Log space transducer

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

A log space transducer, M , has three tapes:

M will be designed to compute a log-space computable function f : Σ ∗ → Σ ∗ (where Σ 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 ⊆ Σ ∗ is said to be log-space reducible to a language B ⊆ Σ ∗ if there exists a log-space computable function, f , which will convert an input from problem A into an input to problem B . I.E. w ∈ A ⟺ f ( w ) ∈ 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.