A Knuth-Bendix-Like Ordering for Orienting Combinator Equations

@article{Bhayat2020AKO,
  title={A Knuth-Bendix-Like Ordering for Orienting Combinator Equations},
  author={Ahmed Bhayat and Giles Reger},
  journal={Automated Reasoning},
  year={2020},
  volume={12166},
  pages={259 - 277}
}
We extend the graceful higher-order basic Knuth-Bendix order (KBO) of Becker et al. to an ordering that orients combinator equations left-to-right. The resultant ordering is highly suited to parameterising the first-order superposition calculus when dealing with the theory of higher-order logic, as it prevents inferences between the combinator axioms. We prove a number of desirable properties about the ordering including it having the subterm property for ground terms, being transitive and… 
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References

SHOWING 1-10 OF 26 REFERENCES
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.
Paramodulation with non-monotonic orderings
TLDR
By a careful further analysis of the technique, this work obtains the first Knuth-Bendix completion procedure that finds a convergent TRS for a given set of equations E and a (possibly non-totalizable) reduction ordering p whenever it exists.
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.
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.
A Higher-Order Iterative Path Ordering
TLDR
An iterative version of HORPO is presented by means of an auxiliary term rewriting system, following an approach originally due to Bergstra and Klop, and well-foundedness of the iterative definition is studied.
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.
How to Prove Higher Order Theorems in First Order Logic
TLDR
This paper presents translations of higher order logics into first order logic with flat sorts and equality and gives a sufficient criterion for the soundness of these translations.
Extensional Paramodulation for Higher-Order Logic and Its Effective Implementation Leo-III
  • A. Steen
  • Computer Science
    KI - Künstliche Intelligenz
  • 2019
TLDR
In this dissertation, both the theoretical and the practical challenges of designing an effective higher-order reasoning system are studied and the resulting system, the automated theorem prover Leo-III, is one of the most effective and versatile systems, in terms of supported logical formalisms, to date.
A Combinator-Based Superposition Calculus for Higher-Order Logic
TLDR
A refutationally complete superposition calculus for a version of higher-order logic based on the combinatory calculus is presented and a novel method of dealing with extensionality is introduced.
Superposition with Lambdas
TLDR
A superposition calculus for a clausal fragment of extensional polymorphic higher-order logic that includes anonymous functions but excludes Booleans is designed and implemented in the Zipperposition prover and evaluated on TPTP and Isabelle benchmarks.
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