A good reading from WikiBooks

Artical from: http://en.wikibooks.org/wiki/Logic_for_Computer_Scientists/Introduction

Please try not to repost this article somewhere else! thanks --gcseed

In order to define datatypes and to derive efficient implementations for it, the concept of abstract datatype definitions is central. The idea is to define the abstract properties of a datatype, instead of giving special realisations. A very trivial example is the definition of a stack: Let  be an alphabet. In order to define a stack  over  we assume  to be a nullary function and we define the following properties of stacks:

Hence, we have the functions  and , which yield stacks and the predicate , furthermore we need the following properties with respect to the -operation.

And we have to give properties with respect of the combinations of functions:

The above formulae state what properties we expect stacks to have. Obviously it contains no hints how to implement such a data type. And indeed this specification is aiming to solve other problems, e.g.

• Is the specification correct? I.e. is there a set  together with the given operations, such that the axioms above hold?
• Is the specification complete. I.e. do the axioms imply all the properties we intuitively assume a stack to have? Are there sets  which do not meet our expectations?

The reader may have already noticed, that the axioms are nothing else than formulas in predicate logic, i.e. all logic where variables like  or  together with so called quantifiers  and  are used. The above two questions, namely correctness and completeness, are very central topics for the design of formal systems in logic. The proof of these properties often is difficult and costly, but on the other hand it is one of the clear advantages of logical systems, that these properties can be proved formally. In the main part of this course we will deal with these questions explicitly.

Missed A2

Since I have missed my 2nd assignment of the course, I will do Q2 here instead

P(n): for every positive integer m, if m < n then T(m) ≤ T(n)
Using complete induction prove that P(n) holds for all integers n ≥ 1
Let k be an arbitrary integer such that k ≥ 1. Now suppose that P(β) holds for all integers β such that 1 ≤ β < k.
Then prove that P(k) holds as well.
By cases:
Case 1. k = 1. P(1) holds trivially since there is no positive integer m < 1.
Case 2. k = 2. T(1) and T(2)
T(1) = 1
T(2) = 1 + max{T(1), T(1)} = 1 + T(1) = 2
Since T(1) < T(2) then P(2) holds!
Case 3. k > 2.
In this case, 1 ≤ (k – 1) < k and, by Lemma 3.3 (in the course notes), 1 ≤ ⌊k/2⌋ ≤ ⌈k/2⌉ < k.
Therefore, by induction hypothesis, P(k − 1), P(⌊k/2⌋) and P(⌈k/2⌉) all hold. Since P(k − 1) holds, to prove that P(k) holds as well, it suffices to prove that T(k − 1) ≤ T(k).
So,
T(k – 1) = 1 + max{ T(⌈(k-1)/2⌉),
T(⌊(k-1)/2⌋) }
≤ 1 + max{ T(⌈k/2⌉), T(⌊k/2⌋) }
= T(k)
T(k – 1) ≤ T(k)

An easy question from the past assignment, and some others

1.      P(n): n³ + n < 3ⁿ. I will prove that P(n) is true for every natural number. (IH)

# I observe that from P(4) and beyond, it is always true.

Proof (by Mathematical Induction)

Base Case: n < 4: P(n) for n  {0,1,2,3}. From the chart above, it is easy to tell that P(n) is false for n {2,3}, so none of these two values can be reached using the logic of the induction step.

Induction Step: Assume n  ℕ - {0,1,2,3} and that P(n) is true.

Then

3 · P(n) = 3 · (n³ + n) < 3 · 3ⁿ  # times 3 on both sides

    n³ + n³ + n³ + 3n < 3ⁿ⁺¹

And from IH, P(n+1) = (n+1)³ + (n + 1) < 3ⁿ⁺¹

= n³ + 3 · n² + 4n +2 < 3ⁿ⁺¹

Since: n³ = n³

                n³ > 3 · n² # n is natural number – {0,1,2,3}, which means n > 3

           n³ > 4n # n is natural number – {0,1,2,3}, which means n > 3

           3n > 2 # n is natural number – {0,1,2,3}, which means n > 3

Then n³ + n³ + n³ + 3n > n³ + 3 · n² + 4n +2 ⇒ 3 · P(n) > P(n+1)

So that we can conclude:  3 · 3ⁿ  > 3 · P(n) > P(n+1)  ⇒ 3ⁿ⁺¹ > P(n+1)

So P(n+1) follows

Since I assumed n is a generic natural number greater than 3, ∀n ℕ - {0,1,2,3}, P(n) ⇒ P(n+1).

I conclude ∀n ℕ - {0,1,2,3}, P(n), by Mathematical Induction.

