Church-Turing Thesis

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He does in fact provide at least one counterexample (around in the video).

One is the real-number GCD algorithm; basically the same as Euclid's standard GCD algorithm, but with real numbers.

These traditional algorithms are known also as classical or sequential.

In the original thesis, effectively computable meant computable by an effective classical algorithm.

Based on an earlier axiomatization of classical algorithms, the original thesis was proven in 2008.

Since the 1930s, the notion of algorithm has changed dramatically.

Slightly more in detail, the (physical) says vaguely that what is computable in the mathematical sense of computation is precisely what is “effectively” computable (physically computable).

In interpreting this one has to be careful which concept of computation is used, there are two different main types: Indeed, there are physical processes (described by the wave equation) which are not type-I computable (Pour-El et al.

He seems to be arguing that since Turing machines cannot represent real numbers, they cannot compute this algorithm.

On the other hand, it can be easily computed with pen and paper, and he gives an example of a ruler-and-compass algorithm for evaluating it.


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