Subclasses of Ptime Interpreted by Programming Languages

We consider the cons-free programming language of Neil Jones, a simple pure functional language, which decides exactly the polynomial-time relations and whose tail recursive fragment decides exactly the logarithmic-space relations. We exhibit a close relationship between the running time of cons-free programs and the running time of logspace-bounded auxiliary pushdown automata. As a consequence, we characterize intermediate classes like NC in terms of resource-bounded cons-free computation. In so doing, we provide the first “machine-free” characterizations of certain complexity classes, like P-uniform NC. Furthermore, we show strong polynomial lower bounds on cons-free running time. Namely, for every polynomial p, we exhibit a relation R ∈Ptime such that any cons-free program deciding R must take time at least p almost everywhere. Our methods use a “subrecursive version” of Blum complexity theory, and raise the possibility of further applications of this technology to the study of the fine structure of Ptime.