Maximal automatic complexity and context-free languages

Let A N denote nondeterministic automatic complexity and L k,c = { x ∈ [ k ] ∗ : A N ( x ) > | x | /c } . In particular, L k, 2 is the language of all k -ary words for which A N is maximal, while L k, 3 gives a rough dividing line between complex and simple. Let CFL denote the complexity class consisting of all context-free languages. While it is not known that L 2 , 2 is infinite, Kjos-Hanssen (2017) showed that L 3 , 2 is CFL -immune but not coCFL -immune. We complete the picture by showing that L 3 , 2 6∈ coCFL . Turning to Boolean circuit complexity, we show that L 2 , 3 is SAC 0 immune and SAC 0 -coimmune. Here SAC 0 denotes the complexity class consisting of all languages computed by (non-uniform) constant-depth circuits with semi-unbounded fanin. As for arithmetic circuits, we show that { x : A N ( x ) > 1 } 6∈ ⊕ SAC 0 . In particular, SAC 0 6⊆ ⊕ SAC 0 , which resolves an open implication from the Complexity Zoo.