Exam 1: Formal Languages and Automata Theory: Part A

Let SHAM3 be the problem of finding a Hamiltonian cycle in a graph G =(V,E)with V divisible by 3 and DHAM3 be the problem of determining if a Hamiltonian cycle exists in such graphs. Which one of the following is true?

S -> aSa| bSb| a| b ; The language generated by the above grammar over the alphabet {a,b} is the set of

Consider the languages: L₁ ={a^n b^n c^m | n,m >01 and L₂ ={a^n b^m c^m |n,m> o) Which one of the following statements is FALSE?

Consider the languagesL₁={0^{i}1^{j}|i != j},L₂={0^{i}1^{j}|i = j},L₃ = {0^{i}1^{j}|i = 2j+1},L4 = {0^{i}1^{j}|i != 2j}.Which one of the following statements is true?

Let L₁ be a recursive language. Let L₂ and L₃ be languages that are recursively enumerable but not recursive. Which of the following statements is not necessarily true?

A minimum state deterministic finite automation accepting the language L = {W |W € {0,1}* , number of 0's and 1's in W are divisible by 3 and 5 respectively has

If s is a string over (0 + 1)* then let n0 (s) denote the number of 0's in s and n₁ (s)the number of l's in s. Which one of the following languages is not regular?

Consider the following two problems on undirected graphs: ?: Given G(V,E), does G have an independent set of size |v|-4? ?: Given G(V,E), does G have an independent set of size 5? Which one of the following is TRUE?

Consider the set of strings on {0,1} in which, every substring of 3 symbols has at most two zeros. For examples, 001110 and 011001 are in the language, but 100010 is not. All strings of length less than 3 are also in the language. A partially completed DFA that accepts this language is shown below. The missing arcs in the DFA are

For S € (0+1)* let d(s)denote the decimal value of s(e.g.d(101)= 5).Let L = {s € (0 + 1) | d (s) mod 5 = 2 and d (s) mod 7 != 4)}Which one of the following statements is true?

Consider the language L₁,L₂,L₃ as given below. L₁={0^{p}1^{q} | p,q \in N} L₂={0^{p}1^{q} | p,q \in N and p=q} L₃={0^{p}1^{q}1^{r} | p,q,r \in N and p=q=r} Which of the following statements is NOT TRUE?

Consider the following statements about the context free grammarG = {S - >SS, S - >ab, S - >ba, S - ?}I. G is ambiguousII. G produces all strings with equal number of a's and b'sIII. G can be accepted by a deterministic PDA.Which combination below expresses all the true statements about G?

The minimum state automaton equivalent to the above FSA has the following number of states

Let L₁ be a recursive language, and let L₂ be a recursively enumerable but not a recursive language. Which one of the following is TRUE?

Definition of a language L with alphabet {a} is given as following. L= { a^{nk} | k > 0, and n is a positive integer constant} What is the minimum number of states needed in a DFA to recognize L?

Let L₁ be a regular language, L₂ be a deterministic context-free language and L₃ a recursively enumerable, but not recursive, language. Which one of the following statements is false?

Consider the regular language L =(111+11111)*. The minimum number of states

Consider the languages: GATE[2005]L₁ = {wwR w €{0, 1} *1L₂ ={w#ww € {O,1}*},where # is a special symbolL₃ ={www € {0,1}*}Which one of the following is TRUE?

Which of the following problems are decidable? 1) Does a given program ever produce an output? 2) If L is a context-free language, then is L' (complement of L) also context-free? 3) If L is a regular language, then is L' also regular? 4) If L is a recursive language, then, is L' also recursive?

Which of the following languages is regular?