Week 11

Of course the professor saves the best for last. Or not.

This week the stuff on halting function and countability I felt like were a little confusing. I don't know if it's summer but I am rather confused. The idea of not having a proof or solution to solve a halting function really baffled me not just conceptually but also mathematical (proof-wise). It really baffled me that no one has yet to create such a function though it seems rather like an easy question. As for the proof of it, I really have no clue on how to use reduction to show that a certain function is non-computable knowing halt is impossible to compute. Maybe I just need to see more examples for this to be clear but definitely just hearing it once in lecture isn't good enough.

And for the problem-solving episode, I will use the product of sums problem. The problem states to find the maximum product for the sum of a certain number. My approach to this solution was through trial and error. At first I knew that number of its square always produce the largest sum and thought this was correct. However, a sum of [2, 3, 3] or (8) is 18 and not 16. From this example, I used the idea of having the longest possible length for your sum of the number and having each number in the list different from one another by exactly one or the same. So then I used example, [3, 3, 3, 3] or (12) and found this to also work for my solution as well as [3, 3, 3, 3, 3] or (15). Here this I knew the general idea to finding the solution to the problem though to be honest I wasn't able to come up with some sort of general expression.

And it was a fun time blogging. And I hope you enjoyed this as much as I did. :)

Week 10

So this week really wasn't a challenge. The stuff on Big-Omega, solving a problem for two functions, bounded functions and the limit property was pretty straight forward. Specifically the contents as Big-Omega as it was literally the same as Big-Oh except one sign was just switched around. Bounded proofs was like using Big-Oh but just solving for an upper and lower bound instead of just one one bound. Limits is just basic calculus.

Hence this week was rather relaxing for me. :)

Week 9

For some reason the contents this week seemed to make more sense to me than last.

At first I was a little bit confused on what and how to do these proofs and I was little confused with the idea of over and under-estimating certain terms when doing this. But all in all it's just really showing for what values of f(n) that is greater than (for big-O) or less than (for delta), therefore we can just overestimate/underestimate and omit certain values to make the equality still true. Proofs with complexity I find is not as bad as I've heard it to be. It seemed to have made more sense when not dealing with the amount of steps in a piece of code (for me at least).

Week 8

So this week we continue doing more of the complexity stuff but this time we introduce proofs into them.

The stuff on proofs using complexity I find is rather confusing. I find it the main problem for me is how to come up with a general expression from a piece of code.

Next, we did stuff on slicing and how it would affect run time. I am actually still rather confused about how the idea of how certain run times work and how many steps are involved in this. And as stated above I find it confusing to form a general expression for number of steps for a piece of code.

Week 7

Reading week was nice and only another half a semester to go. However, remaining of the semester I heard is rather difficult when we begin to do proofs with complexity.

So this week, we finished off the using proofs structures and were introduced a new way called proof by cases. My first thoughts on this was it really wasn't that difficult. It was really simply proofing a certain claim for more than one cases but in one statement. So overall, the proofs I feel relatively fine with.

For rest of the week, we began determining the run time of certain code (otherwise called complexity). I'm still rather slow on how many steps and what is meant by the upper bound and lower bound when doing these; so I am rather a little bit scared for this in the future.

Week 6

And now for the fun part. From doing proof structures, now we begin the actually proofs themselves.

So far this week, the professor has showed us the basic ideas of how to prove a certain proof. The proofs themselves really are just a bunch of algebraic manipulations to make the antecedent and consequent the same; which seems relatively easy for now. Some of the other things we started was using the contrapositive to prove a claim because it simplifies proving the claim as opposed to using direct proof (solving the proof as it is on the question). We also started slightly on the proof structures for the existential. Like the universal proofs, the structure is rather very simple to follow and as for the proofs for the existential, we get to set variable(s) -- which shows that it is in domain -- then substitute that variable into the claim and show that claim is true.

Some of the proof stuff is rather strange, because we need to define certain words when proving a claim. For example, in certain proof it is necessary to show that the definition of an odd number is n = 2k + 1. But as for the proofs so far, they seem a little bit understandable but I'm not so confident in my abilities to solve them.

Week 5

Here is where I hear things begin to get interesting. The beginning of proofs. Several of friends have told me that the content on proof is where things seem to get rather strange.

My thoughts so far on the introduction of proof is rather simple; we started the small details of proofs such as writing them with structure first to get the hang of them before actually proving certain claims. The basic idea of the proof structures are to assume certain parts of the claim such as the what set x (just a random variable) belongs to and the antecedent. This is to apparently introduce the set as well as the implication (which is written at the bottom from basic structuring) which is then eventually proves the consequent is true.

So far, so good but I am rather scared.