Qwiki

Church-Turing Thesis

The Church-Turing thesis is a foundational concept in the philosophy of computer science and theoretical computer science. It posits that any function which can be effectively calculated by a human using an algorithm can also be computed by a Turing machine, a theoretical device that serves as a model of computation.

Historical Background

The thesis is named after two prominent figures in mathematics and logic: Alonzo Church and Alan Turing. In the early 20th century, before a precise definition of computable functions existed, mathematicians used the term "effectively calculable" for functions computable by paper-and-pencil methods.

Church, along with his student Stephen Kleene, and Turing independently developed formal definitions of computable functions. They showed that the classes of functions they defined were equivalent:

  • Lambda Calculus: Developed by Church, this formal system uses function abstraction and application to define computability.
  • Turing Machines: Proposed by Turing as an abstract machine that manipulates symbols on a strip of tape according to a table of rules to simulate the logic of any computer algorithm.
  • General Recursive Functions: Defined by Kurt Gödel and further refined by Kleene, these are functions defined using basic functions and operators.

These equivalences led to the belief that the concept of computability was accurately characterized by these three equivalent processes.

Variations and Implications

Several variations of the Church-Turing thesis have emerged over time:

  • Physical Church-Turing Thesis: This variation extends the thesis to claim that any function that can be physically realized in our universe is computable by a Turing machine.

  • Church-Turing Thesis (Complexity Theory): This concerns what can be efficiently computed, linking the concepts of computability and computational complexity.

  • Church-Turing-Deutsch Principle: Formulated by physicist David Deutsch, this principle connects the thesis with quantum physics, suggesting that a universal quantum computer can simulate any physical system.

Related Concepts

  • Hypercomputation: This refers to theoretical models of computation that can compute functions beyond what is possible with a Turing machine, thus challenging the Church-Turing thesis.

  • Turing Completeness: A system of data-manipulation rules (such as a computer's instruction set, a programming language, or a cellular automaton) is said to be Turing complete if it can be used to simulate any Turing machine.

  • Busy Beaver: A function that represents the maximum number of steps a Turing machine can take before halting, related to the limits of computability and the physical Church-Turing thesis.

The Church-Turing thesis continues to be a central topic in discussions about the nature of computation and the limits of what can be achieved with algorithms and mechanical processes. Its implications extend beyond theoretical computer science, influencing areas such as cognitive science, artificial intelligence, and the philosophy of mind.