Electronic Colloquium on Computational Complexity, Report No. 28 (2005) On the Lattice of Clones Below the Polynomial Time Functions

Abstract. A clone is a set of functions that is closed under generalized substitution. The set FP of functions being computable deterministically in polynomial time is such a clone. It is well-known that the set of subclones of every clone forms a lattice. We study the lattice below FP, which contains other important function complexity classes like FL and FNC i. We show that the lattice is dualatomic and determine some of its dualatoms. We show that no time-complexity class can be a dualatom in this lattice. We show that there are uncountably many subclones of FP.