Murdoch James Gabbay

A New Approach to Abstract Syntax with Variable Binding

Abstract. The permutation model of set theory with atoms (FM-sets), devised by Fraenkel and Mostowski in the 1930s, supports notions of ‘name-abstraction’ and ‘fresh name’ that provide a new way to represent, compute with, and reason about the syntax of formal systems involving variable-binding operations. Inductively defined FM-sets involving the name-abstraction set former (together with Cartesian product and disjoint union) can correctly encode syntax modulo renaming of bound variables. In this way, the standard theory of algebraic data types can be extended to encompass signatures involving binding operators. In particular, there is an associated notion of structural recursion for defining syntax-manipulating functions (such as capture avoiding substitution, set of free variables, etc.) and a notion of proof by structural induction, both of which remain pleasingly close to informal practice in computer science.

Curry-style types for nominal terms

We define a rank 1 polymorphic type system for nominal terms, where typing environments type atoms, variables and function symbols. The interaction between type assumptions for atoms and substitution for variables is subtle: substitution does not avoid capture and so can move an atom into multiple different typing contexts. We give typing rules such that principal types exist and are decidable for a fixed typing environment. a-equivalent nominal terms have the same types; a non-trivial result because nominal terms include explicit constructs for renaming atoms. We investigate rule formats to guarantee subject reduction. Our system is in a convenient Curry-style, so the user has no need to explicitly type abstracted atoms.

A Metalanguage for Programming with Bound Names Modulo Renaming

This paper describes work in progress on the design of an ML-style metalanguage FreshML for programming with recursively defined functions on user-defined, concrete data types whose constructors may involve variable binding. Up to operational equivalence, values of such FreshML data types can faithfully encode terms modulo α-conversion for a wide range of object languages in a straightforward fashion. The design of FreshML is ‘semantically driven’, in that it arises from the model of variable binding in set theory with atoms given by the authors in [7]. The language has a type constructor for abstractions over names ( = atoms) and facilities for declaring locally fresh names. Moreover, recursive definitions can use a form of pattern-matching on bound names in abstractions. The crucial point is that the FreshML type system ensures that these features can only be used in well-typed programs in ways that are insensitive to renaming of bound names.

Nominal rewriting

Nominal rewriting is based on the observation that if we add support for @a-equivalence to first-order syntax using the nominal-set approach, then systems with binding, including higher-order reduction schemes such as @l-calculus beta-reduction, can be smoothly represented. Nominal rewriting maintains a strict distinction between variables of the object-language (atoms) and of the meta-language (variables or unknowns). Atoms may be bound by a special abstraction operation, but variables cannot be bound, giving the framework a pronounced first-order character, since substitution of terms for variables is not capture-avoiding. We show how good properties of first-order rewriting survive the extension, by giving an efficient rewriting algorithm, a critical pair lemma, and a confluence theorem for orthogonal systems.

Nominal (Universal) Algebra

In informal mathematical discourse (such as the text of a paper on theoretical computer science), we often reason about equalities involving binding of object-variables. We find ourselves writing assertions involving meta-variables and captureavoidance constraints on where object-variables can and cannot occur free. Formalizing such assertions is problematic because the standard logical frameworks cannot express capture-avoidance constraints directly. In this article, we make the case for extending the logic of equality with meta-variables and capture-avoidance constraints, to obtain ‘nominal algebra’. We use nominal techniques that allow for a direct formalization of meta-level assertions, while remaining close to informal practice. We investigate proof-theoretical properties, we provide a sound and complete semantics in nominal sets and we compare and contrast our design decisions with other possibilities leading to similar systems.

Foundations of nominal techniques: logic and semantics of variables in abstract syntax

We are used to the idea that computers operate on numbers, yet another kind of data is equally important: the syntax of formal languages, with variables, binding, and alpha-equivalence. The original application of nominal techniques, and the one with greatest prominence in this paper, is to reasoning on formal syntax with variables and binding. Variables can be modelled in many ways: for instance as numbers (since we usually take countably many of them); as links (since they may ‘point’ to a binding site in the term, where they are bound); or as functions (since they often, though not always, represent ‘an unknown’). None of these models is perfect. In every case for the models above, problems arise when trying to use them as a basis for a fully formal mechanical treatment of formal language. The problems are practical—but their underlying cause may be mathematical. The issue is not whether formal syntax exists, since clearly it does, so much as what kind of mathematical structure it is. To illustrate this point by a parody, logical derivations can be modelled using a Godel encoding (i.e. injected into the natural numbers). It would be false to conclude from this that proof-theory is a branch of number theory and can be understood in terms of, say, Peano’s axioms. Similarly, as it turns out, it is false to conclude from the fact that variables can be encoded e.g. as numbers, that the theory of syntax-with-binding can be understood in terms of the theory of syntax-without-binding, plus the theory of numbers (or, taking this to a logical extreme, purely in terms of the theory of numbers). It cannot; something else is going on. What that something else is, has not yet been fully understood. In nominal techniques, variables are an instance of names, and names are data. We model names using urelemente with properties that, pleasingly enough, turn out to have been investigated by Fraenkel and Mostowski in the first half of the 20th century for a completely different purpose than modelling formal language. What makes this model really interesting is that it gives names distinctive properties which can be related to useful logic and programming principles for formal syntax. Since the initial publications, advances in the mathematics and presentation have been introduced piecemeal in the literature. This paper provides in a single accessible document an updated development of the foundations of nominal techniques. This gives the reader easy access to updated results and new proofs which they would otherwise have to search across two or more papers to find, and full proofs that in other publications may have been elided. We also include some new material not appearing elsewhere.

Nominal rewriting systems

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 Klops Combinatory Reduction Systems can be easily defined as nominal rewriting systems. Finally we study confluence properties of nominal rewriting.

One-and-a-halfth-order Logic

The practice of first-order logic is replete with meta-level concepts. Most notably there are meta-variables ranging over formulae, variables, and terms, and properties of syntax such as alpha-equivalence, capture-avoiding substitution and assumptions about freshness of variables with respect to meta-variables. We present one-and-a-halfth-order logic, in which these concepts are made explicit. We exhibit both sequent and algebraic specifications of one-and-a-halfth-order logic derivability, show them equivalent, show that the derivations satisfy cut-elimination, and prove correctness of an interpretation of first-order logic within it. We discuss the technicalities in a wider context as a case-study for nominal algebra, as a logic in its own right, as an algebraisation of logic, as an example of how other systems might be treated, and also as a theoretical foundation for future implementation.

A formal calculus for informal equality with binding

In informal mathematical usage we often reason using languages with binding.We usually find ourselves placing capture-avoidance constraints on where variables can and cannot occur free. We describe a logical derivation system which allows a direct formalisation of such assertions, along with a direct formalisation of their constraints. We base our logic on equality, probably the simplest available judgement form. In spite of this, we can axiomatise systems of logic and computation such as first-order logic or the lambda-calculus in a very direct and natural way. We investigate the theory of derivations, prove a suitable semantics sound and complete, and discuss existing and future research.

A general mathematics of names

We introduce FMG (Fraenkel-Mostowski Generalised) set theory, a generalisation of FM set theory which allows binding of infinitely many names instead of just finitely many names. We apply this generalisation to show how three presentations of syntax-de Bruijn indices, FM sets, and name-carrying syntax-have a relation generalising to all sets and not only sets of syntax trees. We also give syntax-free accounts of Barendregt representatives, scope extrusion, and other phenomena associated to @a-equivalence. Our presentation uses a novel presentation based not on a theory but on a concrete model U.