Step 2. Diagram rewriting

Syntactic derivations in pregroup form can become extremely complicated, which may lead to excessive use of hardware resources and prohibitively long training times. The purpose of the rewrite module is to provide a means to the user to address some of these problems, via rewriting rules that simplify the string diagram. As an example, consider again the sentence “John walks in the park”.

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[1]:

from lambeq import BobcatParser

# Parse the sentence
parser = BobcatParser(verbose='suppress')
diagram = parser.sentence2diagram("John walks in the park")

diagram.draw(figsize=(11,5), fontsize=13)

Note that the representation of the preposition is a tensor of order 5 in the “classical” case, or a state of 5 quantum systems in the quantum case. Applying the prepositional_phrase rewriting rule to the diagram takes advantage of the underlying compact-closed monoidal structure, by using a “cap” to bridge the discontinued subject noun wire within the preposition tensor. Furthermore, the determiner rewriting rule will apply a cap on type \(n \cdot n^l\), eliminating completely the determiner “the”.

[2]:

from lambeq import Rewriter

# Apply rewrite rule for prepositional phrases

rewriter = Rewriter(['prepositional_phrase', 'determiner'])
rewritten_diagram = rewriter(diagram)

rewritten_diagram.draw(figsize=(11,5), fontsize=13)

We will now ask lambeq to normalise the diagram, by “stretching” the wires and re-arranging the boxes if required:

[3]:

normalised_diagram = rewritten_diagram.normal_form()
normalised_diagram.draw(figsize=(9,4), fontsize=13)

In the simplified diagram, the order of the preposition tensor is reduced by 2, which at least for a classical experiment, is a substantial improvement. Note also that the determiner is now eliminated, equating the meaning of the noun phrase “the park” with that of the noun “park”.

Another very useful rewrite rule is the CurryRewriteRule, which allows us to convert adjoint output wires into input wires using map-state duality. For example:

[4]:

curry_functor = Rewriter(['curry'])
curried_diagram = curry_functor(normalised_diagram)
curried_diagram.draw(figsize=(9,4), fontsize=13)

After normalisation the resulting diagram no longer contains any cups, which eliminates :term:post-selection and allows for faster execution.

[5]:

curried_diagram.normal_form().draw(figsize=(5,4), fontsize=13)

These examples clearly demonstrate the flexibility of string diagrams compared to simple tensor networks, which was one of the main reasons for choosing them as lambeq’s representation format. lambeq comes with a number of standard rewrite rules covering auxiliary verbs, connectors, coordinators, adverbs, determiners, relative pronouns, and prepositional phrases.

Rewrite rule	Description
`auxiliary`	Removes auxiliary verbs (such as “do”) by replacing them with caps.
`connector`	Removes sentence connectors (such as “that”) by replacing them with caps.
`coordination`	Simplifies “and” by replacing it with a layer of interleaving spiders.
`curry`	Uses map-state duality to reduce the number of cups in the diagram.
`determiner`	Removes determiners (such as “the”) by replacing them with caps.
`object_rel_pronoun` , `subject_rel_pronoun`	Simplifies relative pronouns (such as “that”) using cups, spiders and a loop.
`postadverb` , `preadverb`	Simplifies adverbs by passing through the noun wire transparently using a cap.
`prepositional_phrase`	Simplifies prepositions by passing through the noun wire using a cap.