5b ContextFree Grammars Parsing

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5b Context-Free Grammars Parsing Ambiguity
5.1,5.2,5.3

• Define Context-Free Grammar (cfg)
• CFG Simulation
• Constructing a Grammar for a Language
• Leftmost and Rightmost Derivations
• Derivation Trees
• Top-Down Parsing
• Membership Testing for Context-Free Languages
• Simple Grammars
• Ambiguous Grammars
• Inherently Ambiguous Languages

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Context-Free Grammar

• G (V,T,S,P) ordered quadruple
• All productions are of the form
• A ? x,
• where A ? V and x ? (V ? T)
• Why Called "Context Free"?
• vs. Context Sensitive...

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Determine Language generated by a Given Grammar

• List all words of length less than or equal to 4
  that are generated by the grammar of Example 5.5
  page 130
• S ? SS aaSb bSaa ?
• Does L(G) x x ? a,b and na(x) 2nb(x)?

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Context-Free Simulator

• http//cs.union.edu/csc140/simulators
• Enter and run the following grammar
• S?SSaSbbSabaab

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and now for something completely different

• Mad Libs on the Context-Free Generator

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Construct Grammar for a Given Language

• Exercise 5.1.7(a) page 134
• S ? A aA aaA aaaA
• A ? aAb Ab ?
• Exercise 5.1.7(f)
• S ? aS aSb SS ?
• aibjcjdie3 i, j ? 0
• aibjckdn i,j,k,n ? 0 , i lt n and j ? k
• aibjbiaj i, j ? 0
• Note bjbi bji bij bibj
• What if it were aibjcidj ?

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CF Grammar for set of all Regular Expressions
over a given Alphabet, say a,b

• S ? ? ? a b
• S ? SS SS S (S)
• Note ? and ? are symbols in ? of grammar
• Does NOT handle precedence of operators see page
  146, i.e. grammar is ambiguous--more on this
  later

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Leftmost Rightmost Derivations

• Leftmost Always apply production to leftmost
  non-terminal at any given point
• Rightmost rightmost non-terminal
• Note Multiple derivations possible for same
  derivation tree, but only one left-most or
  right-most

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Derivation Trees

• No matter which order non-terminals are replaced,
  unless the grammar is ambiguous there is only one
  derivation tree per word
• Example S ? AB A ? bA a B ? aB bA
• Derive w bbaaba
• Leftmost S ? AB ? bAB ? bbAB ? bbaB ? bbaaB
• Rightmost S ? AB ? AaB ? AabA ?
• But SAME derivation tree

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Parsing

• Finding a sequence of productions by which w in
  L(G) is derived
• (i.e. construct a derivation tree)
• Example
• S ? SS aSa' aa' bSb' bb' cSc' cc'
• w cc'bcaa'bb'cc'c'b'aa'
• Note that in this example there is more than one
  derivation tree (ambiguousstay tuned)

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Membership Test for CFLIs word w in L(G)?

• Eliminate productions of the form A?? and A?B.
• Why?
• (more on how to do this later)
• All other productions are "productive" in that we
  make progress toward a word in L(G) by applying
  each production (either lengthen intermediate
  word or convert non-terminal to terminal)
• Therefore, in a "finite length of time" we can
  try ALL possible productions that yield all words
  of w.
• If w is one of them, it is in L.
• If not, w is not in L.

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S-Grammars

• All productions in a Simple Grammar (S-Grammar)
  are of the form
• A ? ax,
• where A ? V, a ? T, x ? V, and any pair (A,a)
  occurs at most once in P.
• The terminal, a, is sometimes called the "handle"
• It determines uniquely which production to use at
  any given point in a derivation
• Good News--Parsing becomes almost trivial with
  S-Grammars
• Bad News--Not all languages are convertible to
  S-Grammars

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Ambiguity

• Two distinct derivation trees for same sentence
  in same grammar
• "Time flies like an arrow." per CJH
• Time marches on (no turning back) as in straight
  path of arrow
• Time is fleeting, as quickly as an arrow
• Use a stopwatch to time arrows the same way you
  time flies
• Time magazine, when properly rolled up and
  thrown, flies like an arrow
• Time flies like to line up and sit on an arrow

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Ambiguous Grammars

• Example, give all derivation trees for w
• S? SS aSa' aa' bSb' bb' cSc' cc'
• w cc'bcaa'bb'cc'c'b'aa'
• Exercise 5.2.13 page 145
• Consider 2 different left-most derivations
• S ? aSbS ? abSabS ? ababS ? abab
• S ? aSbS ? abS ? abaSbS ? ababS ? abab

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Ambiguous Grammar does not imply Ambiguous
Language

• Exercise 5.2.14
• S ? aSb SS ?
• Put two different trees for ababab on board
• Non-ambiguous Grammar for same Language
• S ? AS ? A ? aSb ab(alternate solution in
  text)
• Put (the only) tree for ababab on board

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Programming Languages pages 142-3

• Example 5.11All Expressions treated equally
• Example 5.12Factors vs. Terms vs. Expressions
• Forces evaluation of Factors first, then Terms,
  then Expressions

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Inherently Ambiguous CF Languages

• L1 anbncm, L2 anbmcm
• Each language has a non-ambiguous grammar
• Suppose starting symbols were S1 and S2
  respectively
• and that set of non-terminals was mutually
  disjoint
• However, L L1 ? L2 is inherently ambiguous
• L is CF since we could construct it with S ? S1
  S2
• Consider a3b3c3 ? L, get two distinct derivation
  trees
• Not just different symbols, but different shapes
• Proof that some languages are inherently
  ambiguous is "beyond the scope of this course"

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In-Class ExerciseExercise 5.1.8 (d g) page
134

• Work in groups of 2 or 3
• Enter grammars on simulator and generate words
• "-gt" for arrow, no blank spaces
• Answers on next slide (no peeking)
• First group done with both put (d) on board
• Second group done with both put (g) on board

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In-Class ExerciseAnswers

• 5.1.8(d)
• S ? aSc B
• B ? bBcc ?
• 5.1.8(g)
• S ? aSc aA bB Cc
• A ? aAc aA B
• B ? bBc bB ?
• C ? bCc Cc ?