Constructive Adpositional Grammars, FormallyDIP Colloquium, ILLC, Universiteit van Amsterdam

Marco Benini

marco.benini@uninsubria.itUniversità degli Studi dell’Insubria

13th February 2015

Abstract

Constructive Adpositional Grammars, already presented in this series oflectures by Federico Gobbo, is a way to describe natural languages which hasbeen developed to clarify the relations between syntax, semantics andpragmatics. In this talk, the mathematical model supporting the grammarsis introduced and motivated.

Specifically, it will be shown that, using Category Theory even in a ratherelementary way, it is possible to formalise not only a natural language, butalso, and more significantly, how and why this formalisation is done, makingclear what are the fundamental choices, what are the sound alternatives, andhow variations can be used to obtain essentially equivalent grammars butwith a different orientation towards a specific purpose. So, a formal modelbecomes a powerful way to reason about the structure of a language,making conscious choices on what to emphasise and what to hide, in orderto describe just the aspects of interest.

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Example

“Paul is going to study maths in the library”

�D

←inE

theA

←εOlibrary

O

←εI2

�D

→-ing

O

�D

←εO

mathsO

←εI22

���PaulO

→εI21 study

I2

→toI22

���PaulO

→εI21 go

I2∗

→εI22

PaulO

←εI21 is

I2

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Language as a category

A language is a collection of expressions, which are constructed onefrom another. In the example, “Paul is going to study math in thelibrary” is constructed from “Paul is going to study math”, which, inturn, is constructed from “is”.So, a language is a mathematical category where■ the objects are the expressions■ the arrows are the (concrete) constructions■ identities are the empty constructions■ composition g ◦ f is the rule apply the construction g on the resultof the construction f

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Language as a category

Expressions are abstract entities: they have a unique role in thesyntax and a unique meaning in the semantics, although theirpresentations, i.e., the way they are written, may be ambiguous.A grammar is an abstract description of the constructions, i.e., of thearrows in the category. It is sound when it describes only existingarrows, and it is complete when it describes all the existing arrows.A grammar is built to achieve a purpose, for example, efficientparsing. The adpositional grammars were built to help the analysis ofthe language, to provide a neat description of linguistic phenomena,and to emphasise the structural aspects.

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Grammar characters

Fixed a category C, the collection of grammar characters is a familyG = {Gi }i such that ⋃

i Gi =ObjC, Gi ∩Gj =; when i 6= j , and each Giis inhabited. The collection of possible indexes is called GC .In the example the grammar characters are indexed byGC = {I2, I21 , I22 , I2, I2∗,A,E ,O,U ,D}.In words, G is a partition of expressions, a way to classify them.Although this is the basis of all existing grammars, it is less trivialthan it seems to be. . .

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Grammar characters

. . . in fact, we may define a grammar category G whose objects are thegrammar characters and whose arrows are the abstract constructions

C

G

Thus, concrete constructions are generated by the grammar G assheaf-like structures: C is the étalé space over the fibres on G, and theconcrete constructions are the counter-images of the abstract ones.How effective and useful is this generation process highly depends onthe structure of the grammar category.

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Defining constructions

Naturally, an abstract construction is the collection of the concreteconstructions it generates through the fibres.Assuming that constructions are applied to expressions, the index inthe fibre on the domain is clear. In fact, an abstract construction isdefined by the indexing in the fibre of the codomain.In the example, the abstract rule O → O which associates a noun withan article, generates the concrete construction library → the library.It does so by choosing the element in the fibre on the domain whichsays that library has the O grammar character, and by choosing theelement in the fibre on the codomain which says that the has the Agrammar character thus the library is the concrete instance of theabstract construction.

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Defining constructions

In general, an abstract construction can be specified by a product ofgrammar characters C1×·· ·×Cn so that, given the expression e, theconstruction builds the expression which combines e with an instance(c1, . . . ,cn) of the product.In the adpositional paradigm, we assume that the above combinationis performed by generating a concrete construction of the form

e→ a1(e,a2(c1 . . .an(cn−1,cn)))

with a1, . . . ,an from the distinct U grammar character which groupsthe linking expressions, called adpositions.In the example, go is in the grammar character I(ε,to), so applying itto the pair (Paul,study maths) yields Paul go to study maths.

