2.1 Syntax and semantics of independence logic

To rigorously define the semantics of independence logic—an extension of first-order logic by the independence atom—we need to be more precise about our definitions. For the sake of some technical details later on, we consider team semantics in the context of many-sorted structures (see, e.g., [Reference Manzano, Aranda, Zalta and Nodelman38]).

Definition 2.1. A (many-sorted relational) vocabulary $\tau $ is a tuple $(\mathrm {sor}_\tau ,\mathrm {rel}_\tau ,{\mathfrak {a}}_\tau ,{\mathfrak {s}}_\tau )$ such that

1. (i) $\mathrm {rel}_\tau $ is a set of relation symbolsFootnote ² and $\mathrm {sor}_\tau \subseteq \mathbb {N}$ ,

2. (ii) ${\mathfrak {a}}_\tau \colon \mathrm {rel}_\tau \to \mathbb {N}$ and ${\mathfrak {s}}_\tau \colon \mathrm {rel}_\tau \to \mathbb {N}^{<\omega }$ are functions with ${\mathfrak {s}}_\tau (R)\in \mathrm {sor}_\tau ^{{\mathfrak {a}}_\tau (R)}$ for $R\in \mathrm {rel}_\tau $ , and

3. (iii) if $n_i\in \mathbb {N}$ , $i<k$ , are such that ${\mathfrak {s}}_\tau (R)=(n_0,\dots ,n_{k-1})$ for some ${R\in \mathrm {rel}_\tau }$ , then $n_0,\dots ,n_{k-1}\in \mathrm {sor}_\tau $ .

We call ${\mathfrak {a}}_\tau (R)$ the arity of R and ${\mathfrak {s}}_\tau (R)$ the sort of R. For $n\notin \mathrm {sor}_\tau $ , we say that a vocabulary $\tau '$ is the expansion of $\tau $ by the sort n if

$$\begin{align*}\tau'=(\mathrm{sor}_\tau\cup\{n\},\mathrm{rel}_\tau,{\mathfrak{s}}_\tau,{\mathfrak{a}}_\tau). \end{align*}$$

A (many-sorted) $\tau $ -structure is a function $\mathfrak {A}$ defined on the set $\mathrm {rel}_\tau \cup \mathrm {sor}_\tau $ such that

1. (i) $\mathfrak {A}(n)$ is a nonempty set $A_n$ for $n\in \mathrm {sor}_\tau $ and called the sort n domain of $\mathfrak {A}$ , and

2. (ii) $\mathfrak {A}(R)\subseteq A_{n_0}\times \dots \times A_{n_{k-1}}$ for $R\in \mathrm {rel}_\tau $ , where ${\mathfrak {s}}_\tau (R)=(n_0,\dots ,n_{k-1})$ .

If $\tau '$ is an expansion of $\tau $ by sort n, we call a $\tau '$ -structure $\mathfrak {B}$ an expansion of $\mathfrak {A}$ by the sort n when $\mathfrak {B}\restriction (\mathrm {rel}_\tau \cup \mathrm {sor}_\tau )=\mathfrak {A}$ .

We usually denote $\mathfrak {A}(R)$ simply by $R^{\mathfrak {A}}$ and $\mathfrak {A}(n)$ by $A_n$ . If $\mathfrak {A}$ only has one sort, then we denote the domain of that sort by A and call it the domain of  $\mathfrak {A}$ . When there is no risk for confusion, we write ${\mathfrak {a}}$ and ${\mathfrak {s}}$ for ${\mathfrak {a}}_\tau $ and ${\mathfrak {s}}_\tau $ .

For each sort $n\in \mathbb {N}$ , we designate a set $\{v_i^n \mid i\in \mathbb {N}\}$ of variables of sort n, although for simplicity of notation, we usually use symbols like x, $y,$ and z for variables and indicate the sort by writing ${\mathfrak {s}}(x)$ for the sort of x.

The underlying idea of the new sort quantifiers $\tilde {\exists }$ and $\tilde {\forall }$ will become apparent in Definition 2.4 below.

For an assignment $s\colon D\to \mathfrak {A}$ , a variable $v$ (not necessarily in D) and $a\in A_{{\mathfrak {s}}(v)}$ , we denote by $s(a/v)$ the assignment $D\cup \{v\}\to \mathfrak {A}$ that maps $v$ to a and $w$ to $s(w)$ for $w\in D\setminus \{v\}$ . If $\vec {x}=(x_0,\dots ,x_{n-1})$ is a tuple of variables, we denote by $s(\vec {x})$ the tuple $(s(x_0),\dots ,s(x_{n-1}))$ .

