Parsing conjunctions in D is C o C irc (2021)

Abstract: In distributional compositional models of meaning logical words require special interpretations, that specify the way in which other words in the sentence interact with each other. So far within the DisCoCat framework, conjunctions have been implemented as merging both conjuncts into a single output, however in the new framework of DisCoCirc merging between nouns is no longer possible. We provide an account of conjunction and an interpretation for the word ‘and’ that solves this, and moreover ensures certain intuitively similar sentences can be given the same interpretations.

## introduction the distributional semantics paradigm allows us to model the meanings of words in terms of their surrounding words, as well as the meaning of texts in terms of their constituent words. meanwhile compositional semantics considers how meaning can arise from the grammatical form of a sentence. the distributional compositional (disco-) approach to modelling meaning introduced by @xcite allows the meaning of a sentence to be computed as a function of both the distributional meaning of the words involved, as well as its grammatical form. recent work by @xcite has implemented the approach on quantum hardware, and @xcite ; @xcite have shown the classical model to perform well at various natural language processing tasks. however, there are certain words whose meaning cannot be expressed as a function of their surrounding words -logical words like and, or and not, as well as pronouns such as whom and that -since they tend to appear in all contexts and thus render distributional approaches to modelling them inadequate. on the other hand, such words typically have an impact on the way in which the other words in the sentence interact, and thus take on a more syntactic role, acting as an extension of the grammar. in this paper, we shall focus in particular on the logical connective and. previously, in the disco-formalisms conjunctions have been interpreted as simply mixing the conjuncts involved together to produce a single entity of the appropriate type: as described in @xcite , we can mix the adjectives red and yellow together to make a new adjective that describes things that are both red and yellow. this approach works less well however, when we attempt to mix nouns; when we discuss a hat and a scarf, we are not discussing a single hybrid object that is both a hat and a scarf, but two objects, one of which is a hat, the other of which is a scarf. this difference suggests that there are two different types of conjunctions at play. the discocirc framework allows us to express this differencein the introductory paper coecke (2020), they are denoted as linear and non-linear forms of and. the non-linearity can be seen as arising from an induced duplication: sentence (1a) employs a linear form of and -in this case both adjectives are applied to the socks which then feeds into the rest of the sentence. we can contrast this with the non-linear interpretation of the same sentence in (1b), which does not seem to convey the same meaning. on the other hand, (2a) seems to carry the same meaning as (2b) which exhibits the implicit duplication of wear induced by the conjunction. this distinction was previously lost as conjoined nouns were considered equivalent to a single noun -this meant that any duplication would have occurred at best implicitly. standard disco-approaches are often limited to linear operations only, as they have standardly been implemented using vector spaces and linear maps, however certain phenomena seem to require nonlinear elements: in lewis (2020), negation is modelled as a (non-linear) projection of positive maps onto their orthogonal subspaces, while @xcite shows how modelling duplication explicitly improves performance in sentence similarity and entailment tasks involving ellispsis and anaphora. wijnholds and sadrzadeh (2019b) describes how the duplication is introduced, by augmenting the basic pregroup based grammars with a non-linear reduction rule. here, we will also be introducing a non-linear element, modifying word meanings. in particular, this will allow us to model the duplication in sentences like (1) and ( 2 ), such that the interpretation of sentences we consider equivalent are the same when expressed using the discocirc framework. we will first give a brief overview of this framework, then introduce the required structures and definitions for modelling and, finishing with some examples. ## mathematical background we will give a brief overview of the mathematical background for the disco-approaches to modelling meaning. for a detailed introduction to the category theory see heunen (2020), while @xcite ; @xcite provide a more diagrammatic treatment. @xcite ; coecke (2020) introduce the discocat and disco-circ frameworks respectively. ## new structure in addition to the basic structures presented above, we shall make use of a few extra components in order to model and. ## modelling and having set the scene, we are now equipped to start modelling the word and. the first aspect to note, is that and has a variable type, of the form -x x x -, where x can stand for any (possibly composite) grammatical type. the account we give will hence be equally generic, though we proceed mostly via example. ## related work the non-linearity introduced here has very similar effects to the non-linearity @xcite introduce to deal with ellipsis, however arises rather differently. in particular, they introduce the non-linearity to the grammar parsing, effectively by allowing parts of the derivation to freely be reused elsewhere. in this way, arbitrary parts of the sentence can get copied over to a new location. in contrast, the duplication introduced here is via the words involved, such that the duplication can only occur along the wires given by the (linear) pregroup grammar. the duplication also does not apply to wires themselves -only boxes get copied, making this a strictly weaker notion. though we have not discussed a particular choice of meaning embedding, the approach relates to work encoding logical words using vector-based semantics. within discocirc, we are primarily concerned with characters -treating the nouns involved as individual entities. in some cases, however we might be interested in nouns as concepts, for which conjunction does not seem to imply the presence of multiple characters, but instead suggests a merging of the component concepts. aerts (2009) discusses such concepts, in particular considering the so-called guppy effect (guppy is considered a good example of the concept pet fish despite being a bad example of both fish and pet when the concepts are considered separately). he suggests representing the joint concept pet and fish as a superposition in a fock space to account for this: |pet and fish := (|pet ⊗ |fish ) ⊕ (|pet + |fish ) in the account of conjunction given here, we effectively split the two ways of combining concepts: the tensor is used on 'characters', while everything else is merged -potentially as a superposition. ## conclusion in summary, we have provided an account of and within the framework of discocirc, that allows us to generate diagrams capturing the intuitive meaning of sentences that involve conjunctions. due to the non-linear nature of and, we associate it to a diagram rewriting operation or 2-morphism, that duplicates the relevant parts of our diagrams. the treatment provided allows us to parse the common usage of and, however there is more work to be done when it comes to certain more complex cases. in particular, there are many related words that control how and whether to duplicate words, such as respectively and each, as well as certain phrases like three times which interact with duplication in more complex ways. ellipsis is also a closely related grammatical notion, and it would be interesting to see if the duplication approach explored here can provide a suitable solution.