Provable better quasi orders

It has recently been shown that fairly strong axiom systems such as ${\mathsf{A}\mathsf{C}\mathsf{A}}_0$ cannot prove that the antichain with three elements is a better quasi order ($\mathsf{b}\mathsf{q}\mathsf{o}$). In the present paper, we give a complete characterization of the finite partial orders that are provably $\mathsf{b}\mathsf{q}\mathsf{o}$ in such axiom systems. The result will also be extended to infinite orders. As an application, we derive that a version of the minimal bad array lemma is weak over $\mathsf{A}\mathsf{C}{\mathsf{A}}_{\mathsf{0}}$. In sharp contrast, a recent result shows that the same version is equivalent to ${\mathrm{\Pi }}_2^1$-comprehension over the stronger base theory ${\mathsf{A}\mathsf{T}\mathsf{R}}_0$.