Answer all the problems.

1) Define the following concepts formally:

(a) Finite Automata

(b) Non-Deterministic Finite Automata (NDFA)

(c) Kleene Closure of a set of expressions

(d) Regular Expression

(e) Regular Language

(f) Primitive Recursive Function

(g) Unsolvable Problem

(h) Turing Machine

(i) Universal Turing Machine

(j) Turing-Decidable Problem

(k) Moore Automata

(l) Context-free Language

(m) Pushdown Automata

(n) Halting Problem

(o) NP-Hard Problem

(p) Context-free Language                               

2) Show that language

    L = { a^p : p is positive prime integer} is not regular    

3) Show that each of the following function is primitive recursive function.

(a) f (m, n) = 4mn

(b) fib (n), where fib (n) is defined by

fib (0) = 0
fib (1) =  1
fib ( n + 2) = fib (n) + fib (n + 1) , n ≥ 0

4) Construct one grammer for each of the following languages

(a) { aⁱ b^j c^k | i = 2 j or j = 2k }

(b) { w ∈ { 0, 1}* :    w = w^R}               

5) Construct one Turing Machine for calculating each of the following

(a) Llanguage { 0ⁿ 1ⁿ : n ≥ 1 }

(b) Function f (m, n) = m * n, where ‘*’ denotes multiplication