Given a team X of $\mathfrak {A}$ , a variable $v$ , and a function $F\colon X\to {\mathcal {P}}(A_{{\mathfrak {s}}(v)})\setminus \{\emptyset \}$ , we denote by $X[F/v]$ the (“supplemented”) team $\{s(a/v) \mid s\in X, a\in F(s)\}$ and by $X[A_{{\mathfrak {s}}(v)}/v]$ the (“duplicated”) team $\{s(a/v) \mid s\in X, a\in A_{{\mathfrak {s}}(v)}\}$ .

If we restrict our attention to vocabularies and structures with just one sort, we get exactly the ordinary team semantics of independence logic.

When the underlying structure $\mathfrak {A}$ is clear from the context or is irrelevant to the discussion (e.g., when the formula $\varphi $ does not contain any non-logical symbols or variables of multiple sorts), we simply write $X\models \varphi $ instead of $\mathfrak {A}\models _X\varphi $ .

2.2 Axioms of independence logic

Although logical consequences in team semantics cannot be completely axiomatized (see the beginning of Section 3.4), it makes sense to isolate axioms that suffice for proving as many of the interesting logical consequences as possible. The rules we present here are, of course, not intended to be complete in any sense; rather, they are just what we need in this article. The general question of a more complete set of rules and axioms remains open.

The so-called Armstrong’s Axioms for the dependence atom [Reference Armstrong and Rosenfeld7] follow from the above axioms.

Armstrong’s axioms are complete for dependence atoms, i.e., if $\Sigma $ is a set of dependence atoms and $\varphi $ is another dependence atom, then $\Sigma \models \varphi $ if and only if $\Sigma $ entails $\varphi $ by repeated applications of Armstrong’s axioms.

The notion of a graphoid was introduced in [Reference Pearl and Paz40] after the observation that certain axioms that hold true for conditional independence in probability theory are also satisfied by the vertex separation relation in a(n undirected) graph. A semigraphoid is a weakening of a graphoid, excluding one axiom. The axioms as we present them can be found in [Reference Pearl39].

It is straightforward to show that the semigraphoid axioms are sound in team semantics (a fact which also follows from Proposition 4.23). Next we show that the axioms of independence atom follow from the semigraphoid axioms, save the Reflexivity Rule. It remains open whether the Reflexivity Rule also follows from the semigraphoid axioms.

So it turns out that the semigraphoid axioms, with the Reflexivity Rule added, are sufficient to prove all the others. This will be useful in Section 4. However, we will use all of the above axioms in the sequel.

Next we add rules for conjunction and existential quantifier, as we shall be working mainly with an existential-conjunctive fragment of independence logic.

For the new sort existential quantifier $\tilde \exists $ , we only give the above rather immediate axioms. These rules could possibly be strengthened by allowing variables of sort ${\mathfrak {s}}(x)$ occur in the scope of $\tilde \exists y$ for ${\mathfrak {s}}(y) = {\mathfrak {s}}(x)$ in formulas of  $\Sigma $ . It may also be interesting to look for more axioms for the sort quantifiers. For example, $\tilde \exists x\exists y \neg x=y$ for ${\mathfrak {s}}(x)={\mathfrak {s}}(y)$ is a natural valid sentence (even though the sentence $\exists x\exists y \neg x=y$ is not valid) but apparently not derivable from our current axioms.

If $\varphi $ entails $\psi $ by repeated applications of the above rules, we write $\varphi \vdash \psi $ .

3.1 Empirical and hidden-variable teams

As discussed in Section 2, we consider teams with designated variables for measurements and separate variables for outcomes. An important role in models of quantum physics is played by the so-called hidden variables, variables which are not directly observable, but which play a role in determining the outcomes of measurements, explaining indeterministic or non-local behaviour. The following terminology and notation is helpful in dealing with teams arising in this way in relation to quantum physics.

We use a division of variables into three sorts, defined below. A priori, there is no difference between the variables. This division into three sorts is simply helpful in guiding our intuitions. Our purely abstract results about teams based on these variables help us organize quantum-theoretic concepts. However, it is worth noting that, e.g., the word “measurement” has a meaning that corresponds to a physical event and the assumptions we make in the form of the properties of teams presented in Section 3.2, of course reflect the properties—observed or postulated—of these physical events.

Throughout the article, we will denote by n the number of measurement and outcome variables and by l the number of hidden variables.

