LIN6932 Topics in
Computational Linguistics
Lecture 7
Hana Filip
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Overview
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Historical background
Definition of formal grammars
Chomsky Hierarchy
Computational complexity of natural
language
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Historical Background
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In 1949, Shannon and Weaver published The Mathematical Theory of
Communication, showing that statistical approximations to English based on
Markov processes could be used to encode English efficiently for
transmission in noise. Tasks like machine translation could be solved by
treating e.g. Russian as a noisy encoding of English.
• In 1957, Chomsky published Syntactic Structures, showing that natural
grammars could not be exactly captured by such methods. It seemed to follow
that machine translation could not be modeled as a noisy channel (although
the machines were too small to actually try any of this).
• Chomsky made a point of being open to the idea that STATISTICS could guide
grammatical processing.
• Nevertheless, most work in computational linguistics switched to
linguistically informed high-level SYMBOLIC representations of syntax and semantics,
and small knowledge domains.
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Historical Background
Given the grammar of a language, one can study the use of the language
statistically in various ways; and the development of probabilistic models
for the use of language (as distinct from the syntactic structure of the
language) can be quite rewarding.
Chomsky, 1957:17, note 4
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Historical Background
• Around 1988, the machines got big enough to try both techniques.
• Surprisingly, low level statistical approximations such as Markov processes
worked better than linguistically informed representations on almost all tasks,
such as speech recognition, parsing free text, and MT itself.
• Most work in computational linguistics switched to linguistically uninformed,
low-level statistical approximations and machine learning.
• Around 2000, the process of putting linguistics, statistical models, and
machine learning, back together again.
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Historical Background
• Chomsky’s 1957 book Syntactic Structures, together with certain more
technical papers from around the same time, is one of the most important
documents in linguistics and cognitive science.
• The theory in detail has been completely superseded.
• Nevertheless, surprisingly many of the formal devices that it includes recur,
particularly in the most modern descendents, including: kernel sentences (aka
lexicalized elementary trees or categories etc.); generalized or
“double-based” transformations (aka Merge, combinatory rules,
tree-adjunction, etc.; affix-hopping); the role of statistics
in natural language processing.
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Historical Background
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Affix Hopping as PF Merger and VP Ellipsis*
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Historical Background
One crucial ingredient of the theory was a hierarchy of language types, now
known as CHOMSKY’S HIERARCHY, each type characterized by a class of
rules that are sufficient to specify all languages of that type, an automaton which
is sufficient to recognize whether a sentence is from a given language of that
type, and a class of languages including all those of classes lower in the
hierarchy as a proper subset.
Question: What is the expressive/generative power of natural languages?
Where are human languages located on this hierarchy?
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Formal Grammar (Review)
A formal grammar G is quadruple G = <N, T, S, R>
N a finite set of nonterminal symbols
T a finite set terminal symbols
S start symbol (S  N)
R rules (‘productions’)
– Rules take the form ,
“ rewrites as ”, where ,  are strings of symbols from the
infinite set of strings (TN)* and  must contain at least one
non-terminal symbol.
Other conditions on the rules are imposed for particular classes of grammars.
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Chomsky Hierarchy
Other conditions are imposed for particular classes of grammars.
The Chomsky hierarchy:
Types of grammars defined in terms of additional restrictions on the form of the
rules:
Type 0: No restriction. Each rule is of the form   , and  ≠.
Type 1: Each rule is of the form A  , where A  N and ≠.
Type 2: Each rule is of the form A  . ( may be .)
Type 3: Each rule is of the form A  aB or A  a.
Common names:
Type 0: Unrestricted rewriting systems (Turing Equivalent)
Type 1: Context-sensitive grammars.
Type 2: Context-free grammars.
Type 3: Right-linear, or regular, or finite state automata / finite state grammars.
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Chomsky Hierarchy
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Chomsky Hierarchy
of classes (or families) of languages, originally defined by the form of the rules
needed to generate the languages in those classes, but which can also be
characterized at least in part by "dependencies" between elements that appear
in the strings that make up the languages of those classes.
The smallest infinite class of languages in the Chomsky hierarchy is the class RL
of regular languages. These are the languages that can be represented by regular
expressions.
The next larger class of languages in the Chomsky hierarchy is the class CFL of
context-free languages. Every regular language is also a context-free language,
but the converse is not true.
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Chomsky Chierarchy
TYPE O: UNRESTRICTED REWRITING SYSTEMS
no restrictions on the rules: L(G) = {w  T* | S * w}
a set of strings composed of terminal symbols derived from S
“*” is the reflexive and transitive closure of “”
“*” can be constructed as a set of operators each of which is
obtained by applying  successively 0 or more times.
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Every recursively enumerable language can be described by a rewriting system: an
algorithm that "enumerates" the strings of the language, I.e., this means that its output is
simply a list of the members of L: w1, w2, w3, ... . If necessary, this algorithm may run
forever.
Recursively enumerable languages are languages for which there is a decision procedure
for determining for any arbitrary string that it is a well-defined string in the language, but
not necessarily for determining for any arbitrary string not in the language that it is not.
