This package implements the lambda calculus using interaction nets, providing CLI and API.

Its browserified version is available as an online demo.

Algorithms

The following encodings of the lambda calculus are included:

• closed, the approach of arXiv:1304.2290v8 applied to An Interaction Net Implementation of Closed Reduction by Ian Mackie, this is the default algorithm;

• optimal, an implementation of Lambdascope by Vincent van Oostrom et al;

• abstract, an experimental algorithm described in arXiv:1701.04691v2.

The embedded read-back mechanism is described in Section 7 of 10.4204/EPTCS.225.7.

Benchmarks

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

SAMPLE            CLOSED        OPTIMAL       ABSTRACT
counter             58/6          143/4            N/A
w2eta             137/16          205/7           37/7
22210ii         1740/182        7886/70         731/70
3222ii          5896/545      164197/43        1182/43
1022ii        23026/2085     2489461/59        4298/59
4222ii    1442259/131124            N/A      262401/64
222210ii  6685119/655415            N/A    2359812/201
cfact4         15506/887      56890/691       3711/691
yfact4        24150/1741     526971/760       4168/760
cfact5      799868/16170  3074471/13462    69147/13462
yfact5      857864/22267            N/A    69780/13550

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] (<term> | -f <file>)

Options:
  --algo, -a     Select algorithm         [string]
  --debug, -d    Evaluate step by step   [boolean]
  --file, -f     Read term from file     [boolean]
  --inet, -i     Show interaction net    [boolean]
  --limit, -l    Limit interactions       [number]
  --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:

I = x: x;
K = x, y: x;
S = x, y, z: x z (y z);
T = K;
F = K I;
AND = p, q: p q F;
OR = p, q: p T q;
NOT = p: (a, b: p b a);
C0 = f, x: x;
C1 = f, x: f x;
C2 = f, x: f (f x);
C3 = f, x: f (f (f x));
C4 = f, x: f (f (f (f x)));
C5 = f, x: f (f (f (f (f x))));
C6 = f, x: f (f (f (f (f (f x)))));
C7 = f, x: f (f (f (f (f (f (f x))))));
C8 = f, x: f (f (f (f (f (f (f (f x)))))));
C9 = f, x: f (f (f (f (f (f (f (f (f x))))))));
C10 = f, x: f (f (f (f (f (f (f (f (f (f x)))))))));
SUCC = n: (f, x: f (n f x));
PLUS = m, n: (f, x: m f (n f x));
MULT = m, n: (f: m (n f));
EXP = m, n: n m;
PRED = n: (f, x: n (g, h: h (g f)) (K x) I);
MINUS = m, n: n PRED m;
ZERO = n: n (K F) T;
Y = (a: a a) (self, f: f (self self f));

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 /* [A-Za-z][A-Za-z0-9]* */

%%

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.