computability in europe 2009 heidelberg germany n.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Computability in Europe 2009 Heidelberg Germany PowerPoint Presentation
Download Presentation
Computability in Europe 2009 Heidelberg Germany

Loading in 2 Seconds...

play fullscreen
1 / 11

Computability in Europe 2009 Heidelberg Germany - PowerPoint PPT Presentation


  • 82 Views
  • Uploaded on

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.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Computability in Europe 2009 Heidelberg Germany' - oberon


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
computability in europe 2009 heidelberg germany

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

complexity classes j hartmanis r stearns 1965

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.

the original question

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

the same question at type 2

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)

type 2 time bounds t 2 tb

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

in general the union theorem fails

Theorem:

In general the union theorem fails

There exists such that, for every ,

We need a more manageable sequence of T2TB

manageable sequence of t 2 tb

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:

bff is not a complexity class

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:

non union theorems

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:

union theorems

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).

type 2 big o notations

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).