Unconstrained Church-Turing thesis cannot possibly be true

Yuri Gurevich, The Logic in Computer Science Column by Yuri Gurevich


The Church-Turing thesis asserts that if a partial strings-to-strings func- tion is effectively computable then it is computable by a Turing machine.

In the 1930s, when Church and Turing worked on their versions of the thesis, there was a robust notion of algorithm. 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. New species of algorithms have been and are being introduced. We argue that the generalization of the original thesis, where effectively computable means computable by an effective algorithm of any species, cannot possibly be true.

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