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README.md

Cratylus

Si, como el griego afirma en el Cratilo,
el nombre es arquetipo de la cosa,
en las letras de rosa está la rosa
y todo el Nilo en la palabra Nilo.

Introduction

Cratylus is a polynomial rewriting esolang, designed for submission to December 2012 PLT Games.

A Cratylus program is a sequence of rewrite rules between multivariate polynomials. For instance, the following program has only one rule:

xy => z.

Given a query:

? x^2 y^2.

it gets rewritten to xyz, and this in turn to z^2, which is the normal form. More precisely, a Cratylus program is a sequence of rules, of the form:

R1 => S1.
R2 => S2.
...
Rn => Sn.

where each Ri and Si are multivariate polynomials with integer coefficients. Given a polynomial P, let i be the least index such that Ri divides P, that is, P = Ri Q. Then P gets rewritten to Si Q. If there is no such index, P is in normal form.

Attempts to catch your attention

This Cratylus program computes factorials:

H=>mZ.am=>af.m=>J.af=>aB.f=>k.aB=>u.B=>u.u=>cL.L=>eG.G=>f.
ck=>cr.k=>d.cr=>b.r=>b.bZ=>yZ.b=>j.yZ=>t.y=>t.t=>iK.K=>lE.
E=>b.ij=>is.j=>I.is=>x.s=>x.x=>DZ.D=>j.I=>k.dZ=>wZ.d=>h.
wZ=>o.w=>o.o=>d.hl=>lp.h=>g.lp=>v.p=>v.v=>CZ.C=>h.eg=>en.
g=>q.en=>z.n=>z.z=>aF.F=>g.aq=>A.q=>A.A=>m.J.

For instance, interacting with the Cratylus toplevel:

? H a^5
Z^120

? H a^7
Z^5040

Bear in mind the second query takes a really long time to arrive to the answer. The reason for such slowness is that, internally, additions and multiplications are carried out in unary, and that takes a number of steps that grows exponentially with the length of the numbers involved. By using the Cratylus to C compiler, one can speed up the constant factors to compute up to the factorial of 10 (Z^3628800) in a few seconds.

Although there seems to be no way of improving the algorithm to carry out additions other than in unary, it is still possible to adapt it to behave linearly up to a "big" number. In unary one knows that "1 + 1 = 2". The key to speed up the process is also acknowledging the facts that "10 + 10 = 20", "100 + 100 = 200", etc. If one could represent those facts for arbitrary powers of 10, the program would effectively perform additions in a linear number of steps. Unfortunately, Cratylus is not expressive enough for us to be able to state that equations in general, but only up to a given fixed number (NB: below we describe Cratylus^@, a variant that allows expressing precisely that kind of rewrite rules). For example, one could write a version of factorial improved to work "fast" up to 64-bit integers. See the code.

The following Cratylus program is able to calculate the n-th prime number. The main goal, ? BcE^20., queries for the 20-th prime number:

Bc=>{_}G.EG=>BhE.G=>Bd.Bh=>{_}Aj.{_}Aj=>{_}Am.Aj=>e.
{_}Am=>F.Am=>F.{_}F=>{_}M.F=>N.{_}M=>n.M=>n.n{_}=>{_}Bb.
n=>Be.Bb=>{_}Az.Az=>m{_}.m{_}=>{_}L.m=>y.{_}L=>Y.L=>Y.
Y=>aBa.Ba=>xBk.Bk=>m.xy=>xAp.y=>I.xAp=>Al.Ap=>Al.Al=>{_}Ax.
Ax=>y.aI=>s.I=>s.as=>h.s=>h.ah=>aBi.h=>Bj.Bi=>aq.
q{_}=>{_}An.q=>v.{_}An=>Ab.An=>Ab.Ab=>fAw.Aw=>oBl.Bl=>q.
ov=>oK.v=>u.oK=>Ai.K=>Ai.Ai=>{_}Bg.Bg=>v.au=>aV.u=>D.aV=>Q.
V=>Q.Q=>cAv.Av=>tBp.Bp=>u.tD=>tAf.D=>w.tAf=>Aa.Af=>Aa.
Aa=>aBn.Bn=>D.fw=>fz.w=>B.cz=>cAg.z=>C.cAg=>W.Ag=>W.W=>dAt.
At=>bAy.Ay=>z.dC=>dAq.C=>r.fAq=>fAd.Aq=>i.fAd=>l.Ad=>l.
dl=>H.l=>H.H=>C.br=>bS.r=>Bo.bS=>Ah.S=>Ah.Ah=>cBf.Bf=>r.
Bo=>w.cB=>cP.B=>i.cP=>J.P=>J.J=>B.di=>dO.i=>Ac.dO=>p.O=>p.
bp=>X.p=>X.X=>i.bAc=>bA.Ac=>e.bA=>bAk.A=>Z.bAk=>Ar.Ak=>Ar.
Ar=>A.aZ=>k.Z=>k.ak=>U.k=>U.U=>h.Bj=>gAu.Au=>e.Be=>gBm.
Bm=>{_}N.N=>e{_}.ae=>aAo.e=>Ae.aAo=>T.Ao=>T.T=>e.gAe=>gAs.
Ae=>j.AsE=>j.As=>j.gj=>R.j=>R.R=>G.Bd.

? BcE^20.

Again, the results take a very long time with the standard interpreter. By using the Cratylus to C compiler, the answer is:

{_}^71

which is, in fact, the 20-th prime number.

