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The First-Order Theory of Boolean Diﬀerentiation

. In this contribution we give a ﬁrst-order axiomatisation of boolean diﬀerentiation. More precisely, for each n ∈ N and each complete theory T K of Boolean algebras we will give a complete ﬁrst-order theory T Kn of n boolean derivatives where n (cid:84) i =1 ker( δ 1 ) is a model of the theory T K . These theories can be obtained by adding just a ﬁnite list of axioms to those of T K . We show that any model of T K extends uniquely to a model of T Kn . Moreover, we will also provide a theory of the additive reduct equipped with n boolean derivatives, and we will see that these theories are categorical in every inﬁnite cardinality. We then show that the theories are indeed the asymptotic theories of the class of the algebras of switching functions equipped with any of the ordinarily used notions of derivative. Furthermore, we see that for the case n = 1 they also axiomatise the Fraisse limit of the ﬁnite switching functions with a derivative, and we use this fact to deduce quantiﬁer elimination.