Higher-order unification revisited: Complete sets of transformations

Snyder, Wayne; Gallier, Jean

doi:10.1016/s0747-7171(89)80023-9

Cited by 94 publications

(51 citation statements)

References 17 publications

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“…The language is a version of Church's simple theory of types [5] with an η-conversion rule as presented in [3] and [25] and with implicit α-conversions. Most of the definitions in this section are adapted from [25].…”

Section: Preliminaries 21 Typed Lambda Calculusmentioning

confidence: 99%

“…We assume that all the terms considered in this paper, unless specified otherwise, are in β-normal and η-expanded forms [25]. We further assume that all substitutions are idempotent [26] and contain only terms in β-normal and η-expanded forms in their co-domain.…”

Section: Preliminaries 21 Typed Lambda Calculusmentioning

confidence: 99%

“…We further assume that all substitutions are idempotent [26] and contain only terms in β-normal and η-expanded forms in their co-domain. This allows us to deal with normal forms implicitly (see [25] for more information). Equality between terms is always assumed to be α-equality.…”

Section: Preliminaries 21 Typed Lambda Calculusmentioning

confidence: 99%

“…The majority of the definitions in this section are taken from [25] and [24]. Since some of the definitions in this section are given in an intuitive rather than formal presentation, we recommend to refer to these sources for a better understanding of these definitions.…”

Section: Contexts and Pre-unificationmentioning

confidence: 99%

“…The completing substitution ξS for a system S maps every two variables in S of the same type to the same fresh variable. It is simple to prove that if σ pre-unifies a system, then σ • ξ unifies the system as well [25]. A complete set of pre-unifiers for a system S, denoted by PreUnifiers(S), is a set of substitutions such that {σ • ξS | σ ∈ PreUnifiers(S)} ⊆ Unifiers(S) and for every θ ∈ Unifiers(S) there exists σ ∈ PreUnifiers(S) such that σ| dom(θ) ≤ θ.…”

Section: =mentioning

confidence: 99%

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Bounded Higher-order Unification using Regular Terms

Libal¹

EPiC Series in Computing

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We present a procedure for the bounded unification of higher-order terms [24]. The procedure extends G. P. Huet's pre-unification procedure [11] with rules for the generation and folding of regular terms. The concise form of the procedure allows the reuse of the pre-unification correctness proof. Furthermore, the regular terms can be restricted in order to get a decidable unifiability problem. Finally, the procedure avoids re-computation of terms in a non-deterministic search which leads to a better performance in practice when compared to other bounded unification algorithms.

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Section: Preliminaries 21 Typed Lambda Calculusmentioning

confidence: 99%

Section: Preliminaries 21 Typed Lambda Calculusmentioning

confidence: 99%

Section: Preliminaries 21 Typed Lambda Calculusmentioning

confidence: 99%

Section: Contexts and Pre-unificationmentioning

confidence: 99%

Section: =mentioning

confidence: 99%

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Bounded Higher-order Unification using Regular Terms

Libal¹

EPiC Series in Computing

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Polymorphic Rewriting Conserves Algebraic Confluence

Breazu-Tannen¹,

Gallier²

1994

Information and Computation

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We study combinations of many-sorted algebraic term rewriting systems and polymorphic lambda term rewriting. Algebraic and lambda terms are mixed by adding the symbols of the algebraic signature to the polymorphic lambda calculus, as higher-order constants. We show that if a many-sorted algebraic rewrite system R has the Church-Rosser property (is confluent), then R + β + type-β + type-η rewriting of mixed terms has the Church-Rosser property too. η reduction does not commute with algebraic reduction, in general. However, using long normal forms, we show that if R is canonical (confluent and strongly normalizing) then equational provability from R + β + η + type-β + type-η is still decidable.

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Higher Order Unification via Explicit Substitutions

Dowek¹,

Hardin

Kirchner³

2000

Information and Computation

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Higher order uni cation is equational uni cation for -conversion. But it is not rst order equational uni cation, as substitution has to avoid capture. Thus the methods for equational uni cation (such as narrowing) built upon grafting (i.e. substitution without renaming), cannot be used for higher order uni cation, which needs speci c algorithms. Our goal in this paper is to reduce higher order uni cation to rst order equational uni cation in a suitable theory. This is achieved by replacing substitutionby grafting, but this replacementis not straightforward as it raises two major problems. First, some uni cation problems have solutions with grafting but no solution with substitution. Then equational uni cation algorithms rest upon the fact that grafting and reduction commute. But grafting and -reduction do not commute in -calculus and reducing an equation may change the set of its solutions. This di culty comes from the interaction between the substitutions initiated by -reduction and the ones initiated by the uni cation process. Two kinds of variables are involved: those of -conversion and those of uni cation. So, we need to set up a calculus which distinguishes these two kinds of variables and such that reduction and grafting commute. For this purpose, the application of a substitution of a reduction variable to a uni cation one must be delayed until this variable is instantiated. Such a separation and delay are provided by a calculus of explicit substitutions. Uni cation in such a calculus can be performed by well-known algorithms such as narrowing, but we present a specialised algorithm for greater e ciency. At last we show how to relate uni cation in -calculus and in a calculus with explicit substitutions.Thus we come up with a new higher order uni cation algorithm which eliminates some burdens of the previous algorithms, in particular the functional handling of scopes. Huet's algorithm can be seen as a speci c strategy for our algorithm, since each of its steps can be decomposed into elementary ones, leading to a more atomic description of the uni cation process. Also, solved forms in -calculus can easily be computed from solved forms in -calculus.

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Higher-order unification revisited: Complete sets of transformations

Cited by 94 publications

References 17 publications

Bounded Higher-order Unification using Regular Terms

Bounded Higher-order Unification using Regular Terms

Polymorphic Rewriting Conserves Algebraic Confluence

Higher Order Unification via Explicit Substitutions

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