Skip to main content

gap theorem

gap theorem A theorem in complexity theory that, like the speedup theorem, can be expressed in terms of abstract complexity measures (see Blum's axioms) but will be more understandable in the context of time:

given any total recursive function g(n) ≥ n

there exists a total recursive function S(n) such that DTIME(S(n)) = DTIME(g(S(n)))

(see complexity classes). In other words there is a “gap” between time bounds S(n) and g(S(n)) within which the minimal space complexity of no language lies.

This has the following counter-intuitive consequence: given two universal models of computation, say a Turing machine that makes one move per century and the other a random-access machine capable of performing a million arithmetic operations per second, then there is a total recursive function S(n) such that any language recognizable in time S(n) on one machine is also recognizable within time S(n) on the other.

Cite this article
Pick a style below, and copy the text for your bibliography.

  • MLA
  • Chicago
  • APA

"gap theorem." A Dictionary of Computing. . 21 Mar. 2019 <>.

"gap theorem." A Dictionary of Computing. . (March 21, 2019).

"gap theorem." A Dictionary of Computing. . Retrieved March 21, 2019 from

Learn more about citation styles

Citation styles gives you the ability to cite reference entries and articles according to common styles from the Modern Language Association (MLA), The Chicago Manual of Style, and the American Psychological Association (APA).

Within the “Cite this article” tool, pick a style to see how all available information looks when formatted according to that style. Then, copy and paste the text into your bibliography or works cited list.

Because each style has its own formatting nuances that evolve over time and not all information is available for every reference entry or article, cannot guarantee each citation it generates. Therefore, it’s best to use citations as a starting point before checking the style against your school or publication’s requirements and the most-recent information available at these sites:

Modern Language Association

The Chicago Manual of Style

American Psychological Association

  • Most online reference entries and articles do not have page numbers. Therefore, that information is unavailable for most content. However, the date of retrieval is often important. Refer to each style’s convention regarding the best way to format page numbers and retrieval dates.
  • In addition to the MLA, Chicago, and APA styles, your school, university, publication, or institution may have its own requirements for citations. Therefore, be sure to refer to those guidelines when editing your bibliography or works cited list.