Membership in a type-0 language is undecidable.
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Chomsky Chierarchy
CONTEXT-SENSITIVE GRAMMARS
– Subclass of type-0 grammars
– Restriction: all rules take the form

A  
length()  length()
where AN, , ,   (TN)*,  ≠ 
Membership in a context-sensitive language (CSL) is decidable.
CLSs are languages for which there is a decision procedure for
determining whether an arbitrary string does or does not belong
to the language.
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Chomsky Chierarchy
• Not all decidable languages are contextsensitive (but most are)
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Chomsky Chierarchy
CONTEXT-FREE GRAMMARS
– Subclass of context-sensitive grammars
– Restriction: rules take the form
A
where AN,   (TN)+
- Membership in context-free language (CFL) is
decidable
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Chomsky Chierarchy
REGULAR GRAMMARS
– Subclass of context-free grammars
– Restriction: all rules take the form
Aa
A  aB
or
where A,B  N and a  T
Membership is decidable.
RGs are expressively equivalent to finite state automata, or
Markov process.
Automata viewed as either generators or acceptors.
Grammars viewed as either generators or acceptors.
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Chomsky Chierarchy
REGULAR GRAMMARS
A  a or A  aB
where A,B  N and a  T
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Chomsky Chierarchy
Strong and Weak Equivalence
• Two grammars are said to be “weakly equivalent” if they generate the same
language or string set: cp. categorial grammars and phrase structure grammars.
• Two grammars are “strongly equivalent” if they assign the same tree(s) to
each string in the same language.
• All grammars at a given level in the hierarchy have strongly equivalent
grammars at higher levels, but not vice versa.
• A grammar or class of grammars is said to be “strongly adequate” to the
capture of a language or class of languages if it assigns the “right” trees to
strings. The “right” tree is the one we need for semantic interpretation.
• Weakly equivalent grammars which assign the “wrong” trees are said to be
only “weakly adequate.”
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Chomsky Chierarchy
Categorial Grammar
Phrase Structure Grammar
John loves
Mary
n
<n, <n,t>> n
n
<n,t>
t (semantic type of a sentence)
S
NP
N
John
VP
V
loves
NP
Mary
combinatorially transparent categories
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NL and Chomsky Hierarchy
Where are natural languages located?
Expressive power of human languages
• The first three chapters of Chomsky 1957 show that human languages fall
outside the lowest level of Regular/Finite State languages, and are at least at the level
of Context Free Languages.
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The proof requires a distinction between ideal linguistic capacity, now known
as Competence and the Performance mechanism that actually processes
sentences.
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Competence allows sentences that are so long or convoluted that none of us
will live long enough or have enough memory to process them. Performance
cannot cope with them—but clearly this limitation is accidental, not a fact
about English.
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Syntactic Structures goes on to suggest that the level of human grammars is
still higher in the hierarchy. It raises (but does not answer) the question of
which level is just high enough to contain all human languages.
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NL and Chomsky Hierarchy
Where are natural languages located?
Typical argument for the complexity of NL:
– Find a recursive construction C in a natural language L (English)
– Assume that the construction type in question is theoretically unbounded: i.e., in
theory, speakers could go on producing ever longer instances of the construction.
– Argue that the competence of speakers admits unlimited recursion (while the
performance certainly poses an upper limit; competence vs. performance
distinction!)
– Reduce C to a formal expression of known complexity in language L’ via a
homomorphism (a structure-preserving mapping)
– Make a case that L must be at least as complex L’
– Extrapolate from this one instance to all human languages: if there’s this one
construction C in this one language that has this complexity, then the human
language faculty must allow this in general.
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NL and Chomsky Hierarchy
NL is not regular: Chomsky’s 1957 original argument
Structure of his argument:
Consider 3 hypothetical languages:
1. ab, aabb, aaabbb (anbn)
2. aa, bb, abba, baab, aaaa, bbbb, aabbaa, abbbba, … (palindromic)
3. aa, bb, abab, baba, aaaa, bbbb, aabaab, abbabb, aababaabab (copy
language)
• It can be shown that these are not regular languages
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NL and Chomsky Hierarchy
The Pumping Lemma (a technique for proving that certain
languages are not regular).
If L is an infinite finite automaton language (FAL) over
alphabet A, then there are strings x,y,z  A* such that y ≠ 
and xynz  L for all n ≥ 0.
Why: machines for infinite languages must have loops. The
string y in the lemma corresponds to a string accepted
during a traversal of a loop.
Note that lemma does not say ‘iff’.
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NL and Chomsky Hierarchy
The Pumping Lemma (a technique for proving that certain languages
are not regular).
Example: L = {anbn | n ≥ 0 }
If L were a FAL, then, by Pumping Lemma, there would be x,y,z  A* such that
y ≠  and xynz  L for all n ≥ 0.
