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Topic: On NP Completeness and Intractability
Replies: 90   Last Post: Nov 1, 2005 8:08 PM

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Randy

Posts: 19
Registered: 12/13/04
Re: On NP Completeness and Intractability
Posted: Oct 26, 2005 2:32 PM
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stephen@nomail.com wrote:
> In comp.theory Randy <joe@burgershack.com> wrote:
...
>>I think any algorithm that contains uncertainty (indefinite input) can
>>still be assessed for time or space complexity. Whenever the problem
>>cannot be solved, its complexity must remain undefined. However, it
>>seems to me that when the problem can be solved, its complexity can be
>>characterized and maybe quantified.

>
>
> Noone was talking about indefinite input. The Halting Problem
> is unsolvable and there is nothing indefinite or uncertain about
> the input.


The halting problem is entirely about what is known about: 1) the
program's input or 2) the program's search space. If there is no input,
or the input is completely known, or the search space is completely
known, then the program is deterministic and can be guaranteed to halt.

Halting is only undecidable when the input that is incompletely known
(e.g. infinite), or has an incompletely known effect on the program, as
in external instruction streams that might include or induce infinite
loops. Halting is also undecidable when the program's search space may
not include solutions that satisfy the program's given constraints and
is infinite, potentially causing the search to continue forever.

Not being a mathematician, I can only guess that at its essence, the
halting problem could be regarded as a function that fails because 1)
its domain is incompletely known or 2) its range is incompletely known.
If that simplification is fair, then undecidability is inextricably
linked to incomplete knowledge, ergo uncertainty.

This is where others who have explored the boundary between certainty
and uncertainty come in (in my example, a few AI folks). In practice,
it is possible to manage a program's inability to halt or its
potentially infinite search, just in the way that Operating Systems
detect deadlocks and kill nonresponsive processes that are stuck in
infinite loops. Just as in probabilistic algorithms, the criterion for
"halt" is redefined no longer to be "optimal", but "close enough". Such
algorithms will then have measurable asymptotic complexity measures.

Yes, the imposed limits on the algorithm artificially reshape its
complexity. But so what? Just pretend the core algorithm and its
keeper are one, and voila, you've built a compound program that 1) halts
and 2) has a complexity measure.

For fun, to confirm that input is not central to the halting problem,
try removing the word "input" from the Wiki entry:

http://en.wikipedia.org/wiki/Halting_problem

>
>>For example, a UTM that uses a small number of states will have a lower
>>time complexity than a UTM using (or that must explore) a larger number
>>of states.

>
>
> That seems backwards for UTM. In general the number of states a TM
> has will have nothing to do with the complexity of its running
> time. I really doubt UTM's are an exception.
>
> Stephen


My point is that the size of the problem's input space or search space
or even the UTM's execution space will dictate the effective runtime.
It the UTM is based on a quantum computer or a CA, which must explore
more states than another UTM to get the same job done, perhaps even
nondeterministically, then the speed of the machine will matter too.

