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What does decidable mean?

Looking for the meaning or definition of the word decidable? Here's what it means.

Adjective
  1. capable of being decided.
  2. (computer science) describing a set for which there exists an algorithm that will determine whether any element is or is not within the set in a finite amount of time.
  3. (logic) in intuitionistic logic, a proposition P is decidable in a given theory if it can be proven from the theory that "either P or not P", i.e. in symbols:
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We show that the finite power properly is decidable for rational sets in free group.
Moreover, the geometries axiomatized in the book have consistent and decidable extensions, namely, Euclidean or hyperbolic geometry over real-closed fields.
A milestone open problem is to determine whether the equality of sequences generated by such automata is decidable.
There are uncountably many subsets of N, but since there are only countably many Turing machines, there can be only countably many decidable sets.
We prove that the existence of a coding between two trace monoids is decidable for some families of traces monoids.
We prove here that the finite power property is decidable for rational sets of the free group.

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