Assume that such x,y,z exist so the string xyz is in L. But by
definition of L it should be in the form anbn for some n. What’s y in this case? It
can’t be empty, so it would consist of (1) some number of a’s, or (2) some number
of b’s, or (3) some number of a’s followed by some number of b’s. But it is easy to
see that in any of those cases the strings xyyz, xyyyz, etc. could not belong to L.
But the pumping theorem is not always useful for showing a language to be nonregular.
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NL and Chomsky Hierarchy
NL is not regular - at least context-free power: Chomsky’s 1957 original argument:
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NL and Chomsky Hierarchy
Therefore, Chomsky claims, English cannot be regular
“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, p.21)
Note: Chomsky writes “finite languages”, but he means “regular
languages”.
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NL and Chomsky Hierarchy
Chomsky’s argument: because English contains these
constructions, which are not regular, English is not regular.
As stated, the argument is fallacious.
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NL and Chomsky Hierarchy
How to state the observation correctly
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NL and Chomsky Hierarchy
Similar point about center-embedding/nested dependencies
the cats that the dog chases miau
the dependency between the dog and chases nests within the dependency
between the cats and miau
Assume the following homomorphism:
a = {the cat, the dog, the rat, …}
b = {chase, miau, bite, bark, …}
Then this is an instance of anbn
Chomsky’s argument:
• Any useful syntactic analysis will relate the nouns to their corresponding verbs.
• No FSA is capable of keeping track of center embeddings of arbitrary depth
(which would be required since the grammatical subset of L is infinite). No FSA
can provide a useful syntactic analysis for center-embedding.
Therefore, since English has such constructions, English is non regular language.
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NL and Chomsky Hierarchy
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NL and Chomsky Hierarchy
The language anbn corresponds to the context-free grammar
SaSb
Sab
It gives rise to the following tree for the string aaabbb
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NL and Chomsky Hierarchy
Dissenting view 1:
• all arguments to this effect use center-embedding/nested
dependencies, cp.
the cats that the dog chases miau
the dependency between the dog and chases nests within the
dependency between the cats and miau
• humans are extremely bad at processing center-embedding
• notion of competence that ignores this is dubious
• natural languages are regular after all
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NL and Chomsky Hierarchy
Dissenting view 2:
• Any *finite* language is a regular language.
• If you don't distinguish performance and competence, then English as a language
certainly couldn't contain any sentence longer than the number of words a human
being could utter in a lifetime. (This assumes human lifetimes are finite, but that
seems uncontroversial.)
• This may be a HUGE number, but it is definitely finite, and so without the
distinction English is formally a finite language and therefore regular.
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NL and Chomsky Hierarchy
Are natural languages context-free?
• history of the question: Chomsky’s 1957 conjecture that natural
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languages are not context-free
In the 60’s and 70’s, many attempts to prove that NL is not context-free
Pullum and Gazdar 1982 (Generalized Phrase Structure Grammar):
- all these attempts have failed
- for all we know, natural languages (conceived as string sets) might be
context-free
Huybregts 1984, Shieber 1985:
- proof that Swiss German is not context-free, cross-serial dependencies
Culy 1985: proof that Bambara (a Northwestern Mande language
spoken in Mali) ) is not context-free
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NL and Chomsky Hierarchy
• Nested and Crossing Dependencies
Context-free languages -- unlike regular languages -- can have unbounded
dependencies.
However, these dependencies can only be nested, not crossing.
Example:
anbn has unlimited nested dependencies: context-free
The copy language has unlimited crossing dependencies: not context-free
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NL and Chomsky Hierarchy
• Nested and Crossing Dependencies
Bar-Hillel and Shamir (1960):
– English contains copy-language (crossing dependencies)
– Cannot be context-free
John, Mary, David, ... are a widower, a widow, a widower, ..., respectively.
Claim: the sentence is only grammatical under the condition that if the nth name is
male (female) then the nth phrase after the copula is a widower (a widow)
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NL and Chomsky Hierarchy
• Nested and Crossing Dependencies
suppose the claim is true, intersect English with regular language
L1 =(Paul|Paula)+are(a widower|a widow)+respectively
Result: Copy language L3
{ww|w  (a|b)+}
English  L1 = L2, homomorphism L2  L3
John, David, Paul, …  a
Mary, Paula, Betty, …  b
are, respectively  
a widower  a
a widow  b
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NL and Chomsky Hierarchy
• Result: Copy language L3
{ww|w  (a|b)+}
Copy language is not context-free
Hence L2 is not
Hence English is not
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NL and Chomsky Hierarchy
Cross-serial dependencies in Dutch
Huybregt (1976)
– Dutch has copy-language like structures
– thus Dutch is not context-free
dat Jan Marie Pieter Arabisch laat zien schrijven
that jan marie pieter arabic
let see write
‘that Jan let Marie see Pieter write Arabic’
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NL and Chomsky Hierarchy
Counterargument
Crossing dependencies only concern argument linking,
i.e., semantics
As far as plain strings are concerned, the relevant
fragment of Dutch has the structure
NPnVn
which is context-free
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LIN6932 Topics in Computational Linguistics