The Church-Turing thesis says nothing about the efficiency of with which one model of computation can simulate another. It has been proved for instance that a (multi-tape) only suffers a logarithmic slowdown factor in simulating any Turing machine. No such result has been proved in general for an arbitrary but model of computation. A variation of the Church-Turing thesis that addresses this issue is the (Classical) Strong Church–Turing Thesis (SCTT), which is not due to Church or Turing, but rather was realized gradually in the development of . It states:
Sorry for the catchy title. I want to understand, what should one have to do to disprove the Church-Turing thesis? Somewhere I read it's mathematically impossible to do it! Why?
Church–Turing thesis - Wikipedia, the free encyclopedia
Mark Burgin, Eugene Eberbach, Peter Kugel, and other researchers argue that such as inductive Turing machines disprove the Church–Turing thesis. Their argument relies on a definition of algorithm broader than the ordinary one, so that non-computable functions obtained from some inductive Turing machines are called computable. This interpretation of the Church–Turing thesis differs from the interpretation commonly accepted in computability theory, discussed above. The argument that are indeed algorithms in the sense of the Church–Turing thesis has not found broad acceptance within the computability research community.