The programs above make heavy use of the fact that Cratylus allows working with multivariate polynomials. One may wonder if there is a way of writing programs using only univariate polynomials. Indeed, the following is also a Cratylus program that calculates factorials:

x - 17 => x^2 + 7x.
x^2 + 8x + 7 => x^2 - 2x - 3.
x + 7 => x - 18.
x^2 - 2x - 3 => x^2 - 13x - 14.
x - 3 => x + 6.
x^2 - 13x - 14 => x + 11.
x - 14 => x + 11.
x + 11 => x^2 - 17x - 38.
x - 19 => x^2 + 20x + 51.
x + 17 => x - 3.
x^2 + 8,x + 12 => x^2 - 7x - 18.
x + 6 => x - 2.
x^2 - 7x - 18 => x - 1.
x - 9 => x - 1.
x^2 - x => x^2 + 13x.
x - 1 => x - 5.
x^2 + 13x => x - 10.
x + 13 => x - 10.
x - 10 => x^2 + 24x + 95.
x + 19 => x^2 + 10x - 96.
x + 16 => x - 1.
x^2 - 25 => x^2 + 15x + 50.
x - 5 => x + 18.
x^2 + 15x + 50 => x - 12.
x + 10 => x - 12.
x - 12 => x^2 - 15x.
x - 15 => x - 5.
x + 18 => x + 6.
x^2 - 2x => x^2 + 12x.
x - 2 => x - 4.
x^2 + 12x => x + 8.
x + 12 => x + 8.
x + 8 => x - 2.
x^2 - 10x + 24 => x^2 - 14x + 48.
x - 4 => x + 4.
x^2 - 14x + 48 => x - 11.
x - 8 => x - 11.
x - 11 => x^2 + 15x.
x + 15 => x - 4.
x^2 + 7x + 12 => x^2 - 4x - 21.
x + 4 => x + 9.
x^2 - 4x - 21 => x - 13.
x - 7 => x - 13.
x - 13 => x^2 - 15x - 16.
x - 16 => x + 4.
x^2 + 10x + 9 => x + 14.
x + 9 => x + 14.
x + 14 => x + 7.
x - 18.

For instance, on the following input:

? (x - 17) (x + 1)^3

it calculates 3! = 6:

x^6

For reasons not so evident to me, this version is orders of magnitude slower. This probably relates with the fact that polynomial multiplication and division take quadratic time on the degree.

Even though, superficially, our two factorial programs are very different, in essence they are exactly the same program. They can be put in correspondence by a relation like the following:

H <==> x - 17
m <==> x + 7
Z <==> x
a <==> x + 1
f <==> x - 3
...

See, for example, that the first rewrite rule of the first factorial program was H => m Z, which in the new program can be read as x - 17 => (x + 7) x, i.e. x - 17 => x^2 + 7x.

Features

As part of the Cratylus distribution, the following scripts are provided:

  • cratylus.py -- the Cratylus toplevel interpreter.

  • tools/simp_cr.py -- the Cratylus simplifier. Transforms Cratylus programs into equivalent Cratylus programs with particular restrictions.

  • tools/crc.py -- the Cratylus to C compiler. Compiles a subset of Cratylus into C.

  • tools/s2cr.py -- the S to Cratylus compiler. S is a simple assembler-like language with few instructions: increment and decrement, conditional and unconditional jumps.

  • tools/ss2s.py -- the S-with-macros to S compiler. S-with-macros is a slightly higher level language, with subroutines, and primitive control structures that get macroexpanded into plain S instructions.

Beyond standard Cratylus, the Cratylus^@ dialect is an improved version of Cratylus which allows for more efficient scripts and has an extension for performing I/O. With these extensions, a Cratylus^@ quine is provided.

Turing completeness

Cratylus is a superset of Conway's FRACTRAN. Restricting ourselves to constant polynomials, a FRACTRAN program:

(a1/b1, a2/b2, ..., aN/bN)

gets translated to an equivalent Cratylus program:

b1 => a1.
b2 => a2.
...
bN => aN.

Since FRACTRAN is Turing complete, also is Cratylus.

Moreover, as part of the Cratylus distribution, the script s2cr.py translates a program in the (theoretical) programming language S, which is well-known to be Turing complete, into a Cratylus program, which again shows Cratylus' completeness. See below for details.

Syntax for polynomials

Multivariate polynomials are formed by operating with variables and positive integers.

Variables are either single lowercase characters or a sequence of valid characters started by uppercase.

<variable> ::= [a-z] | [A-Z][_a-z0-9]

Notice that variables can start in uppercase but cannot contain uppercase characters in between. This is for FooBar to be parsed as Foo * Bar. Additionally, any string delimited by braces is a variable:

<variable> ::= ... | {.*}

Numeric literals are restricted to be natural numbers:

<num> ::= [0-9]+

More interesting polynomials are built by operating on variables and numbers. The abstract syntax is given by:

<poly> ::= <poly> + <poly>
         | <poly> - <poly>
         | <poly> * <poly>
         | <poly> ^ <num>

The usual precedence rules apply. All operators are left-associative. More concretely:

<atom> ::= <variable> | <num> | ( <poly> )

<expatom> ::= <expatom>
            | <expatom> ^ <num>

<factor> ::= <expatom>
           | <factor> <expatom>

<term> ::= <factor>
         | <term> * <factor>

<poly> ::= <term>
         | + <term>
         | - <term>
         | <poly> + <term>
         | <poly> - <term>

Notice there are two ways of writing products: <poly><poly> and <poly> * <poly>.

Using Cratylus as a basic polynomial normalizer

Invoke Cratylus with no arguments (no rewrite rules):

$ python cratylus.py

  ____           _         _           
 / ___|_ __ __ _| |_ _   _| |_   _ ___ 
| |   | '__/ _` | __| | | | | | | / __|
| |___| | | (_| | |_| |_| | | |_| \__ \
 \____|_|  \__,_|\__|\__, |_|\__,_|___/
                     |___/             

Copyright (c) 2012 - Pablo Barenbaum <foones@gmail.com>
? 42
42
? x
x
? abracadabra
a^5b^2cdr^2
? x^2 - 1
x^2 - 1
? (x + y)(x - y)
x^2 - y^2
? (Foo + Bar)^2
Bar^2 + 2BarFoo + Foo^2
? -({x}-{y}){x}
-{x}^2 + {x}{y}

Syntax for programs

A program is just a list of rules. Rules of the form p => 1. can be abbreviated as p.. Additionally, a Cratylus script can contain goals which are to be solved as soon as the program is loaded.

<program> ::= <EMPTY>
            | <program> <rule>
            | <program> <goal>

<rule> ::= <poly> .
         | <poly> => <poly> .

<goal> ::= ? <poly> .

Example 1: addition

This example is used to model addition. A polynomial a x^n y^m is rewritten to a polynomial z^(n + m). That is, the exponents of the variables x and y are added resulting in the corresponding exponent of z. When the following program is loaded:

ax => az.
ay => az.
a => 1.

? a x^3 y^2.

Cratylus displays the following final result:

z^5 

Intuitively, given a polynomial of the form a x^n y^m, the first rule transforms all xs into zs, one at a time. When there are no more xs to transform, the second rule applies, which transforms all ys to zs. After that, the third rule applies, which removes the redundant a.

