# On ≤_{1−tt}^{p}-sparseness and nondeterministic complexity classes

## Abstract

For any reduction *r*, a set is called “≤ _{r} ^{p} -sparse” if it is ≤ _{r} ^{p} -reducible to a sparse set. The difficulty of sets in nondeterministic complexity classes is investigated in terms of *non*-≤ _{1−tt} ^{p} -sparseness, i.e., *not* being ≤ _{1−tt} ^{p} -reducible to any sparse set. In particular, nondeterministic complexity classes used to specify various types of one-way functions are mainly considered, i.e., UP, N(poly), UBPP, and *UP*. For each such class, we prove that it contains a non-≤ _{1−tt} ^{p} -sparse set unless it is included in P. Since these classes are included in more general nondeterministic complexity classes such as NP, easy consequences of our observations show the nonsparseness of ≤ _{1−tt} ^{p} -hard sets for such classes.

## Keywords

Polynomial Time Complexity Class Boolean Formula Pseudo Polynomial Time Promise Problem## Preview

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