Definition 5.2 below makes the connection between our concept of an empirical team and the mathematical model predicting what the possible outcomes of experiments could be, namely, the theory of operators of complex Hilbert spaces, explicit.

We will pay special attention to definability of properties of teams. In other words, if P is a property of teams, especially of empirical or hidden-variable teams, we ask whether there is a formula $\varphi $ of independence logic with the free variables $V_{\text {m}}\cup V_{\text {o}}$ (or $V_{\text {m}}\cup V_{\text {o}}\cup V_{\text {h}}$ ) which is satisfied in the sense of team semantics exactly by those teams that have the property P.

A hidden-variable team is a team of the form

where the $\gamma ^i_j$ indicate values which we cannot observe directly. A typical hidden variable is some kind of “state” of the system.

Every team has a background model from which the values of assignments come. In a many-sorted context, the background model has one universe for each sort. The universes may intersect. We assume a universe also for the hidden-variable sort.

Definition 3.2 [Reference Abramsky1]

A hidden-variable team Y realizes an empirical team X if

$$\begin{align*}s\in X \iff \exists s'\in Y \bigwedge_{i<n}(s'(x_i)=s(x_i) \land s'(y_i)=s(y_i)). \end{align*}$$

Two hidden-variable teams are said to be (empirically) equivalent if they realize the same empirical team.

The property of being the realization of a hidden-variable team is definable in independence logic. It can be defined simply by the existential quantifier: If $\varphi (\vec {x},\vec {y},\vec {z})$ is a formula of independence logic, and thereby defines a property of teams, then $\tilde {\exists } z_0\exists z_1\dots \exists z_{l-1}\varphi $ defines the class of empirical teams that are realized by some hidden-variable teams satisfying $\varphi $ . The “hidden” character of the hidden variables is built into the semantics of the sort quantifier.

Realization of an empirical team by a hidden-variable team involves a kind of projection where one projects away the hidden variables. Hidden-variable teams are divided into equivalence classes according to whether they project into the same empirical team or not. This phenomenon can, of course, be thought of more generally: for any set V of variables and $V'\subseteq V$ , one can define a projection mapping $\Pr _{V'}$ such that if X is a team with domain V, then $\Pr _{V'}(X) = \{s\restriction V' \mid s\in X \}$ .

Next we use the resources of independence logic with its team semantics to express properties of empirical and hidden-variable teams. The possible benefits of expressing such properties in the formal language of independence logic are two-fold. First, the quantum-theoretic concepts may suggest interesting new facts about independence logic in general, applicable perhaps also in other fields. Second, concepts, proofs, and constructions of independence logic may shed new light on connections between concepts in quantum physics, and may focus attention on what is particular to quantum physics, and what are merely general logical facts about independence concepts.

3.2 Properties of empirical teams

We observe that the definitions of the simpler properties of empirical teams treated by the first author in [Reference Abramsky1] can be expressed by formulas of independence logic, in fact a conjunction of independence atoms. For the original definitions, we refer to [Reference Bell9, Reference Dickson13, Reference Jarrett32, Reference Shimony42].

As discussed in Section 2, a team is said to support weak determinism if each outcome is determined by the combination of all the measurement variables.

Definition 3.3 (Weak Determinism)

An empirical team X supports weak determinism if it satisfies the formula

(WD) $$ \begin{align} \bigwedge_{i<n} \mathop{=}\hspace{-0.7pt}(\vec{x},y_i). \end{align} $$

Thus weak determinism is expressed simply with a conjunction of dependence atoms. In fact, the meaning of the dependence atom $\mathop {=}\hspace {-0.7pt}(x,y)$ is that x completely determines y. Therefore saying that teams supporting (WD) support weak determinism is appropriate. The only difference to the ordinary dependence atom is that in (WD) we separate the variables into the measurements $x_i$ and the outcomes $y_i$ .

A team is said to support strong determinism if the outcome variable $y_i$ of any measurement is completely determined by the measurement variable $x_i$ .

Definition 3.4 (Strong Determinism)

An empirical team X supports strong determinism if it satisfies the formula

(SD) $$ \begin{align} \bigwedge_{i<n} \mathop{=}\hspace{-0.7pt}(x_i,y_i). \end{align} $$

We now come to the important no-signalling condition. The motivation for this comes from the physical scenario with which we started, in which the parties $i \in n$ are spacelike separated from each other. This means that there can be no information flowing between the measurements performed by each party; in particular, which measurement was performed at party i cannot influence what the possible outcomes of a given measurement are at another party j. More generally, the possible outcomes of given measurements at a set of parties $I \subseteq n$ cannot be influenced by which measurements are performed at the remaining parties $n \setminus I$ . Crucially, although quantum mechanics is non-local, it does satisfy no-signalling, and hence is consistent with relativity theory.

