Time hierarchy theorem: Difference between revisions

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In [[computational complexity theory]], the '''time hierarchy theorems''' are important statements that ensure the existence of certain "hard" problems which cannot be solved in a given amount of time. As a consequence, for every time-bounded [[complexity class]], there is a strictly larger time-bounded complexity class, and so the run-time hierarchy of problems does not completely collapse. One theorem deals with deterministic computations and the other with non-deterministic ones. The equivalentanalogous theorem for space is the [[space hierarchy theorem]].

Both theorems use the notion of a [[constructible function|time-constructible function]]. A [[function (mathematics)|function]] ''f'' : '''N''' → '''N''' is time-constructible if there exists a deterministic [[Turing machine]] such that for every ''n'' in '''N''', if the machine is started with an input of ''n'' ones, it will halt after precisely ''f''(''n'') steps. All [[polynomial]]s with non-negative integral coefficients are time-constructible, as are exponential functions such as 2^''n''.