Correct me if I slept in my computer science studium: If an automaton is Turing-equivalent, the Halting problem shows that there are programs we can not verify (since we can't even predict their output in the first place).  

Now, this is a *theoretical* result; in practice, software writers all over the world try to verify their programs and usually succeed.
Can you give me an actual example of an automaton (Petri net, Turing machine, whatever), either...  

a) one which no human understands at all, or even better (since we could cheat in "a" by simply adding complexity until our brain gives up)  

b) one which is fully understood by human but nobody can *prove* the actual behaviour.  

(It's debatable whether checkmating with King, Rook, and Bishop against King, Knight, and Knight would already constitute a valid example, since I think it falls under the complexity cheat. And maybe "b" is a contradiction in terms.)



Verification of Turing-equivalent automata
(a) I think that "no human understands" what the current busiest $6$-state 
[Busy Beaver](https://en.wikipedia.org/wiki/Busy_beaver/) 
$2$-symbol Turing Machine is doing while it prints
out $3.5 \times 10^{18267}$ $1$'s before halting.


(b) This does not quite address your 2nd question as you phrased it, but
there are several $5$-state quite-Busy Beavers that are believed to loop,
but no one can prove
they loop, or prove they halt. A list of these irregular Turing
Machines is maintained [here](http://skelet.ludost.net/bb/nreg.html).


Incidentally, just to illustrate the complexity of Busy-Beaver questions,
a $4888$-state Turing Machine has been constructed which halts if and only 
if there is a counterexample to Goldbach's conjecture.



> 
> Yedidia, Adam, and Scott Aaronson. "A Relatively Small Turing Machine Whose Behavior Is Independent of Set Theory." arXiv:1605.04343 (2016). ([arXiv Abstract](https://arxiv.org/abs/1605.04343).)
> 
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