2 research outputs found
Complexity Hierarchies and Higher-order Cons-free Term Rewriting
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 1, left-linear cons-free systems
with type order K characterize ETIME 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 (ETIME versus EXPTIME), 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
= ETIME ETIME ... 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.Comment: extended version of a paper submitted to FSCD 2016. arXiv admin note:
substantial text overlap with arXiv:1604.0893
Complexity Hierarchies and Higher-order Cons-free Term Rewriting
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 1, left-linear cons-free systems
with type order K characterize ETIME 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 (ETIME versus EXPTIME), 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
= ETIME ETIME ... 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