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 $O(\log n)$ symbols).
* A write-only, write-once ''output'' tape.

$M$ will be designed to compute a log-space computable function $f\colon \Sigma^\ast \rightarrow \Sigma^\ast$ (where $\Sigma$ is the alphabet of both the ''input'' and ''output'' tapes). If $M$ is executed with $w$ on its ''input'' tape, when the machine halts, it will have $f(w)$ remaining on its ''output'' tape.

A language $A \subseteq \Sigma^\ast$ is said to be log-space reducible to a language $B \subseteq \Sigma^\ast$ if there exists a log-space computable function $f$ that will convert an input from problem $A$ into an input to problem $B$ in such a way that $w \in A \iff f(w) \in B$.

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, . Retrieved on 2008-12-03.
*Sipser, Michael (2012),
Introduction to the Theory of Computation
', Cengage Learning, .

Computational complexity theory
Turing machine

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