Logspace many-one reduction

For languages $A,B\subseteq {\mathrm{\Sigma }}^{\ast }$, write $A{\le }_{\log }B$ if a deterministic read-only-input logspace transducer computes a total function $f$ with $x\in A⟺f(x)\in B\mathrm{.}$ The transducer writes $f(x)$ once on a write-only output stream. Equivalently, there is a fixed polynomial $p$ with $\mathrm{\mid }f(x)\mathrm{\mid }\le p(\mathrm{\mid }x\mathrm{\mid })$ and a deterministic procedure which, on read-only input $(x,j)$ with $j$ in binary, uses $O(\log (\mathrm{\mid }x\mathrm{\mid }+2))$ work cells for every $j$. It returns $f(x{)}_j$ when $0\le j<\mathrm{\mid }f(x)\mathrm{\mid }$ and a distinct end-of-output marker exactly when $j\ge \mathrm{\mid }f(x)\mathrm{\mid }$; thus the first marker occurs at position $\mathrm{\mid }f(x)\mathrm{\mid }$. A streaming transducer can count through positions up to $p(\mathrm{\mid }x\mathrm{\mid })$, emitting bits and stopping at the first marker. Conversely, a streaming transducer can be rerun while counting output bits up to a requested position; a binary position larger than the polynomial bound can be recognized by a logspace comparison and answered with the marker. The stream's output length is polynomially bounded: since the write-only output cannot affect its future behavior and it has only polynomially many internal configurations, repeating one before halting would force an infinite run.