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((Short description|Type of computational algorithm))
((Refimprove|date=April 2009))
In [[computational complexity theory]], a '''log-space reduction''' is a [[reduction (complexity)|reduction]] computable by a [[deterministic Turing machine]] using [[logarithmic space]]. Conceptually, this means it can keep a constant number of [[Pointer (computer programming)|pointer]]s into the input, along with a logarithmic number of fixed-size [[integer]]s.<ref name=AB88>Arora & Barak (2009) p. 88</ref>  It is possible that such a machine may not have space to write down its own output, so the only requirement is that any given bit of the output be computable in log-space. Formally, this reduction is executed via a [[log-space transducer]].

Such a machine has polynomially-many configurations, so log-space reductions are also [[polynomial-time reduction]]s.  However, log-space reductions are probably weaker than polynomial-time reductions; while any non-empty, non-full language in [[P (complexity)|P]] is polynomial-time reducible to any other non-empty, non-full language in P, a log-space reduction from an [[NL (complexity)|NL]]-complete language to a language in [[L (complexity)|L]], both of which would be languages in P, would imply the unlikely L = NL. It is an open question if the [[NP-complete]] problems are different with respect to log-space and polynomial-time reductions.

Log-space reductions are normally used on languages in P, in which case it usually does not matter whether [[many-one reduction]]s or [[Turing reduction]]s are used, since it has been verified that L, [[SL (complexity)|SL]], NL, and P are all closed under Turing reductions((citation needed|date=April 2014)), meaning that Turing reductions can be used to show a problem is in any of these classes. However, other subclasses of P such as [[NC (complexity)|NC]] may not be closed under Turing reductions, and so many-one reductions must be used((citation needed|date=April 2014)).

Just as polynomial-time reductions are useless within P and its subclasses, log-space reductions are useless to distinguish problems in L and its subclasses; in particular, every non-empty, non-full problem in L is trivially L-[[Complete (complexity)|complete]] under log-space reductions. While even weaker reductions exist, they are not often used in practice, because complexity classes smaller than L (that is, strictly contained or thought to be strictly contained in L) receive relatively little attention.

The tools available to designers of log-space reductions have been greatly expanded by the result that L = SL; see [[SL (complexity)|SL]] for a list of some SL-complete problems that can now be used as subroutines in log-space reductions.

==Notes==
((reflist))

==References==
* ((cite book | zbl=1193.68112 | last1=Arora | first1=Sanjeev | authorlink1=Sanjeev Arora | last2=Barak | first2=Boaz |author2link = Boaz Barak| title=Computational complexity. A modern approach | publisher=[[Cambridge University Press]] | year=2009 | isbn=978-0-521-42426-4 ))

==Further reading==
* ((cite book|authorlink=Christos Papadimitriou | first=Christos | last=Papadimitriou | year=1994  | title=Computational Complexity |url=https://archive.org/details/computationalcom00papa |url-access=limited | publisher=Addison Wesley | edition=1st | isbn=0-201-53082-1 | chapter=Chapter 8: Reductions And Completeness | pages=[https://archive.org/details/computationalcom00papa/page/n167 159]–180 | zbl=0833.68049 ))

((DEFAULTSORT:Log-Space Reduction))
[[Category:Reduction (complexity)]]


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