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| :''See also [[Gap theorem (disambiguation)]] for other gap theorems in [[mathematics]].''
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| In [[computational complexity theory]] the '''Gap Theorem,''' also known as the '''Borodin-Trakhtenbrot Gap Theorem,''' is a major theorem about the complexity of [[computable function]]s.<ref>{{Cite journal
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| |first=Lance
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| |last=Fortnow
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| |first2=Steve
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| |last2=Homer
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| |title=A Short History of Computational Complexity
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| |url=http://theorie.informatik.uni-ulm.de/Personen/toran/beatcs/column80.pdf
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| |journal=Bulletin of the European Association for Theoretical Computer Science
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| |pages=95–133
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| |issue=80
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| |date=June 2003
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| |postscript=<!--None-->}}</ref>
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| It essentially states that there are arbitrarily large computable gaps in the hierarchy of [[complexity class]]es. For any [[computable function]] that represents an increase in [[computational resource]]s, 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.
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| The theorem was proved independently by [[Boris Trakhtenbrot]]<ref>{{cite book|last=Trakhtenbrot|first=Boris A.|title=The Complexity of Algorithms and Computations (Lecture Notes)|date=1967|publisher=Novosibirsk University}}</ref>and [[Allan Borodin]].<ref>{{cite journal|author=Allan Borodin|title=Complexity Classes of Recursive Functions and the Existence of Complexity Gaps|journal=Proc. of the 1st Annual ACM Symposium on Theory of Computing|pages=67-78|date=1969}}</ref><ref>{{cite journal|author=Borodin, Allan|title=Computational complexity and the existence of complexity gaps|journal=[[Journal of the ACM]]|volume=19|issue=1|pages=158–174|date=January 1972 |doi=10.1145/321679.321691}}</ref>
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| == Gap theorem ==
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| The general form of the theorem is as follows.
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| :Suppose <math>\Phi</math> is an [[Blum axioms|abstract (Blum) complexity measure]]. For any [[total computable function]] ''g'' for which <math>g(x) \geq x</math> for every <math>\,x</math>, there is a total computable function ''t'' such that with respect to <math>\Phi</math>, the [[complexity class]]es with boundary functions <math>t</math> and <math>g \circ t</math> are identical.
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| The theorem can be proved by using the Blum axioms without any reference to a concrete [[computational model]], so it applies to time, space, or any other reasonable complexity measure.
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| For the special case of time complexity, this can be stated more simply as:
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| :for any total computable function <math>g \, : \, \omega \,\to\, \omega</math> such that <math>g(x) \geq x</math> for all <math>\,x</math>, there exists a time bound <math>T(n)</math> such that <math>DTIME(g(T(n))) = DTIME(T(n))</math>.
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| Because the bound ''T(n)'' may be very large (and often will be [[constructible function|nonconstructible]]) the Gap Theorem does not imply anything interesting for complexity classes such as P or NP, and it does not contradict [[time hierarchy theorem]] or [[space hierarchy theorem]].
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| == See also ==
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| *[[Blum's speedup theorem]]
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| == References ==
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| {{reflist}}
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| [[Category:Theorems in computational complexity theory]]
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