knowledge of the network. The second performs a complete analysis of the network before proceeding with listing.