The chain of reduction steps would be:

   a x^3 y^2
=> a x^2 y^2 z
=> a x   y^2 z^2
=> a     y^2 z^3
=> a     y   z^4
=> a         z^5
=>           z^5

When the -v command line option is used, Cratylus traces the steps of the reduction.

----------------------------------------
Current goal : ax^3y^2
Applying rule: ax => az
Factorization: ax^3y^2 = (ax) * (x^2y^2)
New goal     : ax^2y^2z
----------------------------------------
Current goal : ax^2y^2z
Applying rule: ax => az
Factorization: ax^2y^2z = (ax) * (xy^2z)
New goal     : axy^2z^2
----------------------------------------
Current goal : axy^2z^2
Applying rule: ax => az
Factorization: axy^2z^2 = (ax) * (y^2z^2)
New goal     : ay^2z^3
----------------------------------------
Current goal : ay^2z^3
Applying rule: ay => az
Factorization: ay^2z^3 = (ay) * (yz^3)
New goal     : ayz^4
----------------------------------------
Current goal : ayz^4
Applying rule: ay => az
Factorization: ayz^4 = (ay) * (z^4)
New goal     : az^5
----------------------------------------
Current goal : az^5
Applying rule: a => 1
Factorization: az^5 = (a) * (z^5)
New goal     : z^5
----------------------------------------
Final result:
z^5

The following is a different, more declarative, way of presenting the above rules:

Add X => Add Z.
Add Y => Add Z.
Add.

Toplevel interaction:

? Add X^9 Y^7
Z^16

One could question why define addition in such a convoluted way, given that the interpreter itself is a polynomial calculator (which was already able to add constants).

The key aspect here is that we are representing information, and manipulating a representation. The difference is that this way of doing things "scales", in the sense that allows us to carry out other, more complex, processes.

As in FRACTRAN, each variable can be thought as a register, and rewrite rules can be thought as simple incrementing / decrementing operations on exponents.

Example 2: erasing a variable

The following operation can be used to remove all occurrences of X:

Erase X => Erase.
Erase.

Toplevel interaction:

? Erase X^9 Y^7
Y^7

Example 3: copying a variable

This operation creates two copies of X:

Copy X => Copy Y Z.
Copy.

Toplevel interaction:

? Copy X^9
Y^9Z^9

Example 4: multiplication

The combination of these ideas allow us to write a procedure that does multiplication:

Mul X Y => Copy X Y.        # [1]
Mul => Erase.               # [2]

Copy Y => Copy Y1 Z.        # [3]
Copy   => Rename.           # [4]

Rename Y1 => Rename Y.      # [5]
Rename    => Del1.          # [6]

Del1 X   => Mul.            # [7]
Del1     => Mul.            # [8]

Erase Y  => Erase.          # [9]
Erase.                      # [10]

Toplevel interaction:

? Mul X^10 Y^9
Z^90

Intuitively, given a term of the form Mul X^n Y^m, we want to carry out the product n * m. For doing so, we add up m exactly n times.

The variable Z is used as an accumulator. After the i-th iteration of the algorithm, the partial result will be Mul X^(n - i) Y^m Z^(i * m). When i reaches n, we will have something like Mul Y^m Z^(n * m), after which we only need to "clean" all the Y factors.

More precisely, if n = 0, there are no X factors in the polynomial. So we have a polynomial of the form Mul Y^m ..., i.e. we want to multiply m times zero. Rule [2] applies, which takes us to rules [9] and [10]. These erase, one by one, all occurrences of Y.

If n > 0, rule [1] applies. Rules [3] and [4] are used to transform a term of the form Mul X^(n - i) Y^m Z^(i * m) into a term of the form Mul X^(n - i) Y1^m Z^(i * m + m), that is, they copy the exponent of Y into Y1 and also add the exponent of Y into the accumulator Z.

After doing so, we need to rename Y1 back to Y, to keep on with the iterations. Also, we delete one occurrence of X. After this we reach:

Mul X^(n - i - 1) Y^m Z^((i + 1) * m)

and the invariant is maintained.

Example 5: quotient and remainder

Quotient and remainder algorithm:

DivMod X Y => Copy X Y.
DivMod => End.

Copy Y => Copy Y1 Y2.
Copy => Sub.

Sub X Y2 => Sub.
Sub Y2 => Rem Y2.
Sub => Continue Q.

Continue Y1 => Continue Y.
Continue => DivMod.

Rem Y1 Y2 => Rem.
Rem Y1 => Rem R.
Rem.

End Y => End.
End.

Toplevel interaction:

? DivMod X^62 Y^11           # 62 = 5 * 11 + 7
Q^5R^7

An equivalent obfuscated version:

dxy => cxy.
d => e.
cy => abc.
c => s.
bsx => s.
bs => bl.
s => nq.
an => ny.
n => d.
abl => l.
al => lr.
l.
ey => e.
e.

Toplevel interaction:

? d x^62 y^11
q^5r^7

S to Cratylus compiler

As part of the Cratylus distribution, the script s2cr.py translates a program in the programming language S to Cratylus. The programming language S is introduced in M. Davis' Computability, Complexity, and Languages to study computability.

A program in S works with an infinite set of variables, which hold natural numbers. A program is a list of operations delimited by newlines. Labels can be introduced. At the end of the program, variables are initialized by declarations of the form ! var value.

<program> ::= <instructions> <init>

<instructions> ::= <EMPTY>
                 | <label>: <instructions>
                 | <op> <instructions>

<op> ::= inc <var>
       | dec <var>
       | jmp <label>
       | jz <var> <label>
       | jnz <var> <label>

<init> ::= <EMPTY>
          | ! <var> <num> <init>

For instance, the following S program calculates the product of X and Y, with input X = 11 and Y = 9:

mult:
    jz X mult_end

    # Loop to copy Y to Y1 and Z
    copy:
        jz Y copy_end
        dec Y
        inc Y1
        inc Z
        jmp copy
    copy_end:

    # Loop to rename Y1 to Y 
    rename:
        jz Y1 rename_end
        dec Y1
        inc Y
        jmp rename
    rename_end:

    dec X
    jmp mult
mult_end:

# Loop to erase all copies of Y
erase:
    dec Y
    jnz Y erase

# Initial values    

! X 11
! Y 9

After translation with s2cr.py, we get the following Cratylus program:

{0}{X} => {1}{X}.
{0} => {12}.
{1}{Y} => {2}{Y}.
{1} => {6}.
{2}{Y} => {3}.
{2} => {3}.
{3} => {4}{Y1}.
{4} => {5}{Z}.
{5} => {1}.
{6}{Y1} => {7}{Y1}.
{6} => {10}.
{7}{Y1} => {8}.
{7} => {8}.
{8} => {9}{Y}.
{9} => {6}.
{10}{X} => {11}.
{10} => {11}.
{11} => {0}.
{12}{Y} => {13}.
{12} => {13}.
{13}{Y} => {12}{Y}.
{13} => {14}.
{14}.
? {0}{X}^11{Y}^9.

