# Superposition with First-class Booleans and Inprocessing Clausification

@inproceedings{Nummelin2021SuperpositionWF, title={Superposition with First-class Booleans and Inprocessing Clausification}, author={Visa Nummelin and Alexander Bentkamp and Sophie Tourret and Petar Vukmirovic}, booktitle={CADE}, year={2021} }

We present a complete superposition calculus for first-order logic with an interpreted Boolean type. Our motivation is to lay the foundation for refutationally complete calculi in more expressive logics with Booleans, such as higher-order logic, and to make superposition work efficiently on problems that would be obfuscated when using clausification as preprocessing. Working directly on formulas, our calculus avoids the costly axiomatic encoding of the theory of Booleans into first-order logic…

## 6 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 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 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.

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.

A Comprehensive Framework for Saturation Theorem Proving

- Mathematics, Computer ScienceArch. Formal Proofs
- 2020

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.

Lattice Structure of Some Closed Classes for Three-Valued Logic and Its Applications

- Computer Science, MathematicsMathematics
- 2021

It is shown that it is possible to recover the sublattice of closed classes in the general case of closure of functions with respect to the classical superposition operator and the closure operator R1 for the functions that differ only by dummy variables are considered equivalent.

## References

SHOWING 1-10 OF 43 REFERENCES

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.

Hierarchic Superposition with Weak Abstraction

- Computer ScienceCADE
- 2013

This paper introduces a novel form of clause abstraction, a core component in the hierarchic superposition calculus for transforming clauses into a form needed for internal operation, and argues for the benefits of the resulting calculus and provides a new completeness result for the fragment where all background-sorted terms are ground.

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.

A Clausal Normal Form Translation for FOOL

- Computer ScienceGCAI
- 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.

Non-Clausal Resolution and Superposition with Selection and Redundancy Criteria

- Computer Science, MathematicsLPAR
- 1992

The refutation completeness of the calculi is compatible with a general and powerful redundancy criterion which includes most (if not all) techniques for simplifying and deleting formulas, and the approach applies to constraint theorem proving, including constrained resolution and theory resolution.

A First Class Boolean Sort in First-Order Theorem Proving and TPTP

- Computer ScienceCICM
- 2015

This paper presents an extension FOOL of many-sorted first- order logic, in which the boolean sort is treated as a first-class sort and defines the syntax and semantics of FOOL and its model-preserving translation to first-order logic.

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…

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 and RUE-Resolution

- Computer ScienceCADE
- 1999

Two approaches to primitive equality treatment in higher-order (HO) automated theorem proving are presented: a calculus EP adapting traditional first-orders paramodulation, and a calculus ERUE adapting FO RUE-Resolution to classical type theory, i.e., HO logic based on Church's simply typed λ-calculus.

First-Order Theorem Proving and Vampire

- Computer ScienceCAV
- 2013

The superposition calculus is discussed and the key concepts of saturation and redundancy elimination are explained, present saturation algorithms and preprocessing, and demonstrate how these concepts are implemented in Vampire.