# Restricted Combinatory Unification

@inproceedings{Bhayat2019RestrictedCU, title={Restricted Combinatory Unification}, author={Ahmed Bhayat and Giles Reger}, booktitle={CADE}, year={2019} }

First-order theorem provers are commonly utilised as backends to proof assistants. In order to improve efficiency, it is desirable that such provers can carry out some higher-order reasoning. In his 1991 paper, Dougherty proposed a combinatory unification algorithm for higher-order logic. The algorithm removes the need to deal with \(\lambda \)-binders and \(\alpha \)-renaming, making it attractive to implement in first-order provers. However, since publication it has garnered little interest…

## 16 Citations

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.

Extensional Higher-Order Paramodulation in Leo-III

- Computer ScienceJ. Autom. Reason.
- 2021

Leo-III is an automated theorem prover for extensional type theory with Henkin semantics and choice that supports reasoning in polymorphic first-order and higher-order logic, in all normal quantified modal logics, as well as in different deontic logics.

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.

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 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.

Efficient Full Higher-Order Unification (Technical Report)

- Computer Science
- 2019

This work developed a procedure to enumerate complete sets of higher-order unifiers based on work by Jensen and Pietrzykowski by carefully restricting the search space and tightly integrating decision procedures for fragments that admit a finite complete set of unifiers.

Efficient Full Higher-Order Unification

- Computer ScienceFSCD
- 2020

A procedure to enumerate complete sets of higher-order unifiers based on work by Jensen and Pietrzykowski is developed and implemented in the Zipperposition theorem prover.

Boolean Reasoning in a Higher-Order Superposition Prover

- Computer SciencePAAR+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…

Proceedings of the Second International Workshop on Automated Reasoning: Challenges, Applications, Directions, Exemplary Achievements

- Computer ScienceElectronic Proceedings in Theoretical Computer Science
- 2019

The contributions to automated reasoning made in the context of the project Matryoshka, funded for five years by the European Research Council, are presented, whose general aim is to bridge the gap between ATP and ITP by strengthening higher-order proof automation.

Experimenting with Theory Instantiation in Vampire

- Computer ScienceVampire
- 2019

The challenges when adding instantiation for the theory of arrays in Vampire using an SMT solver are reported on.

## References

SHOWING 1-10 OF 39 REFERENCES

Comparing Unification Algorithms in First-Order Theorem Proving

- Computer ScienceKI
- 2009

Large-scale experiments over the TPTP library containing thousands of problems using the COMPITmethodology confirm that the Robinson algorithm is the most efficient one in practice and reveal main sources of inefficiency in other algorithms.

Embedding Deduction Modulo into a Prover

- Computer ScienceCSL
- 2010

Results show that polarized resolution modulo can be integrated into existing provers, where these restrictions and simplifications are present and some simplification rules, such as strict subsumption eliminations and demodulations, preserve completeness.

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.

Functions-as-Constructors Higher-Order Unification

- Computer ScienceFSCD
- 2016

The main idea behind this extension is that the arguments to a higher-order, free variable can be more than just distinct bound variables: they can also be terms constructed from (sufficient numbers of) such variables using term constructors and where no argument is a subterm of any other argument.

Towards a Substitution Tree Based Index for Higher-order Resolution Theorem Provers

- Computer SciencePAAR@IJCAR
- 2016

This paper tries to handle two difficulties which arise when extending the indexes to higher-order, the need for higher- order anti-unification and the closure of clauses under associativity and com-mutativity.

Extensional Paramodulation for Higher-Order Logic and Its Effective Implementation Leo-III

- Computer ScienceKI - Künstliche Intelligenz
- 2019

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.

Set of Support for Higher-Order Reasoning

- Computer SciencePAAR@FLoC
- 2018

Limiting how axioms introduced during translation can improve proof search with higher-order problems is shown and heuristics based on the set-of-support strategy for minimising the effects are introduced.

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.

Hammer for Coq: Automation for Dependent Type Theory

- Computer Science, MathematicsJournal of Automated Reasoning
- 2018

An architecture of a full hammer for dependent type theory together with its implementation for the Coq proof assistant is presented and 40.8% of the theorems can be proved in a push-button mode in about 40 s of real time on a 8-CPU system.