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Exploring Semantic Incrementality with Dynamic Syntax and Vector Space Semantics

One of the fundamental requirements for models of semantic processing in dialogue is incrementality: a model must reflect how people interpret and generate language at least on a word-by-word basis, and handle phenomena such as fragments, incomplete and jointly-produced utterances. We show that the incremental word-by-word parsing process of Dynamic Syntax (DS) can be assigned a compositional distributional semantics, with the composition operator of DS corresponding to the general operation of tensor contraction from multilinear algebra. We provide abstract semantic decorations for the nodes of DS trees, in terms of vectors, tensors, and sums thereof; using the latter to model the underspecified elements crucial to assigning partial representations during incremental processing. As a working example, we give an instantiation of this theory using plausibility tensors of compositional distributional semantics, and show how our framework can incrementally assign a semantic plausibility measure as it parses phrases and sentences.


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1 Introduction

An incremental, word-by-word view on language processing is motivated by much empirical evidence from human-human dialogue. This evidence includes split, interrupted, and corrective utterances, see e.g. [Howes et al.2011]:

A: Ray destroyed
B: the fuchsia. He never liked it. The roses he spared
A: this time.

In (1), the utterances are either inherently incomplete or potentially complete, with more than one agent contributing to the unfolding of a sequence, with in principle arbitrary speaker switch points and indefinite extendibility. In such cases, speakers and hearers must be processing the structural and semantic information encoded in each utterance incrementally. A second motivation comes from computational dialogue systems, where the ability to process incrementally helps speed up systems and provide more natural interaction [Aist et al.2007]. A third motivation comes from psycholinguistic results, even in individual language processing, which show that hearers can incrementally disambiguate word senses and resolve references, before sentences are complete and even using partial words and disfluent material to do so [Brennan and Schober2001]. In (1a,b), the ambiguous word dribbled can be resolved to a particular sense early on, given the (footballer or baby) subject, without waiting for the rest of the sentence. A fourth comes from cognitive neuroscience and models such as Predictive Processing [Friston and Frith2015, Clark2015] which focus on agents’ incremental ability to generate expectations and judge the degree to which they are met by observed input.

The footballer dribbled   the ball across the pitch.

The baby dribbled   the milk all over the floor.

We use the framework of Dynamic Syntax (DS) for incremental grammatical and semantic analysis [Kempson et al.2001, Cann et al.2005, Kempson et al.2016]. DS has sufficient expressivity to capture the dialogue phenomena in (1) and has been used to provide incremental interpretation and generation for dialogue systems [Purver et al.2011, Eshghi et al.2017]. Yet incremental disambiguation is currently beyond its expressive power; and while its framework is broadly predictive, it does not yet provide an explanation for how specific expectations can be generated or their similarity to observations measured- though see [Hough and Purver2017] for DS’s interface to a probabilistic semantics.

DS does not fix a special form of syntax and instead defines grammaticality directly in terms of incremental semantic tree growth. Symbolic methods are employed for labelling the contents of these trees, via terms either from an epsilon calculus [Kempson et al.2001] or a suitable type theory with records [Purver et al.2010]. These symbolic approaches are not easily able to reflect the non-deterministic content of natural language forms, nor the way any initially unfixable interpretation, polysemy being rampant, can be narrowed down during the utterance interpretation process. For the same reason, the assigned term specifications do not provide a basis for the graded judgements that humans are able to make during processing to assess similarity to (or divergence from) expectations [Clark2015], to incrementally narrow down a word’s interpretation, or disambiguate its sense in the emerging context.

Non-determinisms of meaning and gradient similarity judgements are the stronghold of the so-called distributional or vector space semantics [Salton et al.1975, Schütze1998, Lin1998, Curran2004]. By modelling word meanings as vectors within a continuous space, such approaches directly express graded similarity of meaning (e.g. as distance or angle between vectors) and changes in interpretation (via movements of vectors within a space). Vector space semantics has been extended from words to phrases and sentences using different grammatical formalisms, e.g. Lambek pregroups, Lambek Calculus, and Combinatory Categorial Grammar (CCG) [Maillard et al.2014, Krishnamurthy and Mitchell2013, Coecke et al.2010, Coecke et al.2013]. It has, however, not been extended to incremental and dialogue formalisms such as DS.

