In
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 transducer (LST) is a type of
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 ...
used for
log-space reduction
In computational complexity theory, a log-space reduction is a reduction computable by a deterministic Turing machine using logarithmic space. Conceptually, this means it can keep a constant number of pointers into the input, along with a logari ...
s.
A log space transducer,
, has three tapes:
* A read-only ''input'' tape.
* A read/write ''work'' tape (bounded to at most
symbols).
* A write-only, write-once ''output'' tape.
will be designed to compute a
log-space computable function (where
is the
alphabet
An alphabet is a standardized set of basic written graphemes (called letters) that represent the phonemes of certain spoken languages. Not all writing systems represent language in this way; in a syllabary, each character represents a syll ...
of both the ''input'' and ''output'' tapes). If
is executed with
on its ''input'' tape, when the machine halts, it will have
remaining on its ''output'' tape.
A language
is said to be
log-space reducible to a language
if there exists a
log-space computable function that will convert an input from problem
into an input to problem
in such a way that
.
This seems like a rather convoluted idea, but it has two useful properties that are desirable for a reduction:
# The property of transitivity holds. (A reduces to B and B reduces to C implies A reduces to C).
# If A reduces to B, and B is in
L, then we know A is in
L.
Transitivity holds because it is possible to feed the output tape of one reducer (A→B) to another (B→C). At first glance, this seems incorrect because the A→C reducer needs to store the output tape from the A→B reducer onto the work tape in order to feed it into the B→C reducer, but this is not true. Each time the B→C reducer needs to access its input tape, the A→C reducer can re-run the A→B reducer, and so the output of the A→B reducer never needs to be stored entirely at once.
References
*Szepietowski, Andrzej (1994),
Turing Machines with Sublogarithmic Space ',
Springer Press
Axel Springer SE () is a German digital and popular periodical publishing house which is the largest in Europe, with numerous multimedia news brands, such as '' Bild'', ''Die Welt'', and ''Fakt'' and more than 15,000 employees. It generated to ...
, . Retrieved on 2008-12-03.
*
Sipser, Michael (2012),
Introduction to the Theory of Computation',
Cengage Learning
Cengage Group is an American educational content, technology, and services company for the higher education, K-12, professional, and library markets. It operates in more than 20 countries around the world.(Jun 27, 2014Global Publishing Leaders ...
, .
Computational complexity theory
Turing machine
{{comp-sci-theory-stub