Discrete CoT step complexity for reachability with constant-depth transformers

Determine whether a constant-depth transformer using discrete chain-of-thought can solve directed graph reachability on a graph with n vertices with strictly fewer than O(n^2) discrete CoT decoding steps, and, if so, quantify the minimal number of steps required.

Background

Directed graph reachability is a fundamental reasoning problem closely linked to various practical tasks. Prior work shows that a constant-depth transformer equipped with discrete CoT can solve reachability but currently requires O(n^2) steps, where n is the number of vertices.

This paper demonstrates that continuous CoT can solve reachability in D steps, where D is the graph diameter, highlighting a potential separation. The unresolved question is whether discrete CoT can do better than O(n^2) steps in constant depth.