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- 1. Constructive Adpositional Grammars, Formally DIP Colloquium, ILLC, Universiteit van Amsterdam Marco Benini marco.benini@uninsubria.it Università degli Studi dell’Insubria 13th February 2015
- 2. Abstract Constructive Adpositional Grammars, already presented in this series of lectures by Federico Gobbo, is a way to describe natural languages which has been developed to clarify the relations between syntax, semantics and pragmatics. In this talk, the mathematical model supporting the grammars is introduced and motivated. Speciﬁcally, it will be shown that, using Category Theory even in a rather elementary way, it is possible to formalise not only a natural language, but also, and more signiﬁcantly, how and why this formalisation is done, making clear what are the fundamental choices, what are the sound alternatives, and how variations can be used to obtain essentially equivalent grammars but with a diﬀerent orientation towards a speciﬁc purpose. So, a formal model becomes a powerful way to reason about the structure of a language, making conscious choices on what to emphasise and what to hide, in order to describe just the aspects of interest. (2 of 17)
- 3. Example “Paul is going to study maths in the library” D ← in E the A ← O library O ← I2 D → -ing O D ← O maths O ← I2 2 Paul O → I2 1 study I2 → to I2 2 Paul O → I2 1 go I2∗ → I2 2 Paul O ← I2 1 is I2 (3 of 17)
- 4. Language as a category A language is a collection of expressions, which are constructed one from another. In the example, “Paul is going to study math in the library” is constructed from “Paul is going to study math”, which, in turn, 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 result of the construction f (4 of 17)
- 5. Language as a category Expressions are abstract entities: they have a unique role in the syntax and a unique meaning in the semantics, although their presentations, i.e., the way they are written, may be ambiguous. A grammar is an abstract description of the constructions, i.e., of the arrows in the category. It is sound when it describes only existing arrows, and it is complete when it describes all the existing arrows. A grammar is built to achieve a purpose, for example, eﬃcient parsing. The adpositional grammars were built to help the analysis of the language, to provide a neat description of linguistic phenomena, and to emphasise the structural aspects. (5 of 17)
- 6. Grammar characters Fixed a category C, the collection of grammar characters is a family G = {Gi }i such that i Gi = ObjC, Gi ∩Gj = when i = j, and each Gi is inhabited. The collection of possible indexes is called GC. In the example the grammar characters are indexed by GC = {I2,I2 1 ,I2 2 ,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 trivial than it seems to be. . . (6 of 17)
- 7. Grammar characters . . . in fact, we may deﬁne a grammar category G whose objects are the grammar characters and whose arrows are the abstract constructions C G Thus, concrete constructions are generated by the grammar G as sheaf-like structures: C is the étalé space over the ﬁbres on G, and the concrete constructions are the counter-images of the abstract ones. How eﬀective and useful is this generation process highly depends on the structure of the grammar category. (7 of 17)
- 8. Deﬁning constructions Naturally, an abstract construction is the collection of the concrete constructions it generates through the ﬁbres. Assuming that constructions are applied to expressions, the index in the ﬁbre on the domain is clear. In fact, an abstract construction is deﬁned by the indexing in the ﬁbre of the codomain. In the example, the abstract rule O → O which associates a noun with an article, generates the concrete construction library → the library. It does so by choosing the element in the ﬁbre on the domain which says that library has the O grammar character, and by choosing the element in the ﬁbre on the codomain which says that the has the A grammar character thus the library is the concrete instance of the abstract construction. (8 of 17)
- 9. Deﬁning constructions In general, an abstract construction can be speciﬁed by a product of grammar characters C1 ×···×Cn so that, given the expression e, the construction builds the expression which combines e with an instance (c1,...,cn) of the product. In the adpositional paradigm, we assume that the above combination is 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 groups the linking expressions, called adpositions. In the example, go is in the grammar character I( ,to), so applying it to the pair (Paul,study maths) yields Paul go to study maths. (9 of 17)
- 10. Adtrees The adpositional abstract constructions provide a way to represent concrete 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 go I( ,to) → study math O ↔ to I Paul O ↔ I(to) go I( ,to) Assuming to have a collection of atomic expressions, and that each abstract construction can be reduced to elementary constructions, the adtree presentation allows to recursively write any expression. (10 of 17)
- 11. Adtrees Elementary constructions have the form X G → Y D ↔ a E X G Thus, in formal terms, they are abstract construction G → E from the governor G to the grammar character E of the resulting expression, on the product (D,a), i.e, the pair formed by the dependent D and the adposition a. The concrete expression X denotes the index in the ﬁbre of 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 conjugate construction, which is, of course, equivalent but it moves the focus of the construction. (11 of 17)
- 12. Adtrees Adtrees together with the instances of abstract constructions form a category 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-valid expressions, like “Paul go to study math”, which does not exists in the English language. This extended world is much more regular and so more apt to get analysed. And, it is easy to get back to the world of “sound” expressions by means of redundancy transformations, which are instances of a far more general tool. Appendix B of F. Gobbo, M. Benini, Constructive Adpositional Grammars: Foundations of Constructive Linguistics, Cambridge Scholar Press (2011) contains the technical details omitted here. (12 of 17)
- 13. Transformations A transformation is a functor Ad(C) → Ad(C). Technically, a transformation maps each adtree into another adtree, and constructions into constructions, coherently, preserving identities and composition. Intuitively, a transformation is a way to uniformly establish a correspondence between constructions. For example, changing the tense of a verb from active to passive. Or, adding redundancies wherever appropriate: for example, taking care of according the gender or the number of a noun with the other part 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, all the major linguistic phenomena are modelled through transformations, leading to surprising insights, sometimes. (13 of 17)
- 14. Transformations A grammar functor is a functor F : G → G from the grammar category to itself. As a matter of fact, the interesting transformations are the ones generating grammar functors, simply because these transformations preserve the grammar. By allowing hidden or null expressions in the language, grammar functors can describe very complex and abstract constructions, like in the guiding example: maths O ← I2 2 Paul O → I2 1 study I2 → D ← to O maths O ← I2 2 Paul O → I2 1 study I2 (14 of 17)
- 15. Conclusion Necessarily, many aspects of the mathematical model behind adpositional grammars have been left out of this talk. The important points are: an adpositional grammar is a natural instance of a general paradigm which is based on the categorical notion of sheaves; grammar characters and abstract constructions are design choices in the design of a grammar: sound alternatives are given in an elementary way by conjugate constructions; more complex alternatives can be obtained by applying transformations whose generated grammar functor is isomorphic to identity; hiding expressions by means of transformations leads to a uniform rendering of highly complex constructions, like inﬁnitives, gerunds, and similar. (15 of 17)
- 16. Conclusion Not mentioned, but in the background, is the idea to look at grammars as sheaves, i.e., as Grothendieck toposes: in particular, among the many other things, this would lead to generate a logical system where abstract constructions and transformations become inference rules, allowing to reason about the structures in the language. Of course, this way of reasoning uses the most abstract part of contemporary mathematics and, using the words of P. Johnstone, one of the most respected experts of topos theory, it is not for the faint-hearted. (16 of 17)
- 17. Questions? (17 of 17)