Log-space transducer explained

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} → \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:

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