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https://github.com/codedot/lambda

Macro Lambda Calculus
https://github.com/codedot/lambda

interaction-nets lambda-calculus

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Macro Lambda Calculus

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README 

        
This package implements the lambda calculus using

interaction nets, providing CLI and API.



Its browserified version is available as [an online demo][1].



[1]: https://codedot.github.io/lambda/



# Algorithms



The following encodings of the lambda calculus are included:



* `abstract`, an impure solution to the problem of matching fans

in [Lamping's abstract algorithm][7],

described in [arXiv:1710.07516][6],

this is the default algorithm;



* `closed`, the approach of [arXiv:1304.2290][2] applied to

[_An Interaction Net Implementation of Closed Reduction_][3]

by Ian Mackie;



* `optimal`, an implementation of

[_Lambdascope_][5] by Vincent van Oostrom et al;



* `standard`, an implementation of optimal reduction as in

[_The Optimal Implementation of Functional Programming Languages_][8],

pp. 40-41.



The embedded read-back mechanism is described

in Section 7 of [10.4204/EPTCS.225.7][4].



[2]: https://arxiv.org/abs/1304.2290

[3]: http://dx.doi.org/10.1007/978-3-642-24452-0_3

[4]: http://dx.doi.org/10.4204/EPTCS.225.7

[5]: http://www.phil.uu.nl/~oostrom/publication/pdf/lambdascope.pdf

[6]: https://arxiv.org/abs/1710.07516

[7]: https://doi.org/10.1145/96709.96711

[8]: https://books.google.com/books?id=Bod5HbPh-WwC



# Benchmarks



The following is output of the `test.sh` script provided in the package:



```

SAMPLE          ABSTRACT         CLOSED        OPTIMAL       STANDARD

counter              N/A         162/25         582/14        2322/14

w2eta               37/7         137/16          207/7          374/7

1021              199/55     11871/1088     1599877/55     4307803/55

22210i            494/68       2539/254       58603/68            N/A

3222i            1206/50       8638/819      804530/50            N/A

1022i            4317/69     33369/3139            N/A            N/A

2222101   2621862/327818            N/A            N/A            N/A

facty6nt        1112/210     80562/2436    2790152/210    3013433/210

facty9i         1629/287 3746232/130949            N/A            N/A

33-fact4        3770/704      16114/912      80708/704     234075/704

fibo16nt      24931/3042    134135/5673   5462375/3042   8959455/3042

fact100i      28502/3752   121854/10565            N/A            N/A

35-fact5     72944/13480   805218/16206  4702711/13480            N/A

fibo20i       93534/6863   536843/24626   1961508/6863   4023117/6863

fact1knt 6215039/1353692            N/A            N/A            N/A

```



`T/B` should be read as total of `T` interactions,

of which `B` were β-reductions.



# CLI



This package provides the `lambda` command with the following interface:



```

Usage: lambda [options] ( | -f )



Options:

  --algo, -a     Select algorithm         [string]

  --debug, -d    Evaluate step by step   [boolean]

  --exec, -e     Process m4(1) macros    [boolean]

  --file, -f     Read term from file     [boolean]

  --inet, -i     Show interaction net    [boolean]

  --limit, -l    Limit interactions       [number]

  --macros, -m   Read macros from file    [string]

  --perf, -p     Print benchmarks        [boolean]

  --stats, -s    Save statistics to file  [string]

  --term, -t     Output expanded term    [boolean]

  --help, -h     Show help               [boolean]

  --version, -v  Show version number     [boolean]



```



# Combinators



CLI predefines a number of commonly used combinators:



```

# Common combinators

I = x: x;

K = x, y: x;

S = x, y, z: x z (y z);

Y = (a: a a) (a, f: f (a a f));



# Booleans

T = K;

F = K I;

Not = p, a, b: p b a;

And = p, q: p q F;

Or = p, q: p T q;

Xor = p, q: p Not I q;



# Pairs/lists

[] = K T;

[]? = l: l (h, t: F);

Cons = h, t, x: x h t;

Head = l: l T;

Tail = l: l F;



# Church arithmetic

+1 = n, f, x: f (n f x);

+ = m, n, f, x: m f (n f x);

* = m, n, f: m (n f);

^ = m, n: n m;

-1 = n, f, x: n (g, h: h (g f)) (K x) I;

- = m, n: n -1 m;

0? = n: n (K F) T;



# Church numerals

0 = f, x: x;

1 = f, x: f x;

2 = +1 1;

3 = +1 2;

4 = ^ 2 2;

5 = + 2 3;

6 = * 2 3;

7 = +1 6;

8 = ^ 2 3;

9 = ^ 3 2;

10 = * 2 5;

16 = ^ 2 4;

20 = * 2 10;

30 = * 3 10;

32 = ^ 2 5;

64 = ^ 2 6;

100 = ^ 10 2;

128 = ^ 2 7;

256 = ^ 2 8;

512 = ^ 2 9;

1k = ^ 10 3;

1ki = ^ 2 10;

1m = ^ 10 6;

1mi = ^ 2 20;

1g = ^ 10 9;

1gi = ^ 2 30;



# Recursive functions

FactY = Y (f, n: (0? n) 1 (* (f (-1 n)) n));

Fact = n: n (f, i: * (f (+1 i)) i) (K 1) 1;

Fibo = n: n (f, a, b: f (+ a b) a) F 1 0;

```



# API



`require("@alexo/lambda")` returns a function of a lambda term defined

in a variant of the lambda calculus called Macro Lambda Calculus (MLC)

that allows macro definitions in order to input complex expressions.

The last term in the input is the term whose normal form is to be found.



For developing and testing purposes, the package also exports

two additional functions `.prepare(term)` and `.debug()`.

The `.debug()` function applies a single reduction step to

the interaction net compiled by the previous `.prepare()`

call and returns a human-readable string representation of

the current interaction net state.



# Grammar



Input consists of an optional list of macro definitions and a term:



```

%token NAME



%%



text : defs term

     ;

defs : /* empty */

     | defs NAME '=' term ';'

     ;

term : appl

     | abst

     ;

abst : NAME ',' abst

     | NAME ':' term

     ;

appl :      atom

     | appl atom

     ;

atom : '(' term ')'

     |     NAME

     ;

```



# License



Copyright (c) 2017 Anton Salikhmetov



Permission is hereby granted, free of charge, to any person obtaining a copy

of this software and associated documentation files (the "Software"), to deal

in the Software without restriction, including without limitation the rights

to use, copy, modify, merge, publish, distribute, sublicense, and/or sell

copies of the Software, and to permit persons to whom the Software is

furnished to do so, subject to the following conditions:



The above copyright notice and this permission notice shall be included in

all copies or substantial portions of the Software.



THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR

IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,

FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL THE

AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER

LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,

OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN

THE SOFTWARE.