Extensional Higher-Order Paramodulation and RUE-Resolution

  title={Extensional Higher-Order Paramodulation and RUE-Resolution},
  author={Christoph Benzm{\"u}ller},
This paper presents two approaches to primitive equality treatment in higher-order (HO) automated theorem proving: a calculus EP adapting traditional first-order (FO) paramodulation [RW69], and a calculus ERUE adapting FO RUE-Resolution [Dig79] to classical type theory, i.e., HO logic based on Church's simply typed λ-calculus. EP and ERUE extend the extensional HO resolution approach ER [BK98a]. In order to reach Henkin completeness without the need for additional extensionality axioms both… 

A Resolution Calculus Based on Eager Second-Order Bounded Unification

A modification of the constrained resolution calculus is presented which uses an eager unification algorithm while retaining completeness and is complete with regard to bounded unification only, which for many cases, does not pose a problem in practice.

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

  • A. Steen
  • Computer Science
    KI - 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.

Going Polymorphic - TH1 Reasoning for Leo-III

Modifications to the higher-order automated theorem prover Leo-III are presented for turning it into a reasoning system for rank-1 polymorphic HOL and a suitable paramodulation-based calculus are sketched.

Complete Cut-Free Tableaux for Equational Simple Type Theory

We present a cut-free tableau system for a version of Church’s simple type theory with primitive equality. The system is formulated with an abstract normalization operator that completely hides the

Superposition with Lambdas

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.

Empirically Successful Automated Reasoning in Higher-Order Logic (ESHOL)

The main theorem of the paper shows that the ability to type proofs if the axioms can be typed works for the rules of inference used by Otter-lambda, if type-safe lambda unification is used, and if demodulation and paramodulation from or into variables are not allowed.

Higher-order semantics and extensionality

A methodology of abstract consistency methods is developed by providing the necessary model existence theorems needed to analyze completeness of (machine-oriented) higher-order calculi with respect to these model classes.

The Higher-Order Prover Leo-II

It is crucial that Leo-II returns proof information in a standardised syntax, so that these proofs can eventually be transformed and verified within proof assistants.

Analytic Tableaux for Simple Type Theory and its First-Order Fragment

It is shown that the tableau system yields a decision procedure for three EFO fragments and completeness, compactness, and existence of countable models are proved for STT and EFO with respect to Henkin models and standard models.

Superposition with First-class Booleans and Inprocessing Clausification

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.



Equality and extensionality in automated higher order theorem proving

The three new calculi ER, ERUE, EP and ERUE which improve the mechanisation of defined and primitvie equality in classical type theory and these calculi reach Henkin completeness without requiring additional extensionality axioms are introduced.

System Description: LEO - A Higher-Order Theorem Prover

Leo uses a higher-order Logic based upon Church's simply typed λ-calculus, so that the comprehension axioms are implicitly handled by αβη-equality, and extensionality principles are build in into Leo’s unification, and hence do not have to be axiomatized in order to achieve Henkin completeness.

Extensional Higher-Order Resolution

An extensional higher-order resolution calculus that is complete relative to Henkin model semantics is presented and the long-standing conjecture, that it is sufficient to restrict the order of primitive substitutions to the orders of input formulae is proved.

Resolution in type theory

In [8] J. A. Robinson introduced a complete refutation procedure called resolution for first order predicate calculus. Resolution is based on ideas in Herbrand's Theorem, and provides a very

The Clausal Theory of Types

This book introduces just such a theory, based on a lambda-calculus formulation of a clausal logic with equality, known as the Clausal Theory of Types, which is a concise form of logic programming that incorporates functional programming.

Proofs in Higher-Order Logic

This work resolves the open question of what is a sound definition of skolemization in higher-order logic but also provides a direct, syntactic proof of its correctness.

Resolution by Unification and Equality

In resolution by unification and equality, we recast the theory of binary resolution on the basis of the properties of the equality relationship as stated by the equality axioms. In standard binary

Model Existence for Higher Order Logic

A semantical meta-theory that will support the development of higher-order calculi for automated theorem proving like the corresponding methodology has in first-order logic and establish classes of models that adequately characterize the existing theorem-proving calculi.

An introduction to mathematical logic and type theory - to truth through proof

This volume will be of interest to mathematicians, computer scientists, and philosophers in universities, as well as to computer scientists in industry who wish to use higher-order logic for hardware and software specification and verification.