Randy

--
Randy Crawford http://www.ruf.rice.edu/~rand rand AT rice DOT edu



Date Subject Author
10/25/05
Read On NP Completeness and Intractability
nimmi_srivastav@yahoo.com
10/25/05
Read Re: On NP Completeness and Intractability
Dr. David Kirkby
10/25/05
Read Re: On NP Completeness and Intractability
ianparker2@gmail.com
10/25/05
Read Re: On NP Completeness and Intractability
Pubkeybreaker
10/25/05
Read Re: On NP Completeness and Intractability
osmium
10/25/05
Read Re: On NP Completeness and Intractability
Pubkeybreaker
10/25/05
Read Re: On NP Completeness and Intractability
Willem
10/25/05
Read Re: On NP Completeness and Intractability
Bryan Olson
10/25/05
Read Re: On NP Completeness and Intractability
Willem
10/25/05
Read Re: On NP Completeness and Intractability
Ed Prochak
10/25/05
Read Re: On NP Completeness and Intractability
Dr. David Kirkby
10/25/05
Read Re: On NP Completeness and Intractability
Gene
10/25/05
Read Re: On NP Completeness and Intractability
MartDowd
10/25/05
Read Re: On NP Completeness and Intractability
RobertSzefler
10/25/05
Read Re: On NP Completeness and Intractability
Dr. David Kirkby
10/25/05
Read Re: On NP Completeness and Intractability
Pubkeybreaker
10/25/05
Read Re: On NP Completeness and Intractability
Wim Benthem
10/26/05
Read Re: On NP Completeness and Intractability
RobertSzefler
10/25/05
Read Re: On NP Completeness and Intractability
KP Bhat
10/25/05
Read Re: On NP Completeness and Intractability
stephen@nomail.com
10/25/05
Read Re: On NP Completeness and Intractability
tchow@lsa.umich.edu
10/25/05
Read Re: On NP Completeness and Intractability
googmeister@gmail.com
10/25/05
Read Re: On NP Completeness and Intractability
nimmi_srivastav@yahoo.com
10/25/05
Read Re: On NP Completeness and Intractability
stephen@nomail.com
10/26/05
Read Re: On NP Completeness and Intractability
tchow@lsa.umich.edu
10/27/05
Read Re: On NP Completeness and Intractability
stephen@nomail.com
10/25/05
Read Re: On NP Completeness and Intractability
stephen@nomail.com
10/25/05
Read Re: On NP Completeness and Intractability
Randy
10/25/05
Read Re: On NP Completeness and Intractability
C6L1V@shaw.ca
10/25/05
Read Re: On NP Completeness and Intractability
stephen@nomail.com
10/25/05
Read Re: On NP Completeness and Intractability
Wim Benthem
10/25/05
Read Re: On NP Completeness and Intractability
stephen@nomail.com
10/25/05
Read Re: On NP Completeness and Intractability
Randy
10/25/05
Read Re: On NP Completeness and Intractability
stephen@nomail.com
10/26/05
Read Re: On NP Completeness and Intractability
tchow@lsa.umich.edu
10/26/05
Read Re: On NP Completeness and Intractability
Randy
10/27/05
Read Re: On NP Completeness and Intractability
Arthur J. O'Dwyer
10/27/05
Read Re: On NP Completeness and Intractability
Randy
10/27/05
Read Re: On NP Completeness and Intractability
Willem
10/27/05
Read Re: On NP Completeness and Intractability
Randy
10/27/05
Read Re: On NP Completeness and Intractability
stephen@nomail.com
10/27/05
Read Re: On NP Completeness and Intractability
Randy
10/27/05
Read Re: On NP Completeness and Intractability
stephen@nomail.com
10/27/05
Read Re: On NP Completeness and Intractability
Randy
10/27/05
Read Re: On NP Completeness and Intractability
stephen@nomail.com
10/27/05
Read Re: On NP Completeness and Intractability
tchow@lsa.umich.edu
10/27/05
Read Re: On NP Completeness and Intractability
Randy
10/27/05
Read Re: On NP Completeness and Intractability
tchow@lsa.umich.edu
10/27/05
Read Re: On NP Completeness and Intractability
Joe Hendrix
10/27/05
Read Re: On NP Completeness and Intractability
Randy
10/27/05
Read Re: On NP Completeness and Intractability
Joe Hendrix
10/27/05
Read Re: On NP Completeness and Intractability
Randy
10/27/05
Read Re: On NP Completeness and Intractability
stephen@nomail.com
10/27/05
Read Re: On NP Completeness and Intractability
Arthur J. O'Dwyer
10/27/05
Read Re: On NP Completeness and Intractability
googmeister@gmail.com
10/27/05
Read Re: On NP Completeness and Intractability
tchow@lsa.umich.edu
10/27/05
Read Re: On NP Completeness and Intractability
Randy
10/27/05
Read Re: On NP Completeness and Intractability
Arthur J. O'Dwyer
10/28/05
Read Re: On NP Completeness and Intractability
Joe Hendrix
10/28/05
Read Re: On NP Completeness and Intractability
Ben Rudiak-Gould
10/28/05
Read Re: On NP Completeness and Intractability
Bryan Olson
10/28/05
Read Re: On NP Completeness and Intractability
Pekka Orponen
10/27/05
Read Re: On NP Completeness and Intractability
tchow@lsa.umich.edu
10/27/05
Read Re: On NP Completeness and Intractability
Randy
10/25/05
Read Re: On NP Completeness and Intractability
tchow@lsa.umich.edu
10/25/05
Read Re: On NP Completeness and Intractability
Michael J. Hennebry
10/25/05
Read Re: On NP Completeness and Intractability
stephen@nomail.com
10/25/05
Read Re: On NP Completeness and Intractability
Randy
10/25/05
Read Re: On NP Completeness and Intractability
stephen@nomail.com
10/25/05
Read Re: On NP Completeness and Intractability
Willem
10/25/05
Read Re: On NP Completeness and Intractability
Arthur J. O'Dwyer
10/25/05
Read Re: On NP Completeness and Intractability
googmeister@gmail.com
10/31/05
Read Re: On NP Completeness and Intractability
Oliver Wong
10/31/05
Read Re: On NP Completeness and Intractability
Robert Israel
11/1/05
Read Re: On NP Completeness and Intractability
Willem
11/1/05
Read Re: On NP Completeness and Intractability
googmeister@gmail.com
11/1/05
Read Re: On NP Completeness and Intractability
Willem
11/1/05
Read Re: On NP Completeness and Intractability
Torkel Franzen
11/1/05
Read Re: On NP Completeness and Intractability
googmeister@gmail.com
11/1/05
Read Re: On NP Completeness and Intractability
Oliver Wong
11/1/05
Read Re: On NP Completeness and Intractability
Willem
11/1/05
Read Re: On NP Completeness and Intractability
Robert Israel
10/26/05
Read Re: On NP Completeness and Intractability
Randy
10/26/05
Read Re: On NP Completeness and Intractability
stephen@nomail.com
10/26/05
Read Re: On NP Completeness and Intractability
Randy
10/27/05
Read Re: On NP Completeness and Intractability
Bryan Olson
10/27/05
Read Re: On NP Completeness and Intractability
Randy
10/27/05
Read Re: On NP Completeness and Intractability
Bryan Olson
10/27/05
Read Re: On NP Completeness and Intractability
Randy
10/25/05
Read Re: On NP Completeness and Intractability
Pat Farrell
10/25/05
Read On what "NP-Hard" means; was: Re: On NP Completeness and Intractability
Bryan Olson

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