In this paper, we address these lacunae, by defining an incremental vector space semantic model for DS that can express non-determinism and similarity in word meaning, and yet keep incremental compositionality over conversational exchanges. As a working example, we instantiate this model using the plausibility instance of [Clark2013b] developed for a type-driven compositional distributional semantics, and show how it can incrementally assign a semantic plausibility measure as it performs word-by-word parses of phrases and sentences. We discuss how this ability enables us to incrementally disambiguate words using their immediate contexts and to model the plausibility of continuations and thus a hearer’s expectations.

2 Dynamic Syntax and its Semantics

In its original form, Dynamic Syntax (DS) provides a strictly incremental formalism relating word sequences to semantic representations. Conventionally, these are seen as trees decorated with semantic formulae that are terms in a typed lambda calculus [Kempson et al.2001], chapter 9:

0.752 “In this paper we will take the operation to be function application in a typed lambda calculus, and the objects of the parsing process […] will be terms in this calculus together with some labels; […]”

This allows us to give analyses of the semantic output of the word-by-word parsing process in terms of partial semantic trees, in which nodes are labelled with type and semantic formula , or with requirements for future development (e.g. . ), and with a pointer indicating the node currently under development. This is shown in Figure 1 for the simple sentence Mary likes John. Phenomena such as conjunction, apposition and relative clauses are analysed via Linked trees (corresponding to semantic conjunction). For reasons of space we do not present an original DS tree here; an example of a non-restrictive relative clause linked tree labelled with vectors is presented in Figure 3.

“mary …” “…likes …”
12.5 11.25 levelsep=1cm, treesep=3.5cm
11.5 levelsep=1cm, treesep=4cm
Figure 1: DS parsing as semantic tree development, for the simple sentence “mary likes john”.

However, the DS formalism is in fact considerably more general. To continue the quotation above:

“[…] it is important to keep in mind that the choice of the actual representation language is not central to the parsing model developed here. […] For instance, we may take to be feature structures and the operation to be unification, or to be lambda terms and Application, or to be labelled categorial expressions and Application: Modus Ponens, or to be DRSs and Merging.”

Indeed, in some variants this generality is exploited; for example, Purver.etal10 outline a version in which the formulae are record types in Type Theory with Records (TTR) [Cooper2005]; and Hough.Purver12 show how this can confer an extra advantage – the incremental decoration of the root node, even for partial trees, with a maximally specific formula via type inference, using the TTR merge operation as the composition function. In the latter account, underspecified record types decorate requirement nodes, containing a type judgement with the relevant type (e.g.  at type nodes). HoughPurver17Lattices show that this underspecification can be given a precise semantics through record type lattices: the dual operation of merge, the minimum common super type (or join) is required to define a (probabilistic) distributive record type lattice bound by and . The interpretation process, including reference resolution, then takes the incrementally built top-level formula and checks it against a type system (corresponding to a world model) defined by a record type lattice. Implicitly, the record type on each node in a DS-TTR tree can be seen to correspond to a potential set of type judgements as sub-lattices of this lattice, with the appropriate underspecified record type (e.g.

) as their top element, with a probability value for each element in the probabilistic TTR version. In this paper, we show how equivalent underspecification, and narrowing down of meaning over time — but with the additional advantages inherent in vector space models, e.g. similarity judgements — can be defined for vector space representations with analogous operations to

and .

3 Compositional Vector Space Semantics for DS

Vector space semantics are commonly instantiated via lexical co-occurrence, based on the distributional hypothesis that meanings of words are represented by the distributions of the words around them- this is often described by Firth’s claim that “you shall know a word by the company it keeps” [Firth1957]. This can be implemented by creating a co-occurrence matrix [Rubenstein and Goodenough1965], whose columns are labelled by context words and whose rows by target words; the entry of the matrix at the intersection of a context word and a target word is a function (such as TF-IDF or PPMI) of the number of times occurred in the context of (as defined via e.g. a lexical neighbourhood window, a dependency relation, etc.). The meaning of each target word is represented by its corresponding row of the matrix. These rows are embedded in a vector space, where the distances between the vectors represent degrees of semantic similarity between words [Schütze1998, Lin1998, Curran2004].

