The study of branching programs for the Tree Evaluation Problem, introduced by S. Cook et al. (TOCT 2012), remains one of the most promising approaches to separating L from P. Given a label in $[k]$ at each leaf of a complete binary tree and an explicit function in $[k]^2 \to ... more >>>

We show that the Tree Evaluation Problem with alphabet size $k$ and height $h$ can be solved by branching programs of size $k^{O(h/\log h)} + 2^{O(h)}$. This answers a longstanding challenge of Cook et al. (2009) and gives the first general upper bound since the problem's inception.

more >>>

The Tree Evaluation Problem ($TreeEval$) (Cook et al. 2009) is a central candidate for separating polynomial time ($P$) from logarithmic space ($L$) via composition. While space lower bounds of $\Omega(\log^2 n)$ are known for multiple restricted models, it was recently shown by Cook and Mertz (2020) that TreeEval can be ... more >>>

The input to the Tree Evaluation problem is a binary tree of height $h$ in which each internal vertex is associated with a function mapping pairs of $\ell$-bit strings to $\ell$-bit strings, and each leaf is assigned an $\ell$-bit string.
The desired output is the value of the root, ... more >>>

The input to the Tree Evaluation problem is a binary tree of height $h$ in which each internal vertex is associated with a function mapping pairs of $\ell$-bit strings to $\ell$-bit strings,and each leaf is assigned an $\ell$-bit string.
The desired output is the value of the root, where ... more >>>

We show that for all functions $t(n) \geq n$, every multitape Turing machine running in time $t$ can be simulated in space only $O(\sqrt{t \log t})$. This is a substantial improvement over Hopcroft, Paul, and Valiant's simulation of time $t$ in $O(t/\log t)$ space from 50 years ago [FOCS 1975, ... more >>>

Williams (STOC 2025) recently proved that time-$t$ multitape Turing machines can be simulated using $O(\sqrt{t \log t})$ space using the Cook-Mertz (STOC 2024) tree evaluation procedure. As Williams notes, applying this result to fast algorithms for the circuit value problem implies an $O(\sqrt{s} \cdot \mathrm{polylog}\; s)$ space algorithm for evaluating ... more >>>