A unification algorithm for typed λ-calculus

Huet, Gérard

doi:10.1016/0304-3975(75)90011-0

Cited by 428 publications

(174 citation statements)

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“…In order to understand (and implement) what entailment involving λProlog programs should be, one simply needs to understand intuitionistic reasoning in STT. A large literature also exists that describes that logic and various ways to implement it, e.g., goal-directed search [19], higher-order unification [12,16], backtracking search [20], term representation to support efficient λ-reduction and unification [21], etc. While a particular implementation of logic programming is a particular piece of technology, that logic programming language and the checker implemented in it are not tied to that technology.…”

Section: Logic Foundations For These Featuresmentioning

confidence: 99%

Foundational Proof Certificates in First-Order Logic

Chihani

Miller

Renaud

2013

Automated Deduction – CADE-24

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We present the design philosophy of a proof checker based on a notion of foundational proof certificates. At the heart of this design is a semantics of proof evidence that arises from recent advances in the theory of proofs for classical and intuitionistic logic. That semantics is then performed by a (higher-order) logic program: successful performance means that a formal proof of a theorem has been found. We describe how the λProlog programming language provides several features that help guarantee such a soundness claim. Some of these features (such as strong typing, abstract datatypes, and higher-order programming) were features of the ML programming language when it was first proposed as a proof checker for LCF. Other features of λProlog (such as support for bindings, substitution, and backtracking search) turn out to be equally important for describing and checking the proof evidence encoded in proof certificates. Since trusting our proof checker requires trusting a programming language implementation, we discuss various avenues for enhancing one's trust of such a checker.

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Section: Logic Foundations For These Featuresmentioning

confidence: 99%

Foundational Proof Certificates in First-Order Logic

Chihani

Miller

Renaud

2013

Automated Deduction – CADE-24

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“…Attendant on these lambda terms is a notion of equality given by the α-, β-and η-conversion rules. Second, λProlog uses the higher-order unification procedure [Hue75], which respects this extended notion of equality. Finally, the language extends the collection of goals or queries with two new kinds of expressions, these being of the form ∀x G and D ⊃ G, in which G is a goal and D is a conjunction of clauses.…”

Section: The λProlog Languagementioning

confidence: 99%

Choices in Representation and Reduction Strategies for Lambda Terms in Intensional Contexts

Liang

Nadathur

2004

J Autom Reasoning

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Higher-order representations of objects such as programs, proofs, formulas and types have become important to many symbolic computation tasks. Systems that support such representations usually depend on the implementation of an intensional view of the terms of some variant of the typed lambda calculus. New notations have been proposed for the lambda calculus that provide an excellent basis for realizing such implementations. There are, however, several choices in the actual deployment of these notations the practical consequences of which are not currently well understood. We attempt to develop such an understanding here by examining the impact on performance of different combinations of the features afforded by such notations. Amongst the facets examined are the treatment of bound variables, eagerness and laziness in substitution and reduction, the ability to merge different structure traversals into one and the virtues of annotations on terms that indicate their dependence on variables bound by external abstractions. We complement qualitative assessments with experiments conducted by executing programs in a language that supports an intensional view of lambda terms while varying relevant aspects of the implementation of the language. Our study provides insights into the preferred approaches to representing and reducing lambda terms and also exposes characteristics of computations that have a somewhat unanticipated effect on performance.

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“…Unification problems associated with sets of connections will be analyzed by processors which apply Huet's unification algorithm [23]. Ultimately the unification processors should accommodate any constraints on unification terms which the user wishes to impose and recognize special cases and certain unification problems where processing is guaranteed to terminate (such as those of first-order logic).…”

Section: Matingsearchmentioning

confidence: 99%

“…Naturally, one can generate expansion terms incrementally by using substitutions which introduce single connectives and quantifiers in a general way, thus achieving the effects of Huet's splitting rules [22]. It turns out that in the context of the search process we envision, projections such as those used in [23] are also needed. We shall refer to such substitutions for a single variable as primitive substitutions.…”

Section: Expansionmentioning

confidence: 99%