Distributional semantics has been extended from word level to sentence level, where a compositional operation acts on the vectors of the words to produce a vector for the sentence. Existing models vary from using simple additive and multiplicative compositional operations [Mitchell and Lapata2010] to compositional operators based on fully fledged categorial grammar derivations, e.g. pregroup grammars [Coecke et al.2010, Clark2013b] or CCG [Krishnamurthy and Mitchell2013, Baroni et al.2014, Maillard et al.2014]. However, the work done so far has not been directly compatible with incremental processing: this paper is the first attempt to develop such an incremental semantics, using a framework not based on a categorial grammar, i.e. one in which a full categorial analysis of the phrase/sentence is not the obligatory starting point.

Compositional vector space semantic models have a complementary property to DS. Whereas DS is agnostic to its choice of semantics, compositional vector space models are agnostic to the choice of the syntactic system. Coeckeetal show how they provide semantics for sentences based on the grammatical structures given by Lambek’s pregroup grammars [Lambek1997]; CoeckeGrefenSadr show how this semantics also works starting from the parse trees of Lambek’s Syntactic Calculus [Lambek1958]; Wijnholds shows how the same semantics can be extended to the Lambek-Grishin Calculus; and [Krishnamurthy and Mitchell2013, Baroni et al.2014, Maillard et al.2014] show how it works for CCG trees. These semantic models homomorphically map the concatenation and slashes of categorial grammars to tensors and their evaluation/application/composition operations to tensor contraction.

In DS terms, structures are mapped to general higher order tensors, e.g. as follows:

Each abbreviates the linear expansion of a tensor, which is normally written as follows:

for a basis of and its corresponding scalar value. The operations are mapped to contractions between these tensors, formed as follows:

In their most general form presented above, these formulae are large and the index notation becomes difficult to read. In special cases, however, it is often enough to work with spaces of rank around 3. For instance, the application of a transitive verb to its object is mapped to the following contraction:

This is the contraction between a cube in and a vector in , resulting in a matrix in in .

We take the DS propositional type to correspond to a sentence space , and the entity type a word space . Given vectors in and the (cube) tensor in , the tensor semantic trees of the DS parsing process of become as in Fig. 2.111 There has been much discussion about whether sentence and word spaces should be the same or separate. In previous work, we have worked with both cases, i.e. when and when .

“mary …” “…likes …” “…john”

      11 levelsep=1cm, treesep=2.25cm       11 levelsep=1cm, treesep=2.5cm

Figure 2: A DS with Vector Space Semantics parse of ‘Mary likes John’.

A very similar procedure is applicable to the linked structures, where conjunction can be interpreted by the map of a Frobenius algebra over a vector space, e.g. as in [Kartsaklis2015], or as composition of the interpretations of its two conjuncts, as in [Muskens and Sadrzadeh2016]. The map has also been used to model relative clauses [Clark et al.2013, Sadrzadeh et al.2013, Sadrzadeh et al.2014]. It combines the information of the two vector spaces into one. Figure 2 shows how it combines the information of two contracted tensors and .

“mary, …” “…who …”


12.5 [M]
11.5 [W] linestyle=dashed [ncurv=.6,angleA=270,angleB=160]-¿MW

“…sleeps, …”

12.5 [M]
13.5 [W] [ncurv=.6,angleA=270,angleB=160]-¿MW
“…snores …”

13.5 [M]
13.5 [W] [ncurv=.6,angleA=270,angleB=160]-¿MW

Figure 3: A DS with Vector Space Semantics parse of ‘Mary, who sleeps, snores’.

DS requirements can now be treated as requirements for tensors of a particular order (e.g. , as above). If we can give these suitable vector-space representations, we can then provide an analogue to Hough.Purver12’s incremental type inference procedure, allowing us to compile a partial tree to specify its overall semantic representation (at its root node). One alternative would be to interpret them as picking out an element which is neutral with regards to composition: the unit vector/tensor of the space they annotate. A more informative alternative would be to interpret them as enumerating all the possibilities for further development. This can be derived from all the word vector and phrase tensors of the space under question — i.e. all the word and phrases whose vectors and tensors live in and in in this case — by taking either the sum or the direct sum of these vectors/tensors. Summing will give us one vector/tensor, accumulating the information encoded in the vectors/tensors of each word/phrase; direct summing will give us a tuple, keeping this information separate from each other. This gives us the equivalent of a sub-lattice of the record type lattices described in [Hough and Purver2017], with the appropriate underspecified record type as the top element, and the attendant advantages for incremental probabilistic interpretation.

