  Otto's Theorem characterises the bisimulation-invariant PTIME queries over
graphs as exactly those that can be formulated in the polyadic mu-calculus,
hinging on the Immerman-Vardi Theorem which characterises PTIME (over ordered
structures) by First-Order Logic with least fixpoints. This connection has been
extended to characterise bisimulation-invariant EXPTIME by an extension of the
polyadic mu-calculus with functions on predicates, making use of Immerman's
characterisation of EXPTIME by Second-Order Logic with least fixpoints. In this
paper we show that the bisimulation-invariant versions of all classes in the
exponential time hierarchy have logical counterparts which arise as extensions
of the polyadic mu-calculus by higher-order functions. This makes use of the
characterisation of k-EXPTIME by Higher-Order Logic (of order k+1) with least
fixpoints, due to Freire and Martins.
