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}
}

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

TLDR
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

TLDR
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

TLDR
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

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

TLDR
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
TLDR
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

TLDR
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

TLDR
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 )

TLDR
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

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TLDR
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.

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TLDR
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TLDR
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.

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TLDR
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TLDR
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TLDR
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.