These alternatives all provide the desired compositionality, but differ in the semantic information they contribute. The use of the identity provides no semantic information; the sum gives information about the “average” vector/tensor expected on the basis of what is known about the language and its use in context (encoded in the vector space model); the direct sum enumerates the possibilities. In each case, more semantic information can then arrive later as more words are parsed. The best alternative will depend on task and implementation: in the next section, we give a working example using the sum operation.

4 Incremental Plausibility: a working example

In order to exemplify the abstract tensors and tensor contraction operations of the model and provide a proof of concept for its applicability to semantic incrementality, we characterise the incremental disambiguation of the The footballer dribbled…. example in (1). This example is worked out in the instance of the compositional distributional semantics introduced in [Clark2013b] and implemented in [Polajnar et al.2014], intended to model plausibility. In this instance, is a two dimensional space with basis vectors true and false . Sentences that are highly plausible have a vector representation close to the basis; highly implausible sentences have one close to the basis. As an illustrative example, we take to be the following matrix based on co-occurrence counts:222For illustrative purposes, the co-occurrence counts are taken from random excerpts of up to 100 sentences, taken from the BNC; a full implementation would of course use larger datasets.

infant nappy pitch goal
baby 34 10 0 0
milk 10 1 0 0
footballer 0 0 11 52
ball 0 1 27 49

For an example of a vector representation, consider the row corresponding to baby: this gives us a vector with the linear expansion , for infant,  nappy,  pitch,  goal a basis vector of and its corresponding scalar value. The value represents the number of times baby occurred in the same piece of text as nappy; the value represents the number of times baby occurred in the same excerpt as goal, e.g. as the subjects of wore nappy or crawled into a goal.

Intransitive verbs will have matrix representations with linear expansion with a basis vector of and a basis vector of . A high value for on the basis means that it is highly plausible that has the property ; a high value at the means that it is highly implausible that has property . For example, consider the verbs vomit, score, dribble in their intransitive roles: has a high value at , since it is highly plausible that things that are scored are goals; and a high entry at , since it is highly implausible that things that wear nappies (e.g. babies) score. has an opposite plausibility distribution for infant and nappy wearing agents. is a mixture of these two, since both nappy wearing and goal scoring agents do it, but in different senses. Here, we instantiate the matrix purely from text co-occurrence, approximating plausibility from co-occurrence of verb and entity in the same text excerpt and implausibility from lack thereof, i.e. occurrence of verb without the entity. Other methods could of course be used, e.g. using dependency parse information to show verb-agent relations directly; or learning entries via regression [Polajnar et al.2014]. Note that while this makes our plausibility and implausibility degrees dependent, and the two dimensional can therefore be reduced to a one dimensional one, the theory supports spaces of any dimension, so we present values and computations for both dimensions to illustrate this.

vomit 10 2 9 3 3 9 0 12
score 1 7 0 8 7 1 8 0
dribble 22 2 21 3 14 10 16 8

The interpretation of an intransitive sentence, such as Babies vomit is calculated as follows:

Similar calculations provide us with the following sentence representations:

It follows that Babies vomit is more plausible than Footballers vomit, Footballers score is more plausible than Babies score, but Babies dribble and Footballers dribble have more or less the same degree of plausibility.

A transitive verb such as control will have a tensor representation as follows:

for basis of and either or . Suppose that control has a 1 entry value at pitch and goal with and a low or zero entry everywhere else. It is easy to show that the sentence representation of Footballers control balls is much more plausible than that of Babies control balls.

In an unfinished utterance, such as babies …, parsing will first derive a semantic tree containing the vector for babies and a tensor for ; then we tensor contract the two to obtain a vector in . The underspecified tensor in is computed by summing all known elements of :

The tensor contraction of this with the vector of babies provides us with the meaning of the utterance:

Similar calculations to the previous cases show that plausibility increases when moving from the incomplete utterence to the complete one . Conceptually speaking, the incomplete phrase will be a dense, high-entropy vector with nearly equal values on and , whereas the complete phrase (or the more complete phrase), will result in a sparser vector with more differential values on and . Continuation with a less plausible verb, e.g. score would result in a reduction in plausibility; and different transitive verb phrases would of course have corresponding different effects. We therefore cautiously view this as an initial step towards a model which can provide the “error signal” feedback assumed in models of expectation during language interpretation [Clark2015].

