Complexity Hierarchies and Higher-Order Cons-Free Rewriting

Abstract

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 >=
1, left-linear cons-free systems with type order k characterize
E^kTIME 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^1TIME != E^2TIME != .... Our work confirms prior
results that having full non-determinism (via overlaps of rules) does
not directly allow for 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.