Question: I've heard the motto [**interaction is more powerful than algorithms**](http://www.cs.brown.edu/people/pw/papers/ficacm.ps) from [Peter Wegner](http://www.cs.brown.edu/~pw/). The basis of the idea is that a (classical) Turing Machine cannot handle interaction, that is, communication (input/output) with the outside world/environment.

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> How can this be so? How can something be more powerful than a Turing Machine? What is the essence of this story? Why is it not more well known?
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Answer: 
I do not think models with IO are "more powerful" than Turing machines, they just model different things.

In theory, you could view IO as (noisy?) oracle. With a perfect oracle you can computer Turing-uncomputable functions; but where to get the oracle from? Humans are the only "super-Turing" choice (if there is any) and we are known to be very unreliable.

A class of programs that fit this model are interactive proof assisstants (e.g. [Isabelle/HOL](http://www.cl.cam.ac.uk/research/hvg/Isabelle/), [Coq](http://coq.inria.fr/)). They deal with undecidable proof spaces but (arguably) every proof can be found (and checked) with suitable user input.
Check this out :)
"***Turing's Ideas and Models of Computations***"
<https://www.cs.montana.edu/~elser/turing%20papers/Turing%27s%20Ideas%20and%20Models%20of%20Computation.pdf>