# Superposition with Lambdas

@article{Bentkamp2019SuperpositionWL,
title={Superposition with Lambdas},
author={Alexander Bentkamp and Jasmin Christian Blanchette and Sophie Tourret and Petar Vukmirovi{\'c} and Uwe Waldmann},
journal={ArXiv},
year={2019},
volume={abs/2102.00453}
}
• Published 27 August 2019
• Computer Science
• ArXiv
We designed a superposition calculus for a clausal fragment of extensional polymorphic higher-order logic that includes anonymous functions but excludes Booleans. The inference rules work on $$\beta \eta$$ β η -equivalence classes of $$\lambda$$ λ -terms and rely on higher-order unification to achieve refutational completeness. We implemented the calculus in the Zipperposition prover and evaluated it on TPTP and Isabelle benchmarks. The results suggest that superposition is a…
Superposition for Full Higher-order Logic
• Philosophy
• 2021
This work aims to reach a sound and refutationally complete calculus for higher-order logic with polymorphism, extensionality, Hilbert choice, and Henkin semantics, and its implementation in Zipperposition outperforms all other higher- order theorem provers.
Superposition for Lambda-Free Higher-Order Logic
• Computer Science
IJCAR
• 2018
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 Full Higher-Order Logic (Technical Report)
• Computer Science
• 2021
This work designs a sound and refutationally complete calculus for higher-order logic with polymorphism, extensionality, Hilbert choice, and Henkin semantics, and implements its implementation in Zipperposition on a par with an earlier, pragmatic prototype of Booleans.
Superposition for Higher-Order Logic
• Philosophy
• 2021
We recently designed two calculi as stepping stones towards superposition for full higher-order logic: Boolean-free λ-superposition and superposition for first-order logic with interpreted Booleans.
New Techniques for Higher-Order Superposition
• Computer Science
• 2020
Techniques that address the need for new heuristics to curb the explosion of specific higher-order rules in the Zipperposition theorem prover are described.
Making Higher-Order Superposition Work
• Computer Science
• 2021
Techniques that address issues such as infinitely branching inference rules, new possibilities such as reasoning about formulas, and the need to curb the explosion of specific higher-order rules are described and extensively evaluated in the Zipperposition theorem prover.
A Comprehensive Framework for Saturation Theorem Proving
A framework for formal refutational completeness proofs of abstract provers that implement saturation calculi, such as ordered resolution or superposition, based on modular extensions of lifted redundancy criteria is presented.
Boolean Reasoning in a Higher-Order Superposition Prover
• Computer Science
PAAR+SC²@IJCAI
• 2020
We present a pragmatic approach to extending a Boolean-free higher-order superposition calculus to support Boolean reasoning. Our approach extends inference rules that have been used only in a
A Combinator-Based Superposition Calculus for Higher-Order Logic
• Mathematics, Computer Science
IJCAR
• 2020
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.
Verifying Saturation Provers Modularly
• Mathematics, Computer Science
• 2020
A formalization in Isabelle/HOL of a comprehensive framework for proving the completeness of automatic theorem provers based on resolution, superposition, or other saturation calculi, and re-verified Bachmair and Ganzinger's resolution prover RP to show the benefits of modularity.

## References

SHOWING 1-10 OF 96 REFERENCES
Superposition for Lambda-Free Higher-Order Logic ( Technical Report )
• Computer Science
• 2018
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.
Superposition for Lambda-Free Higher-Order Logic
• Computer Science
IJCAR
• 2018
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.
A Unification Algorithm for Typed lambda-Calculus
• G. Huet
• Computer Science
Theor. Comput. Sci.
• 1975
Encoding Monomorphic and Polymorphic Types
• Computer Science
Log. Methods Comput. Sci.
• 2016
This work extends the approach to rank-1 polymorphism and presents alternative schemes that lighten the translation of polymorphic symbols based on the novel notion of "cover", and finds them vastly superior to previous schemes.
A Comprehensive Framework for Saturation Theorem Proving
A framework for formal refutational completeness proofs of abstract provers that implement saturation calculi, such as ordered resolution or superposition, based on modular extensions of lifted redundancy criteria is presented.
A Focused Sequent Calculus for Higher-Order Logic
A focused intuitionistic sequent calculus for higher-order logic that has primitive support for equality and mixes λ-term conversion with equality reasoning and is proved sound with respect to Church's simple type theory.
Boolean Reasoning in a Higher-Order Superposition Prover
• Computer Science
PAAR+SC²@IJCAI
• 2020
We present a pragmatic approach to extending a Boolean-free higher-order superposition calculus to support Boolean reasoning. Our approach extends inference rules that have been used only in a
HOT: A Concurrent Automated Theorem Prover Based on Higher-Order Tableaux
• Computer Science, Mathematics
TPHOLs
• 1998
An improved variant of the calculus which closely corresponds to the proof procedure implemented in Hot is introduced and the design of Hot's design is discussed that can be characterized as a concurrent blackboard architecture.
A Combinator-Based Superposition Calculus for Higher-Order Logic
• Mathematics, Computer Science
IJCAR
• 2020
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.
Extensional Higher-Order Resolution
• Mathematics