Properties of Context-Free Languages

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Properties of Context-Free Languages

• Juan Carlos Guzmán
• CS 6413 Theory of Computation
• Southern Polytechnic State University

2
Summary

• Normal Forms
• Pumping Lemma
• Closure Properties
• Decision Properties

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Normal Forms

• Recall that many different grammars generate the
  same language
• We would like to restrict the form of the
  productions of the CFG
• Chomsky Normal Form
• Greibach Normal Form
• Tasks to accomplish
• Eliminate useless symbols
• Eliminate e-productions
• Eliminate unit productions

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Grammar Transformations

• We are about to present a series of
  transformations on grammars
• You should consider each of them as a
  transformation function
• T Grammar ? Grammar

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Elimination of Useless Symbols

• Let G(V,T,P,S)
• X?V is useful if there exist ?, ?, and w such
  that
• S ? ? X ? ? w
• Two considerations
• X must generate strings
• X ? v
• X must be reachable from S
• S ? ? X ?

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Elimination of Non-Generating Symbols (Tg)

• Let G(V,T,P,S ) be a CFG
• G (V ? S ,T,P,S ), where
• V A (A??)?P ? ? ? (T ?V )
• P (A??) (A??)?P
• ? A ?V
• ? ? ? (T ?V )
• contains only generating symbols

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Example

• G (S,A,B,C ,a,b ,P ,S ), where
• P S ? a A,
• A ? AB BCA a,
• B ? b,
• C ? ACA BCB
• V S,A,B
• G (S,A,B ,a,b ,P ,S ), where
• P S ? a A,
• A ? AB a,
• B ? b

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Elimination of Non-Reachable Symbols (Tr)

• Let G(V,T,P,S ) be a CFG
• G (V,T,P,S ), where
• V S ? B (A??B ?)?P ? A?V
• P A?? (A??)?P ? A ?V
• contains only reachable symbols

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Example

• G (S,A,B,C ,a,b ,P ,S ), where
• P S ? a A,
• A ? AB a,
• B ? b,
• C ? ACA BCB
• V S,A,B
• G (S,A,B ,a,b ,P ,S ), where
• P S ? a A,
• A ? AB a,
• B ? b

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Useful Symbols

• Remove
• non-generating symbols
• non-reachable symbols

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Elimination of e-Productions (Te)

• Let G(V,T,P,S ) be a CFG
• Ve A (A??)?P ? ??Ve
• G (V-Ve,T,P,S ), where
• P A??0X1 Xk?k
• A??0B1 Bk?k ?P ?
• for all 1?i ?k Bi ? Ve ? Xi ? e,
  Bi ?
• for all 0?i ?k ?i ?(T ?V-Ve) ?
• ?0X1 Xk?k gt 0
• does not contain e-prods and generates L(G) - e

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Example

• G (S ,a,b ,P ,S ), where
• P S ? aSbS bSaS e
• Ve S
• G (S ,a,b ,P ,S ), where
• P S ? aSbS aSb abS ab
• bSaS bSa baS ba
• Note that G does not generate e

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Elimination of Unit Productions (Tu )

• Let G(V,T,P,S ) be a CFG
• Let Up (A,A) A?V
• ? (A,C ) (A,B)?Up ? (B?C )?P
  
• G (V,T,P,S ), where
• P A?? (A,B)?Up ? (B??)?P ? ??V
• does not contain unit prods and generates L(G )

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Example

• G (E,T,F ,,,(,),a ,P ,E ), where
• P E ?ET T, T ?TF F, F ? a (E )
• Up (E,E ),(E,T ),(E,F ),(T,T ),(T,F ),(F,F )
• G (V,T,P,S ), where
• P E ?ET TF a (E ),
• T ?TF a (E ),
• F ? a (E )

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Summary of Transformations

• Given a CFG G, we can obtain a new grammar G
  such that
• no e-productions
• no unit productions
• no useless symbols
• by transforming the original grammar in this
  order
• Tr ? Tg ? Tu ? Te

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Results of the Transformations

• After the transformations
• the grammars do not have useless symbols (and
  associated productions)
• their productions (A??) are not
• e-productions
• Unit productions
• Therefore, ? must satisfy
• ?gt1, or
• ??T

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Implications for Transformed Grammars

• Transformed grammars have some nice properties
• No unit productions
• No e-productions
• However, they produce bushy trees

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Chomsky Normal Form

• Any CFG without e can be transformed so that each
  of its productions is of the form
• A ? BC, where A,B,C ?V
• A ? a, where A ?V ? a ? T
• The idea behind CNF is to obtain grammars whose
  parse trees are binary trees

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Chomsky Normal Form

• Productions of grammars not yet in CNF, but
  already transformed, are of the following forms
• A?X1 Xk k gt1, all Xi ?T ?V, or
• A?a a ?T
• We need to further transform the first kind of
  productions so that
• the right-hand-side consists only of variables,
  and
• break long RHSs into chains of productions
  

