Editing Time hierarchy theorem (section)

==Sharper hierarchy theorems==
The gap of approximately <math>\log f(n)</math> between the lower and upper time bound in the hierarchy theorem can be traced to the efficiency of the device used in the proof, namely a universal program that maintains a step-count. This can be done more efficiently on certain computational models. The sharpest results, presented below, have been proved for:
* The unit-cost [[random-access machine]]<ref>{{cite journal |last1=Sudborough |first1=Ivan H. |last2=Zalcberg |first2=A. |title=On Families of Languages Defined by Time-Bounded Random Access Machines |journal=SIAM Journal on Computing |date=1976 |volume=5 |issue=2 |pages=217–230 |doi=10.1137/0205018}}</ref>
* A [[programming language]] model whose programs operate on a binary tree that is always accessed via its root. This model, introduced by [[Neil D. Jones]]<ref>{{cite journal |last1=Jones |first1=Neil D. |title=Constant factors ''do'' matter |journal=25th Symposium on the Theory of Computing |date=1993 |pages=602–611 |doi=10.1145/167088.167244|s2cid=7527905 }}</ref> is stronger than a deterministic Turing machine but weaker than a random-access machine.
For these models, the theorem has the following form:
<blockquote>If ''f''(''n'') is a time-constructible function, then there exists a decision problem which cannot be solved in worst-case deterministic time ''f''(''n'') but can be solved in worst-case time ''af''(''n'') for some constant ''a'' (dependent on ''f'').</blockquote>
Thus, a constant-factor increase in the time bound allows for solving more problems, in contrast with the situation for Turing machines (see [[Linear speedup theorem]]). Moreover, Ben-Amram proved<ref>{{cite journal |last1=Ben-Amram |first1=Amir M. |title=Tighter constant-factor time hierarchies |journal=Information Processing Letters |date=2003 |volume=87 |issue=1 |pages=39–44|doi=10.1016/S0020-0190(03)00253-9 }}</ref> that, in the above models, for ''f'' of polynomial growth rate (but more than linear), it is the case that for all <math>\varepsilon > 0</math>, there exists a decision problem which cannot be solved in worst-case deterministic time ''f''(''n'') but can be solved in worst-case time <math>(1+\varepsilon)f(n)</math>.