Nominal unification

Abstract

We present a generalisation of ÿrst-order uniÿcation to the practically important case of equations between terms involving binding operations. A substitution of terms for variables solves such an equation if it makes the equated terms -equivalent, i.e. equal up to renaming bound names. For the applications we have in mind, we must consider the simple, textual form of substitution in which names occurring in terms may be captured within the scope of binders upon substitution. We are able to take a "nominal" approach to binding in which bound entities are explicitly named (rather than using nameless, de Bruijn-style representations) and yet get a version of this form of substitution that respects -equivalence and possesses good algorithmic properties. We achieve this by adapting two existing ideas. The ÿrst one is terms involving explicit substitutions of names for names, except that here we only use explicit permutations (bijective substitutions). The second one is that the uniÿcation algorithm should solve not only equational problems, but also problems about the freshness of names for terms. There is a simple generalisation of classical ÿrst-order uniÿcation problems to this setting which retains the latter's pleasant properties: uniÿcation problems involving -equivalence and freshness are decidable; and solvable problems possess most general solutions.

Nominal Algebra and the HSP Theorem

“…For further details see elsewhere [UPG04,Mat07]. 2 Definition 3.6 Let the derivable freshnesses by inductively defined by the rules in Figure 1.…”

“…The term-language of nominal algebra is nominal terms [UPG04]. Nominal terms extend first-order terms (the language of universal algebra) with object-level variables (atoms), and with constructs to support binding (nominal abstraction), α-equivalence (permutations), and capture-avoidance (freshness conditions).…”

“…Nominal algebra has been used to axiomatise substitution [GM08] and firstorder logic [GM07b]. Also in other work [UPG04,FG07,CU03], nominal techniques have proved their ability to be ' away from informal practice', following de Bruijn's philosophy while remaining mathematically completely rigorous.…”

Structural recursion with locally scoped names

“…Odersky's theory may seem too simple: compared with the dynamic allocation interpretation, there is no scope extrusion of local names from function arguments and no sharing of local names between components of a tuple. Nevertheless, combined with name-swapping, it produces a calculus that can represent any function defined by α-structural recursion for Λ at least, and potentially for any nominal signature in the sense of Definition 2.1 in Urban et al (2004).…”

The representational adequacy of Hybrid

“…Central to the formalisation is an inductive set that is bijective with α-equivalence classes of λ-expressions. Further work has studied unification within the nominal framework (Urban et al 2004). One aim of this work is to develop a framework for meta-programming applications, especially for developing operational semantics (see also Miller (2006)).…”