CSI 3104 /Winter 2006: Introduction to Formal Languages Chapter 13: Grammatical Format

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CSI 3104 /Winter 2006 Introduction to Formal
Languages Chapter 13 Grammatical Format

• Chapter 13 Grammatical Format
• I. Theory of Automata
• ? II. Theory of Formal Languages
• III. Theory of Turing Machines

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Chapter 13 Grammatical Format

• Theorem. All regular languages are context-free
  languages.
• Proof. We show that for any FA, there is a CFG
  such that the language generated by the grammar
  is the same as the language accepted by the FA.
• By constructive algorithm.
• Input a finite automaton.
• Output a CF grammar.

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Chapter 13 Grammatical Format

• The alphabet ? of terminals is the alphabet of
  the FA.
• Nonterminals are the state names. (The start
  state is renamed S.)

• For every edge, create a production

X ? aY
X ? aX

• For every final state, create a production
• X ? ?

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Chapter 13 Grammatical Format

• Example

S ? aM bS M ? aF bS F ? aF bF ?
babbaaba
S ? bS ? baM ? babS ? babbS ?babbaM ?babbaaF ?
babbaabF ? babbaabaF ? babbaaba
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Chapter 13 Grammatical Format

• Definition A semiword is a sequence of terminals
  (possibly none) followed by exactly one
  nonterminal.
• (terminal)(terminal)(terminal)(Nonterminal)
• Definition A CFG is a regular grammar if all of
  its productions have the form
• Nonterminal ? semiword
• or
• Nonterminal ? word (a sequence of terminals
  or ?)

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Chapter 13 Grammatical Format

• Theorem. All languages generated by regular
  grammars are regular.
• Proof. By constructive algorithm. We build a
  transition graph.
• The alphabet ? of the transition graph is the set
  of terminals.
• One state for each nonterminal. The state named
  S is the start state. We add one final state .
• Transitions

Nx ? wyNz
Np? wq
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Chapter 13 Grammatical Format
Example S ? aaS S ? bbS S ? L
Alternative Algorithm

1. Np? wq whenever wq is not ?
2. For each transition of the form N? ?, we mark the
   state for N with ..

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Chapter 13 Grammatical Format
Example
S ? aaS bbS abX baX ? X ? aaX bbX
abS baS
EVEN-EVEN
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Chapter 13 Grammatical Format

• If the empty word is in the language, a
  production of the form N ? ? (called a
  ?-production) is necessary.
• The existence of a production of the form N ? ?
  does not necessarily mean that ? is a part of the
  language.
• S ? aX S ? a X ? ?
• Theorem. Let L be a language generated by a CFG.
  There exists a CFG without productions of the
  form X ? ? such that
• If ??L, L is generated by the new grammar.
• If ??L, all words of L except for ? are generated
  by the new grammar.

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• Proof For each N ? ? and all
• X ? smt1 N smt2
• include production X ? smt1 smt2
• and eliminate N ? ?.
• Example S ? aSa bSb ?
• L-? generated with
• S ? aSa bSb aa bb

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Problem Creating new ? productions

• Example S ? a Xb aYa X ? Y ? Y ? b
  X
• S ? a Xb b aYa X ? Y Y ? b X ?
• Definition N nullable if N ?
• In example, X and Y are nullable Y ?X? ?
• Modified replacement rule
• Include X ? smt1 smt2 for each
• X ? smt1 N smt2 and N nullable,
• Delete all (old and new) ?-productions
• Ex S ? a Xb aYa b aa X ? Y Y ? b X
• How to find all nullable nonterminals in a CFG?

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Chapter 13 Grammatical Format

• Definition. A production of the form
• Nonterminal ? Nonterminal is called a unit
  production.
• Theorem. Let L be a language generated by a CFG
  that has no ?-productions. Then there is another
  CFG without ? -productions and without unit
  productions that generates L.
• Proof A ? B B ? S1 S2 .. Replace with A ? S1
  S2
• Does it always work ?
• Example S ? A bb A ? B b

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Modified elimination rule

• Whenever A ? X1? X2? ?B introduce
• A ? S1S2S3 for all B ? S1S2S3 ..
• And eliminate all unit productions
• Example S ? A bb A ? B b B ? S a
• S ? bb b a A ? b a bb B ? a bb b

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• Theorem. Let L be a language generated by a CFG.
  Then there exists another CFG that generates all
  words of L (except ?) such that all the
  productions are of the form
• Nonterminal ? sequence of
  Nonterminals
• Nonterminal ? one terminal
• Proof Introduce A?a B?b and replace a, b,
  with A, B,.. in all productions
• Example S?XXaYaSbb X?YYb Y?aYaaX
• becomes?

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Chapter 13 Grammatical Format

• Definition. A CFG in is said to be in Chomsky
  Normal Form (CNF) if all the productions have the
  form
• Nonterminal ? (Nonterminal)(Nonterminal)
• Nonterminal ? terminal
• Theorem. Let L be a language generated by a CFG.
  There there is another grammar which is in CNF
  that generates all the words of L (except ?).

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CNF construction

• Proof following previous theorem, one can
• Eliminate all ?-productions
• Eliminate all unit productions
• Reduce to X ? X1X2Xn and X?terminal
• Replace first type, if ngt2, with
• X?X1R1 R1?X2R2 R2?X3R3 Rn-3?Xn-2Rn-2 Rn-2 ?
  Xn-1Xn

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Chapter 13 Grammatical Format
Definition. In a sequence of terminals and
nonterminals in a derivation (called a working
string), if there is at least one nonterminal,
then the first one is called the leftmost
nonterminal. Definition. A leftmost derivation is
a derivation where at each step, a production is
applied to the leftmost nonterminal in the
working string. Example. S ? aSX b
X ? Xb a S ? aSX ? aaSXX ? aabXX ? aabXbX ?
aababX ? aababa
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Chapter 13 Grammatical Format
S ? XY X ? XX a Y ? YY b
Example
aaabb
Derivation II
Derivation I
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Chapter 13 Grammatical Format
S ? XY ? XXY ? aXY ? aXXY ? aaXY ?
aaaY ? aaaYY ? aaabY ? aaabb
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Chapter 13 Grammatical Format
S ? XY ? XXY ? XXXY ? aXXY ? aaXY ?
aaaY ? aaaYY ? aaabY ? aaabb
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Chapter 13 Grammatical Format
Theorem. Any word that is in the language
generated by a CFG has a leftmost
derivation. Proof do a preorder traversal in the
derivation tree Example. S ? (S) S ? S S p
q S ? (S) ? (S ? S) ? (p ? S) ? (p ? (S)) ?
(p ? (S ? S)) ? (p ? (S ? S)) ? (p ? (p ?
S)) ? (p ? (p ? q))