What is The Church-Turing Thesis in TOC?

The Church-Turing thesis says that every solvable decision problem can be transformed into an equivalent Turing machine problem.

It can be explained in two ways, as given below −

• The Church-Turing thesis for decision problems.

• The extended Church-Turing thesis for decision problems.

Let us understand these two ways.

The Church-Turing thesis for decision problems

There is some effective procedure to solve any decision problem if and only if there is a Turing machine which halts for all input strings and solves the problem.

The extended Church-Turing thesis for decision problems

A decision problem Q is said to be partially solvable if and only if there is a Turing machine which accepts precisely the elements of Q whose answer is yes.

Proof

A proof by the Church-Turing thesis is a shortcut often taken in establishing the existence of a decision algorithm.

For any decision problem, rather than constructing a Turing machine solution, let us describe an effective procedure which solves the problem.

The Church-Turing thesis explains that a decision problem Q has a solution if and only if there is a Turing machine that determines the answer for every q ϵ Q. If no such Turing machine exists, the problem is said to be undecidable.