I am back in Toronto!

After 12 days, I am finally back to Toronto! Hopefully I will catch up the lectures and tutorials questions fast! From the lecture notes, seems that I have missed the materials based on Master Theorem, and some reviews in the past 2 weeks.

I applied the Master theorem to a recurrence for a running time I encountered 

T(n)=4T(n/2)+O(r)

r is independent of n. Case 1 of the Master theorem applies and tells us that T(n)=O(n2).

However, this hides a constant dependent on r in the big-oh notation: our recurrence has depth O(log2n) so at the final level we have O(4log2n)=O(n2) subproblems, each of which takes O(r)time to be handled. This means the actual running time is O(n2r) (or worse: this analysis only talks about the lowest level).

This is my actual recursion:

T(n)=r2T(n/r)+O(nr2)

Leaving Toronto for awhile!

Since there's some personal issues, I will be leaving Toronto for about 10 days. That since Blogger does not work in China (blocked like facebook and youtube), which means I will not be update my SLOG for about 13 days. See you all when I am back!

Some thoughts after the midterm

First of all, I have to say reading the questions carefully is more important than understanding the question. e.g. Q2 on the midterm, the question was asking number of leaves instead of number of total nodes in the tree, which makes the question lot easier by only asking the lowest level of the tree. From the observation, the worst-case is when we have a full binary tree, where it's lowest level always has 2^(n-1) leaves.

Second, any non-full binary trees will be smaller than full binary trees, therefore we can prove in both cases, P(n) is true!

Finally! My first SLOG update!

After a long and busy week, finally, my SLOG is here live! Assignment #1 of CSC236 was extremely hard, which is mainly because I was trying to "prove" or "explain" everything instead of writing down a clear and easy understanding idea. Already, jump right into the topic: such as question 4 on the assignment, " use complete or mathematical induction  to prove that any binary string that begin and ends with the same bit has an even number of occurrences of substring from {01, 10}."

Prof Heap has mentioned that he prefer mathematical induction in class, so I tried to think in that direction, and then came up with the following:

P(n): any binary string n, if its first digit n[0] is same as the last digit n[-1], then it has even number of occurrence of {01, 10}. (IH)

Proof (by Complete Induction)

Special Case (Base case): When len(n) = 0, then length of the binary string n is 1, which implies that n is a single-digit string {0, 1}. Since a single-digit string can never include {01, 10}, so that the occurrence of {01, 10} is 0, which is even.

E.g. S = 1 or S = 0.

Induction Step: Assume len(n) > 0, and that for every natural number 0 < j < len(n) then (n[:j]) is true.

Case 1: n[0] = 1 and n > 0: then n[0] and n(1) both {0, 1}.

Subcase 1.1: n[0] = n[1] then since n[0]= 1 (mentioned in Case 1), then n[1] = n[0] = 1, which gives the first 2 digits of n is “11” that ∉ {01, 10}, so the first two digits of the binary string n contains 0 of {01, 10}. And we slice n = n[1:] , then move onto the next digit

Subcase 1.2: n[0] ≠ n[1] then since n[0]= 1 (mentioned in Case 1), then n[0] ≠ n[1] = 0, which gives the first 2 digits of n is “10” that {01, 10}. so the first two digits of the binary string n contains 1 of {01, 10}. And we slice n again so that n = n[1:] , then move onto the next digit

Case 2: n[0] = 0 and n > 0: then n[0] and n(1) both {0, 1}.

Subcase 2.1: n[0] = n[1] then since n[0]= 0 (mentioned in Case 1), then n[1] = n[0] = 0, which gives the first 2 digits of n is “00” that ∉ {01, 10}, so the first two digits of the binary string n contains 0 of {01, 10}. And we slice n = n[1:], then move onto the next digit

Subcase 2.2: n[0] ≠ n[1] then since n[0]= 1 (mentioned in Case 1), then n[0] ≠ n[1] = 1, which gives the first 2 digits of n is “01” that {01, 10}. So the first two digits of the binary string n contains 1 of {01, 10}. And we slice n again so that n = n[1:] , then move onto the next digit

    Since binary string n contains 1 of {01, 10} every time there is a different digit in the string, and I assume P(n), which means n[0] = n[-1], and it has to have ∃x ℕ, there are 2x times of occurrence of digit that is not the same as n[0]. In addition, if n[0] = n[1] = n[2] = ··· = n[-1], there will be 0x times of occurrence of digit that is not the same as n[0]. Which 2x and 0x are both even numbers. I conclude that for all non-empty binary string n, P(n).

And as you can see, I used the complete induction instead, which result in having a loop case in the proof! I wish I can find out where it went wrong.