In his famous 1968 demo\footnote{The mother of all demos: \url{https://youtu.be/yJDv-zdhzMY}}, he showcased: a collaborative real-rime word processing editor, the computer mouse, a chorded keyboard, a windowing system, hypertext, file sharing, file annotating, video conferencing and revision control among others.
It was not only mesmerizing because it was a huge list of striking ideas, but also because it was actually functioning live, contrary to the Memex which remained a concept.
The reason why it was not only a technology push forward, but the creation of a whole new scientific area is that computers were not anymore just computing machines, but interactive machines that help us in our everyday tasks.
-This vision changed the way we work, our entertainments, and any other of our activities which uses an interactive system at some point.\\
+This vision changed the way we work, our entertainments, and any other of our activities which uses an interactive system at some point.
\subsection{Wegner}
-Wegner extends the Church-Turing thesis with \defword{interactive computation}~\cite{goldin06}.
-He exposes the limitations of algorithms, which return a specified result given predefined inputs.
-This means algorithms are isolated from the outside world during computation.
+Wegner refutes the strong Church-Turing thesis, which states that TMs can do anything a computer can do~\cite{goldin08}.
+He proves it by showing that interaction has a greater expressiveness than algorithms, and extends the original Church-Turing thesis to \defword{interactive computation}~\cite{goldin06}.
+Knuth specifies \defword{algorithms} as as follows~\cite{knuth68}:%return a specified result given predefined inputs.
+
+\begin{itemize}
+ \item They always terminate after a finite number of steps.
+ \item Every of their steps must be precisely defined.
+ \item They have zero or more inputs taken from specified sets of objects, specified before the algorithm begins, or dynamically as it runs.
+ \item They provide one or more outputs, which have a specified relation to the inputs.
+ \item All of its operations can be done in a finite length of time using a pen and paper.
+\end{itemize}
+
+With this definition, we observe that algorithms cannot deal with infinite input streams.
+They operate on predefined specifications that cannot evolve during computation.
+%This means algorithms are isolated from the outside world during computation.
At the opposite, interaction is a paradigm that observes the environment continuously.
Wegners formalizes the concept of \defword{interaction machines} and proves it is more expressive than TMs~\cite{wegner99}.
\myquote{The equivalent expressiveness of TMs, algorithms, computable functions, and formal systems is due not to their maximal expressiveness but to their common reliance on induction for computation and reasoning.}{Peter Wegner~\cite{wegner99}}
-Instead of static inputs defined beforehand, interaction machines are connected to single (SIM) or multiple (MIM) data streams.
+Instead of finite set of inputs, interaction machines are connected to single (SIM) or multiple (MIM) data streams.
Their expressiveness in not defined by induction, but respectively by non-well founded sets and co-induction.
-However, induction (hence algorithms, TMs, $\lambda$-c, \dots) outputs are recursively enumerable while non-well founded sets and co-induction outputs are non-enumerable~\cite{gordon94}.
+Induction (hence algorithms, TMs, $\lambda$-c, \dots) outputs are recursively enumerable while non-well founded sets and co-induction outputs are non-enumerable~\cite{gordon94}.
This proves that interaction has a greater expressiveness than algorithms, allowing Wegner to extend the Church-Turing thesis for interaction machines.
-Now, Interaction as described by Wegner includes every paradigm that can handle infinite streams, including Object Oriented programming or Neural Networks for example.
+%Now, Interaction as described by Wegner includes every paradigm that can handle infinite streams, including Object Oriented programming or Neural Networks for example.
% is more powerful than algorithms\cite{wegner95,wegner97}.
-As an example, Wegner describes an interaction machine, which simply echoes an input stream to an output stream~\cite{wegner97}.
+As an example, Wegner describes an interaction machine that simply echoes an input stream to an output stream~\cite{wegner97}.
By doing so, this machine wins half of chess games by echoing two simultaneous games one to the other.
Such a performance is inaccessible to algorithms, because they only take into account the current configuration to compute the next play.
They are not aware of future plays, and therefore can only estimate the future as possibilities.
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