4. Parsing

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4. Parsing
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• An Introduction
• 4.1 Leftmost Derivations and Ambiguity
• L (G) is the set of terminal strings that can be
  derived in any manner, from the start symbol.
• A terminal string may be generated by a number
  of different derivations.
• e.g. String abcbaa is derived from 4 distinct
  derivation.
• - derivation of Pascal expression were given in
  a left most form.
• Theorem 4.1.1
• Let G (V,?,P,S) be a context-free grammar.
• A string w is in L (G) if, and only if, then is
  a leftmost derivation of
• from s.

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• Definition 4.1.2
• A context-free grammar G is ambiguous if there
  is a string w ? L (G) that can be derived by two
  distinct leftmost derivations.
• Example 4.1.1
• Let G be the grammar
• S ? aS Sa a
• G is ambiguous since the string as has two
  distinct leftmost derivations.
• S ? aS S ? Sa
• ? aa ? aa
• the language of a G is a . This language is
  generated by the unambiguous grammar
• S ? aS a

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• Example 4.1.2
• L (G) S? bS Sb a
• The language of G b ab.
• The leftmost derivations
• S ? bS S ? Sb
• ? bSb ? bSb
• ? bab ? bab
• ? ambiguity
• Quiz Show that grammar
• G1 S? bS aA
• A? bA ?
• where its language bnabm is an ambiguity

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• 4.2 The Graph of a grammar
• The leftmost derivation of a context-free
  grammar G can be represented by a labeled
  directed graph g(G).
• A string w is adjacent to v in g(G) if v ?Lw ,
• ( i.e w can be obtained from v by one leftmost
  rule application)
• Definition 4.2.1
• Let G (V,?,P,S) be a context-free grammar.
  The left most graph of the grammar G, g(G), is
  the labeled directed graph (N,P,A), where the
  node and arcs are defined by
• i) N w ? (V??) S ?Lw
• ii) A v, w, r ? N x N x P v ?Lw by
  application of rule r
• A path from S to w in g(G) represents a
  derivation of w from S in the grammar G. The
  label on the arc from v to w specifics the rule
  application to v to obtain w.

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S

• S?aS

S?aS
S?Sb
aS
Sb
ab
S?aS
S?aS
S?Sb
S?Sb
S?ab
S?ab
aaS
aSb
aab
Sbb
abb
S?Sb
S?aS
S?Sb
S?aS
S?aS
S?Sb
S?ab
S?ab
aSbb
aaaS
aaSb
aabb
Sbbb
abbb
aaab
S?Sb
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• Ambiguous grammar
• S? aS Sb ab
• 
  ? rules generate cycles

S
S? a
S? ss
S??
ss
a
S? a
S? ss
S??
sss
aS
Ambiguous
S? ss
S? ss a ?
ssss
ass
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• 4.3 Top-Down Parser ( Breadth-First )
• Objective To determine whether an input
  string is derivable from the rules of a grammar.
• The application rule A ? w in the left most
  derivation.
• has the form uAv ?uwv
• string u, consisting solely of terminals, ?
  terminal prefix.
• Algorithm 4.3.1
• Breadth-First Top Down Parsing Algorithm.
• Input context-free grammar G ( V,S,P,S)
  queue Q
• Initialize T with root S, INSERT (S,Q)
• Repeat
• 2.1 q REMOVE (Q)
• 2.2 Q 0
• 2.3 done false

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• Let q uAv where A is the leftmost variable in
  q.
• 2.4 repeat
• 2.4.1 if there is no A rule number greater
  than I then done false
• 2.4.2 if not done then
• Let A w be the first A rule with
  number greater than I and let j be the
  number of this rule.
• 2.4.2.1 if uwv ? S and the terminal prefix of
  uwv matches a prefix of p then
• 2.4.2.1.1 INSERT (uwv, Q)
• 2.4.2.1.2 Add node uwv to T. Set a pointer
  from uwv to q.
• end if
• 2.4.3 i j
• until done or p uwv
• until EMPTY (Q) or p uwv

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• If p uwv then accept else reject
• ? search tree T containing nodes from g(G)
• 4.4 A Depth-First Top-Down Parser
• 
  S
• A
• T
• b (A)
• (T)
  ( AT) ( TT) ( bT)
  (bb)
• (b) ((A))

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• Exercise
• Build the sub graph of the grammar of G
  consisting of the left sentential forms that are
  generated by derivations of length 3 or less
• S ? aS AB B
• A ? abA ab
• B ? BB ba