Generalized complexity cores and levelability of intractable sets

The notion of a complexity core is generalized and an existence theorem that follows from this generalized notion gives necessary and sufficient conditions on a class C such that every infinite set not in C has a proper generalized complexity core. Such conditions will become very simple, i.e., C is closed under union, if C contains all of the finite sets. Also, the cases of existence of a generalized complexity core and a recursive generalized complexity core are distinguished. The relation between notions of complexity core and P-immunity is investigated. A sufficient and necessary condition is given for a recursive P-immune set to be a proper complexity core of a set. This result with many others on complexity cores is also extended to the general setting. The symmetry of the lattice of proper generalized complexity cores of a set is described: for any two proper generalized com- plexity cores H and H' of a set which are not maximal, there is a lattice-automorphism that maps H onto H'. As an extension of polynomial levelability, the notion of poly- nomial levelability with approximation algorithms (PLAA) is introduced. Two conclusions are proved: all of the "natural" NP-complete sets are PLAA unless NP = P; all EXP-complete sets are PLAA. The exponential-time computable complexity core is studied. A new and more significant result on NP-completeness is that every known NP-complete set has an exponential-time computable proper complexity core of non-sparse density unless P = NP, that is, every honestly paddable NP-complete set and every k-creative set with honest product function has an exponential-time computable proper complexity core of non-sparse density unless P = NP. Meanwhile, every (LESSTHEQ)(' )(, (' ))-complete set has an exponential-time computable proper complexity core of exponential density.