Succinctness for Singleton Sets

Abstract

Corollaries 11.7 through 11.16 exhibited progressions of programming systems that are uniformly limiting-recursively more succinct for characteristic functions of finite sets. We bragged that in these corollaries, not only was the difference in succinctness between levels extreme, but the programs witnessing this difference compute rather simple functions. One can reasonably counter that the characteristic functions of finite sets from these corollaries are far from simple since the finite sets involved are immense. For example, as noted in Corollary 11.19, there is no limiting-recursive bound (in a ϕ* -program for the h of the Corollaries) on the cardinality of these finite sets. The question thus arises as to what happens with relative succinctness if one restricts succinctness comparisons to characteristic functions of sets that have an a priori bound on their cardinality. We address this question in this chapter by studying relative succinctness for characteristic functions of singleton sets.