CS 461

1
CS 461 Sept. 2

• Review NFA ? DFA
• Combining FAs to create new languages
• union, intersection, concatenation, star
• We can basically understand how these work by
  drawing a picture.
• Section 1.3 Regular expressions
• A compact way to define a regular set, rather
  than drawing an FA or writing transition table.

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Example 2

• NFA transition table given to the right.
• DFA start state is A.
• DFA accept state would be anything containing D.

inputs inputs inputs
State 0 1 e
?A A A,C -
B D - C
C - B -
D ? B D -
3
continued

• Lets begin.
• d(A, 0) A
• d(A, 1) AC
• We need new state AC.
• d(AC, 0) A
• d(AC, 1) ABC
• Continue from ABC

inputs inputs inputs
State 0 1 e
?A A A,C -
B D - C
C - B -
D ? B D -
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answer

• NFA

• DFA

inputs inputs inputs
State 0 1 e
?A A A,C -
B D - C
C - B -
D ? B D -
inputs inputs
State 0 1
?A A AC
AC A ABC
ABC AD ABC
AD ? ABC ACD
ACD ? ABC ABCD
ABCD ? ABCD ABCD
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Moral

• NFAs and DFAs have same computational power.
• NFAs often have fewer states than corresponding
  DFA.
• Typically, we want to design a DFA, but NFAs are
  good for combining 2 DFAs.
• After doing NFA ? DFA construction, we may see
  that some states can be combined.
• Later in chapter, well see how to simplify FAs.

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U and n

• Suppose M1 and M2 are DFAs. We want to combine
  their languages.
• Union We create new start state. v
• Do you understand formalism p. 60 ?
• How can we also do intersection?
• Hint A n B (A U B)

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Concatenation

• Concat For each happy state in M1, turn it into
  a reject state and add e-trans to M2s start.
• Example
• L1 does not contain 00
• L2 has even of 1s
• Lets draw NFA for L1L2.
• Lets decipher formal definition of d on p. 61.

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Star

• We want to concat the language with itself 0
  times.
• Create new start state, and make it happy.
• Add e-transitions from other happy states to the
  start state.
• Example
• L begins with 1 and ends with 0
• Lets draw NFA for L.
• Formal definition of d on p. 63.

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Regular expression

• A concise way to describe a language
• Text representation, straightforward to input
  into computer programs.
• Use alphabet symbols along with operators
• means or
• means repetition
• Concatenation
• There is no and or complement.

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Examples

• What do you think these regular expressions mean?
• 0 1
• 01
• 0011 (a shorthand would be 01 )
• (0 1)
• Whats the difference between 101 and 1(01)1 ?
  Does this mean anything with 1 at beginning and
  end?

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Practice

• Words with 2 or more 0s. Whats wrong with this
  answer 10101 ?
• Words containing substring 110.
• Every even numbered symbol is 0.
• Whats wrong with ((0 1)0) ?
• Words of even length.
• The last 2 symbols are the same.
• What is the shortest word not in 1(01)0 ?
• True or false (111) (11 111)