# Markov Chain

Markov chain is a relatively simple [stochastic](stochastic.md) (working with probability) mathematical model for predicting or generating sequences of symbols. It can be used to describe some processes happening in the [real world](real_world.md) such as behavior of some animals, Brownian motion or structure of a language. In the world of programming Markov chains are pretty often used for generation of texts that look like some template text whose structure is learned by the Markov chain (Markov chains are one possible model used in [machine learning](machine_learning.md)). Chatbots are just one example.

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](finite_state_automaton.md) which instead of input symbols on its transition arrows has probabilities.

## Example

Let's say we want to create a simple [AI](ai.md) for an NPC in a video [game](game.md). 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 (for example shooting 3 bullets in a row has a probability of 0.1%) 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](machine_learning.md).

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](kiss.md) 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](c.md) 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 learner 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;
}
```

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](raycastlib.md) functions that's widely seen in the original code.