mildly context-sensitive languages
TRANSCRIPT
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Gökalp Demirci
Mildly Context-SensetiveLanguges
Cogs 500 – Introduction to Cognitive Science
by
H. Gökalp Demirci
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Content
• Introduction
• We need another class
• Mildly context sensitive languages
– Definition
– TAG Formalism
– Automaton for MCSL
• Conclusion
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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.
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Introduction
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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].
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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 )
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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.
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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
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MildlyContext-SensitiveLanguages
• 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
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MildlyContext-SensitiveLanguages
• 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)
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• 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)
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MildlyContext-SensitiveLanguages
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MildlyContext-SensitiveLanguages
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MildlyContext-SensitiveLanguages
• Let's see an example of parsing with cross-serial dependencies, Dutch sentence:
“ik haar hem de nijlpaarden zag helpen voeren”
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MildlyContext-SensitiveLanguages
“ik haar hem de nijlpaarden zag helpen voeren”
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MildlyContext-SensitiveLanguages
• 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
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MildlyContext-SensitiveLanguages
• Let's see how EPDA works on an exampleparsing of sentence:
“Jan Piet Marie zag laten zwemmen ”
• Notice the crossed dependencies in thisDutch sentence.
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MildlyContext-SensitiveLanguages
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• 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
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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