Church-Turing Thesis

Church-Turing Thesis-60
is a (mostly informal) statement about the nature of computability.

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.

Equivalently, it holds that a function is recursive if and only if it is computable.

While this thesis is not a precise mathematical statement, and therefore cannot be proved, it is almost universally held to be true.

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.The Church-Turing thesis asserts that if a partial strings-to-strings function 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.More concretely, the kind of experiment proposed there to test whether non-computable sequences of events may appear in experiment has been claimed to have been carried out with quantum process in (CDDS 10).The Church-Turing thesis is the hypothesis that any function which can be computed (by any deterministic procedure) can be computed by a Turing machine.the Church-Turing thesis is not a theorem and cannot be proven true or false This is not quite correct: It cannot be proven true.But it could be proven false with a counterexample.Anyway, broadly, the Church-Turing thesis is not a theorem and cannot be proven true or false (so said my professors and Wikidepia).Effectively, it is a definition of computation that has so far been accepted because nothing fundamentally different and more powerful than a Turing machine has been exhibited (there is a theorem that Turing machines, partial recursive functions and the lambda calculus are equivalent - and then there's the jump to "these are really general and I think that about covers it" but still, not a theorem, not provable). It's one of those talks that seems to talk about things I know but then immediately jumps to theories and definitions I have never heard of."Sequential algorithms were axiomatized in 2000" - This "axiomization" was apparently performed by the author of the talk, my Googling says.

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