When run, the answer is:

{Z}^99

For compiling an S program, we associate a variable {k} for representing the k-th instruction in the source program. For instance, if we are in the initial state, at the beginning of the program, the polynomial has {0} as a factor. This way we can represent an arbitrary state machine, where the only factor of the form {k} represents the instruction pointer / current state.

The simplest example is a non-conditional jump from the i-th instruction to the j-th instruction, which is represented by the following rule:

{i} => {j}.

If the i-th instruction increments the variable X, we get the following rule:

{i} => {i+1} {X}.

Besides incrementing the exponent of {X} this rule also increments the instruction pointer. Decrementation is handled similarly, with the caveat that {X} might have a 0 exponent, in which case we just advance the instruction pointer:

{i} {X} => {i+1}.
{i} => {i+1}.

Conditional jumps combine the previous ideas. A jump from the i-th instruction to the j-th instruction testing if X is zero (jz) is compiled as:

{i} {X} => {i+1} {X}.
{i} => {j}.

The additional {X} in the right-hand side of the first rule is to avoid decrementing the value of X, since we only want to check if it is zero or not. The jnz instruction is symmetric:

{i} {X} => {j} {X}.
{i} => {i+1}.

Cratylus simplifier

Also part of the Cratylus distribution is the script simp_cr.py which normalizes Cratylus programs.

Notice that most of the Cratylus programs above, are written in monomial form. That is, all the polynomials that appear in the source program are monomials with leading coefficient equal to 1.

It is easy to see that a program in monomial form can be translated to an equivalent FRACTRAN program, if the set of variables that occur in the source program is put in 1-1 correspondence with an arbitrary set of primes. (More generally, a program in monomial form can be translated to an equivalent program if the set of variables is put in 1-1 correspondence with a set of different irreducible elements in a unique factorization domain).

For instance, as we saw before, the S program which calculates the product of X and Y gave output to the following Cratylus program:

{0}{X} => {1}{X}.
{0} => {12}.
{1}{Y} => {2}{Y}.
{1} => {6}.
{2}{Y} => {3}.
{2} => {3}.
{3} => {4}{Y1}.
{4} => {5}{Z}.
{5} => {1}.
{6}{Y1} => {7}{Y1}.
{6} => {10}.
{7}{Y1} => {8}.
{7} => {8}.
{8} => {9}{Y}.
{9} => {6}.
{10}{X} => {11}.
{10} => {11}.
{11} => {0}.
{12}{Y} => {13}.
{12} => {13}.
{13}{Y} => {12}{Y}.
{13} => {14}.
{14}.

? {0}{X}^11{Y}^9.

Given a Cratylus program in monomial form, tools/simp_cr.py -f file.cr translates it to an equivalent FRACTRAN program. The output of simp_cr.py -f in this case is:

85 => 65.
17 => 11.
26 => 86.
13 => 3.
86 => 41.
43 => 41.
41 => 427.
61 => 3149.
47 => 13.
21 => 161.
3 => 37.
161 => 31.
23 => 31.
31 => 106.
53 => 3.
185 => 29.
37 => 29.
29 => 17.
22 => 19.
11 => 19.
38 => 22.
19 => 59.
59.

? 425000000000.

Notice that, in the translation, the variable representing the initial state was {0} and got translated into the numeric constant 17. The input variables, {X} and {Y} got translated into 5 and 2. The query 425000000000 is equivalent to 17 * 5^11 * 2^9. That is, we are asking for the product of 11 and 9. The answer will be given as the exponent of {Z}, which got translated into 67. Indeed we get:

6045127530961181628652411227196693490382260383969108811846856679868932815922570165203117478978324258618429876174558448218719550835202670830699753771294118292547613370699765040813403

which is exactly 67^99.

The command tools/simp_cr.py -u file.cr transforms a Cratylus program in monomial form to an equivalent program using univariate polynomials. In our case:

x^2 + 2x - 8 => x^2 - 6x + 8.
x + 4 => x - 3.
x^2 - 3x - 4 => x^2 - 5x - 6.
x - 4 => x + 3.
x^2 - 5x - 6 => x + 7.
x - 6 => x + 7.
x + 7 => x^2 + 11x + 18.
x + 9 => x^2 + x - 90.
x - 9 => x - 4.
x^2 + 5x + 6 => x^2 + 8x + 12.
x + 3 => x + 5.
x^2 + 8x + 12 => x - 7.
x + 6 => x - 7.
x - 7 => x^2 + 9x + 8.
x + 8 => x + 3.
x^2 + 3x - 10 => x - 5.
x + 5 => x - 5.
x - 5 => x + 4.
x^2 - 2x - 3 => x - 1.
x - 3 => x - 1.
x^2 - 1 => x^2 - 2x - 3.
x - 1 => x - 8.
x - 8.

In this case our query can be:

? (x + 4) (x - 2)^2 (x + 1)^3.

which asks for the product of 2 and 3. We get the answer:

x^6 + 60x^5 + 1500x^4 + 20000x^3 + 150000x^2 + 600000x + 1000000

which is (x + 10)^6.

The command tools/simp_cr.py -v file.cr transforms a Cratylus program in monomial form to an equivalent program using variable names as short as possible. In our case we get:

dg => dh.
g => f.
ah => al.
h => e.
al => m.
l => m.
m => cq.
q => rs.
r => h.
ce => ck.
e => i.
ck => n.
k => n.
n => ao.
o => e.
di => j.
i => j.
j => g.
af => b.
f => b.
ab => af.
b => p.
p.

