Is a PDA as powerful as a CPU?

This is a question I have stumbled upon in my exam revision and I find it intriguing:

My computer is blue and it has a massive graphics card and a DVD and every-
thing so which is more powerful: my computer or a Pushdown Automaton?

My Thoughts

When we talk about power I have assumed this to be computational power. I believe that a PDA has the computational power to equal the computational power of a CPU (cpu in this case is the core elements of a computer ie memory and processor). This is because a PDA utilises a stack which is memory(RAM) in a computer. The PDA has states as does a CPU and also the PDA calculates simple logic at each state. I realise that the PDA itself would be a complex series of states to emulate the computational power of the cpu and there would have to be a series of PDAs to compute different functions.

My Question: I know that Turing machines are best used to simulate the logic of a CPU but am I right in saying that a PDA (Or PDA's) can be designed to be as powerful as a cpu?

If you count a CPU (as you imply in your text) as an unbounded memory and the processor, then no you can't model it as a PDA. A push down automaton (with one stack) can only recognize context-free languages, for example $\{a^n b^n c^n | n \geq 0\}$ cannot be recognized by a PDA, but presumably could be recognized by your blue computer with massive graphics card and DVD.

Typically, we think of your blue computer with massive graphics cards and DVD as a Turing Machine, because that often proves to be the best way to model it.

On the other hand, if you model your CPU without unbounded memory, then PDA is still not the right model. If your CPU has no memory or only a small (i.e. bounded) amount of on-chip memory, then it can only be in a finite number of states, and is better modeled as a DFA.

StackExchange Computer Science Q#1868, answer score: 13