# Superposition for Lambda-Free Higher-Order Logic

@inproceedings{Bentkamp2018SuperpositionFL, title={Superposition for Lambda-Free Higher-Order Logic}, author={Alexander Bentkamp and Jasmin Christian Blanchette and Simon Cruanes and Uwe Waldmann}, booktitle={IJCAR}, year={2018} }

We introduce refutationally complete superposition calculi for intentional and extensional \(\lambda \)-free higher-order logic, two formalisms that allow partial application and applied variables. The calculi are parameterized by a term order that need not be fully monotonic, making it possible to employ the \(\lambda \)-free higher-order lexicographic path and Knuth–Bendix orders. We implemented the calculi in the Zipperposition prover and evaluated them on TPTP benchmarks. They appear…

## 33 Citations

### Superposition for Full Higher-order Logic

- PhilosophyCADE
- 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 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.…

### Making Higher-Order Superposition Work

- Computer ScienceCADE
- 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 brainiac prover to lambda-free higher-order logic

- Computer ScienceTACAS
- 2019

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.

### 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.

### Superposition with Lambdas

- Computer ScienceCADE
- 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.

### Formalization of the Embedding Path Order for Lambda-Free Higher-Order Terms

- Computer ScienceArch. Formal Proofs
- 2018

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.

### HoCHC: A Refutationally Complete and Semantically Invariant System of Higher-order Logic Modulo Theories

- Mathematics2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
- 2019

This work presents a simple resolution proof system for higher-order constrained Horn clauses (HoCHC) and proves its soundness and refutational completeness w.r.t. both standard and Henkin semantics.

### Restricted Combinatory Unification

- Computer ScienceCADE
- 2019

A restricted version of Dougherty's algorithm that is incomplete, terminating and does not require polymorphism is presented, including a novel use of a substitution tree as a filtering index for higher-order unification.

## References

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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.

### Extending a brainiac prover to lambda-free higher-order logic

- Computer ScienceTACAS
- 2019

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 with Lambdas

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- 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.

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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.

### Formalization of the Embedding Path Order for Lambda-Free Higher-Order Terms

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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.

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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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This thesis extends E, a state-of-the-art first-order ATP, to a fragment of HOL that is devoid of lambda abstractions (LFHOL), and devise generalizations of E’s indexing data structures to LFHOL, as well as algorithms like matching and unification to support HOL features in an efficient manner.

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A restricted version of Dougherty's algorithm that is incomplete, terminating and does not require polymorphism is presented, including a novel use of a substitution tree as a filtering index for higher-order unification.