Logspace composition by recomputation ($f,g\in \mathrm{FL}⟹g\circ f\in \mathrm{FL}$)

To simulate $g$ on the virtual input $f(x)$, store the simulated input-head position and rerun the logspace transducer for $f$ whenever an input symbol is requested. Count emitted symbols until the requested position is reached. The intermediate output has polynomial length, so its positions take $O(\log n)$ bits, and the work space of $g$ is also $O(\log n)$. Storage used by successive recomputations is reused. This proves closure of logspace-computable functions under composition and transitivity of logspace many-one reductions.