Probabilities on a Heyting Algebra

1. Introduction

Probabilities are usually defined over Boolean algebras. In the finite case, it is sufficient to give the values for the atoms. We show here that it is possible to generalize them to Heyting algebras, starting from the prime filters, in other words on the spectrum of the algebra. Surprisingly, it is also possible to define them in a quantum manner, using projectors.

2. Affine Measures

We place ourselves in a finite Heyting algebra $\mathfrak{H}$. The set of its filters will be denoted $\mathcal{F}$ and the subset of its prime filters $\Phi$, which is the co-spectrum of $\mathfrak{H}$1. The finiteness of the algebra implies that all filters are principal. G will be the subset of prime filter infs. A filter can be seen as a set of propositions validated at a certain time, or as the intension of a formal object.

Definition

An affine measure on $\mathfrak{H}$ is an application $\mu :\mathfrak{H}\to \mathbb{R}$ that verifies

$$\left(i\right)\mu (a\vee b)+\mu (a\wedge b)=\mu \left(a\right)+\mu \left(b\right)$$

If, in addition, we have $\mu (\perp )=0$, $\mu (\top )=1$ and $a⩽b⟹\mu \left(a\right)⩽\mu \left(b\right)$, it becomes a probability.

Let $t=(a,b)$. It is useful to introduce a deviation matrix translating relation (i) with Kronecker indices:

$${\Delta }_c^t={\delta }_c^{a\vee b}+{\delta }_c^{a\wedge b}-{\delta }_c^a-{\delta }_c^b$$

We can see that the affine measures verify $\Delta \mu =0$2, i.e. $\mu \in Ker\Delta$.

Let $O_a^b=(a⩽b)=\left\{\begin{array}{cc}1& ifa⩽b\\ 0& otherwise\end{array}\right.$ be the matrix of the order relation of $\mathfrak{H}$.

Proposition

$$\forall t.{\Delta }_a^t{O}_f^a=0⟺f\in Gorf=\perp orf=\top$$

Demonstration

The relation on the left is $O_f^{a\wedge b}+O_f^{a\vee b}-O_f^a-O_f^b=0$, or $(a\wedge b⩾f)+(a\vee b⩾f)=(a⩾f)+(b⩾f)$.

If $\uparrow f$ is prime, we have $f⩽a\vee b⟺f⩽aorf⩽b$, and $f⩽a\wedge b⟺f⩽aandf⩽b$.

Apart from the trivial cases $f=\perp or\top$, let’s look at the various possibilities:

$$\begin{array}{cccc}f⩽a& f⩽b& (f⩽a)+(f⩽b)& (f⩽a\vee b)+(f⩽a\wedge b)\\ 0& 0& 0+0& 0+0\\ 0& 1& 0+1& 1+0\\ 1& 0& 1+0& 1+0\\ 1& 1& 1+1& 1+1\end{array}$$

And if $f\notin G$, we find $a\ngeqq f,b\ngeqq f,a\vee b⩾f,a\wedge b\ngeqq f$ and the relation is false.

qed

Let’s denote by ${\left(O\right|G)}_g^{g^{\prime }}=O_g^{g^{\prime }}$ the restriction of the order relation to G. The strict order implies that $\exists n.{\left(O\right|G-Id)}^n=0$. Let $A=Id-O|G$ and $S^n={\sum }_{i\in n}{A}^i$. We see easily that $S^n·O|G=Id$. $O|G$ is therefore invertible. The matrix $O_a^b$ is then of rank $\left|G\right|$ and all affine measures are obtained by ${\mu }^a=O_g^a{\pi }^g$ (summation restricted to G). Suppose ${\pi }^g$ is a classical probability on G. Since the elements of O and $\pi$ are positive or zero, the growth of $\mu$ is guaranteed. For $a=\perp$, the sum is empty, i.e. zero, and if $a=\top$ the sum is total and equals 1.

