MT4512 Automata, Langugages and Complexity

Class Test

数学测试代写 This class test forms 10% of the assessment for this module. It will be marked out of 20 marks. Please answer all five questions,

This class test forms 10% of the assessment for this module. It will be marked out of 20 marks.

Please answer all five questions, fully justifying your answers. You may cite results from the course without proof, but you must state the name or number of the result you are using.

1.  数学测试代写

Let G₁be the context free grammar (S, {0, 1}, S → 0S |S1 | 0 | 1, S).

(a) For each of the following two words, either give a derivation or prove that it is not in L(G₁): 0²1, 10. [2 marks]

(b) Describe L(G₁) and fully justify your answer. [3 marks]

(c) Construct a PDA P s.t. L(P) = L(G₁). [2 marks]

3.

Let L₁= {(ab)ⁿcⁿaⁿ: n ≥ 0}. Prove that L₁ is not context free. [3 marks]

4.  数学测试代写

Let L2 be the language

{0^a1^b#0^a+b : a, b ∈Z_≥0} 

over the alphabet {0, 1, #}. Give an implementation-level description of a Turing machine M to recognise L₂. [4 marks]

5.  数学测试代写

Let M₁and M₂ be Turing machines, and let L₃= L(M₁) and L₄ = L(M₂).

(a) Prove that L₃ ∪ L₄ is Turing recognisable. [2 marks]

(b) If both M₁ and M₂ are deciders, is L₃ ∩ L₄ decidable? Justify your answer. [2 marks]