Pure Pointer Programs and Tree Isomorphism

In a previous work, Hofmann and Schöpp have introduced the programming language PURPLE to formalise the common intuition of LOGSPACEalgorithms as pure pointer programs that take as input some structured data (e.g. a graph) and store in memory only a constant number of pointers to the input (e.g. to the graph nodes). It was shown that PURPLE is strictly contained in LOGSPACE, being unable to decide st-connectivity in undirected graphs. In this paper we study the options of strengthening PURPLE as a manageable idealisation of computation with logarithmic space that may be used to give some evidence that PTIME-problems such as Horn satisfiability cannot be solved in logarithmic space. We show that with counting, PURPLE captures all of LOGSPACE on locally ordered graphs. Our main result is that without a local ordering, even with counting and nondeterminism, PURPLE cannot solve tree isomorphism. This generalises the same result for Transitive Closure Logic with counting, to a formalism that can iterate over the input structure, furnishing a new proof as a by-product.