Final answer:
A PDA cannot recognize a language where three variables need to be matched in count such as a^m b^m c^m, but it can handle the language a^m b^n c^n d^m because only two variables need to be matched and this can be handled sequentially with a stack.
Step-by-step explanation:
The question posed is regarding the capability of a pushdown automaton (PDA), a computational model used in the field of automata theory and formal languages within computer science. A PDA is equipped with a stack, which gives it more computational power than a finite automaton but less than a Turing machine.
A PDA cannot recognize the language defined by ambmcm, where m is the same for all three variables (i). This is because PDAs can only push or pop one symbol on the stack per move, so they cannot handle the comparison of three unbounded variables where each relies on the count of the others.
However, a PDA can recognize the language defined by ambncndm (ii). Here, the PDA can keep pushing 'a's onto the stack and then pop them when reading 'd's to ensure the counts are equal, while also comparing the counts of 'b's and 'c's by pushing and popping 'b's as it transitions from reading 'b's to 'c's.