What is #P (Sharp P)

#P is the set of counting problems whose corresponding decision problem is in NP.  Or more formally (from here):

A function [math]f:left{0,1right}^*rightarrow mathbb{N}[/math] is in #[math]bf{P}[/math] if there exists a polynomial [math]p: mathbb{N}rightarrow mathbb{N}[/math] and a polynomial-time TM (turing machine) such that for every [math]xin left{0,1right}^*[/math]

[math]f(x)=left|{left{yin left{0,1right}^{p^{(|x|)}}:M(x,y)=1right}}right|[/math].

I later intend to find a proof or develop a proof myself that shows that the number of subset sums problem is in #P.

No Free Lunch (NFL)

My friend Indranil introduced me to the theory of “No Free Lunch”.  From Wikipedia article:

A conventional, but not entirely accurate, interpretation of the NFL results is that “a general-purpose universal optimization strategy is theoretically impossible, and the only way one strategy can outperform another is if it is specialized to the specific problem under consideration”

 

It is definitely something that I must keep in mind whenever doing research.

OpenCV update and Octave

Well I had fun learning some computer vision stuff and going through the openCV tutorials yesterday.  Realized just how little we know about vision. Pretty much every idea I had was either done (since it was simple) or too complicated and could be a thesis.

Today I am going to brush up on my octave and maybe write a machine learning algorithm.