Some prefer the name Turing-Church thesis.

In addition to simpler proofs of existing theorems, new theorems by Landauinclude important facts about Riemann's Hypothesis;facts about Dirichlet series;key lemmas of analysis;a result in Waring's Problem;a generalization of the Little Picard Theorem;a partial proof of Gauss' conjecture about the density of classesof composite numbers;and key results in the theory of pecking orders, e.g.

invalidate the Church–Turing thesis.

Every single one of these satisfies the Church-Turing hypothesis, i.e.

The Church-Turing thesis is a statement of type 2.

There are various equivalent formulations of the Church-Turing thesis.A common one is that every effective computation can be carried out bya Turing machine. The Church-Turing thesis is often misunderstood,particularly in recent writing in the philosophy of mind.

Now, applying Gödel’s completeness theorem to this yieldsin turn:

Some of the philosophical questions raised by the theory of quantum computation are discussed. First it is considered whether the possibility of exponential speed-up in quantum computation provides an argument for a more substantive notion of quantum information than so far allowed. It is concluded that this is not so. Then some questions regarding the status of the Church-Turing hypothesis in the light of quantum computation are considered. In particular, Deutsch’s claim that a physical principle, the Turing Principle, underlies the Church-Turing hypothesis is rebutted. Finally, the question of whether the Church-Turing hypothesis might serve as a constraint on the laws of physics is briefly considered.


What would it mean to disprove Church-Turing thesis?

Since it can also be shown that there are no functions in other than ones whose values can be obtained by a methodsatisfying the above conditions for effectiveness, the Church-Turingthesis licences replacing the informal claim “There is aneffective method for obtaining the values of function ”by the formal claim “ is a member of”—or by any other formal claim equivalent to thisone.

Church Turing Thesis - SlideShare

In the Church–Turing thesis (also known as the Church-Turing conjecture, Church's thesis, Church's conjecture, and Turing's thesis) is a combined ("thesis") about the nature of (computable) functions by (Church's Thesis), by mechanical device equivalent to a (Turing's Thesis) or by use of Church's . The three computational processes (recursion, λ-calculus, and Turing machine) were shown to be equivalent by , , (1934–6) and (1936–7).

Church-Turing thesis - Psychology Wiki

Informally the Church–Turing thesis states that if an (a procedure that terminates) exists then there is an equivalent , -definable function, or , for that algorithm. A more simplified but understandable expression of it is that "everything computable is computable by a Turing machine." Though not formally proven, today the thesis has near-universal acceptance.

There are various equivalent formulations of the Church-Turing thesis

The Church-Turing hypothesis assertsBecause we can (in principle) write a Turing machine program whichcan exactly simulate the behaviour of given ADA program.

Church-Turing Thesis -- from Wolfram MathWorld

One of Alan Turing’s achievements, in his famous paper of 1936,was to present a formally exact predicate with which the informalpredicate “can be done by means of an effective method”may be replaced (Turing 1936). Alonzo Church, working independently,did the same (Church 1936a).

The history of the Church–Turing ..

The replacement predicates that Turing and Church proposed were, onthe face of it, very different from one another. However, thesepredicates turned out to be equivalent, in the sense thateach picks out the same set, call it , of mathematicalfunctions. The Church-Turing thesis is the assertion that this set contains every function whose values can be obtainedby a method satisfying the above conditions for effectiveness.