Computability in Europe 2009 Heidelberg Germany. Union Theorems in Type-2 Computation. Chung-Chih Li School of Information Technology Illinois State University Normal, IL 61790, USA. Complexity classes (J. Hartmanis & R. Stearns 1965). resource bound.
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Union Theorems in Type-2 Computation
School of Information Technology
Illinois State University
Normal, IL 61790, USA
Most natural complexity class can be better understood as a union of precise complexity classes defined as above.
Others like big-O notations.
Yes, with some moderate restrictions, they have.
Union Theorem (E. McCreight & A. Meyer, 1969)
Given any sequence of recursive functions
is recursive and for all
then, there is a recursive function g such that
Do those natural type-2 complexity classes have precise type-2 complexity classes?
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)
Useful (not tricky)
Convergent (may not at the same fragment)
Strongly convergent (there is a computable detector)
There is a uniform and ascending such that
For any uniform and ascending , there is a total continuous functional H such that
There is a uniform, ascending, useful, and convergentisuch that C(i) is not a type-2 complexity class.
There is a uniform, ascending, and uniformly convergent isuch that C(i) is not a type-2 complexity class.
There is a uniform, ascending, useful, and uniformly convergent isuch that C(i) is not a type-2 complexity class.
If iis uniform, ascending, useful, and strongly convergent, then there is a T2TB such that C() = C(i).
“Strongly convergent” in the theorem above can be replaced by “bounded by a continuous functional”.
The in the theorem above is not strong (locking detectable).