HELP! Theoretical Computation

What does the following Mealy machine do? Construct a Mealy machine with input alphabet {a, b} for each of the following tasks: On input string w, produce 1^n where n is the number of a's in w On input string w, produce 1^n where n is half of the number of a's in w On input string w, produce 1^n where n is the number of occurrences of the substring aba in w In a rural intersection, the light on the main highway (East-West) remains green unless a car comes from the perpendicular lane (North-South) and a sensor detects it. A certain time delay after detection, the signal changes and stays that way for a fixed period of time. Design a controller using Finite State Transducer concept (say Moore Machine) for this four light system. Use G, Y, and R for the three states of light and 0 for no traffic detected from N-S direction and 1 for traffic detected. The output will have two color symbols one for the E-W traffic and the other for N-S traffic Using Pumping lemma, prove that the {0^n 1^n 2^n|n greterthanequalto 0} language over the alphabet {a, b} is not a regular language. Using pumping lemma, prove that the following language over the alphabet {a, b} is not regular. L = {a^n ba^3n|n notequalto 0}