CSA405: Advanced Topics in NLP

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CSA405 Advanced Topics in NLP

• Xerox Notation

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3 Related Perspectives
LANGUAGE
NOTATION
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3 Related Perspectives
REGULAR LANGUAGES
denotes
encodes
compiles into
REGULAR EXPRESSIONS
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Xerox R.E. Notation

• Atomic Expressions
• Normal Symbol
• Special Symbol
• Complex Expressions
• Union
• Intersection
• Concatenation
• Closure
• Other Operators
• Abbreviations

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Atomic Expressions

• The simplest kind of RE is a symbol. Typically, a
  symbol is the sort of item that can appear on the
  arc of a network.
• For example, the symbol a is an RE that
  designates the language containing the string "a"
  and nothing else
• Multicharacter symbols such as Plur are also
  symbols, but they happen to have multicharacter
  print names.

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Special Atomic Expressions

• The epsilon (e) symbol 0 denotes the empty string
  language "".
• The ANY symbol ? denotes the language of all
  single symbol strings. The empty string is not
  included in ?.

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Empty and UniversalLanguage Machines
C
A
S
A
set of strings
I
N
Q
U
E
empty language
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Brackets

• A denotes the same language as A
• can also be used to denote the empty string
  language
• Brackets ensure unique syntax but can sometimes
  be dropped.
• Brackets not the same as ( ) which are used for
  optional elements
• Checkpoint what are the FSAs
• for a
• for (a)

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Complex REs Union

• If A and B are arbitrary REs, A B is the
  union of A and B which denotes the union of the
  languages denoted by A and B respectively.
• Union is associative and commutative
• Checkpoint Write down the strings in the
  language denoted by a b ab.

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Complex REs Intersection

• If A and B are arbitrary REs, A B is the
  intersection of A and B which denotes the
  intersection of the languages denoted by A and B
  respectively.
• Intersection is associative and commutative
• Checkpoint Write down the strings in the
  language denoted by
• a b c d e ab d e f g

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Complex REs Concatenation

• If A and B are arbitrary REs A B is the
  concatenation of A and B
• Checkpoint do the following denote the same
  languages?
• d o g
• dog
• d og
• What are the strings in the language denoted by
  ab cd

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Concatenation of 2 Networks
a
c

b
d
a
c

b
d
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Complex REs Closures

• A denotes the concatenation of A with itself 0
  or more times.
• What is the FSA for a ?
• A (Kleene Star) denotes A 0.
• What is the FSA for a ?

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Complex REs Closures

• A denotes the concatenation of A with itself 0
  or more times.
• A (Kleene Star) denotes A 0.

a
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Other Operations

• Complementation A denotes the complement
  language of A the set of strings not in A
• Minus A - B denotes the set difference of the
  languages denoted by A and B. (A-B A B)
• Checkpoint Write a definition of complementation
  involving minus

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Abbreviations

• A Closure (Kleene Star)
• (A) Optional Element
• ? Any symbol
• \b Any symbol other than b
• A Complement ( ? - A )
• 0 Empty string language
• A ? A ?

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String Relations

• Ordered pair set having two members lta,bgt
  (distinct from ltb,agt).
• A relation is simply a set of ordered pairs.
• Some familiar relations over integers
• lt0,1gt,lt1,2gt,lt2,3gt,lt3,4gt,lt4,5gt,.
• lt0,0gt,lt1,1gt,lt2,4gt,lt3,6gt,lt4,8gt,.

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Relations and Morphology

• In morphological analysis and generation, we are
  typically interested in relations made up of
  ordered pairs of strings over lexical and surface
  languages, e.g.
• lt"dogs","dogPL"gt, lt"walked","walk"ED"gt.

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Describing Relations

• Notation Regular Expressions with extra
  operations including
• Cross Product
• Composition
• Mechanism Finite State Transducers (i.e. FS
  networks whose arcs are labelled with ordered
  pairs of symbols).

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3 Related Perspectives
REGULAR RELATIONS
denotes
encodes
compilesinto
XEROX R.E. NOTATION
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Cross Product

• A .x. B denotes the relation that pairs every
  string of language A with every string of
  language B.
• Example c a t .x. c h a t
• Special case and special notation when A and B
  are symbols a .x. b ab

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Symbol Pairs

• Any pair of symbols ab denotes a relation that
  consists of the corresponding pair of strings,
  i.e.
• lt"a,"b"gt
• The left symbol a is the upper, lexical, symbol
  the right symbol b is the lower, surface symbol.
• No distinction between aa and a

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Checkpoint

• A abc
• B bcd
• What are the elements in the relation A .x. B?

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Composition

• A .o. B denotes the composition of relations A
  and B.
• Definition
• If A contains ltx,ygt
• And B contains lty,zgt
• Then A .o. B contains ltx,zgt
• A and B must be relations. If either is just a
  language, it is assumed to abbreviate the
  identity relation.

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Examples
Expression
Language/Relation
Network

?

""
a
"a"
a
(a)
"","a"
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Exercise fill in the blanks
Language/Relation
Expression
Network
a
"","a","aa",..
a
a 0 b
a0 ba
a b0
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Issue

• The class of Regular Languages is closed under
  the operations union, intersection,
  concatenation, and complementatation.
• Is the same true of Regular Relations?
• Is the same true if we include the operations of
  cross product and composition?

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Closure PropertiesDefinition of P

• Consider P ab 0c.
• P is a relation that maps a string of zero or
  more a to an equal length string of b followed by
  zero or more c
• lt" "," "gt, lt" ","c"gt, lt" ","cc"gt, ...
  lt"a","b"gt, lt"a","bc"gt, lt"a","bcc"gt, ...
  lt"aa","b"gt, lt"aa","bbc"gt, lt"aa","bbcc"gt, ...
• P is a regular relation

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Closure PropertiesDefinition of Q

• Consider Q 0b ac.
• Q is a relation that maps a string of zero or
  more a to an equal length string of c preceded by
  zero or more b
• lt" "," "gt, lt" ","b"gt, lt" ","bb"gt, ...
  lt"a","c"gt, lt"a","bc"gt, lt"a","bbc"gt, ...
  lt"aa","cc"gt, lt"aa","bcc"gt, lt"aa","bbcc"gt, ...
• N.B. Q is a regular relation

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Closure PropertiesP Q

• Let us now consider the intersection of P and Q.
• lt"", ""gt, lt"a","bc"gt, lt"aa","bbcc"gt,
  lt"aaa","bbbccc"gt, ....
• The lower side language is clearly bncn
• Is this finite state?

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bncn

• This language is generated by the following
  context free grammarS ? eS ? aSb
• It cannot be generated by a regular grammar whose
  rules must take the form A ? aA ? aB

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Closure Propertiesbncn

• Consequently, the language cannot be generated by
  a FSA, and the same goes for any relation
  involving that language.
• Therefore, there is no operation on the FSTs for
  P and Q that yields their intersection.
• If not closed under intersection, then not closed
  under complementation nor subtraction.

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Closure Properties of Regular Languages and
Relations
Operation Regular Languages Regular
Relations Union yes
yes Concatenation
yes yes Iteration
yes
yes Intersection yes
no Subtraction yes
no Complementation yes
no Composition
n/a yes