# Grr

Grr is a pure textual programming language that composes rules and functions inspired in macro and high-order programming. Exist two ways of define rules in Grr, first using a simple substitution in x to y (x <- y) or make a function f that receive x and return y (f(x) <- y), however in the interpreter language they are all same substitution procedure. Grr is by default a recursive language, then rules can be replaced by the same rules (y <- y).

## Examples

### Hello World

``` [Ώ <- T
T <- HelloWorld]
```

### Naturals Numbers

The code below encodes natural numbers through of recursive functions.

```[
Ώ <- MUL(FOUR)(FOUR)
FOUR <- INC(INC(INC(INC(N))))
N* <- *
INC(Y) <- Y*
DEC(K) <- K-
*- <- λ
MUL(X)(Y) <- Y-REPEATZ(X)
*REPEATZ(X) <- REPEATZ(X)X
REPEATZ(X) <- X
]
```

## Reserved Notations

In grr there are some reserved notations that formalize some conceptions that grr leak, Ώ is the main function where the rules can be decomposed initially, λ is an empty string and brackets construct nested rules that control backing recursive rules. Comments can be inserted with space after the replaced rule.

```[ (1)
P <- F(F(x))x (x <- λ) (F(y) <- y) (only two rules) rules only of (1 + 1) = (2)
x <- λ
[ if a string down in a rule here (2 + 1) = (3)
F(y) <- y  (x <- K) (if a string down here the string gain a new rule of x)
[ (3)
x <- K
]
]
]
```

## Corresponding calculation

Grr rules are just string substitution, therefore are high-order objects like lambda functions, then it's possible to implement easily a lambda term, for example, SK combinators.

``` s(x)(y)(z) <- x(z)(y(z))
k(x)(y) <- x
i(x) <- s(k)(k)(x)
```

### Turing Completeness

This program below is Brainfuck interpreter (10 cells limit) :

```[
Ώ <- BRAINFUCK(>+<+++{-})
Δ* <- * *Δ <- * *α <- λΔ α* <- λΔ INC(Y) <- Y* DEC(K) <- Kα
ADD(X)(Y) <- XY MUL(X)(Y) <- Y-REPEATZ(X) *REPEATZ(X) <- REPEATZ(X)X REPEATZ(X) <- X
FIVE <- INC(INC(INC(INC(INC(Δ)))))
HUNDRED <- MUL(TEN)(TEN)
THOUSAND <- MUL(HUNDRED)(HUNDRED)
MULK(_)(X)(Y) <- _DEC(DEC(Y))REPEATZ(X)
BRAINFUCK(x) <- 'BF(TAPE(x))'G___K
[
TAPE(x) <- MULK(kPOINTER(x)(Δ))(POINTER)(TEN)
kPOINTER(x)(y) <- yx|T(x)(y)(λ)(Δ)
POINTER <- T(#)(Δ)(λ)(Δ)
ç> <- move_cell ç< <- back_cell ç+ <- incre_cell ç- <- decr_cell ç{ <- init_loop ç} <- cond_loop
Δ> <- ç>P Δ< <- ç<P Δ+ <- ç+P Δ- <- ç-P Δ{ <- ç{P Δ} <- ç}P
P> <- P P< <- P P+ <- P P- <- P P{ <- P P} <- P
*> <- *α *< <- *α *+ <- *α *- <- *α *{ <- *α *} <- *α
*S| <- SX| *SX <- SX
P| <- λ| PT <- T
ΔS| <- CON_| SX| <- REP_|
AA(D)l <- l AA(_)0 <- l_
l* <- * lΔ <- Δ Δ| <- |
move_cell|T(x)(y)(z)(\$)T(X)(Y)(Z)(_) <- T(#)(Δ)(z)(\$)y*x|T(x)(y*)(Z)(_)
T(X)(Y)(Z)(_)back_cell|T(x)(y)(z)(\$) <- y*x|T(x)(y*)(Z)(_)T(#)(Δ)(z)(\$)
incre_cell|T(X)(Y)(Z)(_) <- Y*X|T(X)(Y*)(Z)(_*)
decr_cell|T(X)(Y)(Z)(_) <- Y*X|T(X)(Y*)(Z)(_α)
init_loop|T(X)(Y)(Z)(_) <- Y*X|T(X)(Y*)(ZAA(Y))(_)
cond_loop|T(X)(Y)(Z)(_) <- _S|T(X)(Y)(Z)(_)
CON_|T(X)(Y)(Z)(_) <- Y*X|T(X)(Y*)(Zl)(_)
REP_|T(X)(Y)(Z)(_) <- Z0X|T(X)(Z0)(Zl)(_)
G___K <- ___END
[
'BF(x)'___END <- x
[
T(x)(y)(z)(_) <- (_)
]
]
]
]
```

Grr is able to compute a brainfuck interpreter with infinite numbers of cells, once brainfuck is Turing completeness, by consequence, Grr is Turing complete too.

## Implementation

Grr interpreter is avaliable in : https://github.com/caotic123/Grr-Programming-Language