A Completeness Theorem for 'Total Boolean Functions'

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 nontrivial 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 a + b = 1; and a pair of polynomials is total if it sends total values to total values. This means that a pair (P 1 , P 2 ) of polynomials in 2n variables is total iff

We first restrict our attention to the case of an infinite field k: the above condition is then equivalent to the stronger condition (a pair of polynomials satisfying this condition is called strongly total)

The proof of this is easy but interesting: refer to any algebra textbook (“Algebra” by Lang, corollary 1.7 in chapter IV for example) if you are in a hurry… The constructions presented below also work for finite fields, but give a weaker result: see the remark at the end of section 5.

• Strongly total polynomials form an affine subspace of k

2. The centroidal calculus for boolean functions. The centroidal calculus produces pairs of polynomials (P 1 , P 2 ) using

• constants: T := (1, 0) and F := (0, 1);

• pairs of variables: (X 1 , X 2 );

where

A pair of polynomials is centroidal if it is generated by the above operations.

A note on terminology: “affine calculus” would be a much better name than “centroidal calculus”; but in the context of linear logic, this would lead to endless confusion.

The following proposition answers the natural question that was raised by Christine Tasson and Thomas Ehrhard:

Proposition. Suppose the field k is infinite; then the spaces of centroidal polynomials and of total polynomials coincide.

That centroidal polynomials are total is a direct consequence of their definition: all centroidal polynomials are in fact strongly total, in the sense of ( * ). The rest of this note is devoted to the converse.

Here is a collection of recipes for constructing centroidal polynomials:

• ¬(P 1 , P 2 ) = (P 2 , P 1 ) := if (P 1 , P 2 ) then F else T;

Using those, we can get more complex centroidal polynomials:

(a) suppose P 1 is any polynomial; we can always get a centroidal term (P 1 , P 2 ) for some polynomial P 2 :

-using " * “, we can get any monomial (M, . . .), -if M is such a monomial, α its coefficient in P 1 and m the total number of monomials in P 1 , (mαM, . . .) = if (mα, 1 -mα) then (M, . . .) else F, -we can then sum those monomials using coefficients 1/m to get (P 1 , . . .).

• This operation is neither commutative nor associative!

The last point implies in particular that it is equivalent to show that (P 1 , P 2 ) is centroidal and to show that (P 1 + P 2 , 0) is centroidal.

4. An interesting vector space. Write k[X 1 , . . . , X n ] d for the vector space of polynomials of degree at most d. The operator ϕ

It is easy to see that the following polynomials are all in the kernel of ϕ:

where ( k i k ) + ( k j k ) ≤ d and at least one of the i k is non zero.

Lemma. The above polynomials are linearly independent.

Proof: suppose α k P k = 0 where each P k is one of the above vectors. We show that the coefficient of any (X 1 + X 2 ) i1 . . . (X 2n-1 + X 2n ) in -1 X j1 1 . . . X jn 2n-1 is zero by induction on k j k .

• If k j k = 0: since the linear combination is zero, this implies that the global coefficient of each monomial is zero. Since (X 1 + X 2 ) i1 . . . (X 2n-1 + X 2n ) in -1 is the only polynomial contributing to the monomial X i1 2 . . . X in 2n , its coefficient must be zero.

• The polynomial (X 1 + X 2 ) i1 . . . (X 2n-1 + X 2n ) in -1 X j1 1 . . . X jn 2n-1 is the only polynomial contributing to X i1 2 . . . X in 2n X j1 1 . . . X jn 2n-1 because, by induction hypothesis, all the polynomials with fewer X 2k-1 ’s have zero for coefficient. This implies that the above coefficient is also zero…

Proof: the number of those polynomials is exactly 2n+d 2n -n+d n : • the first term accounts for the polynomials with

• the second term removes the polynomials where all the i k ’s are zero. We have a family of 2n+d 2n -n+d n linearly independent polynomials in a space of the same dimension: they necessarily form a basis.

5. Back to total polynomials. Abusing our terminology, we say that a single polynomial P is total [resp. centroidal]

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