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In this paper we ask, “Can anything be said about a smallest circuit that computes a given function.” We are able to show that for a wide class of functions, which includes all graph problems, an optimal circuit has a restricted structure. For instance, the input wires in an optimal circuit for Hamiltonian Path have at most linear fan-out. This is analogous to the possibly counter-intuitive statement...
In this paper we investigate the structure of sets which are complete for various classes. We show, for example, that sets complete for deterministic time classes contain infinite polynomial time recognizable subsets, thus showing that they are not complex almost everywhere. We show by a related technique that any set complete for NEXP_TIME contains an infinite subset in DEXP_TIME, thereby showing...
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