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## Quine theory

As stated before, any Turing-complete language admits (at least) one quine.

The complete proof is quite long, but if we rely on some theorems, we can introduce a kind of short proof that would help us to understand how to code a quine, including a BF quine.

## Definitions and theorems

*Note: these definitions and theorems might not be the exact ones but a reduced version, enough to understand the proof*

- A computable function
*N*is a function that can be computed using a Turing-complete language. We will use functions with integer parameters (0 or more) - Any computable function
*N*can be related to an integer*n*in an*injective*way (one function : one integer; and one integer : at most one function)- E.g. our source code can be considered as a set of chars, themselves a huge amount of bits, aka a number in its binary form

- The
*Universality theorem*states that a computable function exists, named*U*, such that*U(n)*does the same job than*N()*- This is what we called an
*interpreter*. - In other words, from a number, we can execute the corresponding program, if it exists (remember:
*injective*way, so some integers may be invalid)

- This is what we called an
- The
*parameterization theorem*, also called*smn theorem*states that a computable function*S*such that, for any integers*n*and*m*,*s(n,m)*does the same job than*N(m)*- In this application,
*m*is a fixed parameter of*N*

- In this application,
- Let's call
*H*a computable function over (valid) integers, called transformation- In other words, we transform a program identification number into another program's one

## Let's start

Let's first prove there H admits a fixed point

- Consider a computable function
*N*and its identification number*n* - Consider
*S(n,n)*, the computable function obtained when executing*N(n)*,*N*with a fixed parameter value which is its one identification number*n* - Consider
*H(S(n,n))*, the computable function obtained when transforming by*H*the computable function*N(n)* - Let's have an integer
*m = H(S(n,n))*. Such*H(S(n,n))*is a computable function, with a single parameter*n*, and this function will be called*M*, identified by its number*m* - Thanks to the universality theorem, if we know
*m*, we can execute*M* - Now, consider
*M(m)*- By definition,
*M(m) = S(m,m)* - By construction,
*M(m) = H(S(m,m))*

- By definition,
- Conclusion :
*S(m,m)*is a fixed point of computable function*H*

## Apply to quine

Let's now say *H* is the function with a parameter *n*, that prints source code of program *N*.

We proved that there is at least one computable function, named *quine* that is a fixed point of *H*, so *quine* and *H(quine)* do the same job.

What does *H(quine)* does ? It displays *quine* source code. So *quine* does the same, and prints its own source code.

## Quine construction

In order to get a quine, we saw that we need a function *M*, given by its value *m*. This means we needs in theory an interpreter to do that.

Actually, we can restrict this to an interpreter able to interpret *m* which is *H(...)*. So our source code needs to be able to interpret *H* at least - other functions are not really mandatory.

And as our *H* is "Print source code of given program", then it's quite easy to implement.

- First, we need to declare a
*data*part in our code - Then a function H to print the
*data*as*data declaration* - Finally, print the
*data*as*contents*