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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 prove that the existence of a coding between two trace monoids is decidable for some families of traces monoids.
The aim of this article is to prove that the variety generated by the piecewise testable languages and the band languages is decidable.
We prove here that the finite power property is decidable for rational sets of the free group.
Which mathematical theory best describes real space is an empirical issue, not decidable independently of physical considerations.
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.

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