The result is

Theorem

Any probability is of the form

$${\mu }^a=O_g^a{\pi }^g=\sum _{g⩽a}{\pi }^g$$

where $\pi$ is a classical probability on the set G, i.e. ${\pi }^g⩾0,{\sum }_g{\pi }^g=1$ .

3. Antichains

It is also possible to express these probabilities using the antichains of the algebra.

Proposition

$$\mu \left(a\right)=\sum _{g\in Max(G\cap \downarrow a)}\mu \left(g\right)$$

Demonstration

$Max(G\cap \downarrow a)$ is the set of maximal elements of $\{g⩽a\}$. This is an antichain, since none of these maxima is comparable to any other.

If $a=\perp$, the antichain is empty, and we have $\mu (\perp )$ = 0. If $a\in G$, we get the definition of $\mu$ on prime filter infs. For other elements, the $Max$ ensures that ${\pi }^g$ are not counted more than once.

qed

Notes

The probability $\mu$ is first defined on G, which is generally much smaller than $\mathfrak{H}$. It is extended to the other elements by the sum of its values on the antichains of G.

Let’s denote the antichains of G by $AC\left(G\right)$. An order relation is defined by

$$\alpha ,\beta \in AC\left(G\right)\alpha ⩽\beta ⟺Max(\alpha \cup \beta )=\beta$$

This order makes it a distributive lattice isomorphic to ${\mathfrak{H}}^{op}$, via

$$i\left(\alpha \right)=\bigvee \alpha i^{-1}\left(a\right)=Max(G\cap [\perp ,a])$$

4. Projectors

Let $\mathcal{H}$ be a Hilbert space of dimension $\left|G\right|$ and $\mathcal{JH}$ be the set of its orthogonal projectors. If $J,K\in \mathcal{JH}$, let $J\vee K$ be the projector on the subspace generated by the union of their images, and $J\wedge K$ that on their intersection. We also introduce a natural order relation $J⩽K$ given by $ImJ\subseteq ImK$.

We know that if $[J,K]=0$, then $J\vee K=J+K-JK$ and $J\wedge K=JK$ .

Let $|{\beta }_g\rangle$ be an orthonormal basis of $\mathcal{H}$ and $|{\beta }_g\rangle \langle {\beta }_g|$ be the associated projectors. The latter commutate and verify exactly (i).

Hence the

Theorem

Let $J:\mathfrak{H}\to \mathcal{JH}$ be given by

$$J\left(a\right)=\sum _{g⩽a}|{\beta }_g\rangle \langle {\beta }_g|$$

J is a monomorphism of distributive lattices.

Any probability on $\mathfrak{H}$ can be written as

$${\mu }^a=\langle \psi \left|J\left(a\right)\right|\psi \rangle$$

where $\psi$ is a vector of norm 1, in other words a amplitude of probability in the quantum sense.

Notes

If we denote by ${\psi }^g$ the coordinates of $\psi$ in the ${\beta }_g$ basis, we obtain ${\pi }^g={\left|{\psi }^g\right|}^2$.

Generally speaking, we don’t have $J(\neg a)=Id-J\left(a\right)$. Better still, if $Id-J\left(a\right)=j\left(b\right)$, we can easily show that $a=\neg b$, $b=\neg a$, $a\vee \neg a=b\vee \neg b=\top$, $a=\neg a$ and $b=\neg b$.

5. Conditioning vs. Quantum Reduction

The notion of conditional probability can be traced back to the classical definition:

$$\mu \left(a\right|b)={\displaystyle \frac{\mu (a\wedge b)}{\mu \left(b\right)}}$$

Theorem

Let $j_b:\psi \mapsto {\displaystyle \frac{J\left(b\right)|\psi \rangle }{\parallel J\left(b\right)|\psi \rangle \parallel }}$, then $\mu \left(a\right|b)=\langle j_b\psi \left|J\left(a\right)\right|j_b\psi \rangle$

The demonstration is obvious, given the commutativity of the $J\left(a\right)$ projectors.

