# Superposition for Full Higher-order Logic

@inproceedings{Bentkamp2021SuperpositionFF,
title={Superposition for Full Higher-order Logic},
author={Alexander Bentkamp and Jasmin Christian Blanchette and Sophie Tourret and Petar Vukmirovic},
year={2021}
}
• Philosophy
We recently designed two calculi as stepping stones towards superposition for full higher-order logic: Boolean-free $$\lambda$$ λ -superposition and superposition for first-order logic with interpreted Booleans. Stepping on these stones, we finally reach a sound and refutationally complete calculus for higher-order logic with polymorphism, extensionality, Hilbert choice, and Henkin semantics. In addition to the complexity of combining the calculus’s two predecessors, new challenges arise…
9 Citations

### 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.

### Superposition with First-class Booleans and Inprocessing Clausification

• Computer Science
• 2021
A complete superposition calculus for first-order logic with an interpreted Boolean type to lay the foundation for refutationally complete calculi in more expressive logics with Booleans, and to make superposition work efficiently on problems that would be obfuscated when using clausification as preprocessing.

### 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.

### Extending a High-Performance Prover to Higher-Order Logic

• Computer Science
• 2022
This work extends E to full higher-order logic, and finds the resulting prover is the strongest one on benchmarks coming from a proof assistant, and the second strongest on TPTP benchmarks.

### SAT-Inspired Higher-Order Eliminations

• Computer Science
• 2022
Several propositional preprocessing techniques to higher-order logic are generalized, building on existing ﬁrst-order generalizations, and a new technique is introduced, which is called quasipure literal elimination, that strictly subsumes pure literal elimination.

### 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.

### Mechanical Mathematicians A new generation of automatic theorem provers eliminate bugs in software and mathematics.

• Computer Science
• 2022
Higher-order automatic theorem provers based on higher-order logics, which support functions as arguments, quantification over functions, and binders, are seen as more suitable than first-order logic for expressing a wide range of mathematics, and they are also useful for hardware and software verification.

### Mechanical Mathematicians A new generation of automatic theorem provers eliminate bugs in software and mathematics

• Computer Science
• 2022
Higher-order automatic theorem provers based on higher-order logics, which support functions as arguments, quantification over functions, and binders, are seen as more suitable than first-order logic for expressing a wide range of mathematics, and they are also useful for hardware and software verification.

### The Logic Languages of the TPTP World

An overview of the logic languages of the TPTP World, from classical first- order form (FOF), through typed FOF, up to typed higher-order form, and beyond to non-classical forms is provided.

## References

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### 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 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 with First-class Booleans and Inprocessing Clausification

• Computer Science
• 2021
A complete superposition calculus for first-order logic with an interpreted Boolean type to lay the foundation for refutationally complete calculi in more expressive logics with Booleans, and to make superposition work efficiently on problems that would be obfuscated when using clausification as preprocessing.

### Superposition with Lambdas

• Computer Science
• 2019
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.

### 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.

### The Higher-Order Prover Leo-III

• Computer Science
IJCAR
• 2018
The automated theorem prover Leo-III for classical higher-order logic with Henkin semantics and choice is presented and natively supports almost every normal higher- order modal logic.

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• 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.

### A Clausal Normal Form Translation for FOOL

• Computer Science
GCAI
• 2016
A new CNF translation algorithm for FOOL that is friendly and efficient for superposition-based first-order provers, implemented in the Vampire theorem prover and evaluated on a large number of problems coming from formalisation of mathematics and program analysis.

### 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.