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Adtrees

The adpositional abstract constructions provide a way to representconcrete expressions: adpositional trees (adtrees, for short). Evidently,this representation can be extended to constructions as well.For instance, the example in the previous slide can be rewritten as

goI(ε,to)

7→ study mathO

↔toI

PaulO

↔ε

I(to)

goI(ε,to)

Assuming to have a collection of atomic expressions, and that eachabstract construction can be reduced to elementary constructions, theadtree presentation allows to recursively write any expression.

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Adtrees

Elementary constructions have the form

XG

→YD

↔aE

XG

Thus, in formal terms, they are abstract construction G →E from thegovernor G to the grammar character E of the resulting expression, onthe product (D,a), i.e, the pair formed by the dependent D and theadposition a. The concrete expression X denotes the index in the fibreof the domain G , while Y is the concrete instance of the product D.By swapping X and Y , and G and D, we get the conjugateconstruction, which is, of course, equivalent but it moves the focus ofthe construction.

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Adtrees

Adtrees together with the instances of abstract constructions form acategory Ad(C). There is an evident embedding of C into Ad(C),given by the realisation of abstract constructions in C.In general, Ad(C) extends C: it contains adtrees for non-validexpressions, like “Paul go to study math”, which does not exists inthe English language.This extended world is much more regular and so more apt to getanalysed. And, it is easy to get back to the world of “sound”expressions by means of redundancy transformations, which areinstances of a far more general tool.Appendix B of F. Gobbo, M. Benini, Constructive AdpositionalGrammars: Foundations of Constructive Linguistics, CambridgeScholar Press (2011) contains the technical details omitted here.

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Transformations

A transformation is a functor Ad(C)→ Ad(C). Technically, atransformation maps each adtree into another adtree, andconstructions into constructions, coherently, preserving identities andcomposition.Intuitively, a transformation is a way to uniformly establish acorrespondence between constructions. For example, changing thetense of a verb from active to passive.Or, adding redundancies wherever appropriate: for example, takingcare of according the gender or the number of a noun with the otherpart of the discourse, thus realising the retraction from Ad(C) to C.As a matter of fact, as my colleague has already shown in his talk, allthe major linguistic phenomena are modelled through transformations,leading to surprising insights, sometimes.

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Transformations

A grammar functor is a functor F : G→G from the grammar categoryto itself. As a matter of fact, the interesting transformations are theones generating grammar functors, simply because thesetransformations preserve the grammar.By allowing hidden or null expressions in the language, grammarfunctors can describe very complex and abstract constructions, like inthe guiding example:

mathsO

←εI22

���PaulO

→εI21 study

I2

7→�D

←��toO

mathsO

←εI22

���PaulO

→εI21 study

I2

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Conclusion

Necessarily, many aspects of the mathematical model behindadpositional grammars have been left out of this talk.The important points are:■ an adpositional grammar is a natural instance of a generalparadigm which is based on the categorical notion of sheaves;

■ grammar characters and abstract constructions are design choicesin the design of a grammar: sound alternatives are given in anelementary way by conjugate constructions; more complexalternatives can be obtained by applying transformations whosegenerated grammar functor is isomorphic to identity;

■ hiding expressions by means of transformations leads to a uniformrendering of highly complex constructions, like infinitives, gerunds,and similar.

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Conclusion

Not mentioned, but in the background, is the idea to look atgrammars as sheaves, i.e., as Grothendieck toposes: in particular,among the many other things, this would lead to generate a logicalsystem where abstract constructions and transformations becomeinference rules, allowing to reason about the structures in thelanguage.Of course, this way of reasoning uses the most abstract part ofcontemporary mathematics and, using the words of P. Johnstone, oneof the most respected experts of topos theory, it is not for thefaint-hearted.

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Questions?

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