A Completeness Theorem for "Total Boolean Functions"

In [3], Christine Tasson introduces an algebraic notion of totality for a denotational model of linear logic. The notion of total boolean function is, in a way, quite intuitive. This note provides a positive answer to the question of completeness of the " boolean centroidal calculus " w.r.t. total boolean functions. 0. Introduction. Even though the question answered in this note has its roots in denotational semantics for the differential λ-calculus ([2] and [1], see also [4]), no background in proof-theory is necessary to understand the problem. In the end, it boils down to a question about a special kind of polynomials in 2n variables over an arbitrary field k. This note is almost " self-contained " , assuming only mild knowledge about polynomials and vector spaces (and a modicum about affine spaces). The only exotic (??) technology is the following formula for counting monomials or multi-sets. The number of different monomials of degree d over n variables is usually denoted n d. A simple counting argument shows that the number of monomials of degree at most d in n variables is n+1 d. A closed formula for n d in terms of the usual binomial coefficient is given by: n d = n + d − 1 n. Thus, the number of monomials of degree at most d in n variables is given by n+d n. 1. Total boolean polynomials. The category of finite dimensional vector spaces give a denotational model for multiplicative additive linear logic. Adding the exponential is a non-trivial task and requires infinite dimensional spaces and thus, topology. Moreover, we need to find a subclass of spaces satisfying E ≃ E * *. Finiteness spaces (see [1]) give a solution. We won't need the details of this technology, but it is interesting to note that objects are topological vector spaces, and that morphisms (in the co-Kleisli category of the !-comonad) are " analytic functions " , i.e. power series. Of particular interest is the space B used to interpret the booleans: this is the vector space k 2 , where k is the ambient field. A morphism from B n to B is a pair (P 1 , P 2) of " finite " power series (polynomials) in 2n variables, where each pair (X 2i−1 , X 2i) of variables corresponds to the i-th argument of the function. A boolean value (a, b) is total if …