Editing Complexity class (section)

===Hierarchy theorems===
{{main article|Time hierarchy theorem|Space hierarchy theorem}}
By definition of '''DTIME''', it follows that <math>\mathsf{DTIME}(n^{k_1})</math> is contained in <math>\mathsf{DTIME}(n^{k_2})</math> if <math>k_1 \leq k_2 </math>, since <math>O(n^{k_1}) \subseteq O(n^{k_2})</math> if <math>k_1 \leq k_2</math>. However, this definition gives no indication of whether this inclusion is strict. For time and space requirements, the conditions under which the inclusion is strict are given by the time and space hierarchy theorems, respectively. They are called hierarchy theorems because they induce a proper hierarchy on the classes defined by constraining the respective resources. The hierarchy theorems enable one to make quantitative statements about how much more additional time or space is needed in order to increase the number of problems that can be solved.

The [[time hierarchy theorem]] states that
:<math>\mathsf{DTIME}\big(f(n) \big) \subsetneq \mathsf{DTIME} \big(f(n) \sdot \log^{2}(f(n)) \big)</math>.

The [[space hierarchy theorem]] states that
:<math>\mathsf{DSPACE}\big(f(n)\big) \subsetneq \mathsf{DSPACE} \big(f(n) \sdot \log(f(n)) \big)</math>.

The time and space hierarchy theorems form the basis for most separation results of complexity classes. For instance, the time hierarchy theorem establishes that '''P''' is strictly contained in '''EXPTIME''', and the space hierarchy theorem establishes that '''L''' is strictly contained in '''PSPACE'''.