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# Markov Chain

Markov chain is a relatively simple stochastic (working with probability) mathematical model that captures a "statistical structure" of a sequence of symbols (words, letters, numbers, musical notes, ...) and is typically used for predicting or generating such sequences. It can very well describe certain processes that occur in the real world such as behavior of some animals, Brownian motion or structure of human language. In programming Markov chains are pretty often used for generating "random" text that resembles given sample text -- Markov chain learns the structure of the sample text by computing probabilities of words following other words and then generates a new text so that these probabilities are preserved (of course, the new text will be a nonsensical gibberish from linguistic point of view, but it will resemble the template text statistically and "visually"). Chatbots and procedural generation are just some examples of where we might encounter Markov chains. This is however not where the use cases end -- an advanced use of Markov chain may be for example identification of the text author by the writing style, since it can be assumed that different authors prefer to use different words, phrases and idioms which will show in the generated Markov chain, and comparing these to the chain generated from the anonymous writing may give away the true author.

There are different types of Markov chains. Here we will be focusing on discrete time Markov chains with finite state space as these are the ones practically always used in programming. They are also the simplest ones.

Such a Markov chain consists of a finite number of states S0, S1, ..., Sn. Each state Si has a certain probability of transitioning to another state (including transitioning back to itself), i.e. P(Si,S0), P(Si,S1), ..., P(Si,Sn); these probabilities have to, of course, add up to 1, and some of them may be 0. These probabilities can conveniently be written as a n x n matrix.

Basically Markov chain is like a finite state automaton which instead of input symbols on its transition arrows has probabilities.

## Example

Let's say we want to create a simple AI for an NPC in a video game. At any time this NPC is in one of these states:

- Taking cover (state A):
  - 50% chance to stay in cover
  - 50% chance to start looking for a target
- Searching for a target (state B):
  - 50% chance to remain searching for a target
  - 25% chance to start shooting at what it's looking at
  - 25% chance to throw a grenade at what it's looking at
- Shooting a bullet at the target (state C):
  - 70% chance to remain shooting
  - 10% chance to throw a grenade
  - 10% chance to start looking for another target
  - 10% chance to take cover
- Throwing a grenade at the target (state D):
  - 50% chance to shoot a bullet
  - 25% chance to start looking for another target
  - 25% chance to take cover

Now it's pretty clear this description gets a bit tedious, it's better, especially with even more states, to write the probabilities as a matrix (rows represent the current state, columns the next state):

| A | B | C | D
--- | --- | --- | --- | ---
A | 0.5 | 0.5 | 0 | 0
B | 0 | 0.5 | 0.25 | 0.25
C | 0.1 | 0.1 | 0.7 | 0.1
D | 0.25 | 0.25 | 0.5 | 0

We can see a few things: the NPC can't immediately attack from cover, it has to search for a target first. It also can't throw two grenades in succession etc. Let's note that this model will now be yielding random sequences of actions such as [cover, search, shoot, shoot, cover] or [cover, search, search, grenade, shoot] but some of them may be less likely and some downright impossible (e.g. two grenades in a row). Notice a similarity to for example natural language: some words are more likely to be followed by some words than others (e.g. the word "number" is more likely to be followed by "one" than for example "cat").

## Code Example

Let's write an extremely primitive Markov bot that will work on the level of individual text characters. It will take a training text on input, for example a book, and learn the probabilities with which any letter is followed by another letter. Then it will generate a random output according to these probabilities, something that should resemble the training text. Yes, you may say we are doing a super simple machine learning.

Keep in mind this example is really extremely simple, it only looks one letter back and makes some further simplifications, for example it only approximates the probabilities with kind of a KISS hack -- we won't record any numeric probability, we'll only hold a table of letters, each one having a "bucket" of letters that may possibly follow; during training we'll always throw a preceding letter's follower to a random place in the preceding letter's bucket, with the idea that once we finish training, statistically in any bucket there will remain more letters that are more likely to follow given letter, just because we simply threw more such letters in. Similarly when generating the output text we will choose a letter to follow the current one by looking into the table and pulling out a random follower from that letter's bucket, again hoping that letters that have greater presence in the bucket will be more likely to be randomly selected. This approach has issues, for example regarding the question of ideal bucket size, and it introduces statistical biases (maximum probability is limited by bucket size, order matters, later letters are kind of privileged), but it kind of works. Try to think of how we could make a better text generator -- for starters it might work on the level of words and could take into account a history of let's say three letters, i.e. it would record triplets of words and then list of words that likely follow, along with each one's probability that we would record as an actual number to make the probabilities accurate.

