A Higher-Order Iterative Path Ordering

@inproceedings{Kop2008AHI,
  title={A Higher-Order Iterative Path Ordering},
  author={Cynthia Kop and Femke van Raamsdonk},
  booktitle={LPAR},
  year={2008}
}
The higher-order recursive path ordering ( HORPO ) defined by Jouannaud and Rubio provides a method to prove termination of higher-order rewriting. We present an iterative version of HORPO by means of an auxiliary term rewriting system, following an approach originally due to Bergstra and Klop. We study well-foundedness of the iterative definition, discuss its relationship with the original HORPO, and point out possible ways to strengthen the ordering. 
View via Publisher
research.vu.nl
7 Citations
Highly Influential Citations
2
Background Citations
2
Methods Citations
2

7 Citations

Citation Type

Citation Type

A Lambda-Free Higher-Order Recursive Path Order
TLDR
This new order fully coincides with the standard RPO on first-order terms also in the presence of currying, distinguishing it from previous work and appears promising as the basis of a higher-order superposition calculus.
Formalization of the Embedding Path Order for Lambda-Free Higher-Order Terms
TLDR
The embedding path order is a variant of the recursive path order for untyped λ-free higher-order terms that is a groundtotal and well-founded simplification order, making it more suitable for the superposition calculus.
A Transfinite Knuth-Bendix Order for Lambda-Free Higher-Order Terms
TLDR
The Knuth–Bendix order is generalized to higher-order terms without \(\lambda \)-abstraction and appears promising as the basis of a higher- order superposition calculus.
WANDA – a Higher Order Termination Tool
TLDR
The methodology used in Wanda is discussed, which includes a higher-order dependency pair framework and a variation of the higher- order recursive path ordering, as well as some non-termination analysis techniques and delegation to a first-order tool.
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…
A Knuth-Bendix-Like Ordering for Orienting Combinator Equations
TLDR
A number of desirable properties about the KBO are proved including it having the subterm property for ground terms, being transitive and being well-founded, and the ordering fails to be a reduction ordering as it lacks compatibility with certain contexts.
Star Games and Hydras
TLDR
This work revisits its presentation by means of some simple rules on trees equipped with a `star' as control symbol, signifying a command to make that tree (or term) smaller in the order being defined, and obtains a very direct proof of termination of the famous Hydra battle.

References

SHOWING 1-10 OF 23 REFERENCES
Higher-Order Orderings for Normal Rewriting
TLDR
The termination proof methods based on reduction orderings are extended to higher-order rewriting systems a la Nipkow using higher- order pattern matching for firing rules, and any use of eta, as a reduction, as an expansion or as an equation is accommodate.
Coq formalization of the higher-order recursive path ordering
  • A. Koprowski
  • Computer Science
    Applicable Algebra in Engineering, Communication and Computing
  • 2009
TLDR
This paper describes the undertaking of providing a complete, axiom-free, fully constructive formalization of results in the proof assistant Coq of the higher-order recursive path ordering, HORPO with a proof of its well-foundedness.
The Computability Path Ordering: The End of a Quest
TLDR
This paper focuses on the higher-order recursive path ordering, for which an improved definition is provided, the Computability Path Ordering, which appears to capture the essence of computability arguments a la Tait and Girard.
The higher-order recursive path ordering
  • J. Jouannaud, A. Rubio
  • Mathematics
    Proceedings. 14th Symposium on Logic in Computer Science (Cat. No. PR00158)
  • 1999
This paper extends the termination proof techniques based on reduction orderings to a higher-order setting, by adapting the recursive path ordering definition to terms of a typed lambda-calculus…
Polymorphic higher-order recursive path orderings
TLDR
A family of recursive path orderings for terms of a typed lambda-calculus generated by a signature of polymorphic higher-order function symbols is defined, which can be generated from two given well-founded orderings, on the function symbols and on the type constructors.
Iterative Lexicographic Path Orders
TLDR
A presentation, which will refer to as iterative lexicographic path orders (ILPO), based on Bergstra and Klop’s definition of recursive path orders by way of an auxiliary term rewriting sytem is presented.
On Termination of Higher-Order Rewriting
We discuss the termination methods using the higher-order recursive path ordering and the general scheme for higher-order rewriting systems and combinatory reduction systems.
Higher-order critical pairs
  • T. Nipkow
  • Computer Science
    [1991] Proceedings Sixth Annual IEEE Symposium on Logic in Computer Science
  • 1991
TLDR
The notion of critical pair is generalized to higher-order rewrite systems, and the analog of the critical pair lemma is proved.
Orderings for Term-Rewriting Systems
Methods of proving that a term-rewriting system terminates are presented. They are based on the notion of "simplification orderings", orderings in which any term that is homeomorphically embeddable…
Horizontal Composability Revisited
TLDR
A view of implementation is given which is believed to provide a more adequate reection of the rather subtle interplay between implementation, specication structure and program structure.
...
...