Names and Binding in Type Theory

Names and name-binding are useful concepts in the theory and practice of formal systems. In this thesis we study them in the context of dependent type theory. We propose a novel dependent type theory with primitives for the explicit handling of names. As the main application, we consider programming and reasoning with abstract syntax involving variable binders. Gabbay and Pitts have shown that Fraenkel Mostowski (FM) set theory models a notion of name using which informal work with names and binding can be made precise. They have given a number of useful constructions for working with names and binding, such as a syntax-independent notion of freshness that formalises when two objects do not share names, and a freshness quantifier that simplifies working with names and binding. Related to FM set theory, a number of systems for working with names have been given, among them are the first-order Nominal Logic, the higher-order logic FM-HOL, the Theory of Contexts as well as the programming language FreshML. In this thesis we study how dependent type theory can be extended with primitives for working with names and binding. Our choice of primitives is different from that in FM set theory. In FM set theory the fundamental primitive for working with names is swapping. All other concepts such as \alpha-equivalence classes and binding are constructed from it. For dependent type theory, however, this approach of constructing everything from swapping is not ideal, since it requires us to make strong assumptions on the type theory. For instance, the construction of \alpha-equivalence classes from swapping appears to require quotient types. Our approach is to treat constructions such as \alpha-equivalence classes and name-binding directly, turning them into primitives of the type theory. To do this, it is convenient to take freshness rather than swapping as the fundamental primitive. Based on the close correspondence between type theories and categories, we approach the design of the dependent type theory by studying the categorical structure underlying FM set theory. We start from a monoidal structure capturing freshness. By analogy with the definition of simple dependent sums \Sigma and products \Pi from the cartesian product, we define monoidal dependent sums \Sigma^* and products \Pi^* from the monoidal structure. For the type of names N, we have an isomorphism \Sigma^*_N\iso\Pi^*_N generalising the freshness quantifier. We show that this structure includes \alpha-equivalence classes, name binding, unique choice of fresh names as well as the freshness quantifier. In addition to the set theoretic model corresponding to FM set theory, we also give a realizability model of this structure. The semantic structure leads us to a bunched type theory having both a dependent additive context structure and a non-dependent multiplicative context structure. This type theory generalises the simply-typed \alpha\lambda-calculus of O'Hearn and Pym in the additive direction. It includes novel monoidal products \Pi^* and sums \Sigma^* as well as hidden-name types H for working with names and binding. We give examples for the use of the type theory for programming and reasoning with abstract syntax involving binders. We show that abstract syntax can be handled both in the style of FM set theory and in the style of Weak Higher Order Abstract Syntax. Moreover, these two styles of working with abstract syntax can be mixed, which has interesting applications such as the derivation of a term for the unique choice of new names

Separation and Renaming in Nominal Sets

Nominal sets provide a foundation for reasoning about names. They are used primarily in syntax with binders, but also, e.g., to model automata over infinite alphabets. In this paper, nominal sets are related to nominal renaming sets, which involve arbitrary substitutions rather than permutations, through a categorical adjunction. In particular, the left adjoint relates the separated product of nominal sets to the Cartesian product of nominal renaming sets. Based on these results, we define the new notion of separated nominal automata. We show that these automata can be exponentially smaller than classical nominal automata, if the semantics is closed under substitutions

A dependent nominal type theory

Nominal abstract syntax is an approach to representing names and binding pioneered by Gabbay and Pitts. So far nominal techniques have mostly been studied using classical logic or model theory, not type theory. Nominal extensions to simple, dependent and ML-like polymorphic languages have been studied, but decidability and normalization results have only been established for simple nominal type theories. We present a LF-style dependent type theory extended with name-abstraction types, prove soundness and decidability of beta-eta-equivalence checking, discuss adequacy and canonical forms via an example, and discuss extensions such as dependently-typed recursion and induction principles

Nominal techniques

This is the author accepted manuscript. The final version is available from the Association for Computing Machinery via http://dx.doi.org/10.1145/2893582.2893594 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]. </jats:p

A Dependent Type Theory with Abstractable Names

This paper describes a version of Martin-Löf's dependent type theory extended with names and constructs for freshness and name-abstraction derived from the theory of nominal sets. We aim for a type theory for computing and proving (via a Curry-Howard correspondence) with syntactic structures which captures familiar, but informal, ‘nameful’ practices when dealing with binders.Partially supported by the UK EPSRC program grant EP/K008528/1, Rigorous Engineering for Mainstream Systems (REMS). Supported by the UK EPSRC leadership fellowship (Peter Sewell) grant EP/H005633/1, Semantic Foundations for Real-World Systems.This is the final published version of the article. It was originally published in Electronic Notes in Theoretical Computer Science (Pitts AM, Matthiesen J, Derikx J, Electronic Notes in Theoretical Computer Science 2015, 312, 19–50, doi:10.1016/j.entcs.2015.04.003) http://dx.doi.org/10.1016/j.entcs.2015.04.00

Nominal Game Semantics.

