Top/EntranceQuizSolns

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Problem 1

T - regular languages are closed under concatenation

F - context-free languages are not closed under intersection

F - language is not regular

F - context-free languages not closed under complement

T - if all words in L have length < 1000 then L is finite, thus regular

T - e.g. with dynamic programming

T - see def'n of Chomsky Normal Form

T - any finite language is regular

F - every regular language is context free

T

F - for some NFA's with n states, there are no equivalent DFA's with less than 2ⁿ states

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Problem 2

states: (each state's name is what we need to see to have seen 0010)

0010 - accept

010 - accept

10 - accept

0 - accept

reject

transitions:

state	symbol	new state
0010	0	010
0010	1	0010
010	0	10
010	1	0010
10	0	10
10	1	0
0	0	reject
0	1	0010
reject	0	reject
reject	1	reject

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Problem 3

S -> a S d | B | C

B -> b B d | T

C -> a C c | T

T -> b T c | epsilon

explanation:

T -> { bⁱ cⁱ }

B -> { b^j T d^j } = {b^j bⁱ cⁱ d^j } = { b^k cⁱ d^j : i+j = k }

C -> { a^j T c^j } = { a^j bⁱ cⁱ c^j } = { a^j bⁱ c^k : i+j = k }

S -> { aⁿ B dⁿ } ∪ { aⁿ B dⁿ } = { aⁿ b^k cⁱ d^j dⁿ : i+j=k } ∪ {aⁿ a^j bⁱ c^k dⁿ : i+j = k}

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Problem 4

The language is regular. Here is a DFA:

states:

00 -- accept

01

02

10

11

12

transition function:

δ(ij, 0) = i'j where i' = i+1 mod 2

δ(ij, 1) = ij' where j' = j+1 mod 3