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Computer Science
Theoretical Computer Science

Nominal unification

Profile image of AW Pitts IVAW Pitts IVProfile image of Christian UrbanChristian Urban

2004, Theoretical Computer Science

Abstract

We present a generalisation of first-order unification to the practically important case of equations between terms involving binding operations. A substitution of terms for variables solves such an equation if it makes the equated terms α-equivalent, i.e. equal up to renaming bound names. For the applications we have in mind, we must consider the simple, textual form of substitution in which names occurring in terms may be captured within the scope of binders upon substitution. We are able to take a "nominal" approach to binding in which bound entities are explicitly named (rather than using nameless, de Bruijn-style representations) and yet get a version of this form of substitution that respects α-equivalence and possesses good algorithmic properties. We achieve this by adapting two existing ideas. The first one is terms involving explicit substitutions of names for names, except that here we only use explicit permutations (bijective substitutions). The second one is that the unification algorithm should solve not only equational problems, but also problems about the freshness of names for terms. There is a simple generalisation of classical first-order unification problems to this setting which retains the latter's pleasant properties: unification problems involving α-equivalence and freshness are decidable; and solvable problems possess most general solutions.

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We study nominal anti-unification, which is concerned with computing least general generalizations for given terms-in-context. In general, the problem does not have a least general solution, but if the set of atoms permitted in generalizations is finite, then there exists a least general generalization which is unique modulo variable renaming and alpha-equivalence. We present an algorithm that computes it. The algorithm relies on a subalgorithm that constructively decides equivariance between two terms-in-context. We prove soundness and completeness properties of both algorithms and analyze their complexity. Nominal anti-unification can be applied to problems where generalization of first-order terms is needed (inductive learning, clone detection, etc.), but bindings are involved.

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For finite convergent term-rewriting systems it is shown that the equational unification problem is recursively independent of the equational matching problem, the word matching problem, and the 2nd-order equational matching problem. Apart from the latter these results are derived by considering term-rewriting systems on signatures that contain unary function symbols only (i.e., string-rewriting systems). Also for this special case 2nd-order equational matching is shown to be reducible to lst-order equational matching. In addition, we present some new decidability results for simultaneous equational matching and unification. Finally, we compare the word unijication problem to the 2nd-order equational unzjication problem.

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Nominal Syntax with Atom Substitutions: Matching, Unification, Rewriting
Jesús Domínguez

Lecture Notes in Computer Science, 2019

Unification and matching algorithms are essential components of logic and functional programming languages and theorem provers. Nominal extensions have been developed to deal with syntax involving binding operators: nominal unification takes into account α-equivalence; however, it does not take into account non-capturing substitutions, which are not primitive in nominal syntax. We consider an extension of nominal syntax with non-capturing substitutions and show that matching is decidable and finitary but unification is undecidable. We provide a matching algorithm and characterise problems for which matching is unitary, giving rise to expressive and efficient rewriting systems.

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The use of types in designing unification algorithms: two case studies
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We discuss the use of type systems in a non-strict sensewhen designing unification algorithms. We first give anew (rule-based) algorithm for an equational theory which representsa property of El-Gamal signature schemes and show howa type system can be used to prove termination of the algorithm.Lastly, we reproduce termination result for theory of partialexponentiation given earlier.

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Nominal rewriting systems
Ian Mackie

Proceedings of the 6th ACM SIGPLAN international conference on Principles and practice of declarative programming, 2004

We present a generalisation of first-order rewriting which allows us to deal with terms involving binding operations in an elegant and practical way. We use a nominal approach to binding, in which bound entities are explicitly named (rather than using a nameless syntax such as de Bruijn indices), yet we get a rewriting formalism which respects α-conversion and can be directly implemented. This is achieved by adapting to the rewriting framework the powerful techniques developed by Pitts et al. in the FreshML project. Nominal rewriting can be seen as higher-order rewriting with a first-order syntax and built-in α-conversion. We show that standard (first-order) rewriting is a particular case of nominal rewriting, and that very expressive higher-order systems such as Klop's Combinatory Reduction Systems can be easily defined as nominal rewriting systems. Finally we study confluence properties of nominal rewriting.

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A New Higher-order Unification Algorithm for λKanren
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Is there a world where true is equal to false? Indeed, there is, if higher-order unification [13] and miniKanren [9] meet each other. Even if all components are already well-established, it may still end up with unnatural or even absurd results, if we combine them naively. This paper unveils the difficulties in handling binders with standard higher-order unification. To solve these problems, we propose a new unification formulation. Higher-order unification [13] solves equations among λ-terms modulo αβη-equivalence [3]. Miller [19] finds a decidable fragment of higher-order unification problems by restricting the application forms of the input terms. For an application form F a, if F is a unification variable, then a must be a list of zero or more distinct bound variables. Such restricted application forms are called patterns and we hereafter refer to the problem identified by Miller as pattern unification. Pattern unification then becomes the core of λProlog [20]. λProlog implement...

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Nominal Unification is an extension of first-order unification where terms can contain binders and unification is performed modulo α-equivalence. Here we prove that the existence of nominal unifiers can be decided in quadratic time. First, we linearly-reduce nominal unification problems to a sequence of freshness and equalities between atoms, modulo a permutation, using ideas as Paterson and Wegman for first-order unification. Second, we prove that solvability of these reduced problems may be checked in quadratic time. Finally, we point out how using ideas of Brown and Tarjan for unbalanced merging, we could solve these reduced problems more efficiently.

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Nominal Unification from a Higher-Order Perspective
Jordi Levy

ACM Transactions on Computational Logic, 2012

Nominal Logic is a version of first-order logic with equality, name-binding, renaming via nameswapping 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.

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(Nominal) Unification by Recursive Descent with Triangular Substitutions
ramana kumar

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We mechanise termination and correctness for two unification algorithms, written in a recursive descent style. One computes unifiers for first order terms, the other for nominal terms (terms including α-equivalent binding structure). Both algorithms work with triangular substitutions in accumulator-passing style: taking a substitution as input, and returning an extension of that substitution on success. This style of algorithm has performance benefits and has not been mechanised previously. The algorithms use nested recursion so the termination proofs are non-trivial; the termination relation is also slightly different from usual.

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