A Log-Sensitive Encoding of Turing Machines in the $\lambda$-Calculus

This note modifies the reference encoding of Turing machines in the $\lambda$-calculus by Dal Lago and Accattoli, which is tuned for time efficiency, as to accommodate logarithmic space. There are two main changes: Turing machines now have two tapes, an input tape and a work tape, and the input tape is encoded differently, because the reference encoding comes with a linear space overhead for managing tapes, which is excessive for studying logarithmic space.