Logical Calculi for Reasoning with Binding

Abstract

In informal mathematical usage we often reason about languages 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 thesis we make the case for extending logical frameworks with metavariables and capture-avoidance constraints. We use nominal techniques that allow for a direct formalisation of meta-level assertions, while remaining close to informal practice. Our focus is on derivability and we show that our derivation rules support the following key features of meta-level reasoning: • instantiation of meta-variables, by means of capturing substitution of terms for meta-variables; • ??-renaming of object-variables and capture-avoiding substitution of terms for object-variables in the presence of meta-variables; • generation of fresh object-variables inside a derivation. We apply our nominal techniques to the following two logical frameworks: • Equational logic. We investigate proof-theoretical properties, give a semantics in nominal sets and compare the notion of ??-renaming to existing notions of ??-equivalence with meta-variables. We also provide an axiomatisation of capture-avoiding substitution, and show that it is sound and complete with respect to the usual notion of capture-avoiding substitution. • First-order logic with equality. We provide a sequent calculus with metavariables and capture-avoidance constraints, and show that it represents schemas of derivations in first-order logic. We also show how we can axiomatise this notion of derivability in the calculus for equational logic.