computability

logic and mathematics

Also known as: computability theory, solvability

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June 24, 2026

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Computability, a core concept in logic and mathematics, explores the limits of what can be computed or calculated. It is deeply intertwined with computer science and the theory of recursive functions.123

At its heart is the idea of effective computability, which asks if a function's value can be determined by an algorithm or a mechanical procedure. This concept was formalized by Alan Turing with the invention of the Turing machine. This hypothetical automaton, with its infinite tape and finite states, serves as a model for all digital computers and defines what it means for a function to be computable. The equivalence of various definitions of computability, including Turing's, supports Church's thesis, which posits that these definitions capture the intuitive notion of effective calculability.3142

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history of logic
Effective computability
One of the starting points of recursion theory was the decision problem for first-order logic—i.e., the problem of finding an algorithm or repetitive procedure that would mechanically (i.e., effectively) decide whether a given formula of first-order logic is logically true. A positive solution to…
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One of the starting points of recursion theory was the decision problem for first-order logic—i.e., the problem of finding an algorithm or repetitive procedure that would mechanically (i.e., effectively) decide whether a given formula of first-order logic is logically true. A positive solution to…
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automata theory
The generalized automaton and Turing’s machine
…automaton is then said to compute the statement (polynomial) or the statement is said to be computable. A wider class of computable statements is introduced with the general automaton, yet to be defined, as with the more general Turing machine.
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…automaton is then said to compute the statement (polynomial) or the statement is said to be computable. A wider class of computable statements is introduced with the general automaton, yet to be defined, as with the more general Turing machine.
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