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Time hierarchy theorem
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===Statement=== <blockquote>'''Time Hierarchy Theorem.''' If ''f''(''n'') is a time-constructible function, then there exists a [[decision problem]] which cannot be solved in worst-case deterministic time ''o''(''f''(''n'')) but can be solved in worst-case deterministic time ''O''(''f''(''n'')log ''f''(''n'')). Thus :<math>\mathsf{DTIME}(o(f(n))) \subsetneq \mathsf{DTIME}\left (f(n)\log f(n) \right).</math> Equivalently, if <math>f, g</math> are time-constructable, and <math>f(n) \ln f(n) = o(g(n))</math>, then <math display="block">\mathsf{DTIME}(f(n)) \subsetneq \mathsf{DTIME} (g(n))</math></blockquote> '''Note 1.''' ''f''(''n'') is at least ''n'', since smaller functions are never time-constructible.<br> '''Example.''' <math>\mathsf{DTIME}(n) \subsetneq \mathsf{DTIME} (n (\ln n)^2) </math>.
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