Proper complexity function

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A proper complexity function is a function f mapping a natural number to a natural number such that:

  • f is nondecreasing;
  • there exists a k-string Turing machine M such that on any input of length n, M halts after O(n + f(n)) steps, uses O(f(n)) space, and outputs f(n) consecutive blanks.

If f and g are two proper complexity functions, then f + g, fg, and 2f, are also proper complexity functions.

Similar notions include honest function, space-constructible function, and time-constructible function.



  1. ^ Alexei Myasnikov, Vladimir Shpilrain, Alexander Ushakov. Group-based Cryptography. Birkhäuser Verlag, 2008, p.28