discopy 0.3.4

Distributional Compositional Python

Distributional Compositional Python

DisCoPy is a tool box for computing with monoidal categories.

Features

Diagrams & Recipes

Diagrams are the core data structure of DisCoPy, they are generated by the following grammar:

diagram ::= Box(name, dom=type, cod=type)
    | diagram @ diagram
    | diagram >> diagram
    | Id(type)

type ::= Ty(name) | type.l | type.r | type @ type | Ty()

String diagrams (also known as tensor networks or Penrose notation) are a graphical calculus for computing with monoidal categories. For example, if we take ingredients as types and cooking steps as boxes then a diagram is a recipe:

from discopy import Ty, Box, Id, Swap

egg, white, yolk = Ty('egg'), Ty('white'), Ty('yolk')
crack = Box('crack', egg, white @ yolk)
merge = lambda x: Box('merge', x @ x, x)

crack_two_eggs = crack @ crack\
    >> Id(white) @ Swap(yolk, white) @ Id(yolk)\
    >> merge(white) @ merge(yolk)
crack_two_eggs.draw(path='docs/_static/imgs/crack-eggs.png')

Snakes & Sentences

Wires can be bended using two special kinds of boxes: cups and caps, which satisfy the snake equations, also called triangle identities.

from discopy import Cup, Cap

x = Ty('x')
left_snake = Id(x) @ Cap(x.r, x) >> Cup(x, x.r) @ Id(x)
right_snake =  Cap(x, x.l) @ Id(x) >> Id(x) @ Cup(x.l, x)
assert left_snake.normal_form() == Id(x) == right_snake.normal_form()

In particular, DisCoPy can draw the grammatical structure of natural language sentences encoded as reductions in a pregroup grammar (see Lambek, From Word To Sentence (2008) for an introduction).

from discopy import pregroup, Word

s, n = Ty('s'), Ty('n')
Alice, Bob = Word('Alice', n), Word('Bob', n)
loves = Word('loves', n.r @ s @ n.l)

sentence = Alice @ loves @ Bob >> Cup(n, n.r) @ Id(s) @ Cup(n.l, n)
pregroup.draw(sentence, path='docs/_static/imgs/alice-loves-bob.png')

Functors & Rewrites

Monoidal functors compute the meaning of a diagram, given an interpretation for each wire and for each box. In particular, tensor functors evaluate a diagram as a tensor network using numpy. Applied to pregroup diagrams, DisCoPy implements the distributional compositional (DisCo) models of Clark, Coecke, Sadrzadeh (2008).

from discopy import TensorFunctor

F = TensorFunctor(
    ob={s: 1, n: 2},
    ar={Alice: [1, 0], loves: [[0, 1], [1, 0]], Bob: [0, 1]})

assert F(sentence) == 1

Free functors (i.e. from diagrams to diagrams) can fill each box with a complex diagram. The result can then be simplified using diagram.normalize() to remove the snakes.

from discopy import Functor

def wiring(word):
    if word.cod == n:  # word is a noun
        return word
    if word.cod == n.r @ s @ n.l:  # word is a transitive verb
        return Cap(n.r, n) @ Cap(n, n.l)\
            >> Id(n.r) @ Box(word.name, n @ n, s) @ Id(n.l)

W = Functor(ob={s: s, n: n}, ar=wiring)


rewrite_steps = W(sentence).normalize()
sentence.to_gif(*rewrite_steps, path='autonomisation.gif', timestep=1000)

Getting Started

pip install discopy

Documentation

The tool paper is now available on arXiv:2005.02975, it was presented at ACT2020.

The documentation is hosted at readthedocs.io, you can also checkout the notebooks for a demo!