# Automata Theory | Set 1

Following questions have been asked in GATE CS exam.**1. Let S and T be language over ={a,b} represented by the regular expressions (a+b*)* and (a+b)*, respectively. Which of the following is true? (GATE CS 2000)**

(a) ScT (S is a subset of T)

(b) TcS (T is a subset of S)

(c) S=T

(d) SnT=Ø

**Answer:** (c).

2. Let L denotes the language generated by the grammar S – OSO/00. Which of the following is true? (GATE CS 2000)

(a) L = O

(b) L is regular but not O

(c) L is context free but not regular

(d) L is not context free

**Answer:** (b)**Explanation:** Please note that grammar itself is not regular but language L is regular as L can be represented using a regular grammar, for example S -> S00/00.**References:**

http://en.wikipedia.org/wiki/Regular_grammar

**3. Consider the following two statements:S1: { 0^2n |n >= l} is a regu1ar languageS2: { 0^m 0^n 0^(m+n) l m >= 1 and n >= 2} is a regu1ar languageWhich of the following statements is correct? (GATE CS 2001)**a) Only S1 is correct

b) Only S2 is correct

c) Both S1 and S2 are correct

d) None of S1 and S2 is correct

**Answer:** (c)**Explanation:**

S1 can be written as (00)^n where n >= 1. And S2 can be written as (00)^(m+n) where m >=2 and n >= 1. S2 can be further reduced to (00)^x where x >= 3.

We can easily write regular grammars for both S1 and S2.

G1 -> G100/00 (For S1)

G2 -> G200/000000 (For S2)

4. Which of the following statements in true? (GATE CS 2001)

(a) If a language is context free it can always be accepted by a deterministic push-down automaton

(b) The union of two context free languages is context free

(c) The intersection of two context free languages is context free

(d) The complement of a context free language is context free

**Answer: **(b)**Explanation:**

Context-free languages are closed under the following operations. That is, if L and P are context-free languages and D is a regular language, the following languages are context-free as well:

• the Kleene star L * of L

• the image Ø(L) of L under a homomorphism Ø

• the concatenation of L and P

• the union of L and P

• the intersection of L with a regular language D (L n D).

Context-free languages are not closed under complement, intersection, or difference.**Why a) is not true?**

The language recognized by deterministic pushdown automaton is deterministic context free language. Not all context-free languages are deterministic. This is unlike the situation for deterministic finite automata, which are also a subset of the nondeterministic finite automata but can recognize the same class of languages (as demonstrated by the subset construction).**References:**

http://en.wikipedia.org/wiki/Context-free_language

http://en.wikipedia.org/wiki/Deterministic_pushdown_automaton

**5. Given an arbitrary non-deterministic finite automaton (NFA) with N states, the maximum number of states in an equivalent minimized DFA is at least. (GATE CS 2001)**(a) N^2

(b) 2^N

(c) 2N

(d) N!

**Answer:** (b)**References:**

http://en.wikipedia.org/wiki/Powerset_construction