# Logspace Transducer

I know that a logspace transducer is a deterministic Turing machine that enables us to use log-space complexity. I do not understand though why that is correct. Whatever algorithms can be implemented in that machine are automatically of log-space complexity? Why is that?

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The definition of "log-space complexity" is that the problem can be solved by a deterministic Turing machine that uses logarithmic space; the definition of a log-space transducer is that it's a Turing machine that uses logarithmic space. –  David Richerby Feb 23 '14 at 21:05

The composition property of logspace transducers is important for the following theorem: if $L$ is P-complete and $L$ reduces to $M$, then $M$ is P-complete. If you follow the proof carefully, you will see that it is crucial that the composition of two logspace transducers is another logspace transducer.