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 $f:{\mathrm{\Sigma }}^{\ast }\to {\mathrm{\Sigma }}^{\ast }$ (where $\mathrm{\Sigma }$ 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 $A\subseteq {\mathrm{\Sigma }}^{\ast }$ is said to be log-space reducible to a language $B\subseteq {\mathrm{\Sigma }}^{\ast }$ 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 $w\in A⟺f(w)\in B$.

This seems like a rather convoluted idea, but it has two useful properties that are desirable for a reduction:

1. The property of transitivity holds. (A reduces to B and B reduces to C implies A reduces to C).
2. 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, ISBN 3-540-58355-6. Retrieved on 2008-12-03.
• Sipser, Michael (2012), Introduction to the Theory of Computation, Cengage Learning, ISBN 978-0-619-21764-8.