Nominal Logic Programming

Nominal logic is an extension of first-order logic which provides a simple foundation for formalizing and reasoning about abstract syntax modulo consistent renaming of bound names (that is, alpha-equivalence). This article investigates logic programming based on nominal logic. We describe some typical nominal logic programs, and develop the model-theoretic, proof-theoretic, and operational semantics of such programs. Besides being of interest for ensuring the correct behavior of implementations, these results provide a rigorous foundation for techniques for analysis and reasoning about nominal logic programs, as we illustrate via examples.Comment: 46 pages; 19 page appendix; 13 figures. Revised journal submission as of July 23, 200

Constrained Polymorphic Types for a Calculus with Name Variables

We extend the simply-typed lambda-calculus with a mechanism for dynamic rebinding of code based on parametric nominal interfaces. That is, we introduce values which represent single fragments, or families of named fragments, of open code, where free variables are associated with names which do not obey alpha-equivalence. In this way, code fragments can be passed as function arguments and manipulated, through their nominal interface, by operators such as rebinding, overriding and renaming. Moreover, by using name variables, it is possible to write terms which are parametric in their nominal interface and/or in the way it is adapted, greatly enhancing expressivity. However, in order to prevent conflicts when instantiating name variables, the name-polymorphic types of such terms need to be equipped with simple {inequality} constraints. We show soundness of the type system

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

Generative Unbinding of Names

This paper is concerned with the form of typed name binding used by the FreshML family of languages. Its characteristic feature is that a name binding is represented by an abstract (name,value)-pair that may only be deconstructed via the generation of fresh bound names. The paper proves a new result about what operations on names can co-exist with this construct. In FreshML the only observation one can make of names is to test whether or not they are equal. This restricted amount of observation was thought necessary to ensure that there is no observable difference between alpha-equivalent name binders. Yet from an algorithmic point of view it would be desirable to allow other operations and relations on names, such as a total ordering. This paper shows that, contrary to expectations, one may add not just ordering, but almost any relation or numerical function on names without disturbing the fundamental correctness result about this form of typed name binding (that object-level alpha-equivalence precisely corresponds to contextual equivalence at the programming meta-level), so long as one takes the state of dynamically created names into account