Anyway with all this said, below is a C code implementing the above described text generator. To use it just pipe some input ASCII text to it, however make it reasonably sized (a few thousand lines maybe, please don't feed it whole Britannica, the output won't be better), keep in mind the program always trains itself from scratch (in practice we might separate training from generation, as serious training might take very long, i.e. we would have a separate training program that would output a trained model, i.e. the learned probabilities, and then a generator that would only take the trained model and generate output text). Here is the code:

    #include <stdio.h>
    #include <stdlib.h>

    #define OUTPUT_LEN 10000   // length of generated text
    #define N 16               // bucket size for each letter
    #define SEED 123456
    #define IGNORE_NEWLINES 1

    unsigned char charFollowers[256][N];

    int main(void)
    {
      srand(SEED);

      for (int i = 0; i < 256; ++i)
        for (int j = 0; j < N; ++j)
           charFollowers[i][j] = ' ';

      unsigned char prevChar = 0;

      while (1)
      {
        int c = getchar();

        if (c == EOF)
          break;

    #if IGNORE_NEWLINES
        if (c == '\n')
          c = ' ';
    #endif

        charFollowers[prevChar][rand() % N] = c; // put char at random place
        prevChar = c;
      }

      prevChar = ' ';

      for (int j = 0; j < OUTPUT_LEN; ++j) // now generate the output
      {
        prevChar = charFollowers[prevChar][rand() % N]; // take random follower
        putchar(prevChar);
      }

      puts("\n");
      return 0;
    }

Trying it out on the text of this wiki may output something like this:

    Ther thellialy oris threstpl/pifragmediaragmagoiby s agmexes, den
    atss pititpenaraly d thiplio s ts gs, tis wily NU gmarags gos
    aticel/EEECTherixed atstixedells, s s ores agolltixes tixe. TO: N
    s, s, TOpedatssth NUCAPorag: puffrits, pillly ars agmen No tpix abe
    aghe. aragmed ssh titixen plioix ag: Th tingoras TOD s wicipixe d
    tpllifr.edarenexeramed Thecospix ts ts s osth s pes ovipingor
    g: agors agass s TOnamand s aghech th wopipistalioiaris agontibuf
    ally Thrixtply tiaceca th oul/EEEEEEEECPU), wicth NU athed wen
    aragag athichipl Thechixthass s gmelliptilicex th ostunth gmagh
    atictpixe. ar Th on wipixexepifrag gman g: sthabopl/te.

We see at first glance it looks a bit like English text, even with some quirks specific to this wiki, for example here and there having FULL CAPS words (due to acronyms and also rants that often appear here). It even generated the word "CPU". Notice the algorithm correctly learned punctuation, i.e. it knows that after commas and periods there is almost always space and after space there is usually not another space. For comparison here is a Spanish-like text generated from Don Quixote (with accents removed):

    Diloma Dadro hacaci gua usta lesano strore sto do diaco; ma ro
    hiciso stue ue dita. do que menotamalmeci ma quen do gue lo;
    denestajo qucos rdo horor Da que qunca. quadombuce que queromiderbre
    hera ha rlabue F de querdos Dio macino; dombidrompo mi ste derdiba
    l, mbiolo Ferbes l ste s lolo que ha Du hano quenore Dio ueno que
    hala F uano he Dorame de qus rl, ha didesa que halanora Fla quco
    dil qucio ue do mestostaloste hados de gusta querana. stuce F s s
    Do lo dre s hal Fro gue sa sa. la sido la dico; hado mbuno Do.
    mororo; rdenaja. qunolole Diba. do. Fa gor stamestamo ha quno
    unostabro quero mue s Diado Didota. quencoralor dio sotomo Fuen
    que halora. gunore quabrbe rol gostuno hadolmbe Da que unendor
    que le di so; qunta rajos s F de qucol

We see some shorter words like lo, le, de, he, que and sido are real Spanish words. Though punctuation is quite nice, the algorithm fails to learn that after period the word of the next sentence should start with a capital letter (it only does so sometimes by pure chance) -- this is due to the algorithm only seeing one character back; after a period there is also one space which already makes the algorithm forget about the period. This would be addressed by keeping longer history, as said above. Now let's try a difference kind of text altogether, let's try to feed in the source code of Anarch:

      2  camechererea = 20;
    #erereppon.xereponioightFuaighe16_ARABEIUnst
    chtreraySqua->rarepL_RCL_CL_PE;
      caminsin.yDINeramaxer = costRCL_PERCL_ditsins->pL_ime1
       = 0;

      * = RCL_dime1,y 1)
      0;
    }
    }
      ck;
      camererayDimameaxSqua ca  = ca->ra caininin.xS_UAME;
      caminstFua-> 0 0;
    }   ca->ponstramiomereaxSquts  chts 154;
      1)

Here it's pretty clear the code won't work but its structure really does resemble the original source: curly brackets and semicolons are correctly followed by newlines, assignments look pretty correct as well, dereference arrows (`->`) appear too -- the code even generated the `RCL_` prefix of the raycastlib functions that's widely seen in the original code.

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