Polynomial-time computability of the n-th tent code bit

Determine whether there exists an algorithm that, for a fixed rational μ ∈ Q and a rational initial value x = p/q, computes the n-th bit b_n of the tent code ^n_μ(x) in time polynomial in the input size log p + log q + log n; equivalently, establish the time complexity class of predicting whether f_μ^n(x) < 1/2 from rational inputs.

Background

The paper resolves space complexity aspects for recognizing valid tent codes under smoothed analysis but leaves the time complexity of computing individual bits of the tent code open. This task amounts to predicting the n-th iterate’s relation to a threshold for rational inputs.

The authors raise the explicit question of polynomial-time solvability with respect to the natural input size and suggest the problem might be NP-hard; however, they report no known result. Pinpointing the time complexity would bridge algorithmic practice and theory for chaotic sequence prediction.