Nominal Unification from a Higher-Order Perspective

Nominal Logic is a version of first-order logic with equality, name-binding, renaming via name-swapping and freshness of names. Contrarily to higher-order logic, bindable names, called atoms, and instantiable variables are considered as distinct entities. Moreover, atoms are capturable by instantiations, breaking a fundamental principle of lambda-calculus. Despite these differences, nominal unification can be seen from a higher-order perspective. From this view, we show that nominal unification can be reduced to a particular fragment of higher-order unification problems: Higher-Order Pattern Unification. This reduction proves that nominal unification can be decided in quadratic deterministic time, using the linear algorithm for Higher-Order Pattern Unification. We also prove that the translation preserves most generality of unifiers

Higher Order Unification via Explicit Substitutions

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

E-Unification for Second-Order Abstract Syntax

Higher-order unification (HOU) concerns unification of (extensions of) ?-calculus and can be seen as an instance of equational unification (E-unification) modulo ??-equivalence of ?-terms. We study equational unification of terms in languages with arbitrary variable binding constructions modulo arbitrary second-order equational theories. Abstract syntax with general variable binding and parametrised metavariables allows us to work with arbitrary binders without committing to ?-calculus or use inconvenient and error-prone term encodings, leading to a more flexible framework. In this paper, we introduce E-unification for second-order abstract syntax and describe a unification procedure for such problems, merging ideas from both full HOU and general E-unification. We prove that the procedure is sound and complete

Nominal Narrowing

Nominal unification is a generalisation of first-order unification that takes alpha-equivalence into account. In this paper, we study nominal unification in the context of equational theories. We introduce nominal narrowing and design a general nominal E-unification procedure, which is sound and complete for a wide class of equational theories. We give examples of application

Metaconfluence of Calculi with Explicit Substitutions at a Distance

Confluence is a key property of rewriting calculi that guarantees uniqueness of normal-forms when they exist. Metaconfluence is even more general, and guarantees confluence on open/meta terms, i.e. terms with holes, called metavariables that can be filled up with other (open/meta) terms. The difficulty to deal with open terms comes from the fact that the structure of metaterms is only partially known, so that some reduction rules became blocked by the metavariables. In this work, we establish metaconfluence for a family of calculi with explicit substitutions (ES) that enjoy preservation of strong-normalization (PSN) and that act at a distance. For that, we first extend the notion of reduction on metaterms in such a way that explicit substitutions are never structurally moved, i.e. they also act at a distance on metaterms. The resulting reduction relations are still rewriting systems, i.e. they do not include equational axioms, thus providing for the first time an interesting family of lambda-calculi with explicit substitutions that enjoy both PSN and metaconfluence without requiring sophisticated notions of reduction modulo a set of equations

Towards Correctness of Program Transformations Through Unification and Critical Pair Computation

Correctness of program transformations in extended lambda calculi with a contextual semantics is usually based on reasoning about the operational semantics which is a rewrite semantics. A successful approach to proving correctness is the combination of a context lemma with the computation of overlaps between program transformations and the reduction rules, and then of so-called complete sets of diagrams. The method is similar to the computation of critical pairs for the completion of term rewriting systems. We explore cases where the computation of these overlaps can be done in a first order way by variants of critical pair computation that use unification algorithms. As a case study we apply the method to a lambda calculus with recursive let-expressions and describe an effective unification algorithm to determine all overlaps of a set of transformations with all reduction rules. The unification algorithm employs many-sorted terms, the equational theory of left-commutativity modelling multi-sets, context variables of different kinds and a mechanism for compactly representing binding chains in recursive let-expressions.Comment: In Proceedings UNIF 2010, arXiv:1012.455

Set Unification

The unification problem in algebras capable of describing sets has been tackled, directly or indirectly, by many researchers and it finds important applications in various research areas--e.g., deductive databases, theorem proving, static analysis, rapid software prototyping. The various solutions proposed are spread across a large literature. In this paper we provide a uniform presentation of unification of sets, formalizing it at the level of set theory. We address the problem of deciding existence of solutions at an abstract level. This provides also the ability to classify different types of set unification problems. Unification algorithms are uniformly proposed to solve the unification problem in each of such classes. The algorithms presented are partly drawn from the literature--and properly revisited and analyzed--and partly novel proposals. In particular, we present a new goal-driven algorithm for general ACI1 unification and a new simpler algorithm for general (Ab)(Cl) unification.Comment: 58 pages, 9 figures, 1 table. To appear in Theory and Practice of Logic Programming (TPLP