Fun in mathematics!

Are we rational? Better rephrase it, am I rational? Its a very good question. But if I try to investigate from philosophical point of view then the very next question is “What is rationality?”. Couple of days ago I was working with free will paradox or Newcomb's paradox. It is a famous old problem. Do we have free will? Well, why not approach this problem in a completely different fashion. Let say, we got a guy, very religious. To be more specific we allow a religious Muslim guy to play the game where the fictitious being is God. Obviously we know the answer. This is because our player believes that the being knows his moves pretty accurately. Arguments for choosing both of the boxes not gonna work here. Because in our case the being is pretty accurate and more strong argument is our player believes that the being accurate.

Now the question becomes, is our player rational? Well, I am gonna argue that our player is completely rational and he made the maximum return out of the game. To prove that our player made the maximum return out of the game all we have to show is his decision was rational. If he is a rational person then obviously the return is maximum [Game Theory]. Now the whole things turns into rationality of our fictitious player. If I take the favor defining rationality in such a way that the model of rationality proves our player rational then we are done.

Every human being can be characterize by few parameters. The value of those parameters may vary from man to man. Some try to achieve materialistic goal some go for the spiritual. It differs from person to person. No one is more rational than other. Everyone is rational from their own perspective. Hence, by carefully choosing those parameters for our fictitious player we can make him religious which is obviously rational also. Now using this model for rationality we can go a step ahead of it. Every interaction between human and other entity can be define by their individual rationality. Hence if we can model rationality for each of the entity in this universe then a suitable mathematical model of this universe can be obtained. One the other way if we can model the primitive stage then the current world model can be derived after finite number of derivation steps. Therefore, there is a possibility for the mathematical model of this universe.

[All models are wrong but some are useful. - George E. P.] :D

Reversible Computing & Cellular Automata

As part of Formal Languages course I had to read a paper, present it and write a report on it. The title of the paper was, “Reversible Computing and Cellular Automata – A survey” by Kenichi Morita from Hiroshima University. He is one of the top most scientists of theoretical computer science.

The paper was very interesting and I knew very little about these topics before. It was a survey paper on reversible computing and cellular automata. I would like to share my presentation and report here. As this is my first presentation here outside my research group. Anyway the topics were very interesting. I hope you enjoy the topic.

Here they have reports with almost every course. This is kind of new thing for me. CSE BUET undergrad courses are in very good standard (I believe more than they do here in undergrad) but they don’t have any report along with the courses. The only report I had submitted in my undergrad is my thesis report.

Turing Machine

I knew about Turing Machine from my first semester. But for some reason I feel like I should compile the basic characteristics of Turing Machine. It is worth to share here, because Turing Machine is one of the most fundamental concepts of computer science. I tried to put the very basic things about it.

Turing machines, first described by Alan Turing, are simple abstract computational devices intended to help investigate the extent and limitations of what can be computed [1].
1. A Turing machine is a kind of state machine. At any time the machine is in any one of a finite number of states. Instructions for a Turing machine consist in specified conditions under which the machine will transition between one state and another.
2. A Turing machine has an infinite one-dimensional tape divided into cells. The tape has one end, at the left say, and stretches infinitely far to the right. Each cell is able to contain one symbol, either ‘0’ or ‘1’.
3. The machine has a read-write head, which at any time scanning a single cell on the tape. This read-write head can move left and right along the tape to scan successive cells.
4. The action of a Turing machine is determined completely by
      a. The current state of the machine
      b. The symbol in the cell currently being scanned by the head and
      c. A table of transition rules, which serve as the “program” for the machine.
5. Each transition rule is a 4-tuple: < State_curr, Symbol, State_next, Action > which can be read as saying “if the machine is in state State_curr and the current cell contains Symbol then move into state State_next taking Action”.
6. The actions available to a Turing machine are either to write a symbol on the tape in the current cell, or to move the head one cell to the left or right, which we will denote by the symbols « and » respectively.
7. If the machine reaches a situation in which there is not exactly one transition rule specified, i.e., none or more than one, then the machine halts.

Some of the best sources:
[1] Turing Machine, Stanford Encyclopedia of Philosophy.
[2] Turing Machine, Wolfram Math World.
[3] Turing Machine, Famous or Infamous Wikipedia.
[4] Turing Machine, Lego Pages.
[5] Turing Machine, Encyclopedia Britannica.