Log-space transducer

A log space transducer (LST) is a type of Turing machine used for log-space reductions.

A log space transducer, ${\displaystyle M}$, has three tapes:

• A read-only input tape.
• A read/write work tape (bounded to at most ${\displaystyle O(\log n)}$ symbols).
• A write-only, write-once output tape.

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

A language ${\displaystyle A\subseteq \Sigma ^{\ast }}$ is said to be log-space reducible to a language ${\displaystyle B\subseteq \Sigma ^{\ast }}$ if there exists a log-space computable function, ${\displaystyle f}$, which will convert an input from problem ${\displaystyle A}$ into an input to problem ${\displaystyle B}$. I.E. ${\displaystyle w\in A\iff f(w)\in B}$.

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

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.