A Constructive Mathematics approach for NL formal grammars

From philosophy to computer science How to formalise the cognitive linguistics results? Adpositional grammars Conclusions

A Constructive Mathematics approach for Natural
Language formal grammars
An Introduction to Adpositional Grammars (AdGrams)

Federico Gobbo and Marco Benini
{federico.gobbo,marco.benini}@uninsubria.it
University of Insubria, Varese, Italy
CC Some rights reserved.

ECAP09, UAB, Barcelona, July 2009

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Philosophers and natural language formalization

How to clean natural languages (NLs) from ambiguity?

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Philosophers and natural language formalization

How to clean natural languages (NLs) from ambiguity?

Leibniz: characteristica universalis to catch the laws of human
thought and lingua generalis as a quasi-natural Latin to be
used as a written medium for scholars.
Frege: Begriﬀsschrift and the deﬁnition of unsaturated
expressions.
Husserl: the meaning categories as the formal constituents of
a logical grammar.

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Computer science and logic-based approaches to NL

Goal: formalize NL grammars through mathematical formulae
proved through computation. Some formalisms in use today (the
list is not complete!):
Based on Chomsky’s constituency and transformation notions:
Minimalism, Tree-Adjoining Grammar (TAG), Head-Driven
Phrase Structure Grammar (HPSG).

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Based on categorial calculus: Type-Logical Grammar (TLG),
Thinking Through Grammar (TTG), Combinatory Categorial
Grammar (CCG), Pre-group.

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Based on Tesni`re Dependency and Valency: Extensible
e
Dependency Grammar (XDG), Algebraic Syntax, Functional
Generative Description (FGD), Mel‘cuk’s Dependency Syntax.

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Based on Tesni`re Dependency and Valency: Extensible
e
Dependency Grammar (XDG), Algebraic Syntax, Functional
Generative Description (FGD), Mel‘cuk’s Dependency Syntax.
How to choose the best one, i.e., the most expressive, in
linguistic terms? 3/19


Psychological interpretation of NL formalisms

In recent years scholars got interested in strong psychological
interpretations of their formalisms:

“if I succeed to give a clear account of more
psychological phenomena thanks to my formalism, this
means that my formalism is better.”

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Psychological interpretation of NL formalisms

In recent years scholars got interested in strong psychological
interpretations of their formalisms:

“if I succeed to give a clear account of more
psychological phenomena thanks to my formalism, this
means that my formalism is better.”

Surprisingly, formal grammarians didn’t read the 20-year long
results of cognitive linguistics, where linguistic phenomena are
explained in psychological and cognitive terms.

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Cognitive linguistics and formalisation

Cognitive linguistics (e.g., Taylor, Cruse) are not interested in
formalisation, as their primary interest is in metaphor
interpretation: concepts are not formalised per se.

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Nonetheless, Langacker borrowed the dichotomy
trajector/landmark from the German school of Gestalt (e.g., Kurt
Koﬀka and Max Wertheimer) into syntax.

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Nonetheless, Langacker borrowed the dichotomy
trajector/landmark from the German school of Gestalt (e.g., Kurt
Koﬀka and Max Wertheimer) into syntax.

In other words, a cognitive account is inside the analytical
framework, instead of being a serendipity.

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Trajector vs. landmark and the Tesnerian dependency

a trajector is the most salient participant put in the focused
position;
the landmark is the reference point of observation performed
by the trajector.

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Trajector vs. landmark and the Tesnerian dependency

a trajector is the most salient participant put in the focused
position;
the landmark is the reference point of observation performed
by the trajector.
Our hypothesis is that the trajector/landmark relation is conveyed
in NLs either by prepositions (most Hindo-European languages, like
English or Catalan) or by postpositions (e.g., Turkish, Japanese) –
i.e., adpositions, and from this term adpositional grammar.

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Trajectors, landmarks and the Tesnerian dependency

Adpositional grammars (AdGrams) retain the concept of valency
from Tesni`re, but it reconﬁgures the concept of dependency
e
thanks to the dichotomy trajector/landmark.