5 Nondeterminism of Meaning and Incremental Disambiguation

5.1 Incremental Disambiguation

Distributional semantics comes with a straightforward algorithm for acquisition of word meaning, but when a word is ambiguous its vector representation becomes a mixture of the representations of its different senses. Post processing of these vectors is needed to obtain different representations for each sense [Schütze1998, Kartsaklis and Sadrzadeh2013]. Given vectors for individual senses, our setting can incrementally disambiguate word meanings as the sentence is processed. For instance, we can incrementally determine that in Footballers dribble, dribble means ‘control the ball’; while in Babies dribble it means something closer to ‘drip’. This is done by computing that Babies dribble is more plausible than Babies dribble, and also that Footballers dribble is more plausible than Footballers dribble. Note that this disambiguation can be made before the sentence is complete: in

Her fingers tapped on her i-pad

, or The police tapped his phone, the combination of subject and verb alone can (given suitable vectors and tensors) give information about the relative plausibility of the readings of tapped as ‘knocked’ or ‘intercepted’. This can then be strengthened when the object is parsed (or, indeed, weakened or even reversed, depending on the object).

The above examples are taken from the disambiguation dataset of [Kartsaklis et al.2013]. Parts of this dataset has been tested on the plausibility model of [Clark2013b] by [Polajnar et al.2014], where it has been shown that plausibility implementations of verb tensors do a better job in disambiguating them. Repeating this task in our model to experimentally validate the incremental disambiguation hypothesis constitutes work in progress.

5.2 Incremental Expectation

Using our model on examples such as the above, we can also incrementally compute plausibility of possible continuations. Consider the “dribble” example in (1) again: after parsing Footballers dribble, we can calculate not merely that the verb’s interpretation can be narrowed down in the presence of the subject, but also that the continuation ball would be very plausible, and the continuation milk very implausible. A similar computation provides us with the plausible continuations for Police intercept vs Fingers knock. If we are using the (direct) sum method to assign overall plausibility to the unfinished sentence, the plausibility values of the possible continuations have already been calculated; here we need only inspect the particular values of interest. Using this method, we can therefore explain how people assign shifting expectations as parsing proceeds, and make interim probabilistic evaluations on the basis thereof – giving us a basis for a model embodying the ‘predictive processing’ stance of [Clark2013a]. We leave experiments into evaluating this hypothesis for future work.

6 Discussion

Our distributional DS model gives us a basis for incremental interpretation via compositional, grammar-driven vector space semantics. The particular instantiation outlined above assigns sentence representations in only a two-dimensional plausibility space, but the framework generalises to any vector space. Our intention is to extend this to more informative spaces, and integrate with the incremental probabilistic approaches to interpretation such as HoughPurver17Lattices’s approach to reference resolution.

One important step will be to adapt the model for incremental generation. In the original formulation by PurverKempson04Generation, DS generation is defined as a process of DS parsing, along with a check against a goal tree. At each generation step, every word in the vocabulary is tested to check if it is parseable from the current parse state; those which can be parsed are tested, with the resulting DS tree being checked to see if it subsumes the goal tree. If it does subsume it, then the parsed word can be generated as output; when the current tree and goal tree match, generation is complete and the process halts. Hough.Purver12 updated this to use a goal concept as a TTR record type, with the subsumption check now testing whether a DS-TTR tree’s top-level record type is a proper supertype of (i.e. subsumes) the current goal record type. Given the equivalence of our proposed model to Hough.Purver12’s parsing process described above, the only additional apparatus required for generation for DS with Vector Space Semantics is the use of a goal tensor, and a characterisation of subsumption between two tensors. For the latter, we intend to look into a distributional characterisation of inclusion [Kartsaklis and Sadrzadeh2016], in the spirit of a real-valued measure of relevance proposed in probabilistic type theory by [Hough and Purver2017]. Other approaches to this are exploring type theory and vector space semantics hybrids such as [Asher et al.2017].


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