Extend the square-root–log space simulation to arbitrary random-access models

Extend the simulation TIME[t(n)] ⊆ SPACE[√(t(n) log t(n)] for multitape Turing machines to arbitrary random-access machine models (with non-oblivious access patterns), by developing a method that overcomes the high indegree of the corresponding computation graph so it can be stored and navigated within o(t(n)) space.

Background

The main theorem shows that any time-t multitape Turing machine computation can be simulated in O(√(t log t)) space via a reduction to Tree Evaluation and the Cook–Mertz algorithm. This relies critically on low-indegree computation graphs arising from sequential tape access.

Random-access models (e.g., RAM) can have very high indegree in their computation graphs, making the current approach unsuitable. The authors note an extension is possible for oblivious random-access models (where access patterns can be computed) but state that a general extension is unknown.