Computability in Europe 2009 Heidelberg Germany

1. Computability in Europe 2009Heidelberg Germany Union Theorems in Type-2 Computation Chung-Chih Li School of Information Technology Illinois State University Normal, IL 61790, USA

2. Complexity classes (J. Hartmanis & R. Stearns 1965) resource bound Most natural complexity class can be better understood as a union of precise complexity classes defined as above. Others like big-O notations.

3. Do those natural complexity classes have precise complexity classes? The original question: Yes, with some moderate restrictions, they have. Union Theorem (E. McCreight & A. Meyer, 1969) Given any sequence of recursive functions such that, is recursive and for all then, there is a recursive function g such that

4. Do those natural type-2 complexity classes have precise type-2 complexity classes? The same question at type-2: Does this question make sense at all? Basic Feasible Functional (BFF) at type-2. A natural type-2 analog to PTIME (S. Cook & B Kapron 1989)

5. Dynamic resource bound for type-2 computations Computable Nontrivial Bounded Convergent F-monotone (for strong T2TB) Type-2 Time Bounds (T2TB): • Some appropriate clocking scheme • Some appropriate definition of small sets (compact) Then, is a workable notion Now, we are almost ready to go

6. Theorem: In general the union theorem fails There exists such that, for every , We need a more manageable sequence of T2TB

7. Uniform (the sequence is computable) Ascending Useful (not tricky) Convergent (may not at the same fragment) Uniformly convergent Strongly convergent (there is a computable detector) Manageable sequence of T2TB There is a uniform and ascending such that Theorem: For any uniform and ascending , there is a total continuous functional H such that Lemma:

8. Lemma: Let be useful, If there is an   T2TBF such that than, is convergent. BFF is not a complexity class There is no   T2TBF such that = BFF Theorem:

9. There is a uniform, ascending, useful, and convergentisuch that C(i) is not a type-2 complexity class. Theorem: Non-Union Theorems There is a uniform, ascending, and uniformly convergent isuch that C(i) is not a type-2 complexity class. Corollary: There is a uniform, ascending, useful, and uniformly convergent isuch that C(i) is not a type-2 complexity class. Conjecture:

10. If iis uniform, ascending, useful, and strongly convergent, then there is a  T2TB such that C() = C(i). Theorem: Union Theorems “Strongly convergent” in the theorem above can be replaced by “bounded by a continuous functional”. Conjecture: The  in the theorem above is not strong (locking detectable).

11. Definition: Let   T2TB. Define Type-2 Big-O notations Theorem If  is strong and useful, there is   T2TB such that C() = O(). Corollary 1. If 1 and  2 are strong and useful, there is   T2TB such that C() = O(1 +2). 2. If 1 and  2 are strong and useful and 1* 2, then C(1 +2) = O(2).