Question

The Alphabet: a, b, c
I’m trying to define a PDA which accepts

 a^n b^m c^p : n + p = 2k for some integer k, m = k, and n, m, p, k >= 0

I think some strings that would be accepted are: #abc#; #aabbcc#; #aaabbbccc#; #abbccc#; #aaabbc# etc
The number of a’s, b’s and c’s are not necessarily equal.

Start the head of the push down automata on the black space that is right most.

Usually I write my PDAs in columns:

State:    Symbol Read:    Next State:    Head Instruction:    
s         #               r1             Left
r1        c               r2             #

and so on…

Answer 1

I think the language you describe is not context-free, and therefore cannot be
recognized with a PDA. The problem is that you need to enforce a constraint
(n+p = 2m) that spans an arbitrarily long substring, yet is not allowed to “pump” (when
attempting to construct a proof using the pumping lemma for context-free languages).