image transcription:

an image incorporating two famous memes. on top is the title “learning about Σ* in theory of computation.”

in the centre is a close-up of Chad face – often used when talking about sigma males – cropped in a five-pointed star shape.

below are two soyjaks pointing towards the aforementioned Chad face. those soyjaks are labeled “me” and “my brain”.

• This is just a representation of your feelings in Big-O notation.

• neither an understatement nor an overstatement, a Big-Theta

• Sum asterisk or something? 🙃

• sigma star.

if you haven’t heard about theory of computation, let me define some keywords:

• symbol: smallest unit. denoted by any character(a,b,c, etc.) or number (0,1, etc.)
• alphabet: set of symbols. denoted by Σ(sigma). e.g.: {a, b, c}
• string: sequence of symbols. eg: a, aa, aaab, etc.
• language: set of strings. e.g.: {a, as, aaab, …}

now, sigma has powers. Σ² is set of all strings of length 2. e.g.: {aa, ab, bb, …}. you can generalise this to Σ^n.
Σ* is union of all powers of sigma. i.e., Σ¹ + Σ² + …

so, a language is basically a subset of Σ*.

as for why theory of computation even exists, you basically try to define what a computer can/cannot do.
and you try to mathematically define a computer. then you try to define what a language is(in case of programming , you need it to form languages and compilers). hence the need for this.

• It’s funny, I never associated formal languages as part of the theory of computation. We only learnt about them from the perspective of automata/state machine theory

• Automata and formal languages were pretty much my entire “Theory of Computation” class. It’s what’s in Sipser.

• Didn’t you go into Turing machines and the Halting problem from that?

That was my intro into computation: regex, automatas, state machines, stack state machines, formal languages, grammars, Turing machines, Hanting Problem, P NP.

• No, we went Automata, Finite State Machines, regex, grammars, set-theoretical and other mathematical formalisms

• Can images work as formal Languages?

• A tree can be seen as a formal language. Look into L-systems.

If you generalize what a symbol is (the rgb value of a pixel) you can write a grammar that ends producing a list of pixels. You can then place it in a 2d matrix and you have an image.

I guess a better approach would be wave function colapse, but seems to me like it could be formally described as a grammar (CS or CF, dunno, would have to look into it)

• wait until you learn about sigma-algebras in measure theory