Time hierarchy theorem

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 timebounded complexity class, there is a strictly larger timebounded complexity class, and so the runtime hierarchy of problems does not completely collapse. One theorem deals with deterministic computations and the other with nondeterministic ones.
Both theorems use the notion of a timeconstructible function. A function f : N → N is timeconstructible 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 polynomials with nonnegative integral coefficients are timeconstructible, as are exponential functions such as 2^{n}.
Contents 
Deterministic time hierarchy theorem
Statement
If f(n) is a timeconstructible function, then there exists a decision problem which cannot be solved in worstcase deterministic time f(n) but can be solved in worstcase deterministic time f(n)^{2}. In other words, the complexity class TIME(f(n)) is a strict subset of TIME(f(n)^{2}).
Proof
We include here a proof that TIME(f(n)) is a strict subset of TIME(f(2n + 1)^{3}) as it is simpler. See the bottom of this section for information on how to extend the proof to f(n)^{2}.
To prove this, we first define a language as follows:
 <math> H_f = \left\{ ([M], x)\ \ M \ \mbox{accepts}\ x \ \mbox{in}\ f(x) \ \mbox{steps} \right\} <math>
Here, M is a deterministic Turing machine, and x is its input (the initial contents of its tape). [M] denotes an input that encodes the Turing machine M. Let m be the size of the tuple ([M], x).
We know that we can decide membership of H_{f} by way of a deterministic Turing machine that first calculates f(x), then writes out a row of 0s of that length, and then uses this row of 0s as a "clock" or "counter" to simulate M for at most that many steps. At each step, the simulating machine needs to look through the definition of M to decide what the next action would be. It is safe to say that this takes at most f(m)^{3} operations, so
 <math> H_f \in \mathsf{TIME}(f(m)^3) <math>
The rest of the proof will show that
 <math> H_f \notin \mathsf{TIME}(f( \left\lfloor m/2 \right\rfloor )) <math>
so that if we substitute 2n + 1 for m, we get the desired result. Let us assume that H_{f} is in this time complexity class, and we will attempt to reach a contradiction.
If H_{f} is in this time complexity class, it means we can construct some machine K which, given some machine description [M] and input x, decides whether the tuple ([M], x) is in H_{f} within <math> \mathsf{TIME}(f( \left\lfloor m/2 \right\rfloor )) <math>.
Therefore we can use this K to construct another machine, N, which takes a machine description [M] and runs K on the tuple ([M], [M]), and then accepts only if K rejects, and rejects if K accepts. If now n is the length of the input to N, then m (the length of the input to K) is twice n plus some delimiter symbol, so m = 2n + 1. N's running time is thus <math> \mathsf{TIME}(f( \left\lfloor m/2 \right\rfloor )) = \mathsf{TIME}(f( \left\lfloor (2n+1)/2 \right\rfloor )) = \mathsf{TIME}(f(n)) <math>.
Now if we feed [N] as input into N itself (which makes n the length of [N]) and ask the question whether N accepts its own description as input, we get:
 If N accepts [N] (which we know it does in at most f(n) operations), this means that K rejects ([N], [N]), so ([N], [N]) is not in H_{f}, and thus N does not accept [N] in f(n) steps. Contradiction!
 If N rejects [N] (which we know it does in at most f(n) operations), this means that K accepts ([N], [N]), so ([N], [N]) is in H_{f}, and thus N does accept [N] in f(n) steps. Contradiction!
We thus conclude that the machine K does not exist, and so
 <math> H_f \notin \mathsf{TIME}(f( \left\lfloor m/2 \right\rfloor )) <math>
Extension
The reader may have realised that the proof is simpler because we have chosen a simple Turing machine simulation for which we can be certain that
 <math> H_f \in \mathsf{TIME}(f(m)^3) <math>
It has been shown [1] (http://www.cs.berkeley.edu/~luca/cs172/noteh.pdf) that a more efficient model of simulation exists which establishes that
 <math> H_f \in \mathsf{TIME}(f(m) \log f(m)) <math>
but since this model of simulation is rather involved, it is not included here.
Nondeterministic time hierarchy theorem
If g(n) is a timeconstructible function, and f(n+1) = o(g(n)), then there exists a decision problem which cannot be solved in nondeterministic time f(n) but can be solved in nondeterministic time g(n). In other words, the complexity class NTIME(f(n)) is a strict subset of NTIME(g(n)).
Consequences
The time hierarchy theorems guarantee that the deterministic and nondeterministic version of the exponential hierarchy are genuine hierarchies: in other words P ⊂ EXPTIME ⊂ 2EXP ⊂ ..., and NP ⊂ NEXPTIME ⊂ 2NEXP ⊂ ...
However, the time hierarchy theorems provide no means to relate deterministic and nondeterministic complexity, or time and space complexity, so they cast no light on the great unsolved questions of complexity theory: whether P and NP, NP and PSPACE, PSPACE and EXPTIME, or EXPTIME and NEXPTIME are equal or not.