# Higher-Order Unification via Combinators

@article{Dougherty1993HigherOrderUV,
title={Higher-Order Unification via Combinators},
author={Daniel J. Dougherty},
journal={Theor. Comput. Sci.},
year={1993},
volume={114},
pages={273-298}
}
34 Citations

### A Combinatory Logic Approach to Higher-order E-unification (Extended Abstract)

• Mathematics
• 1992
The case in which E admits presentation as a convergent term rewriting system is treated in detail: in this situation, a modification of ordinary narrowing is shown to be a complete method for enumerating higher-order E-unifiers.

### Higher-Order and Semantic Unification

• Computer Science
FSTTCS
• 1993
It is shown that this standard unification procedure, with slight modifications, can be used to solve the satisfiability problem in combinatory logic with a convergent set of algebraic axioms R, thus resulting in a complete higher-order unification procedure for R.

### Generating Compressed Combinatory Proof Structures: An Approach to Automated First-Order Theorem Proving

This “combinator term as proof structure” approach to automated first-order proving is introduced, an implementation and first experimental results are presented and features known from the connection structure calculus are realized.

### First-order unification using variable-free relational algebra

• Computer Science
Log. J. IGPL
• 2011
We present a framework for the representation and resolution of first-order unification problems and their abstract syntax in a variable-free relational formalism which is an executable variant of

### Unification for \lambda -calculi Without Propagation Rules

The unification procedure was developed for the calculi with ES that belong to the paradigm known as “act at a distance”, and explicit substitutions are not propagated to the level of variables, as usual.

### A Functional Logic Language Based on Higher Order Narrowing

• H. Kuchen
• Computer Science
Functional Programming
• 1995
This work presents the functional logic language Higher Order Babel which provides higher order unification for parameter passing and solving equations and replaces the expensive β-reduction by the more efficient combinator reduction.

### 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 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.

### Superposition for Lambda-Free Higher-Order Logic ( Technical Report )

• Computer Science
• 2018
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.

## References

SHOWING 1-10 OF 28 REFERENCES

### Higher-Order Unification, Polymorphism, and Subsorts (Extended Abstract)

This paper analyzes the problems that arise in extending Huet's higher-order unification algorithm from the simply typed λ-calculus to one with type variables and suggests a new type system for the λ -calculus which may pave the way to a complete unification algorithm for polymorphic terms.

### Introduction to combinators and λ-calculus

• Computer Science
Acta Applicandae Mathematicae
• 1990
These notes form a simple introduction to the two topics, suitable for a reader who has no previous knowledge of combinatory logic, but has taken an undergraduate course in predicate calculus and recursive functions.

### Mechanizing omega-Order Type Theory Through Unification

• Mathematics, Computer Science
Theor. Comput. Sci.
• 1976

### Higher Order E-Unification

An investigation of Higher-Order E-Unification, which consists of unifying typed lambda terms in the context of a first-order set of equations E, and a set of inference rules for higher-order E-unification and a proof of its soundness and completeness.

### An Efficient Unification Algorithm

• Computer Science
TOPL
• 1982
A new unification algorithm, characterized by having the acyclicity test efficiently embedded into it, is derived from the nondeterministic one, and a PASCAL implementation is given.

### Combining algebra and higher-order types

• V. Breazu-Tannen
• Computer Science, Mathematics
[1988] Proceedings. Third Annual Information Symposium on Logic in Computer Science
• 1988
The author shows that provability in the higher-order equational proof system obtained by adding the simply typed beta and eta axions to some many-sorted algebraic proof system is effectively reducible to Provability in that algebraicProof system.

### A Complete Mechanization of Second-Order Type Theory

A generalization of the resolution method for higher order logic is presented and it is established that the author's generalized resolution procedure is complete with respect to a natural notion of validity based on Henkin's general validity for type theory.

### Higher-order Unification with Dependent Function Types

In spite of the undecidability of this problem, Huet's algorithm for unification in the simply typed λ-calculus (λ→) is quite usable in practice.