Constructing NPDA for Languages a) and b)

How to construct an NPDA for language a) L = {anbm | 0 ≤ m ≤ n}?

Follow the steps to construct an NPDA for language a):

a) Define the states, input alphabet, stack alphabet, transition function, start state, and final states.

b) What are the steps to construct an NPDA for language b) L3 = {w: na (w) + nb (w) = nc (w)}?

c) Follow the same steps as above to construct an NPDA for language b).

Answer:

To construct an NPDA for language a) L = {anbm | 0 ≤ m ≤ n}, follow the steps:

Define the states: Q = {q0, q1, q2}

Define the input alphabet: Σ = {a, b}

Define the stack alphabet: Γ = {Z, X}

Define the transition function:
δ(q0, a, Z) = {(q0, XZ)}
δ(q0, a, X) = {(q0, XX)}
δ(q0, b, X) = {(q1, ε)}
δ(q1, b, X) = {(q1, ε)}
δ(q1, ε, Z) = {(q2, Z)}

Define the start state: q0

Define the final states: F = {q2}

How to construct an NPDA for language b) L3 = {w: na (w) + nb (w) = nc (w)}?

Answer:

To construct an NPDA for language b) L3 = {w: na (w) + nb (w) = nc (w)}, follow the steps:

Define the states: Q = {q0, q1, q2, q3}

Define the input alphabet: Σ = {a, b, c}

Define the stack alphabet: Γ = {Z, X}

Define the transition function:
δ(q0, a, Z) = {(q0, XZ)}
δ(q0, a, X) = {(q0, XX)}
δ(q0, b, X) = {(q1, ε)}
δ(q1, b, X) = {(q1, ε)}
δ(q1, c, X) = {(q2, ε)}
δ(q2, c, X) = {(q2, ε)}
δ(q2, ε, Z) = {(q3, Z)}

Define the start state: q0

Define the final states: F = {q3}