Is TreeEval in L?

Determine whether the Tree Evaluation Problem (binary arity d=2, height h=log n, leaf value length ℓ=log n) belongs to the deterministic logarithmic-space complexity class L; equivalently, establish whether there exists a deterministic algorithm that computes the root value of such TreeEval instances using O(log n) space.

Background

The Tree Evaluation Problem (TreeEval) was proposed as a candidate for separating P from L. For the canonical parameterization d=2 and h=ℓ=log n, natural pebbling-based algorithms use roughly Ω(log2 n) space.

Cook and Mertz (2024) showed TreeEval can be solved in O(log n·log log n) space but with super-polynomial time, and Goldreich provided a slight improvement. These results suggest TreeEval could be close to L, but do not establish membership.

This paper introduces a polynomial-time algorithm that uses O(log n) free space plus O(log{1+ε} n) catalytic space, thereby achieving almost log-space in a more restrictive model but still falling short of establishing TreeEval ∈ L in the standard sense. The question of whether TreeEval lies in L remains a central conjecture.

References

One way of viewing our results is as an approach that is complementary to that of Cook-Mertz for analyzing the conjecture that $TreeEval\in\LClass$.

— Polynomial-Time Almost Log-Space Tree Evaluation by Catalytic Pebbling  (2604.02606 - Asadi et al., 3 Apr 2026) in Section 1: Introduction and statement of results