 
Summary: PROPERTIES OF NPCOMPLETE SETS
CHRISTIAN GLAßER
, A. PAVAN¡ , ALAN L. SELMAN¢ , AND SAMIK SENGUPTA£
Abstract. We study several properties of sets that are complete for ¤¦¥ . We prove that if § is an ¤¨¥ complete
set and © § is a pselective sparse set, then §© is ¨ hard for ¤¦¥ . We demonstrate existence of a sparse
set ©!¦"¨#%$'&¨(0)214365 such that for every §7'¤¦¥89¥ , §9@© is not hard for ¤¦¥ . Moreover, we prove for
every §@'¤¨¥!A¥ , that there exists a sparse ©B&DCE¥ such that §F!© is not ¨ hard for ¤¨¥ . Hence, removing
sparse information in ¥ from a complete set leaves the set complete, while removing sparse information in &GCE¥
from a complete set may destroy its completeness. Previously, these properties were known only for exponential
time complexity classes.
We use hypotheses about pseudorandom generators and secure oneway permutations to derive consequences
for longstanding open questions about whether ¤¨¥ complete sets are immune. For example, assuming that pseudo
random generators and secure oneway permutations exist, it follows easily that ¤¨¥ complete sets are not pimmune.
Assuming only that secure oneway permutations exist, we prove that no ¤¦¥ complete set is E"H#%$'&¦(I)QPQRS5 
immune. Also, using these hypotheses we show that no ¤¦¥ complete set is quasipolynomialclose to ¥ .
We introduce a strong but reasonable hypothesis and infer from it that disjoint Turingcomplete sets for ¤¨¥ are
not closed under union. Our hypothesis asserts existence of a T¦¥ machine U that accepts VXW such that for some
V`YabYdc , no ) P R timebounded machine can correctly compute infinitely many accepting computations of U .
We show that if T¦¥FegfihpT¦¥ contains E"H#%$'&¨(0) P Rq5 biimmune sets, then this hypothesis is true.
