Lecture Notes on Unification

The difficulty here is to know which term M of type τ to use to instantiante the quantifier. This problem is not specific to linear logic but arises in all logics that permit quantification. The general idea is simply to postpone the choice of M , just as we postponed splitting the resources in the multiplicative rules. M , in effect, becomes a metavariable or logic variable or existential variable whose value will be determined by constraints imposed upon it later on during proof search. Solving equational constraints imposed upon metavariables during proof search is called unification. By the rule above we can see that the values of metavariables are constrained already by the first premise of both rules, Ψ ` M : τ . For this lecture, we assume that this constraint will always be respected, but we do not explicitly discuss mechanisms to do so. Some care is needed: not only must the eventual value of X have the right type, but it must make sense in the right context Ψ. Relevant techniques are discussed in the literature [Mil92], including extensions to rich type theories where terms include binders [Mil91, NPP08].