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Concept# Constructible function

Summary

In complexity theory, a time-constructible function is a function f from natural numbers to natural numbers with the property that f(n) can be constructed from n by a Turing machine in the time of order f(n). The purpose of such a definition is to exclude functions that do not provide an upper bound on the runtime of some Turing machine.
Time-constructible definitions
There are two different definitions of a time-constructible function. In the first definition, a function f is called time-constructible if there exists a positive integer n0 and Turing machine M which, given a string 1n consisting of n ones, stops after exactly f(n) steps for all n ≥ n0. In the second definition, a function f is called time-constructible if there exists a Turing machine M which, given a string 1n, outputs the binary representation of f(n) in O(f(n)) time (a unary representation may be used instead, since the two can be interconverted in O(f(n)) time).
There is also a notion of

Official source

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