  Constructor rewriting systems are said to be cons-free if, roughly,
constructor terms in the right-hand sides of rules are subterms of constructor
terms in the left-hand side; the computational intuition is that rules cannot
build new data structures. It is well-known that cons-free programming
languages can be used to characterize computational complexity classes, and
that cons-free first-order term rewriting can be used to characterize the set
of polynomial-time decidable sets.
  We investigate cons-free higher-order term rewriting systems, the complexity
classes they characterize, and how these depend on the order of the types used
in the systems. We prove that, for every k $\geq$ 1, left-linear cons-free
systems with type order k characterize E$^k$TIME if arbitrary evaluation is
used (i.e., the system does not have a fixed reduction strategy).
  The main difference with prior work in implicit complexity is that (i) our
results hold for non-orthogonal term rewriting systems with possible rule
overlaps with no assumptions about reduction strategy, (ii) results for such
term rewriting systems have previously only been obtained for k = 1, and with
additional syntactic restrictions on top of cons-freeness and left-linearity.
  Our results are apparently among the first implicit characterizations of the
hierarchy E = E$^1$TIME $\subseteq$ E$^2$TIME $\subseteq$ .... Our work
confirms prior results that having full non-determinism (via overlaps of rules)
does not directly allow characterization of non-deterministic complexity
classes like NE. We also show that non-determinism makes the classes
characterized highly sensitive to minor syntactic changes such as admitting
product types or non-left-linear rules.
