The barrier of objects: From dynamical systems to bounded organizations

ionAx ∪ {x : τ ′} e : τA λx.e : τ ′ → τ LetA e : σ Ax ∪ {x : σ} e′ : τA (let x = e in e′) : τ The meaning of the Taut-rule is simply that a free variable has the type assignedto it in the boundary condition. If there is no assignment the expression x is nottypable, and is barred from the universe.The meaning of rule App is also clear. It is useful, however, to know how therule is implemented, since it introduces an important concept. Suppose that wehave an object e whose type has been established to be σ, and that we want toapply it to an object e′ of type τ ′. For this to be possible e must have a type ofthe form τ ′ → τ where τ stands for a generic unknown type of (e)e′ that needsto be determined. Hence, for the interaction (e)e′ to be possible e’s establishedtype σ and the required type τ ′ → τ must be made equal. This may be possible,since σ and τ ′ may contain type variables which can be made more specific inorder to satisfy the equality. This means we must look for some type substitutionT of the free variables in σ and in τ ′ → τ such that Tσ = T (τ ′ → τ). T is calleda unifier, and the procedure for finding T is called unification. It boils down tosolving a set of equations. For details about how this procedure is carried out thereader is referred to any standard textbook on type theory. The point is that a