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Chomsky Normal Form

• Transformations
• For every terminal a that appears on a RHS of
  length 2 or more
• Create a production A?a
• Replace a in all such productions with A
• Replace every production A?B1 Bk (k gt2) with
• A?B1C1
• C1?B2C2
• Ck-2?Bk-1Bk

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Example
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Greibach Normal Form

• All productions must be of the form
• A?aB1 Bk k ?0
• Note that each derivation step is associated with
  the generation of a terminal
• This translates nicely to PDAs where each
  movement of the automaton will be guided by the
  recognition of an input character
• To convert to GNF
• Order the variables (A1 An)
• Modify the production set so that
• Ai ? Aj? implies that i ? j
• remove left recursion i.e., Ai ? Aj? implies that
  i lt j
• Ai ? a?
• Ai ? a?, ? ?V
• The algorithm resembles matrix triangularization
• It appears in 1st edition of our book

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Relation Between Height and Yield of a CNF Parse
Tree

• Note that tree nodes of grammars in CNF are
• binary nodes for productions (A ? BC)
• unit terminal nodes for productions (A?a)
• The yield of a complete CNF parse tree of height
  n is of size 2n-1 or less

S
height n-1 At most 2n-1
height n
a1 a2 a3 at
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Pumping Lemma

• Let L be a context-free language. Then there
  exists a constant n (which depends on L) such
  that for every string z in L such that z?n, we
  can break z into five strings, z uvwxy, such
  that
• vwx ? n
• vx ? e
• For all i ? 0, the string uviwxiy is also in L

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Pumping Lemma

• In plain words
• For any context-free language
• Words of large size will contain a substring
• Somewhere in the middle
• Not null, not too big
• That substring can itself be broken into three
  pieces vwx
• v not null or x not null
• v and x can be pumped (together) over and over
  again
• The new words are guaranteed in the language
• How large the words must be in order to be
  considered large depends on the actual language

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Pumping Lemma Proof

• Find a CNF for the language
• The size of the word relates to the height of the
  tree

A0
A1
A2
Ak
a
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Pumping Lemma Proof

• Find a CNF for the language
• For large words, a variable must be repeated

S
Ai
Aj
Note Ai Aj , i lt j
v
u
x
y
w
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Related Strings

• The strings
• uwy
• uvvwxxy
• uvnwxny
• are also in the language

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How about e?

• If the language contains e
• The transformations remove e from the grammar
• Therefore you get a different language!!!
• CNF is not defined for languages with e
• If a language contains e
• A new grammar can be given, which generates the
  same language
• e will be generated in one derivation
• All other productions comply with CNF

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Closure Properties

• Context-free languages are closed under
• Substitution
• Regular Operators
• Homomorphism
• Reversal
• Intersection with regular language
• Inverse homomorphism

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Substitution

• A substitution is an operation which replaces
  characters with strings
• These strings are pulled from a particular
  language

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SubstitutionFormally

• Let S be an alphabet
• Let La a language associated to a ? S
• s(a) La
• s(a1a2an) s(a1)s(a2)s(an) La1 La2 Lan
• s(L) s(w) w ? L

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Substitution

• CFLs are closed under substitution with CFLs
• Let G (V, S,P,S ), such that L(G ) L
• Let Ga (Va,Ta,Pa,Sa), such that L(Ga) La
• Let G (V,T,P,S ) where
• V V ? (?a?S Va )
• T (?a?S Ta )
• P (?a?S Pa ) ? P, where
• P is all productions of P, where each terminal
  a was replaced by the corresponding Sa
• G generates s(L)

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Example

• G (S,0,1,P,S), where
• P S ? SS 0S1 e
• L0 (
• L1 )
• Or
• L0 0
• L11

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Closure Under Regular Operators

• CFLs are closed under
• Union
• Concatenation
• Closure (), and positive closure()

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Closure Under Homomorphism

• CFLs are closed under homomorphism
• This is a special case of substitution
• Substitution with a single string

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Reversal

• CFLs are closed under reversal
• Just reverse all productions

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Intersection with a Regular Language

• CFLs are not closed under intersection
• They are closed under intersection with a regular
  language

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Inverse Homomorphism

• CFLs are closed under inverse homomorphism

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Decision Properties of CFLs

• Complexity to transform grammars to PDAs, and
  within PDAs
• Complexity of transformation to CNF
• Testing Emptyness of CFLs
• Testing Membership in a CFL

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Undecidable Problems

• Is a given CFG G ambiguous?
• Is a given CFL L inherently ambiguous?
• Is the intersection of two CFLs empty?
• Are two CFLs the same?
• Is a given CFL equal to S, where S is the
  alphabet of the language?