That's good because then when we run imagine that we have some kind of satisfiability algorithm that runs in polynomial time it's running on a polynomial size input, so it's not like we're running a polynomial time algorithm on an exponential size input, which would take exponential time to run.

Let's say this is my polynomial, let me call my polynomial p of x.

NP hard and therefore a polynomial time solution for that problem solves everything in NP.

Formerly the class NP is the set of non deterministic polynomial time decidable problems that is to say that it's a problem that can be dissolved by a program that runs in polynomial time that has non deterministic elements in it.

First, it has to be NP hard, meaning that nothing in NP is harder then X. Which is to say that if you could solve X in polynomial time, you could use it to solve everything in NP in polynomial time.

We know that every problem that's in P is in NP and every problem that is NP is in EXP, that is to say any problem that you can solve in polynomial time, we can certainly solve in non deterministic polynomial time, and any problem that we can solve in a non deterministic polynomial time, we can also solve in exponential time, but here's what we don't know.

I have this polynomial in the denominator here.

That problems that are in NP don't necessarily require exponential time, but they may not be solvable in polynomial time either we don't know.

low order polynomial such as a plus one, when we really needed a higher order polynomial to fit the data.

All right, so now we're in a good position to define the set p of polynomial time decidable problems.

Sometimes a quadratic polynomial, or just a quadratic itself, or quadratic expression, but all it means is a second degree polynomial.

So the problems that we can solve in a non deterministic polynomial time might be exactly the same as the ones that we can solve in polynomial time, which both would be then different from exponential time or could very well be that there are really three different categories here.

Let's say I'm defining, so this is a polynomial.

An accepting certificate is information that we can use to quickly show that the answer to the decision problem is yes, (if it actually is yes) and here, short means polynomial size and quickly means polynomial time.

There's one other thing that we know, we do know that there really is a difference between polynomial and exponential time.

So a binomial is just a polynomial with two terms.

And I don't know exactly what this third degree polynomial

I've chosen the degree d of polynomial using the test set.

And this polynomial we're going to do, we're going to keep adding terms to the polynomial, so that we can better and better approximate this function.

That is to say the set of problems that we can solve in a non deterministic polynomial time might be exactly the same as the ones that we can solve in exponential time.

because you can then include all those polynomial terms of x1 and x2.

And that's what's actually called a quadratic equation, or this second degree polynomial.

Let's say you try to choose what degree polynomial to fit to data.

And they tell us a couple of the 0's of this polynomial.

Now a third degree polynomial can have as many as three 0's.

So the total running time is like n to the k, which is a polynomial times 2 to the n to the c, which is exponential.

The other choice is no, P is not contained within NP because if the problem is NP, it means it's decidable in polynomial time and we don't need any kind of certificate and so, it's not an NP and then finally the opposite of that is, yes if it can be decided in polynomial time, no certificate is needed.

So there's a sort of outer set and that's distinct from say this inner set, which is the set of problems that are solvable in polynomial time.

A root of a polynomial is the x coordinate of one of its x intercepts.

After every step we're canceling out the largest degree of the polynomial we're dividing into.

look at how well my fifth order polynomial hypothesis, had done on my test set.

Might want to review factoring a polynomial, if you found that a little confusing, that step.

The simplest polynomial is just a constant, and it would just be a horizontal line someplace.

And by large the algorithms for which that we can actually solve in a reasonable amount of time, all seem to be in this category of having a polynomial bound.

For example, if you use polynomials this might be a high degree polynomial over here and maybe a linear function over here which is a low degree polynomial your training data error tends to go like this.

In that case, given that we're fitting a high order polynomial, this is a usual overfitting setting.

And so this degree of polynomial, so the parameter is no longer fit to the test set.

like the degree polynomial to use with the learning algorithm or choose the regularization parameter for learning algorithm.

If we have a very high degree of polynomial, our training error is going to be really low.

That is, if d the degree of polynomial was too large for the data set that we have.

If you have a polynomial, you could have more than one values of x that satisfy this equation.

We can also try adding polynomial features things like x2 square x2 square and product features x1, x2.

And that is lead theoreticians at least to find efficient in one particular way sometimes and that is that an algorithm is efficient if it has a polynomial bound on its running time.

All right, so to do the last little bit of this proof to show that 3 colorability is NP hard, we're going to show that if we had the ability to solve 3 colorability problems in polynomial time, then we could solve three SAT problems in polynomial time as well, and so what we need to be able to do to show that is if you walk up to me with any 3 CNF formula,

So the easiest way to approximate it is to say, well, the simplest polynomial is just a constant, right?