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Macpherson Delva
Sean Morgan
Section 1.4  Nonregular Languages
Not all languages can be represented by a regular expression or recognized by a NFA. These
languages are said to be nonregular. Nonregular languages can be very simple such as L = { 0
n
1
n
 n
`
N}, to recognize this language a machine would have to keep track of the number of 0’s and
1’s and a NFA has no way of counting. We can proof that L is a nonregular language formally by
proving that there is no NFA which accepts it.
Proof by contradiction that L is irregular
1 Assume a DFA exist that accepts L, the number of states within this DFA is denoted by k
2 Take a case with k zeros.
To accept k zeros there must be at least k+1 transition, so at least one
repeated transition. Call the repeated transition q.
3. Assume it take it takes ii zeros to get to q the first time ( 0
i
) and that it takes j zeros to get to q
the second time. Where i<j
4.) Therefore 0
i
1
i
and 0
j
1
i
go to the same place since they start at the same place and follow the
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 Spring '08
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