MIX is a 2-MCFL and the word problem in Z is solved by a third-order collapsible pushdown automaton

In this work we establish that the language MIX = {w ∈ {a; b; c}||w|a = |w|b = |w|c} and the language O2 = {w ∈ {a; a; b; b}||w|a = |w|a ∧ |w|b = |w|b} are 2-MCFLs. As 2-MCFLs form a class of languages that is included in both the IO and OI hierarchies, and as O2 is the group language of a simple presentation of Z we exhibit here the first, to our knowledge, non-virtually-free group language (i.e. non-context-free group language) that is captured by the IO and OI hierarchies. Moreover, it was a long-standing open problem whether MIX was a mildly context sensitive language or not, and it was conjectured that it was not, so we close this conjecture by giving it a negative answer.