# On the structure and complexity of infinite sets with minimal perfect hash functions

## Abstract

This paper studies the class of *infinite* sets that have minimal perfect hash functions—one-to-one onto maps between the sets and Σ^{*}—computable in polynomial time. We show that all standard NP-complete sets have polynomialtime computable minimal perfect hash functions, and give a structural condition, E=Σ _{2} ^{E} , sufficient to ensure that *all* infinite NP sets have polynomial-time computable minimal perfect hash functions. On the other hand, we present evidence that some infinite NP sets, and indeed some infinite P sets, do not have polynomial-time computable minimal perfect hash functions: if an infinite NP set *A* has polynomial-time computable perfect minimal hash functions, then *A* has an infinite sparse NP subset, yet we construct a relativized world in which some infinite NP sets lack infinite sparse NP subsets. This world is built upon a result that is of interest in its own right; we determine optimally—with respect to any relativizable proof technique—the complexity of the easiest infinite sparse subsets that infinite P sets are guaranteed to have.

## Keywords

Hash Function Ranking Function Random Oracle Polynomial Hierarchy Relativize World## Preview

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