Extensions of nominal terms

This thesis studies two major extensions of nominal terms. In particular, we study an extension with λ-abstraction over nominal unknowns and atoms, and an extension with an arguably better theory of freshness and α-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-specified 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 significant ways: adding λ-abstraction over nominal unknowns and atoms— facilitating functional programing—and improving the theory of α-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notions of scope, intuited from working with syntax possessing a non-capturing substitution, such as the λ-calculus, is no longer applicable. As a result, notions of λ-abstraction and α-equivalence must be carefully reconsidered. In particular, the first research contribution of this thesis is the two-level λcalculus, intuitively an intertwined pair of λ-calculi. As the name suggests, the two-level λ-calculus has two level of variable, modelled by nominal atoms and unknowns, respectively. Both levels of variable can be λ-abstracted, and requisite notions of β-reduction are provided. The result is an expressive context-calculus. The traditional problems of handling α-equivalence and the failure of commutation between instantiation and β-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’ first-order flavour 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 infinite and coinfinite sets of atoms. This infinite-coinfinite 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 α-equivalence on permissive nominal terms closer to ‘informal practice’. The reader may consider λ-abstraction and α-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 λ-abstraction and α-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.