mildly context-sensitive languages

8/6/2019 Mildly Context-Sensitive Languages

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Gökalp Demirci

Mildly Context-SensetiveLanguges

Cogs 500 – Introduction to Cognitive Science

by

H. Gökalp Demirci



Gökalp Demirci

Content

• Introduction

• We need another class

• Mildly context sensitive languages

– Definition

– TAG Formalism

– Automaton for MCSL

• Conclusion



Gökalp Demirci

Introduction

• Formal language: Set of words over analphabet.

• Mainly interested in classification of formallanguages and formalism describinglanguages.

• Chomsky Hierarchy (1956) described majority of knowledge we have today on formal languagetheory.



Gökalp Demirci

Introduction



Gökalp Demirci

Introduction

• Natural languages are set of strings over certain alphabets.

• So they are also formal languages.• Where do they stand in Chomsky Hierarchy?

• Which class do they belong?“The main problem of immediate relevance to the theory of lan-

guage is that of determining where in the hierarchy of devices the

grammars of natural languages lie.” [Chomsky, 1959, p. 138].



Gökalp Demirci

We need anew class

• Natural languages are not regular.

Proof: Palindromic strings occur in natural lang.

“It is clear, then that in English we can find a sequence a + S1 + b,

where there is a dependency between a and b, and we can select as

S1 another sequence c + S2 + d, where there is a dependency

between c and d ... etc. A set of sentences that is constructed in this

way...will have all of the mirror image properties of [2] which exclude

[2] from the set of finite languages.”

(Chomsky 1957)


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We need anew class

• Natural languages are not context-free.

• Chomsky asked this question in 1957

• Proven by Huybregts (1984) and Shieber (1985)

Proof: Cross-serial dependencies occur innatural languages like Swiss-German, Dutch..

• dat Jan Piet Marie de Kinderen zag helpen laten zwemmen

(That Jan Piet Marie the children saw help make swim )

(That Jan saw Piet help Marie make the children swim )



We need anew class

• It is clear that recursively enumerablelanguages properly contains natural languages.

• Turing Machines are able to parse naturallanguages.

• It is also true that natural languages are contextsensitive (may be parsed by a TM inpolynomial space).

• So how much extra power do context freegrammars need to parse natural languages.

• We need a new class in the hierarchy.



MildlyContext-SensitiveLanguages

• First introduced by Aravind Joshi in 1985.

• Here is an informal definition of MCSL: – Beside that it contains all context-free

languages, it should have following properties:

• It admits limited cross-serial dependencies.

• Languages in MCSG are parsable in

polynomial time. • It has constant-growth property.

Definition




• Structures that formally define MCSL:

– Tree adjoining grammars (TAG)

– Combinatory categorial Grammars (CCG)

– Linear indexed grammars (LIG)

– Head grammars (HG)

– Linear context-free rewriting

systems(LCFRS)

Definition




• Developed by Joshi in the 70's and 80's.

• It is a tree generating system rather than a

string generator.

• Consists of initial and auxiliary trees.

• Trees can be combined by substitution andadjunction.

• (note the difference from other rewritingsystems: adjunction!! )

Tree Adjoining Grammars (TAG)




• Formal definition:A tree-adjoining grammar (TAG) G is a quintuple {Vn, Vt,Tini,

Taux, S}, whereVn is a finite set of non-terminals

Vt is a finite set of terminals

Tini is a finite set of trees, called the initial trees

Taux is a finite set of trees, called the auxiliary trees and S⋲Vn is the

start symbol.

Tree Adjoining Grammars (TAG)




• Let's see an example of parsing with cross-serial dependencies, Dutch sentence:

“ik haar hem de nijlpaarden zag helpen voeren”




“ik haar hem de nijlpaarden zag helpen voeren”




• All classes in the hierarchy has automatonsthat basically simulate the grammar in a

more machine fashionable way.• TAG's has an automaton called embedded

pushdown automaton (EPDA) defined by Vijay-Shanker in 1988.

• Basically it differs from PDA by its ability tocreate new stacks on the left and right of thecurrent stack.

Automaton for MCSL




• Let's see how EPDA works on an exampleparsing of sentence:

“Jan Piet Marie zag laten zwemmen ”

• Notice the crossed dependencies in thisDutch sentence.



• We have seen that the hierarchy defined byChomsky is not enough for fitting naturallanguages into the picture.

• So Joshi added one more level to the hierarchy,mildly context-sensitive languages, betweencontext-free languages and context-sensitivelanguages.

• MCF grammars are widely used for naturallanguage processing.

• We could process by algorithms recognizingcontext-sensitive languages then we will haveproblems on complexity of algorithm andunlimited cross-serial dependencies.

Conclusion



Thank you for your time

References1.On the Learnability of Mildly Context-Sensitive Languages using Positive Data and

Correction Queries, Doctoral dissertation, L. B. Bona

2. Mildly Context Sensitive Grammar Formalisms, Petra Schmidt

3. Tree-Adjoining Grammars, A. K. Joshi, Yves Schabes

4. A model-theoretic approach to Mildly Context-sensitive Grammars, Ippei Ukai

5. The Convergence of Mildly Context-Sensetive Grammar Formalism, Joshi et. al.

6. Restricting Grammatical Complexity, Robert Frank

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