church turing thesis

Church-Turing Thesis

The Church-Turing thesis (formerly commonly known simply as Church's thesis) says that any real-world computation can be translated into an equivalent computation involving a Turing machine . In Church's original formulation (Church 1935, 1936), the thesis says that real-world calculation can be done using the lambda calculus , which is equivalent to using general recursive functions .

The Church-Turing thesis encompasses more kinds of computations than those originally envisioned, such as those involving cellular automata , combinators , register machines , and substitution systems . It also applies to other kinds of computations found in theoretical computer science such as quantum computing and probabilistic computing.

There are conflicting points of view about the Church-Turing thesis. One says that it can be proven, and the other says that it serves as a definition for computation. There has never been a proof, but the evidence for its validity comes from the fact that every realistic model of computation, yet discovered, has been shown to be equivalent. If there were a device which could answer questions beyond those that a Turing machine can answer, then it would be called an oracle .

Some computational models are more efficient, in terms of computation time and memory, for different tasks. For example, it is suspected that quantum computers can perform many common tasks with lower time complexity , compared to modern computers, in the sense that for large enough versions of these problems, a quantum computer would solve the problem faster than an ordinary computer. In contrast, there exist questions, such as the halting problem , which an ordinary computer cannot answer, and according to the Church-Turing thesis, no other computational device can answer such a question.

The Church-Turing thesis has been extended to a proposition about the processes in the natural world by Stephen Wolfram in his principle of computational equivalence (Wolfram 2002), which also claims that there are only a small number of intermediate levels of computing power before a system is universal and that most natural systems are universal.

This entry contributed by Todd Rowland

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Church’s Thesis for Turing Machine

In 1936, A method named as lambda-calculus was created by Alonzo Church in which the Church numerals are well defined, i.e. the encoding of natural numbers. Also in 1936, Turing machines (earlier called theoretical model for machines) was created by Alan Turing, that is used for manipulating the symbols of string with the help of tape.

Church Turing Thesis :

Turing machine is defined as an abstract representation of a computing device such as hardware in computers. Alan Turing proposed Logical Computing Machines (LCMs), i.e. Turing’s expressions for Turing Machines. This was done to define algorithms properly. So, Church made a mechanical method named as ‘M’ for manipulation of strings by using logic and mathematics. This method M must pass the following statements:

  • Number of instructions in M must be finite.
  • Output should be produced after performing finite number of steps.
  • It should not be imaginary, i.e. can be made in real life.
  • It should not require any complex understanding.

Using these statements Church proposed a hypothesis called

Church’s Turing thesis

that can be stated as: “The assumption that the intuitive notion of computable functions can be identified with partial recursive functions.”

Or in simple words we can say that “Every computation that can be carried out in the real world can be effectively performed by a Turing Machine.”

In 1930, this statement was first formulated by Alonzo Church and is usually referred to as Church’s thesis, or the Church-Turing thesis. However, this hypothesis cannot be proved. The recursive functions can be computable after taking following assumptions:

  • Each and every function must be computable.
  • Let ‘F’ be the computable function and after performing some elementary operations to ‘F’, it will transform a new function ‘G’ then this function ‘G’ automatically becomes the computable function.
  • If any functions that follow above two assumptions must be states as computable function.

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COMMENTS

  1. The Church-Turing Thesis

    The Church-Turing thesis (or Turing-Church thesis) is a fundamental claim in the theory of computability. It was advanced independently by Church and Turing in the mid 1930s. There are various equivalent formulations of the thesis.

  2. Church-Turing Thesis -- from Wolfram MathWorld

    The Church-Turing thesis (formerly commonly known simply as Church's thesis) says that any real-world computation can be translated into an equivalent computation involving a Turing machine. In Church's original formulation (Church 1935, 1936), the thesis says that real-world calculation can be done using the lambda calculus, which is ...

  3. The Church-Turing Thesis > The Rise and Fall of the

    The Church-Turing result Gödel’s famous incompleteness theorems of 1931 placed unexpected new obstacles in the way of Hilbert’s desired consistency proof for arithmetic (Gödel 1931). Suspicion also began to build that Gödel’s incompleteness results might further imply the unsolvability of the Entscheidungsproblem .

  4. The Church-Turing Thesis

    The Church-Turing Thesis First published Wed Jan 8, 1997; substantive revision Fri Nov 10, 2017 There are various equivalent formulations of the Church-Turing thesis. A common one is that every effective computation can be carried out by a Turing machine.

  5. Church’s Thesis for Turing Machine

    Church Turing Thesis : Turing machine is defined as an abstract representation of a computing device such as hardware in computers. Alan Turing proposed Logical Computing Machines (LCMs), i.e. Turing’s expressions for Turing Machines. This was done to define algorithms properly.