And the query is:

? g a^9 d^11
s^99

S-with-macros to S compiler

Also part of the Cratylus distribution is the script ss2s.py which translates an S program with macros to a regular S program.

An S program with macros allows defining subroutines with parameters, and using some basic control constructs (if and while), besides allowing labels, inc, dec, jmp, jz and jnz. The grammar of S-with-macros is a superset of that of S:

<program> ::= <instructions> <init>

<instructions> ::= <EMPTY>
                 | <sub> \n <instructions>
                 | <label>: \n <instructions>
                 | <op> \n <instructions>

<op> ::= inc <var>
       | dec <var>
       | jmp <label>
       | jz <var> <label>
       | jnz <var> <label>
       | <sub_name> <var1> ... <varN>
       | WHILEZ var \n <instructions> \n END
       | WHILENZ var \n <instructions> \n END
       | IFZ var \n <instructions> \n END
       | IFNZ var \n <instructions> \n END

<sub> ::= SUB <sub_name> <var1> ... <varN> \n <instructions> \n END

<init> ::= <EMPTY>
          | ! <var> <num> <init>

For instance, the following S-with-macros program calculates factorials:

SUB rename X Y
    WHILENZ X
        dec X
        inc Y
    END
END

SUB bicopy X Y Z
    WHILENZ X
        dec X
        inc Y
        inc Z
    END
END

SUB zero X
    WHILENZ X
        dec X
    END
END

SUB mult X Y Z
    WHILENZ X 
        dec X
        bicopy Y Y2 Z
        rename Y2 Y
    END
    zero Y
END

SUB fact X Y
    inc Y
    WHILENZ X
        bicopy X X1 X2
        mult X1 Y T
        rename T Y
        rename X2 X
        dec X
    END
END

fact X Z

! X 5

The ss2s.py script translates the S-with-macros program to the following plain S program:

    # fact X Z
    inc Z
:l:1:
    jz X :l:2
    # bicopy X X1 X2
:l:3:
    jz X :l:4
    dec X
    inc fact:1:X1
    inc fact:1:X2
    jmp :l:3
:l:4:
    # mult X1 Y T
:l:5:
    jz fact:1:X1 :l:6
    dec fact:1:X1
    # bicopy Y Y2 Z
:l:7:
    jz Z :l:8
    dec Z
    inc mult:1:Y2
    inc fact:1:T
    jmp :l:7
:l:8:
    # rename Y2 Y
:l:9:
    jz mult:1:Y2 :l:10
    dec mult:1:Y2
    inc Z
    jmp :l:9
:l:10:
    jmp :l:5
:l:6:
    # zero Y
:l:11:
    jz Z :l:12
    dec Z
    jmp :l:11
:l:12:
    # rename T Y
:l:13:
    jz fact:1:T :l:14
    dec fact:1:T
    inc Z
    jmp :l:13
:l:14:
    # rename X2 X
:l:15:
    jz fact:1:X2 :l:16
    dec fact:1:X2
    inc X
    jmp :l:15
:l:16:
    dec X
    jmp :l:1
:l:2:
    ! X 5

By using the S to Cratylus compiler (s2cr.py script), this in turn gets compiled to the following Cratylus program:

{0} => {1}{Z}.
{1}{X} => {2}{X}.
{1} => {32}.
{2}{X} => {3}{X}.
{2} => {7}.
{3}{X} => {4}.
{3} => {4}.
{4} => {5}{fact:1:X1}.
{5} => {6}{fact:1:X2}.
{6} => {2}.
{7}{fact:1:X1} => {8}{fact:1:X1}.
{7} => {19}.
{8}{fact:1:X1} => {9}.
{8} => {9}.
{9}{Z} => {10}{Z}.
{9} => {14}.
{10}{Z} => {11}.
{10} => {11}.
{11} => {12}{mult:1:Y2}.
{12} => {13}{fact:1:T}.
{13} => {9}.
{14}{mult:1:Y2} => {15}{mult:1:Y2}.
{14} => {18}.
{15}{mult:1:Y2} => {16}.
{15} => {16}.
{16} => {17}{Z}.
{17} => {14}.
{18} => {7}.
{19}{Z} => {20}{Z}.
{19} => {22}.
{20}{Z} => {21}.
{20} => {21}.
{21} => {19}.
{22}{fact:1:T} => {23}{fact:1:T}.
{22} => {26}.
{23}{fact:1:T} => {24}.
{23} => {24}.
{24} => {25}{Z}.
{25} => {22}.
{26}{fact:1:X2} => {27}{fact:1:X2}.
{26} => {30}.
{27}{fact:1:X2} => {28}.
{27} => {28}.
{28} => {29}{X}.
{29} => {26}.
{30}{X} => {31}.
{30} => {31}.
{31} => {1}.
{32}.
? {0}{X}^5.

Finally, by using the Cratylus simplifier (script simp_cr.py with the -v -t "{Z}" Z command-line switch, we get our first example back:

H => mZ.
am => af.
m => J.
af => aB.
f => k.
aB => u.
B => u.
u => cL.
L => eG.
G => f.
ck => cr.
k => d.
cr => b.
r => b.
bZ => yZ.
b => j.
yZ => t.
y => t.
t => iK.
K => lE.
E => b.
ij => is.
j => I.
is => x.
s => x.
x => DZ.
D => j.
I => k.
dZ => wZ.
d => h.
wZ => o.
w => o.
o => d.
hl => lp.
h => g.
lp => v.
p => v.
v => CZ.
C => h.
eg => en.
g => q.
en => z.
n => z.
z => aF.
F => g.
aq => A.
q => A.
A => m.
J.
? a^5H.

Cratylus to C compiler

The script crc.py compiles a Cratylus program in monomial form to an equivalent C program. The GNU Multiple Precision Library is used to represent polynomial exponents.

The factorial program above gets compiled into a C program.

Cratylus^@: a more efficient variant of Cratylus

As said before, Cratylus programs are very inefficient, since we are restricted to representing and working with natural numbers in unary.

To mitigate this problem, we propose the Cratylus^@ variant of Cratylus. The new programs must be written in monomial form. Besides the usual kinds of rules, the exponents can also be the special symbol @.

A rule containing an occurrence of @ is equivalent to an infinite number of rules, where @ is to be replaced by each of the non-zero natural numbers.

For instance the Cratylus^@ program:

x^@ => y^@.

