Nominal (Universal) Algebra: Equational Logic with Names and Binding

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 capture-avoidance constraints on where object-variables can and cannot occur free. Formalising such assertions is problematic because the standard logical frameworks cannot express capture-avoidance constraints directly. In this paper 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 formalisation 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. ∗http://www.gabbay.org.uk †[email protected]