STOC24 7 C 2 Tree Evaluation is in Space O(log n log log n)

A brief introduction to space lower bounds via composition, and how they fail.


ABSTRACT
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 Ω(log^2 n) are known for multiple restricted models, it was recently shown by Cook and Mertz (2020) that TreeEval can be solved in space O(log^2 n/log log n). Thus its status as a candidate hard problem for L remains a mystery.

Our main result is to improve the space complexity of TreeEval to O(log n · log log n), thus greatly strengthening the case that Tree Evaluation is in fact in L.

We show two consequences of these results. First, we show that the KRW conjecture (Karchmer, Raz, and Wigderson 1995) implies L ⊈ NC1; this itself would have many implications, such as branching programs not being efficiently simulable by formulas. Our second consequence is to increase our understanding of amortized branching programs, also known as catalytic branching programs; we show that every function f on n bits can be computed by such a program of length poly(n) and width 2^O(n).


CHAPTERS
00:00 Introduction: Space & Time
02:02 I: Tree Evaluation
07:22 II: Reusing Space
18:50 III: The Full Proof
22:51 IV: What Now?