A New Approach to Abstract Syntax with Variable Binding

Lecture Notes in Computer Science, 2006

It is possible to represent the terms of a syntax with binding constructors by a family of types, indexed by the free variables that may occur. This approach has been used several times for the study of syntax and substitution, but never for the formalization of the metatheory of a typing system. We describe a recent formalization of the metatheory of Pure Type Systems in Coq as an example of such a formalization. In general, careful thought is required as to how each definition and theorem should be stated, usually in an unfamiliar 'big-step' form; but, once the correct form has been found, the proofs are very elegant and direct.

2018

We introduce a universe of regular datatypes with variable binding information, for which we define generic formation and elimination (i.e. induction /recursion) operators. We then define a generic alpha-equivalence relation over the types of the universe based on name-swapping, and derive iteration and induction principles which work modulo alpha-conversion capturing Barendregt's Variable Convention. We instantiate the resulting framework so as to obtain the Lambda Calculus and System F, for which we derive substitution operations and substitution lemmas for alpha-conversion and substitution composition. The whole work is carried out in Constructive Type Theory and machine-checked by the system Agda.

Lecture Notes in Computer Science

In this paper we define an unranked nominal language, an extension of the nominal language with tuple variables and term tuples. We define the unification problem for unranked nominal terms and present an algorithm solving the unranked nominal unification problem.

Soft Computing, 2017

The category Nom, of all finitely supported Gsets, called G-nominal sets, where G = Perm f (D) is the group of all finitary permutations over a countable infinite set D, is a subject of interest by both set theorists and computer scientists. The category 01-Nom, of all G-nominal sets equipped with the source and the target operations, was introduced by Pitts. He has shown that this category is isomorphic to the category of sets whose elements have a finite support property with respect to an action of the monoid Cb of name substitutions. The latter category is a coreflective subcategory of the category Set Cb , of sets with the action of the monoid Cb. For a functorial relation between the categories Nom and Set Cb , we study the existence of the free objects in the category 01-Nom. More precisely, we construct the left adjoint to the forgetful functor from the category of 01-Gnominal sets to the category of G-nominal sets, where G is a suitable subgroup of the group of all permutations over a countable infinite set D.

Theory and Practice of Logic Programming, 2017

The problem of mechanically formalizing and proving metatheoretic properties of programming language calculi, type systems, operational semantics, and related formal systems has received considerable attention recently. However, the dual problem of searching for errors in such formalizations has attracted comparatively little attention. In this article, we present αCheck, a bounded model checker for metatheoretic properties of formal systems specified using nominal logic. In contrast to the current state of the art for metatheory verification, our approach is fully automatic, does not require expertise in theorem proving on the part of the user, and produces counterexamples in the case that a flaw is detected. We present two implementations of this technique, one based on negation-as-failure and one based on negation elimination, along with experimental results showing that these techniques are fast enough to be used interactively to debug systems as they are developed.

Journal of the ACM, 2014

When defining computations over syntax as data, one often runs into tedious issues concerning α -equivalence and semantically correct manipulations of binding constructs. Here we study a semantic framework in which these issues can be dealt with automatically by the programming language. We take the user-friendly “nominal” approach in which bound objects are named. In particular, we develop a version of Scott domains within nominal sets and define two programming languages whose denotational semantics are based on those domains. The first language, λν -PCF, is an extension of Plotkin’s PCF with names that can be swapped, tested for equality and locally scoped; although simple, it already exposes most of the semantic subtleties of our approach. The second language, PNA, extends the first with name abstraction and concretion so that it can be used for metaprogramming over syntax with binders. For both languages, we prove a full abstraction result for nominal Scott domains analogous to...

Journal of Automated Reasoning, 2011

This paper introduces a variant of nominal abstract syntax in which bindable names are represented by normal meta-variables as opposed to a separate class of globally fresh names. Distinct meta-variables can be instantiated with the same concrete name, which we call aliasing. The possible aliasing patterns are controlled by explicit constraints on the distinctness (freshness) of names. This approach has already been used in the nominal meta-programming language αML. We recap that language and develop a theory of contextual equivalence for it. The central result of the paper is that abstract syntax trees (ASTs) involving binders can be encoded into αML in such a way that α-equivalence of ASTs corresponds with contextual equivalence of their encodings. This is novel because the encoding does not rely on the existence of globally fresh names and fresh name generation, which are fundamental to the correctness of the pre-existing encoding of abstract syntax into FreshML.

