Log-space transducer

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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, which will convert an input from problem A into an input to problem B. I.E. 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.