Polynomial Interpretations for Higher-Order Rewriting

The termination method of weakly monotonic algebras, which has been defined for higher-order rewriting in the HRS formalism, offers a lot of power, but has seen little use in recent years. We adapt and extend this method to the alternative formalism of algebraic functional systems, where the simply-typed lambda-calculus is combined with algebraic reduction. Using this theory, we define higher-order polynomial interpretations, and show how the implementation challenges of this technique can be tackled. A full implementation is provided in the termination tool WANDA

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 $\geq$ 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.Comment: extended version of a paper submitted to FSCD 2016. arXiv admin note: substantial text overlap with arXiv:1604.0893

Dynamic Dependency Pairs for Algebraic Functional Systems

We extend the higher-order termination method of dynamic dependency pairs to Algebraic Functional Systems (AFSs). In this setting, simply typed lambda-terms with algebraic reduction and separate {\beta}-steps are considered. For left-linear AFSs, the method is shown to be complete. For so-called local AFSs we define a variation of usable rules and an extension of argument filterings. All these techniques have been implemented in the higher-order termination tool WANDA

Complexity Hierarchies and Higher-Order Cons-Free Rewriting

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.Comment: Extended version (with appendices) of a paper published in FSCD 201