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Question

Consider the following types of languages L1: Regular, L2: Contextfree, L3: Recursive, L4: Recursivelyenumerable.
Which of the following is/are TRUE?
I. ¯L3 L4 is recursively enumerable.
II. ¯L2 L3 is recursive.
III. ˙L1 L2 is context-free.
IV. L1 ¯L2 is context-free.

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Solution

L1 Regular, L2CFL
L3 REC
L4 RE
I. ¯L3L4 is RE
¯L3L4=¯¯¯¯¯¯¯¯¯¯¯¯RECRE=RECRE
=RE RE=RE
So I is true
II. ¯L2L3 is recursive
¯L2L3=¯¯¯¯¯¯¯¯¯¯¯¯CFLREC=¯¯¯¯¯¯¯¯¯¯¯CSLREC
=CSL REC
=REC REC=REC
So II is True.
III. L1L2 is CFL

L1L2 = (REG)CFL
=REGCFL=CFL
So III is True
IV. L1¯L2 is CFL
L1¯L2=REG¯¯¯¯¯¯¯¯¯¯¯¯CFL=REG¯¯¯¯¯¯¯¯¯¯¯CSL
=REGCSL=CSL
Since, A CSL need not be a CFL,So IV is False
So only I,II and III are true.

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