Description of Inherently Ambiguous Context-Free Grammars

1
Description of Inherently Ambiguous Context-Free
Grammars

• Ryan Villalpando
• April 7, 2003
• CPSC 627

2
Introduction

• Review of Context-Free Grammars
• Definition of Ambiguity
• Examples of Ambiguity
• Parse Trees
• Leftmost Derivations
• Removing Ambiguity
• Definition of Inherent Ambiguity
• Proof of Inherent Ambiguity

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Review of Context-Free Grammars

• A context-free grammar is a 4-tuple (V, S, R, S)
• V is the finite set of variables
• S is the finite set of terminals
• R is a set of rules
• S ? V is the start variable

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Example of a CFG

• The language 1n0n n gt 0 gives the grammar
• S-gtS1 S2
• S1-gt0S11 e
• S2-gt1S20 e

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Ambiguity

• A CFG is ambiguous if it can generate the same
  string in different ways
• These CFGs will have several different parse
  trees and several meanings.
• This result may cause issues with some
  programming applications

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Example of Ambiguity in English

• The girl touches the boy with the flower.
• Meaning 1
• The flower is the object being used to touch the
  boy.
• Meaning 2
• The boy is holding the flower.

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Example of Ambiguity in Math

• a a a
• Meaning 1
• (a a) a
• Meaning 2
• a (a a)

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Parse Trees

• Derivations
• E gt E E gt E E E
• E gt E E gt E E E

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Leftmost Derivation

• A derivation of a string is a leftmost derivation
  if at every step, the leftmost remaining variable
  is the one replaced.
• If a grammar is unambiguous, it will have unique
  leftmost derivations.

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The leftmost derivations of a a a

• E
• gt E E
• gt a E
• gt a E E
• gt a a E
• gt a a a

• E
• gt E E
• gt E E E
• gt a E E
• gt a a E
• gt a a a

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Removing Ambiguity

• There is no algorithm that can tell whether a CFG
  is ambiguous.
• There are techniques for eliminating ambiguity
• Adding non-terminals or dividing the variables
  into factors, terms, and expressions.

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Removing Ambiguity from a b c

• Grammar
• I -gt abIaIbI0I1
• F -gt I(E)
• T -gt FT F
• E -gt TE T
• Terms (T) cannot be broken by the operator.
• A factor (F) cannot be broken by a or a .
• Is are identifiers
• Es are expressions

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Inherently Ambiguous Context-Free Grammar

• A context-free language L is said to be
  inherently ambiguous if all its grammars are
  ambiguous.
• If even one grammar for L is unambiguous, then L
  is an unambiguous language.

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Example of an Inherently Ambiguous CFL

• L anbncmdm ngt1, mgt1 U anbmcmdn ngt1,
  mgt1
• Thus,
• There are as many as as bs and as many cs as
  ds, or
• There are as many as as ds and as many bs as
  cs.

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A Grammar for L

• S -gt AB C
• A -gt aAb ab
• B -gt cBd cd
• C -gt aCd aDd
• D -gt bDc bc

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Leftmost Derivations for L with string aabbccdd

• S gt AB gt aAbB gt aabbB gt aabbcBd gt aabbccdd
• S gt C gt aCd gt aaDdd gt aabDcdd gt aabbccdd

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Parse Trees for aabbccdd
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Proving Inherent Ambiguity

• The previous parse trees and leftmost derivations
  prove that language L has at least some ambiguous
  grammars.
• This doesnt prove that all of Ls grammars are
  ambiguous, so we dont know if L is ambiguous.

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Proof Idea

• We argue that all but a finite number of the
  strings, whose counts of the four symbols a, b,
  c, d are all equal, must be generated in two
  different ways
• as and bs are equal and cs and ds are equal
• as and ds are equal and bs and cs are equal

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Proof Idea Continued

• The only way to generate strings where as and
  bs are equal is with a variable like A.
• Variations can occur, but we must follow the
  mechanism for generating as in a way that
  matches the bs.
• This idea carries for the variables B, C, and D.
• This argument will show that no matter what
  modifications we make to the basic grammar, it
  will generate at least some of the strings of the
  form anbncndn.
• Following this proof idea will show that L is
  inherently ambiguous, because no matter how the
  grammar is changed, it will remain ambiguous.

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Summary

• Ambiguity can cause problems in programming
  applications, yet sometimes ambiguity can be
  removed.
• If all the grammars of a context-free language
  are ambiguous, then that language is ambiguous.
• A CFL can be proved ambiguous by showing that all
  the grammars generated by the CFL fall into
  certain categories.

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Citations

• Hopcroft, John E. Introduction to automata
  theory, languages, and computation. Reading,
  Addison-Wesley, 1979.
• Sipser, Michael Introduction to the Theory of
  Computation. Boston, PWS Publishing Company, 1997

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The End

• Questions and Answers