Rewrites x^42 in one step to y^42. That allows us to write an efficient addition routine:

Add X^@ => Z^@.
Add Y^@ => Z^@.
Add.

Note that if there is more than one occurrence of @ at the left-hand side, Cratylus^@ will bind @ to the least of the exponents. For instance, in this case:

X^@ Y^@ => Z^@.

Cratylus^@ will rewrite X^10 Y^8 by the rule X^8 Y^8 => Z^8, which results in X^2 Z^8.

With this addition to the language, it is possible to write an actually efficient factorial program:

b^@s=>s.sx^@=>b^@x^@K.s=>K.l^@K=>Q.K=>Q.Q=>lW.bW=>r.W=>r.br=>bAb.r=>R.Ab=>bq.
a^@q=>q.b^@q=>a^@b^@k.q=>k.e^@k=>k.kl^@=>e^@l^@C.k=>C.p^@C=>w.C=>w.aw=>aV.w=>P.
u^@V=>o.V=>o.ao=>aZ.o=>g.Z=>c^2dE.c^@E=>c^@f^@i^@Aa.f^@Aa=>i^@Y.i^@Y=>f^@i^@L.
a^@L=>a^@v^@M.L=>M.f^@v^@M=>m.f^@m=>I.mv^@=>m.c^@m=>D.m=>D.i^@D=>c^@H.D=>H.
d^@H=>f^@hv^@.H=>h.f^@h=>d^@h.hv^@=>d^@E.h=>E.i^@I=>I.a^@c^@I=>O.c^@O=>gu^@.
d^@O=>ot^@.d^@g=>g.gt^@=>a^@g.g=>X.uX=>uN.X=>B.e^@N=>e^@p^@B.N=>B.B=>nt.
e^@nt=>c^@d^@F.nt^@=>T.n=>T.c^@F=>e^@F.d^@F=>e^@n.T=>w.u^@P=>J.P=>J.e^@J=>A.
J=>A.a^@A=>j.A=>j.jl^@=>j.jp^@=>l^@p^@y.j=>y.by=>G.y=>G.bG=>U.G=>U.U=>r.p^@R=>z.
R=>z.x^@z=>S.z=>S.S.

? s x^1000.

The result being:

l^402387260077093773543702433923003985719374864210714632543799910429938512398629020592044208486969404800479988610197196058631666872994808558901323829669944590997424504087073759918823627727188732519779505950995276120874975462497043601418278094646496291056393887437886487337119181045825783647849977012476632889835955735432513185323958463075557409114262417474349347553428646576611667797396668820291207379143853719588249808126867838374559731746136085379534524221586593201928090878297308431392844403281231558611036976801357304216168747609675871348312025478589320767169132448426236131412508780208000261683151027341827977704784635868170164365024153691398281264810213092761244896359928705114964975419909342221566832572080821333186116811553615836546984046708975602900950537616475847728421889679646244945160765353408198901385442487984959953319101723355556602139450399736280750137837615307127761926849034352625200015888535147331611702103968175921510907788019393178114194545257223865541461062892187960223838971476088506276862967146674697562911234082439208160153780889893964518263243671616762179168909779911903754031274622289988005195444414282012187361745992642956581746628302955570299024324153181617210465832036786906117260158783520751516284225540265170483304226143974286933061690897968482590125458327168226458066526769958652682272807075781391858178889652208164348344825993266043367660176999612831860788386150279465955131156552036093988180612138558600301435694527224206344631797460594682573103790084024432438465657245014402821885252470935190620929023136493273497565513958720559654228749774011413346962715422845862377387538230483865688976461927383814900140767310446640259899490222221765904339901886018566526485061799702356193897017860040811889729918311021171229845901641921068884387121855646124960798722908519296819372388642614839657382291123125024186649353143970137428531926649875337218940694281434118520158014123344828015051399694290153483077644569099073152433278288269864602789864321139083506217095002597389863554277196742822248757586765752344220207573630569498825087968928162753848863396909959826280956121450994871701244516461260379029309120889086942028510640182154399457156805941872748998094254742173582401063677404595741785160829230135358081840096996372524230560855903700624271243416909004153690105933983835777939410970027753472000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

For using the Cratylus^@ variant, use files with .crm extensions. Optionally, the -m option can be passed to the cratylus.py script. For even faster execution, use tools/crc.py to compile Cratylus^@ programs into C.

Extension for input/output facilities

The Cratylus^@ dialect has an extension for performing input/output. In a Cratylus^@ program, beyond the usual lowercase, uppercase and braced variables, two more variables are allowed: < (which stands for "input") and > (which stands for "output").

For the output feature, whenever a goal contains a factor of the form >^n, the ASCII character with numeric code n is written to standard output and the goal is automatically rewritten, factoring >^n out. (If n is not in the range 0..255, Cratylus^@ uses n mod 256).

For the input feature, when the left-hand side of a rule contains a factor of the form <^@ and all of the remaining factors match, a single character is read from standard input, and its character code is bound to the maximal power. On EOF, the value 256 is read.

The simplest output program:

a => >^64.
? a.

Prints an @ (ASCII code 64) when run.

The simplest input program:

a<^@ => b^@.
? a.

Reads a character from standard input and stores its ASCII code as a power of b.

A very simple program that outputs Hello world! followed by a newline:

a => >^10n.
b => >^111c.
c => >^114d.
d => >^108e.
e => >^100h.
f => >^32g.
g => >^119b.
h => >^33a.
i => >^108j.
j => >^111f.
k => >^101l.
l => >^108i.
m => >^72k.
n => 1.

? m.

The following example is a braindead version of cat in Cratylus^@:

# Simple version of cat in Cratylus^@
#
# Usage:
#
#    echo Hello world | python cratylus.py -s cat.crm
#
# (redirect 2>/dev/null for seeing output only)

I<^@  => X^@.
X^256 => 1.
X^@   => I>^@.

? I.

The following example reads a number in unary and stores it as a power of X:

# Read a number in unary from stdin in the form sss...s0
# Example:
#
#    echo ssssssss0 | python cratylus.py -s read_number.crm
#    ==> X^8

I<^@ => a^@.
a^115 => IX.
a^48  => 1.

? I.

The following is a generic string output routine. In this example, strings are encoded as a big enough power of X. More precisely, each character code is represented by a "codon" (group of three digits) in base 10. The first character in the string corresponds to the least significant codon. For instance the string "A" is represented by X^065 (i.e. X^65). The string "AB" is represented by X^066065.

