• Corpus ID: 2923361

Implementation of Lambda-Free Higher-Order Superposition

@inproceedings{Vukmirovi2018ImplementationOL,
  title={Implementation of Lambda-Free Higher-Order Superposition},
  author={Petar Vukmirovi{\'c} and Stephan Schulz},
  year={2018}
}
In the last decades, first-order logic (FOL) has become a standard language for describing a large number of mathematical theories. Numerous proof systems for FOL which determine what formulas are universally true emerged over time. On the other hand, higher-order logic (HOL) enables one to describe more theories and to describe existing theories more succinctly. Due to more complicated higher-order proof systems, higher-order automatic theorem provers (ATPs) are much less mature than their… 

Figures and Tables from this paper

Extending a Brainiac Prover to Lambda-Free Higher-Order Logic
TLDR
This work proposes to start with the state-of-the-art superposition prover E and gradually enrich it with higher-order features, explaining how to extend the prover’s data structures, algorithms, and heuristics to higher- order logic, a formalism that supports partial application and applied variables.
Superposition for Lambda-Free Higher-Order Logic
TLDR
Refutationally complete superposition calculi for intentional and extensional \(\lambda \)-free higher-order logic, two formalisms that allow partial application and applied variables, appear promising as a stepping stone towards complete, efficient automatic theorem provers for full higher- order logic.
Superposition for Lambda-Free Higher-Order Logic ( Technical Report )
TLDR
Refutationally complete superposition calculi for intentional and extensional λ-free higher-order logic, two formalisms that allow partial application and applied variables, appear promising as a stepping stone towards complete, efficient automatic theorem provers for full higher- order logic.
Higher-Order SMT Solving (Work in Progress)
TLDR
This report presents a pragmatic extension of the cvc4 solver in which existing data structures and algorithms are generalized to HOSMT, thus leveraging the extensive research and implementation efforts dedicated to efficient FO solving and discussing an alternative extension being implemented in veriT.
Extending SMT solvers to Higher-Order Logic ( Technical Report )
TLDR
This work proposes a pragmatic extension of SMT solvers to natively support higher-order reasoning without compromising their performance on FOL problems, thus leveraging the extensive research and implementation efforts dedicated to efficient FOL solving.

References

SHOWING 1-10 OF 22 REFERENCES
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.
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.
System Description: E 1.8
TLDR
This work reduces first-order problems to clause normal form and employs a saturation algorithm based on the equational superposition calculus to solve TPTP-5.4.0 FOF and CNF problems in automatic mode.
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.
Extensional Higher-Order Resolution
TLDR
An extensional higher-order resolution calculus that is complete relative to Henkin model semantics is presented and the long-standing conjecture, that it is sufficient to restrict the order of primitive substitutions to the orders of input formulae is proved.
On Restrictions of Ordered Paramodulation with Simplification
TLDR
A new and comparatively simple technique for completeness proofs based on the use of canonical rewrite systems to represent equality interpretations is introduced, which covers most simplification and elimination techniques used in practice yet preserves completeness of the proposed calculi.
Basic Superposition is Complete
We define equality constrained equations and clauses and use them to prove the completeness of what we have called basic superposition: a restricted form of superposition in which only the subterms
E - a brainiac theorem prover
TLDR
E is a sound and complete prover for clausal first order logic with equality and has a very flexible interface for specifying search control heuristics, and an efficient inference engine.
The TPTP Problem Library and Associated Infrastructure
  • G. Sutcliffe
  • Computer Science
    Journal of Automated Reasoning
  • 2017
TLDR
The TPTP problem library and associated infrastructure is described, from its use of Clause Normal Form via the First-Order Form (FOF) and Typed First-order Form (TFF), through to the monomorphic Typed Higher- order Form (TH0).
Simple and Efficient Clause Subsumption with Feature Vector Indexing
  • S. Schulz
  • Computer Science
    Automated Reasoning and Mathematics
  • 2013
TLDR
By restricting the selection of features used in the index, this technique immediately adapts to indexing modulo arbitrary AC theories with only minor loss of efficiency and enables us to integrate new simplification techniques making heavy use of subsumption.
...
...