Dense and Non-Dense Families of Complexity Classes

Let t be any abstract measure of computational complexity, and let L denote the specific measure of memory resource (tape) on one tape Turing machines. De~ote by R:() the class of all total functions whose t-complexity is bounded by the function t() almost everywhere. Call such classes t-complexity classes. We are interested in relationships among these classes, under proper set inclusion (C). In other words, we are interested in the partially ordered structure <I·,.~ where I· = {R:() It() is re-cursive} is called the family of t-complexity classes. Of special interest is the subfamily O· {R:. () It i () is 1 total} , called the family of e'xact .~complexity classes. We show that E L and OL are dense under C for sufficiently large bounds t(),. L but n L is not dense in L • We also construct measures ~ for which L~ and ot are non-dense, for which L~ is dense but Qt is not, for which O~ is dense but L t · ~. d · ~t is not and for wh1ch.0 1S ense 1n ~ Thus density is not a measure invariant t ~ property of L or 0 • These are the first examples of important structural properties of these families which are not measure invariant. I. Preliminaries We assume at least cursory familiarity with the axiomatic approach to computational complexity theory as initiated in Blum [1] and developed recently in [2], [6], and [9]. To establish our notation, we list the following defini-"·tions. Given an acceptable indexing {~i()} ~f the partial recursive functions (of one argument), see Rogers [8], an abs~ract complexity measure over {~i()} 1S a set (t. ()} of partial recursive functions for 1 ~hich there exists a 0,1 valued recursive 7 function M() satisfying Axiom 1: ~i(n) is defined iff ti(n) is defined. 1 iff ti(n) = m. We say that ~ {t i ()} satisfying the axioms is a complexity measure, and the individual t. () are called exact com-1 plexity functions. [The t i () have also been called "step-counting" functions or "run-time'" mnctions or "difficulty" functions.] Given a complexity measure t , define a t-complexity class R:()= {~'i () 1 ~. () total and t. (n) < ten) for almost all 1 1-n (a.e.n.)}. (We use lower case English letters, t,f,g,h, in denoting total as opposed to partial functions.) When t is clear from the context we …