$j_b$ is a quantum reduction operator. It is unitary, but not linear.

6. Integration and Differentiation

Since ${\pi }^g$ is a classical probability on G, we can introduce random variables $X:G\to \mathbb{R}$ and calculate integrals as $\int Xd\mu =X_g{\pi }^g$. But $O_g^{g^{\prime }}$ is invertible, and we get $\int Xd\mu =X_g{\left(\stackrel{-1}{O|G}\right)}_{g^{\prime }}^g{\mu }^{g^{\prime }}$. This expression uses only the values of $\mu$ on G. ${\left(O\right|G)}^{-1}$ appears as a differentiation operator. This confirms that $\mu$ is in fact a cumulative probability.

7. Discussion

7.1. Commutative Windows

The set of projectors $J{\left(a\right)}_{a\in \mathfrak{H}}$ with connectors $\vee ,\wedge$ is a commutative subalgebra of $\mathcal{JH}$ . It is a classical window on a quantum space in the sense of Isham-Doering [1], a context in the sense of Kochen-Specker [2]. The above theorems open a door to quantum modeling based on intuitionistic logic.

7.2. Square Roots

The introduction of the probability amplitude as a square root is closely related to the search for non-informative a priori in Bayesian statistical decision theory. For example, if the aim of a sequence of experiments is to estimate a probability ${\pi }^g$ by realization frequencies ${\displaystyle \frac{N^g}{N}}$, we can define, thanks to the Fisher information, an a priori of Jeffreys which has interesting invariance properties [3]. This a priori has a density on the simplex of dimension $\left|G\right|-1$. The vector ${\left(\sqrt{{\pi }^g}\right)}_{g\in G}$ has quadratic norm 1, and the density image by $\sqrt{ }$ is the uniform measure [4] on the sphere ${\mathbb{S}}^{\left|G\right|-1}$.

7.3. Non Commutative Extension

The $\beta$ basis can be replaced by a partition of the identity ${\sum }_g{B}_g=Id$, where the $B_g$ are orthogonal projectors of a Hilbert of sufficient dimension. Such a partition can be, for example, that of the eigenprojectors of a Hermitian operator whose eigenvalues are arbitrary, provided they are distinct for distinct projectors. This allows the coexistence of several operators that may only commute on certain subspaces, i.e. the coexistence of algebras for which the inter-algebra conjunction (∧) is only defined under certain conditions.

7.4. Superpositions

The non-prime filters f of $\mathcal{F}$ also qualify as validating systems of propositions, but they admit disjunctions that have the following property

$$\exists a,b.a\vee b\in fanda\notin fandb\notin f$$

These disjunctions, which we might call floating, are related to superpositions of quantum states. In Kripke’s semantics for intuitionistic logic, these filters are not among the admissible worlds [5].

7.5. Boole Algebras

When the algebra is Boolean, we retrieve classical probability because all prime filters are ultrafilters.

7.6. Stone Representation

The idea of replacing atoms by prime filters is obviously directly inspired by the famous Stone representation theorem.

7.7. Similarities

The linking of $\Phi$ prime filters, $J\left(a\right)$ projectors and G antichains emerges from the following considerations:

If the intersection of two prime filters is prime, one is in the other.

If two orthogonal projectors commute, either they are orthogonal, or one is less than the other.

If the union of two maximal antichains is an antichain, one is in the other.

7.8. Code

All these results have been confirmed by computer. The code is written in C++, and the matrix calculations are performed by the eigen3 library. [6]

8. Example

These notions have been implemented and used to present Figure 1, Figure 2, Figure 3 and Figure 4. The Heyting algebra is obtained by the antichains of an arbitrary ordered set G, and the probability $\mu$ by a uniform random draw of the probability amplitude $\psi$.