# Grr

Grr is a purely textual programming language that composes rules and functions, inspired by macros and metaprogramming. There exist two ways to define rules in Grr: firstly using a simple substitution in x to y (x <- y), or secondly making a function f that receives x and returns y (f(x) <- y). However in the reference interpreter they are all same substitution procedure. Grr is by default a recursive language, so rules can be replaced by the same rules (y <- y).

## Contents

## Examples

### Hello World

[Ώ <- T T <- HelloWorld]

### Naturals Numbers

The code below encodes natural numbers through recursive functions.

[ Ώ <- MUL(FOUR)(FOUR) FOUR <- INC(INC(INC(INC(N)))) N* <- * INC(Y) <- Y* DEC(K) <- K- *- <- λ ADD(X)(Y) <- XY 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 concepts, such as Ώ which is the main function, where the rules are initially decomposed, λ which is an empty string, and brackets are for constructing nested rules that refer to other, possibly recursive, rules. Comments can be inserted with space after the replacement 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 ] ] ]

## Representation of calculation

Grr rules are just string substitution, and therefore are higher-order objects like lambda calculus functions. Hence it's possible to easily implement lambda terms; 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 a Brainfuck interpreter (with a limit of 10 cells):

[ 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))))) TEN <- ADD(FIVE)(FIVE) 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 simulate a Brainfuck interpreter with infinite numbers of cells. As Brainfuck is Turing complete, Grr is consequently Turing complete too.

## Implementation

The Grr interpreter is available at: https://github.com/caotic123/Grr-Programming-Language