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A property of sets for which one can determine whether something is a member or not in a finite number of computational steps.

Decidability is an important concept in computability theory. A set (e.g. "all numbers with a 5 in them") is said to be "decidable" if I can write a program (usually for a Turing Machine) to determine whether a number is in the set and the program will always terminate with an answer YES or NO after a finite number of steps.

Most sets you can describe easily are decidable, but there are infinitely many sets so most sets are undecidable, assuming any finite limit on the size (number of instructions or number of states) of our programs. I.e. how ever big you allow your program to be there will always be sets which need a bigger program to decide membership.

One example of an undecidable set comes from the halting problem. It turns out that you can encode every program as a number: encode every symbol in the program as a number (001, 002, ...) and then string all the symbol codes together. Then you can create an undecidable set by defining it as the set of all numbers that represent a program that terminates in a finite number of steps.

A set can also be "semi-decidable" - there is an algorithm that is guaranteed to return YES if the number is in the set, but if the number is not in the set, it may either return NO or run for ever.

The halting problem's set described above is semi-decidable. You decode the given number and run the resulting program. If it terminates the answer is YES. If it never terminates, then neither will the decision algorithm.
References in periodicals archive ?
As a consequence, computability, decidability, verification, program generation and search for solution can be exercised on "top level model" and supply designer with valuable data.
On the decidability of query containment under constraints.
Decidability in the syntax of verbs of (not necessarily) West-Germanic languages.
I doubt, for example, if a nonspecialist reader could grasp the difference between what Godel proved about completeness in 1930 and what Alonzo Church proved about decidability in 1936.
One of these papers terminates in completeness and decidability results for noniterative logics--systems of modal logic the formulae of which do not contain a modal operator within the scope of a modal operator.
His real genius was in presenting his material not as reasoned argument but in a voice or series of voices which put always in doubt issues of reliability and decidability.