Regular languages are a subset of LFSA

1
Lecture 23

• Regular languages are a subset of LFSA
• algorithm for converting any regular expression
  into an equivalent NFA
• Builds on existing algorithms described in
  previous lectures

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Regular languages are a subset of LFSA
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Reg. Lang. subset LFSA

• Let L be an arbitrary regular language
• Let R be the regular expression such that L(R)
  L
• R exists by definition of L is regular
• Construct an NFA-l M such that L(M) L
• M is constructed from regular expression R
• Argue L(M) L
• There exists an NFA-l M such that L(M) L
• L is in LFSA
• By definition of L in LFSA and equivalence of
  LFSA and LNFA-l

4
Visualization

• Let L be an arbitrary regular language
• Let R be the regular expression such that L(R)
  L
• R exists by definition of L is regular
• Construct an NFA-l M such that L(M) L
• M is constructed from regular expression R
• Argue L(M) L
• There exists an NFA-l M such that L(M) L
• L is in LFSA
• By definition of L in LFSA and equivalence of
  LFSA and LNFA-l

Regular Languages
LFSA
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Algorithm Specification

• Input
• Regular expression R
• Output
• NFA M such that L(M) L(R)

NFA-l M
Regular expression R
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Recursive Algorithm

• We have an inductive definition for regular
  languages and regular expressions
• Our algorithm for converting any regular
  expression into an equivalent NFA is recursive in
  nature
• Base Case
• Recursive or inductive Case

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Base Case

• Regular expression R has zero operators
• No concatenation, union, Kleene closure
• For any alphabet S, only S 2 regular
  languages can be depicted by any regular
  expression with zero operators
• The empty language f
• The language l
• The S languages consisting of one string a
  for all a in S

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Table lookup

• Finite number of base cases means we can use
  table lookup to handle them

f
l
a
b
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Recursive Case

• Regular expression R has at least one operator
• This means R is built up from smaller regular
  expressions using the union, Kleene closure, or
  concatenation operators
• More specifically, there are 3 cases
• R R1R2
• R R1R2
• R R1

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Recursive Calls
1) R R1 R22) R R1 R23) R R1

• The algorithm recursively calls itself to
  generate NFAs M1 and M2 which accept L(R1) and
  L(R2)
• The algorithm applies the appropriate
  construction
• union
• concatenation
• Kleene closure
• to NFAs M1 and M2 to produce an NFA M such that
  L(M) L(R)

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Pseudocode Algorithm

• NFA RegExptoNFA(regular expression R)
• regular expression R1, R2
• NFA M1, M2
• Modify R by removing unnecessary enclosing
  parentheses
• / Base Case /
• If R a, return (NFA for a) / include l
  here /
• If R f, return (NFA for )
• / Recursive Case /
• Find last operator O of regular expression R
• Identify regular expressions R1 (and R2 if
  necessary)
• M1 RegExptoNFA(R1)
• M2 RegExptoNFA(R2) / if necessary /
• return (OP(M1, M2)) / OP is chosen based on O
  /

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Example
A R (ba)a Last operator is
concatenation R1 (ba) R2 a Recursive
call with R1 (ba) B R (ba) Extra
parentheses stripped away Last operator is
union R1 b R2 a Recursive call with R1 b
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Example Continued
C R b Base case NFA for b returned B
return to this invocation of procedure Recursive
call where R R2 a D R a Base case NFA
for a returned B return to this invocation of
procedure return UNION(NFA for b, NFA for
a) A return to this invocation of
procedure Recursive call where R R2 a
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Example Finished
E R a Last operator is Kleene closure R1
a Recursive call where R R1 a F R a Base
case NFA for a returned E return to this
invocation of procedure return (NFA for a) A
return to this invocation of procedure return
CONCAT(NFA for b,a, NFA for a)
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Pictoral View
concatenate
(ba)a
union
Kleene Closure
a
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Parse Tree
We now present the parse tree for regular
expression (ba)a