Gap theorem

See also <a href="/facts/Gap_theorem_(disambiguation)/fKJmnFEM">Gap theorem (disambiguation)</a> for other gap theorems in <a href="/facts/Mathematics/pxTouaz4">mathematics</a>.
In <a href="/facts/Computational_complexity_theory/etHuVF3U">computational complexity theory</a>, the Gap Theorem, also known as the Borodin–Trakhtenbrot Gap Theorem, is a major theorem about the complexity of <a href="/facts/Computable_function/dzwBlLGn">computable functions</a>.
It essentially states that there are arbitrarily large computable gaps in the hierarchy of <a href="/facts/Complexity_class/IIKrcTbP">complexity classes</a>. For any <a href="/facts/Computable_function/dzwBlLGn">computable function</a> that represents an increase in <a href="/facts/Computational_resource/YyleNx71">computational resources</a>, one can find a resource bound such that the set of functions computable within the expanded resource bound is the same as the set computable within the original bound.
The theorem was proved independently by <a href="/facts/Boris_Trakhtenbrot/oftNXKRM">Boris Trakhtenbrot</a> and <a href="/facts/Allan_Borodin/4JrWPDAL">Allan Borodin</a>.
Although Trakhtenbrot's derivation preceded Borodin's by several years, it was not known nor recognized in <a href="/facts/Western_world/UQl4e7du">the West</a> until after Borodin's work was published.

Gap theorem open-in-new

Gap theorem