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e

The structure of NLs (morphology + syntax) can be expressed
with the following triple:

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e

there is a Governor (G),

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e

there is a Dependent (D),

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e

D and G have their Relation (R).

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e

D and G have their Relation (R).
Remark: R is often a phrasal or sentence adposition.

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Standing on the shoulder of which giants?

AdGrams distinguish two directions of dependency:

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Dependency, when D is the trajector, and G is the landmark;

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Government, when the trajector is G, and consequentially the
landmark is D.

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Government, when the trajector is G, and consequentially the
landmark is D.

The advantage is that no assumption on semantics is made, i.e.,
we follow a strict world-model agnosticism: e.g., in some
possibly SF world the hotel killed the rabbit is true.

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This is the prototypical Dependency-based tree...

q
¡e
¡→e
¡ R e
¡ e
¡ e
D G
Notation: D is always on the left branch, while G is always on the
right.

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...while this one is the prototypical Government-based tree.

q
¡e
¡←e
¡ R e
¡ e
¡ e
D G
Let’s see a couple of examples.

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A Dependency-based phrase...

q
¡e
¡→e
¡ e
¡ e
¡ e
∆ ∆
The torpedo sank...
Figure: Adtree of The torpedo sank the ship.

The trajector is the torpedo, while the landmark is sank the ship.

Notation: delta (∆) means that some information is hidden.

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...and a Government-based phrase

q
¡e
¡←e
¡ e
¡ e
¡ e
∆ ∆
The ship sank
Figure: Adtree of The ship sank.

The landmark is the ship, while the trajector is the act of sinking
(in fact, the Agent and the Instrument are unexpressed here.)

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A Dependency-based sentence...

q
¡e
¡→e
¡ so e
¡ e
¡ e
∆ ∆
she can... A. is...
Figure: Adtree for Alice is rich so she can pay

The most salient information (trajector) is the fact that Alice can
pay.

Here, the adposition is so, which gives the Dependency structure.

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...and a Government-based sentence

q
¡e
¡←e
because
¡ e
¡ e
¡ e
∆ ∆
she is... A. can...
Figure: Adtree for Alice can pay because she is rich

Again, the most salient information is the fact that Alice can pay:
in this case the trajector is the Governor.

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Why constructive mathematics for formalisation

Unlike most other Dependency formal grammar frameworks,
AdGrams use techniques of constructive mathematics, since
constructive logics are the natural framework to model
computation, as argumented by Troelstra and Barendregt.

Surprisingly, mostly if not all formal NL grammars based on
combinatorial calculus take Chomsky’s constituency for grant,
using calculus “only” to build non-transformational grammars.

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The novelties of adpositional grammars

Our approach is different: constructivism permit to hide
information in a very natural and precise way, and it defines the
grammar as a specification in the language of logic, while the
semantics of logic act as a computational engine, so that we can
parse, generate and manipulate sentences essentially for free.

Furthermore, we use a formal method for a non-constituency based
grammar, ant this is the first attempt ever, at least as far as the
authors know.

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Provisional results

We have written an instance of AdGrams in the appropriate logical
formalism, based on intuitionistic logic, together with specialized
semantics.

This ﬁrst test is ﬁt for the quasi-natural language Esperanto and it
proved to cover approxiamtely 95% of the available corpora of
language-in-use, except of the well-known open issues of quoting
and name-entity recognition.

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Further directions

We are currently work on a more general model – i.e., an instance
of adpositional grammars which is cross-linguistically valid since the
beginning – where the engine captures the structure of every NL.

The intricacies are put in the lexicon – the part of semantics which
is rightly computable in terms of meaning components and
pragmatic participants.

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Thanks. Any questions?

Download these slides here:
http://www.slideshare.net/goberiko/

C
CC BY: $ Federico Gobbo 2009. Pubblicato in Italia.

Attribuzione – Non commerciale – Condividi allo stesso modo 2.5

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A Constructive Mathematics approach for NL formal grammars

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