Tutorial notes presenting nominal game semantic

A Simple Nominal Type Theory

Abstract. Nominal logic is an extension of first-order logic with features useful for reasoning about abstract syntax with bound names. For computational applications such as programming and formal reasoning, it is desirable to develop constructive type theories for nominal logic which extend standard type theories for propositional, first- or higher-order logic. This has proven difficult, largely because of complex interactions between nominal logic’s name-abstraction operation and ordinary functional abstraction. This difficulty already arises in the case of propositional logic and simple type theory. In this paper we show how this difficulty can be overcome, and present a simple nominal type theory which enjoys properties such as type soundness and strong normalization, and which can be soundly interpreted using existing nominal set models of nominal logic. We also sketch how recursion combinators for languages with binding structure can be provided. This is an important first step towards understanding the constructive content of nominal logic and incorporating it into existing logics and type theories.

FULL ABSTRACTION FOR NOMINAL GENERAL REFERENCES

Copyright for articles published in Logical Methods in Computer Science is retained by the authors. Logical Methods in Computer Science is an open-access journal. All journal content is licensed under a Creative Commons license (http://creativecommons.org/licenses/by-nd/2.0/)Published in Logical Methods in Computer Science Vol. 5 (3:8) 2009, pp. 1–69 www.lmcs-online.orgResearch financially supported by the Engineering and Physical Sciences Research Council, the Eugenides Foundation, the A. G. Leventis Foundation and Brasenose College

Extensions of nominal terms

This thesis studies two major extensions of nominal terms. In particular, we study an extension with -abstraction over nominal unknowns and atoms, and an extension with an arguably better theory of freshness and -equivalence. Nominal terms possess two levels of variable: atoms a represent variable symbols, and unknowns X are `real' variables. As a syntax, they are designed to facilitate metaprogramming; unknowns are used to program on syntax with variable symbols. Originally, the role of nominal terms was interpreted narrowly. That is, they were seen solely as a syntax for representing partially-speci ed abstract syntax with binding. The main motivation of this thesis is to extend nominal terms so that they can be used for metaprogramming on proofs, programs, etc. and not just for metaprogramming on abstract syntax with binding. We therefore extend nominal terms in two signi cant ways: adding -abstraction over nominal unknowns and atoms| facilitating functional programing|and improving the theory of -equivalence that nominal terms possesses. Neither of the two extensions considered are trivial. The capturing substitution action of nominal unknowns implies that our notions of scope, intuited from working with syntax possessing a non-capturing substitution, such as the -calculus, is no longer applicable. As a result, notions of -abstraction and -equivalence must be carefully reconsidered. In particular, the rst research contribution of this thesis is the two-level - calculus, intuitively an intertwined pair of -calculi. As the name suggests, the two-level -calculus has two level of variable, modelled by nominal atoms and unknowns, respectively. Both levels of variable can be -abstracted, and requisite notions of -reduction are provided. The result is an expressive context-calculus. The traditional problems of handling -equivalence and the failure of commutation between instantiation and -reduction in context-calculi are handled through the use of two distinct levels of variable, swappings, and freshness side-conditions on unknowns, i.e. `nominal technology'. The second research contribution of this thesis is permissive nominal terms, an alternative form of nominal term. They retain the `nominal' rst-order avour of nominal terms (in fact, their grammars are almost identical) but forego the use of explicit freshness contexts. Instead, permissive nominal terms label unknowns with a permission sort, where permission sorts are in nite and coin nite sets of atoms. This in nite-coin nite nature means that permissive nominal terms recover two properties|we call them the `always-fresh' and `always-rename' properties that nominal terms lack. We argue that these two properties bring the theory of -equivalence on permissive nominal terms closer to `informal practice'. The reader may consider -abstraction and -equivalence so familiar as to be `solved problems'. The work embodied in this thesis stands testament to the fact that this isn't the case. Considering -abstraction and -equivalence in the context of two levels of variable poses some new and interesting problems and throws light on some deep questions related to scope and binding

Validating Brouwer's Continuity Principle for Numbers Using Named Exceptions

This paper extends the Nuprl proof assistant (a system representative of the class of extensional type theories with dependent types) withnamed exceptionsandhandlers, as well as a nominalfreshoperator. Using these new features, we prove a version of Brouwer's continuity principle for numbers. We also provide a simpler proof of a weaker version of this principle that only uses diverging terms. We prove these two principles in Nuprl's metatheory using our formalization of Nuprl in Coq and reflect these metatheoretical results in the Nuprl theory as derivation rules. We also show that these additions preserve Nuprl's key metatheoretical properties, in particular consistency and the congruence of Howe's computational equivalence relation. Using continuity and the fan theorem, we prove important results of Intuitionistic Mathematics: Brouwer's continuity theorem, bar induction on monotone bars and the negation of the law of excluded middle.</jats:p