# Divmod algorithm
# input:  Divmod X^n Y^m
# output: Q^(n div m) R^(n mod m)
#
DivmodDm_c^@=>Divmod.DivmodX^@=>Dm__hDm_c^@X^@.Divmod=>Dm__h.Dm__hQ^@=>Dm_z.
Dm__h=>Dm_z.Dm_zR^@=>Dm_s.Dm_z=>Dm_s.Dm_cDm_s=>Dm_cDm_l.Dm_s=>Dm_r.
Dm_d^@Dm_l=>Dm_l.Dm_lY^@=>Dm__fDm_d^@Y^@.Dm_l=>Dm__f.Dm__fDm_e^@=>Dm_eDm_f.
Dm__f=>Dm_eDm_f.Dm_b^@Dm_f=>Dm_f.Dm_d^@Dm_f=>Dm__sDm_b^@Dm_d^@.Dm_f=>Dm__s.
Dm__s=>Dm_aDm_j.Dm_aDm_b^@Dm_j=>Dm__gDm_g^@Dm_h^@.Dm_a^@Dm_j=>Dm_i.Dm_j=>Dm_i.
Dm__gDm_h^@=>Dm__gDm_b^@.Dm__gDm_g^@=>Dm_b^@Dm_j.Dm_iDm_k^@=>Dm_i.
Dm_c^@Dm_i=>Dm__dDm_c^@Dm_k^@.Dm_i=>Dm__d.Dm__dDm_b^@=>Dm__pDm_a^@Dm_b^@.
Dm__d=>Dm_q.Dm__pDm_a^@Dm_k^@=>Dm__i.Dm__p=>Dm__i.Dm__iDm_a^@=>Dm_q.Dm__i=>Dm_q.
Dm_kDm_q=>Dm__uDm_k.Dm_q=>Dm_o.Dm__u=>Dm_aDm_t.Dm_aDm_e^@Dm_t=>Dm_g^@Dm_h^@Dm_w.
Dm_a^@Dm_t=>Dm__l.Dm_t=>Dm__l.Dm_h^@Dm_w=>Dm_e^@Dm_w.Dm_g^@Dm_w=>Dm_e^@Dm_t.
Dm__l=>Dm_aDm_u.Dm_aDm_d^@Dm_u=>Dm_g^@Dm_h^@Dm_x.Dm_a^@Dm_u=>Dm__n.Dm_u=>Dm__n.
Dm_h^@Dm_x=>Dm_d^@Dm_x.Dm_g^@Dm_x=>Dm_d^@Dm_u.Dm__n=>Dm_f.Dm_b^@Dm_o=>Dm_o.
Dm_d^@Dm_o=>Dm__kDm_b^@Dm_d^@.Dm_o=>Dm__k.Dm__kDm_c^@=>Dm__rDm_a^@Dm_c^@.
Dm__k=>Dm_p.Dm__rDm_a^@Dm_b^@=>Dm__c.Dm__r=>Dm__c.Dm__cDm_a^@=>Dm_p.Dm__c=>Dm_p.
Dm_bDm_p=>Dm_bDm_m.Dm_p=>Dm__o.Dm__oDm_d^@=>Dm__tDm_a^@Dm_d^@.Dm__o=>Dm_n.
Dm__tDm_a^@Dm_c^@=>Dm__e.Dm__t=>Dm__e.Dm__eDm_a^@=>Dm_n.Dm__e=>Dm_n.
Dm_e^@Dm_n=>Dm__mDm_e^@Q^@.Dm_n=>Dm__m.Dm__m=>Dm_s.Dm_mR^@=>Dm_m.
Dm_c^@Dm_m=>Dm_c^@Dm_rR^@.Dm_m=>Dm_r.Dm_d^@Dm_r=>Dm__a.Dm_r=>Dm__a.
Dm__aDm_b^@=>Dm_v.Dm__a=>Dm_v.Dm_e^@Dm_v=>Dm__b.Dm_v=>Dm__b.Dm__bDm_c^@=>Dm__j.
Dm__b=>Dm__j.Dm__jX^@=>Dm_y.Dm__j=>Dm_y.Dm_yY^@=>Dm__q.Dm_y=>Dm__q.Dm__q.

# Output routine
IX^@ => Divmod X^@ Y^1000 J. 
I    => >^10.
JR^@ => J>^@.
JQ^@ => IX^@.
J    => I.

# Output an encoded string
? I X^33100108114111119032111108108101072. # Hello world!

Reversing the input is a bit trickier:

