Let L = L₁ ∩ L₂ where L₁ and L₂ are language defined below :

L₁ = {a^mb^mc aⁿ bⁿ m, n ≥ 0}

L₂ = {aⁱb^j c^k | i, j, k ≥ 0}

Then L is : 

The correct answer is Not Recursive.

Key Points

• Let L = L1 ∩ L2 where L1 and L2 are languages defined below :
  • L1 = {a^mb^mc^anbⁿ | m, n ≥ 0}
  • L2 = {aⁱb^jc^k | i, j, k ≥ 0}
• The intersection of these two languages is a language L where the number of a's is equal to the number of b's and the number of c's is equal to the number of b's.
• This language is not recursive because it cannot be decided by a Turing machine in a finite amount of time.

Additional Information

• A recursive language is a type of formal language that can be decided by a Turing machine that always halts.
• A language is considered recursively enumerable if there is a Turing machine that will halt and accept the strings in the language but may run forever for strings not in the language.
• Regular languages can be recognized by finite automata and are a subset of context-free languages.
• Context-free languages can be recognized by pushdown automata and include most programming languages.