computational complexity theory In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved by ...

, a log-space reduction is a reduction computable by a

deterministic Turing machine A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algori ...

using

logarithmic space In computational complexity theory, L (also known as LSPACE or DLOGSPACE) is the complexity class containing decision problems that can be solved by a deterministic Turing machine using a logarithmic amount of writable memory space., Definition& ...

. Conceptually, this means it can keep a constant number of pointers into the input, along with a logarithmic number of fixed-size

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

s.Arora & Barak (2009) p. 88 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 In computational complexity theory, a log space transducer (LST) is a type of Turing machine used for log-space reductions. A log space transducer, M, has three tapes: * A read-only ''input'' tape. * A read/write ''work'' tape (bounded to at most O ...

. Such a machine has polynomially-many configurations, so log-space reductions are also

polynomial-time reduction In computational complexity theory, a polynomial-time reduction is a method for solving one problem using another. One shows that if a hypothetical subroutine solving the second problem exists, then the first problem can be solved by transforming ...

s. However, log-space reductions are probably weaker than polynomial-time reductions; while any non-empty, non-full language in P is polynomial-time reducible to any other non-empty, non-full language in P, a log-space reduction from an NL-complete language to a language in L, both of which would be languages in P, would imply the unlikely L = NL. It is an open question if the

NP-complete In computational complexity theory, a problem is NP-complete when: # it is a problem for which the correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by tryi ...

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 In computability theory and computational complexity theory, a many-one reduction (also called mapping reduction) is a reduction which converts instances of one decision problem L_1 into instances of a second decision problem L_2 where the instan ...

s or

Turing reduction In computability theory, a Turing reduction from a decision problem A to a decision problem B is an oracle machine which decides problem A given an oracle for B (Rogers 1967, Soare 1987). It can be understood as an algorithm that could be used to s ...

s are used, since it has been verified that L, SL, NL, and P are all closed under Turing reductions, 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 may not be closed under Turing reductions, and so many-one reductions must be used. 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 Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...

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 for a list of some SL-complete problems that can now be used as subroutines in log-space reductions.

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