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About Me



So You want to know about me. I am Soham Chatterjee. A 20-year-old Math and Theoretical Computer Science enthusiast Bengali guy. Doing my Bachelor's Degree in Math and Computer Science at Chennai Mathematical Institute. I am not a very interesting guy. I love Anime and songs from retro Bengali, Hindi, western, and J-pop songs. Also, I am a devoted Linux user. This is my blog where I post on math, theoretical computer science, LaTeX, and some Linux stuff.



My favorite Animes are Death Note, Fullmetal Alchemist Brotherhood, Steins Gate, Naruto, Eighty-Six, and Bakuman. Apart from this, I like many animes. They are all special to me in different ways. You check my Anilist account to see which animes I watched till now. You can also follow me there.

My favorite musics are Shiritakunakatta by  Sawai Miku and Avid by Sawano Hiroyuki. But apart from this, I like Yoasobi, Kobukuro Backstreet Boys, and Charlie Puth a lot. For music, you may think I am old-fashioned but I like to keep an offline collection of all the songs I like to listen to. 

Among Movies my favorites are Kimi no Suizou Wo Tabetai and Koe No Katachi. Then also comes Wolf Children, Your Name.

Apart from this, I like to write beautiful documents in LaTeX and theme my desktop in Linux. I have talked about my setup in detail in the respective LaTeX and Linux section. You may go there or take the .tex file and the dotfiles from my GitHub. 

I also had an Instagram Handle named Creative Math Solving where I used to post Math Problems based on Olympiads and a little on Real Analysis. You can follow that page. 

Since you have wanted to know about me this much then you can also follow me.





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Function Automata: A Technique to Solve Finite State Automata Problems

Here I will describe a strategy or technique to solve some finite state automata problems. This strategy was first taught to us by my Theory of Computation Professor. The term 'Function Automata' is given to me since we use function-like structures in this.     Suppose you are given a language that is a regular set \(L\). So there exists a \(DFA\) for \(L\), \(L_D=(Q,\Sigma,\delta, q_1, F)\) where \(Q\) is the set of states, \(\Sigma\) is the set of alphabets, \(\delta\) is the transition function, \(S\) is the starting state and \(f\) is the set of final states. Now let \(Q=\{q_1,q_2,\dots,q_n\) where \(n\in\mathbb{N}\). Now we will do a kind of subset construction but every new state will have \(n\) states of \(Q\) but they can repeat. So $$(\underbrace{q_1,q_1,\dots,q_1}_{n\text{ times}})$$ is a state of the new automata. Now, what is the meaning of these new states? Let \(f=(q_{k_1},q_{k_2},\dots,q_{k_n})\) be a new state where \(q_{k_i}\)'s are not necessary to be diff...

Hierarchy Theorems (Time, Space, Alternation)

There are many hierarchy theorems based on Time, Space, Nondeterminism, Randomization etc. Time Hierarchy Theorem (Version 1) : If $f, g$ are time-constructible functions satisfying $f(n) \log f(n)=o(g(n))$, then $$ DTIME(f(n)) \subsetneq DTIME (g(n)) $$ Proof:  Consider the following Turing machine $D$: "On input $x$, run for $f(|x|)\log (f(|x|))$ steps the Universal $TM$ $\mathcal{U}$ to simulate the execution of $M_x$ on $x$. If $\mathcal{U}$ outputs some bit $b \in\{0,1\}$ in this time, then output the opposite answer (i.e., output $1-b$ ). Else output 0.” Here $M_x$ is the machine represented by the string $x$.      By definition, $D$ halts within $f(n)\log (f(n))$ steps and hence the language $L$ decided by $D$ is in $DTIME(g(n))$. Claim:  $L \notin$ DTIME $(f(n))$ Proof:  For contradiction's sake, assume that there is some $TM$ $M$ and constant $c$ such that $TM$ $M$, given any input $x \in\{0,1\}^*$, halts within $cf(|x|)$ steps and outputs $D(x)$....

Gauss-Lucas Theorem

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