# Reverse standard input.
# Example:
#     $ echo -n ab | python cratylus.py rev.crm -s 2>/dev/null
#     ba

# Divmod algorithm
# input:  Divmod X^n Y^m
# output: Q^(n div m) R^(n mod m)
#
DivmodDm_c^@=>Divmod.DivmodX^@=>Dm__hDm_c^@X^@.Divmod=>Dm__h.Dm__hQ^@=>Dm_z.
Dm__h=>Dm_z.Dm_zR^@=>Dm_s.Dm_z=>Dm_s.Dm_cDm_s=>Dm_cDm_l.Dm_s=>Dm_r.
Dm_d^@Dm_l=>Dm_l.Dm_lY^@=>Dm__fDm_d^@Y^@.Dm_l=>Dm__f.Dm__fDm_e^@=>Dm_eDm_f.
Dm__f=>Dm_eDm_f.Dm_b^@Dm_f=>Dm_f.Dm_d^@Dm_f=>Dm__sDm_b^@Dm_d^@.Dm_f=>Dm__s.
Dm__s=>Dm_aDm_j.Dm_aDm_b^@Dm_j=>Dm__gDm_g^@Dm_h^@.Dm_a^@Dm_j=>Dm_i.Dm_j=>Dm_i.
Dm__gDm_h^@=>Dm__gDm_b^@.Dm__gDm_g^@=>Dm_b^@Dm_j.Dm_iDm_k^@=>Dm_i.
Dm_c^@Dm_i=>Dm__dDm_c^@Dm_k^@.Dm_i=>Dm__d.Dm__dDm_b^@=>Dm__pDm_a^@Dm_b^@.
Dm__d=>Dm_q.Dm__pDm_a^@Dm_k^@=>Dm__i.Dm__p=>Dm__i.Dm__iDm_a^@=>Dm_q.Dm__i=>Dm_q.
Dm_kDm_q=>Dm__uDm_k.Dm_q=>Dm_o.Dm__u=>Dm_aDm_t.Dm_aDm_e^@Dm_t=>Dm_g^@Dm_h^@Dm_w.
Dm_a^@Dm_t=>Dm__l.Dm_t=>Dm__l.Dm_h^@Dm_w=>Dm_e^@Dm_w.Dm_g^@Dm_w=>Dm_e^@Dm_t.
Dm__l=>Dm_aDm_u.Dm_aDm_d^@Dm_u=>Dm_g^@Dm_h^@Dm_x.Dm_a^@Dm_u=>Dm__n.Dm_u=>Dm__n.
Dm_h^@Dm_x=>Dm_d^@Dm_x.Dm_g^@Dm_x=>Dm_d^@Dm_u.Dm__n=>Dm_f.Dm_b^@Dm_o=>Dm_o.
Dm_d^@Dm_o=>Dm__kDm_b^@Dm_d^@.Dm_o=>Dm__k.Dm__kDm_c^@=>Dm__rDm_a^@Dm_c^@.
Dm__k=>Dm_p.Dm__rDm_a^@Dm_b^@=>Dm__c.Dm__r=>Dm__c.Dm__cDm_a^@=>Dm_p.Dm__c=>Dm_p.
Dm_bDm_p=>Dm_bDm_m.Dm_p=>Dm__o.Dm__oDm_d^@=>Dm__tDm_a^@Dm_d^@.Dm__o=>Dm_n.
Dm__tDm_a^@Dm_c^@=>Dm__e.Dm__t=>Dm__e.Dm__eDm_a^@=>Dm_n.Dm__e=>Dm_n.
Dm_e^@Dm_n=>Dm__mDm_e^@Q^@.Dm_n=>Dm__m.Dm__m=>Dm_s.Dm_mR^@=>Dm_m.
Dm_c^@Dm_m=>Dm_c^@Dm_rR^@.Dm_m=>Dm_r.Dm_d^@Dm_r=>Dm__a.Dm_r=>Dm__a.
Dm__aDm_b^@=>Dm_v.Dm__a=>Dm_v.Dm_e^@Dm_v=>Dm__b.Dm_v=>Dm__b.Dm__bDm_c^@=>Dm__j.
Dm__b=>Dm__j.Dm__jX^@=>Dm_y.Dm__j=>Dm_y.Dm_yY^@=>Dm__q.Dm_y=>Dm__q.Dm__q.

# Multiplication algorithm
# input:  Mult X^n Y^m
# output: Z^(n * m)
#
Mul_a^@Mult=>Mult.MultX^@=>Mul_a^@Mul_jX^@.Mult=>Mul_j.Mul_d^@Mul_j=>Mul_j.
Mul_jY^@=>Mul_d^@Mul_vY^@.Mul_j=>Mul_v.Mul_vZ^@=>Mul_o.Mul_v=>Mul_o.
Mul_aMul_o=>Mul__hMul_a.Mul_o=>Mul__j.Mul__hMul_n^@=>Mul_k.Mul__h=>Mul_k.
Mul_aMul_k=>Mul__lMul_a.Mul_k=>Mul_f.Mul__l=>Mul_b^2Mul_cMul_z.
Mul_b^@Mul_z=>Mul__kMul_b^@Mul_e^@Mul_i^@.Mul__kMul_e^@=>Mul__mMul_i^@.
Mul__mMul_i^@=>Mul__eMul_e^@Mul_i^@.Mul__eMul_a^@=>Mul__iMul_a^@Mul_p^@.
Mul__e=>Mul__i.Mul__iMul_e^@Mul_p^@=>Mul_g.Mul_e^@Mul_g=>Mul_x.
Mul_gMul_p^@=>Mul_g.Mul_b^@Mul_g=>Mul_t.Mul_g=>Mul_t.Mul_i^@Mul_t=>Mul_b^@Mul_y.
Mul_t=>Mul_y.Mul_c^@Mul_y=>Mul_e^@Mul_hMul_p^@.Mul_y=>Mul_h.
Mul_e^@Mul_h=>Mul_c^@Mul_h.Mul_hMul_p^@=>Mul_c^@Mul_z.Mul_h=>Mul_z.
Mul_i^@Mul_x=>Mul_x.Mul_a^@Mul_b^@Mul_x=>Mul__d.Mul__dMul_b^@=>Mul_fMul_n^@.
Mul__dMul_c^@=>Mul_kMul_m^@.Mul_c^@Mul_f=>Mul_f.Mul_fMul_m^@=>Mul_a^@Mul_f.
Mul_f=>Mul__c.Mul__cMul_n=>Mul__bMul_n.Mul__c=>Mul_q.
Mul__bMul_d^@=>Mul_d^@Mul_qZ^@.Mul__b=>Mul_q.Mul_q=>Mul_lMul_m.
Mul_d^@Mul_lMul_m=>Mul_b^@Mul_c^@Mul_r.Mul_lMul_m^@=>Mul__f.Mul_l=>Mul__f.
Mul_b^@Mul_r=>Mul_d^@Mul_r.Mul_c^@Mul_r=>Mul_d^@Mul_l.Mul__f=>Mul_o.
Mul__jMul_n^@=>Mul__a.Mul__j=>Mul__a.Mul__aMul_d^@=>Mul_w.Mul__a=>Mul_w.
Mul_a^@Mul_w=>Mul_s.Mul_w=>Mul_s.Mul_sX^@=>Mul_u.Mul_s=>Mul_u.Mul_uY^@=>Mul__g.
Mul_u=>Mul__g.Mul__g.

# Input until EOF, accumulate string in X^...
Reverse  => I.
I<^@     => MaA^@.
MaA^256  => Rv.
MaA^@    => MbB^@MultY^1000.
Ma       => MbMultY^1000.
MbZ^@    => MbX^@.
MbB^@    => IX^@.
Mb       => I.

# Output X^...
RvX^@ => DivmodSX^@Y^1000.
SR^@ => S>^@.
SQ^@ => RvX^@.

? Reverse.

By using this kind of string manipulation routines, one can write a a Cratylus^@ quine. To run it, compile it with tools/crc.py and wait a couple of hours :)