Journal of Functional Programming, 2013

Correct handling of names and binders is an important issue in meta-programming. This paper presents an embedding of constraint logic programming into the αML functional programming language, which provides a provably correct means of implementing proof search computations over inductive definitions involving names and binders modulo α-equivalence. We show that the execution of proof search in the αML operational semantics is sound and complete with regard to the model-theoretic semantics of formulae, and develop a theory of contextual equivalence for the subclass of αML expressions which correspond to inductive definitions and formulae. In particular, we prove that αML expressions, which denote inductive definitions, are contextually equivalent precisely when those inductive definitions have the same model-theoretic semantics. This paper is a revised and extended version of the conference paper (Lakin, M. R. & Pitts, A. M. (2009) Resolving inductive definitions with binders in high...

Journal of Functional Programming, 2011

This paper introduces a new recursion principle for inductively defined data modulo α-equivalence of bound names that makes use of Odersky-style local names when recursing over bound names. It is formulated in simply typed λ-calculus extended with names that can be restricted to a lexical scope, tested for equality, explicitly swapped and abstracted. The new recursion principle is motivated by the nominal sets notion of ‘α-structural recursion’, whose use of names and associated freshness side-conditions in recursive definitions formalizes common practice with binders. The new calculus has a simple interpretation in nominal sets equipped with name-restriction operations. It is shown to adequately represent α-structural recursion while avoiding the need to verify freshness side-conditions in definitions and computations. The paper is a revised and expanded version of Pitts (Nominal System T. InProceedings of the 37th ACM SIGACT-SIGPLAN Symposium on Principles of Programming Languages...

ACM SIGPLAN Notices, 2003

FreshML extends ML with elegant and practical constructs for declaring and manipulating syntactical data involving statically scoped binding operations. User-declared FreshML datatypes involving binders are concrete, in the sense that values of these types can be deconstructed by matching against patterns naming bound variables explicitly. This may have the computational effect of swapping bound names with freshly generated ones; previous work on FreshML used a complicated static type system inferring information about the 'freshness' of names for expressions in order to tame this effect. The main contribution of this paper is to show (perhaps surprisingly) that a standard type system without freshness inference, coupled with a conventional treatment of fresh name generation, suffices for FreshML's crucial correctness property that values of datatypes involving binders are operationally equivalent if and only if they represent a-equivalent pieces of object-level syntax. ...

ACM SIGLOG News, 2016

Programming languages abound with features making use of names in various ways. There is a mathematical foundation for the semantics of such features which uses groups of permutations of names and the notion of the support of an object with respect to the action of such a group. The relevance of this kind of mathematics for the semantics of names is perhaps not immediately obvious. That it is relevant and useful has emerged over the last 15 years or so in a body of work that has acquired its own name: nominal techniques. At the same time, the application of these techniques has broadened from semantics to computation theory in general. This article introduces the subject and is based upon a tutorial at LICS-ICALP 2015 [Pitts 2015a].

Proceedings of the ACM on Programming Languages

This paper provides a new mathematical foundation for the locally nameless representation of syntax with binders, one informed by nominal techniques. It gives an equational axiomatization of two key locally nameless operations, "variable opening" and "variable closing" and shows that a lot of the locally nameless infrastructure can be defined from that in a syntax-independent way, including crucially a "shift" functor for name binding. That functor operates on a category whose objects we call locally nameless sets . Functors combining shift with sums and products have initial algebras that recover the usual locally nameless representation of syntax with binders in the finitary case. We demonstrate this by uniformly constructing such an initial locally nameless set for each instance of Plotkin's notion of binding signature. We also show by example that the shift functor is useful for locally nameless sets of a semantic rather than a syntactic charact...