NPDA for accepting the language L = {a^mb^(m+n)cⁿ| m,n ≥ 1}

A solid understanding of pushdown automata and their input acceptance via final states is essential before proceeding with this topic.

Problem

Design a non deterministic PDA for accepting the language L = {a^m b^(m+n) cⁿ | m,n ≥ 1}.

The strings of given language will be: 

L = {abbc, abbbcc, abbbcc, aabbbbcc, ......}

In each of the string, the total sum of the number of 'a’ and 'c' is equal to the number of 'b’s. And all c's are come after 'a' and 'b'. 

Explanation 

Here, we need to maintain the order of a’s, b's and c’s. That is, all the a’s  are coming first and then all the b’s are coming after that all the c's. Thus, we need a stack along with the state diagram. The count of a’s, b's and c’s is maintained by the stack. We will take 3 stack alphabets: 

𝚪 = { a, b, z }

Where, 𝚪 = set of all the stack alphabet 
z = stack start symbol 

Approach Used in the Construction of PDA 

As we want to design a NPDA, thus every time 'a' comes before 'b' and 'b' comes before 'c'. First we have to count number of a's and that number should be equal to number of b's. When all a's are finished by b's, then count number of b's and that should be equal to number of c's. 

For all the 'a' we will push 'a' into the stack each time and then start popping them when 'b's are coming. After finishing all the popping of all the 'a's we will start pushing 'b' for the rest 'b's. And when 'c's are coming we will pop these 'b' from the stack each time. So, at the end if the stack becomes empty then we can say that the string is accepted by the PDA. 

Stack Transition Functions

\delta(q₀, a, z) \vdash (q₀, az) [push a]
\delta(q₀, a, a) \vdash(q₀, aa) [push a for every a]
\delta(q₀, b, a) \vdash (q₁, \epsilon) [ change state and for every b pop a]
\delta(q₁, b, a) \vdash(q₁, \epsilon) [for every b pop a]
\delta(q₁, b, z) \vdash(q₁, bz) [reached end push b]
\delta(q₁, b, b) \vdash (q₁, bb) [for every b push b]
\delta(q₁, c, b) \vdash (q₂, \epsilon) [ change state and pop b for c ]
\delta(q₂, c, b) \vdash (q₂, \epsilon) [pop b for every c]
\delta(q₂, \epsilon, z) \vdash (q_f , z) [reached final state and stack is empty]

Where, q₀ = Initial state  q_f = Final state 
\epsilon   = indicates pop operation