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sav08:recursive_set [2008/03/18 18:03]
vkuncak created
sav08:recursive_set [2008/03/18 18:04] (current)
vkuncak
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====== Recursive Set ====== ====== Recursive Set ======

-We say that a set $S$ is //​recursive//​ if there exists an algorithm that, given an element $x$ determines whether $x \in S$ holds.+We say that a set $S$ is //​recursive// ​(decidable) ​if there exists an algorithm that, given an element $x$ determines whether $x \in S$.

Example: the set of even numbers is recursive. ​ The algorithm takes $x$ represented in binary and checks whether its least signifficant digit is 0. Example: the set of even numbers is recursive. ​ The algorithm takes $x$ represented in binary and checks whether its least signifficant digit is 0.
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Example: the set of all pairs $(T,x)$ such that $T$ is a representation of Turing machine and $x$ is an input such that $T$ terminates on input $x$, is not recursive. Example: the set of all pairs $(T,x)$ such that $T$ is a representation of Turing machine and $x$ is an input such that $T$ terminates on input $x$, is not recursive.

+Example: the set of encodings of all valid first-order logic formulas is not recursive.
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+Example: the set of all valid formulas of Presburger arithmetic is recursive.