Complexity Hierarchies and Higher-order Cons-free Term Rewriting

TL;DR

This paper extends the understanding of complexity hierarchies by analyzing higher-order cons-free term rewriting systems, demonstrating their ability to characterize exponential time classes without restrictions on reduction strategies.

Contribution

It proves that non-orthogonal, higher-order cons-free term rewriting systems characterize exponential time classes, broadening prior results limited to first-order or orthogonal systems.

Findings

Higher-order cons-free TRSs characterize E^KTIME for all K ≥ 1.

Results hold for non-orthogonal TRSs with no fixed reduction strategy.

Non-determinism and syntactic features affect complexity class characterization.

Abstract

Constructor rewriting systems are said to be cons-free if, roughly, constructor terms in the right-hand sides of rules are subterms of the left-hand sides; the computational intuition is that rules cannot build new data structures. In programming language research, cons-free languages have been used to characterize hierarchies of computational complexity classes; in term rewriting, cons-free first-order TRSs have been used to characterize the class PTIME. We investigate cons-free higher-order term rewriting systems, the complexity classes they characterize, and how these depend on the type order of the systems. We prove that, for every K $\geq$ 1, left-linear cons-free systems with type order K characterize E$^K$TIME if unrestricted 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 no assumptions on reduction strategy, (ii) we consequently obtain much larger classes for each type order (E$^K$TIME versus EXP$^{K-1}$TIME), and (iii) results for cons-free term rewriting systems have previously only been obtained for K = 1, and with additional syntactic restrictions besides cons-freeness and left-linearity. Our results are among the first implicit characterizations of the hierarchy E = E$^1$TIME $\subsetneq$ E$^2$TIME $\subsetneq$ ... Our work confirms prior results that having full non-determinism (via overlapping 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 like admitting product types or non-left-linear rules.