This condition is formalized as follows. Suppose the team X has two possible measurement-outcome combinations s and $s'$ with inputs $x_i$ , $i\in I$ , the same. So now $s(\{y_i \mid i\in I\})$ is a possible outcome of the measurements $\{x_i \mid i\in I\}$ in view of X. We demand that $s(\{y_i \mid i\in I\})$ is also a possible outcome if the inputs $s(x_j)$ , $j\notin I$ , of the other experiments are changed to $s'(x_j)$ .

Definition 3.5 (No-Signalling)

An empirical team X supports no-signalling if it satisfies the formula

(NS) $$ \begin{align} \bigwedge_{I\subseteq n}\{x_i \mid i\notin I\}\perp_{\{x_i \mid i\in I\}}\{y_i \mid i\in I\}. \end{align} $$

In [Reference Abramsky1], a weaker version of no-signalling is presented where the subsets I are singletons, so the corresponding formula would be $\bigwedge _{i<n}\{ x_j \mid j\neq i \}\perp _{x_i}y_i$ .

In principle, supporting no-signalling means just satisfying a conjunction of independence atoms. But the atoms are of a particular form because of our division of variables into different sorts. The atom $\{x_i \mid i\notin I\}\perp _{\{x_i \mid i\in I\}}\{y_i \mid i\in I\}$ says that the outcomes $y_i$ , $i\in I$ , are meant to be related to the measurements $x_i$ , $i\in I$ , and be totally independent of the measurements $x_j$ , $j\notin I$ .

3.3 Properties of hidden-variable teams

For hidden-variable teams, the hidden variables are added in the definition of determinism as extra variables that determine the outcomes of the system.

Definition 3.6 (Weak Determinism)

A hidden-variable team X supports weak determinism if it satisfies the formula

(WD) $$ \begin{align} \bigwedge_{i<n} \mathop{=}\hspace{-0.7pt}(\vec{x}\vec{z},y_i). \end{align} $$

Definition 3.7 (Strong Determinism)

A hidden-variable team X supports strong determinism if it satisfies the formula

(SD) $$ \begin{align} \bigwedge_{i<n} \mathop{=}\hspace{-0.7pt}(x_i\vec{z},y_i). \end{align} $$

A team X is said to support single-valuedness if each hidden variable $z_k$ can only take one value.

Definition 3.8 (Single-Valuedness)

A hidden-variable team X supports single-valuedness if it satisfies the formula

(SV) $$ \begin{align} \mathop{=}\hspace{-0.7pt}(\vec{z}). \end{align} $$

The formula $\mathop {=}\hspace {-0.7pt}({\vec {z}})$ is a so-called constancy atom [Reference Abramsky and Väänänen5], a degenerate form of the dependence atom $\mathop {=}\hspace {-0.7pt}({\vec {x}},{\vec {y}})$ , where ${\vec {x}}$ is the empty tuple.

A team X is said to support ${\vec {z}}$ -independence if the following holds: Suppose the team X has two measurement-outcome combinations s and $s'$ . Now the hidden variables $\vec {z}$ have some value $s(\vec {z})$ in the combination s. We demand that $s(\vec {z})$ should occur as the value of the hidden variable also if the inputs $s(\vec {x})$ are changed to $s'(\vec {x})$ .

Definition 3.9 ( ${\vec {z}}$ -Independence)

A hidden-variable team X supports ${\vec {z}}$ -independence if it satisfies the formula

(zI)

Parameter independence is the hidden-variable version of no-signalling. A team X is said to support parameter-independence if the following holds: Suppose the team X has two measurement-outcome combinations s and $s'$ with the same input data about ${x_i}$ , $i\in I$ , and the same hidden variables $\vec {z}$ , i.e., $s(\{x_i \mid i\in I\})=s'(\{x_i\mid i\in I\})$ and $s(\vec {z})=s'(\vec {z})$ . We demand that the outcome data $s(\{y_i \mid i\in I\})$ should occur as a possible outcome also if the inputs $s(\{x_j \mid j\notin I\})$ are changed to $s'(\{x_j \mid j\notin I\})$ .

Definition 3.10 (Parameter Independence)

A hidden-variable team X supports parameter independence if it satisfies the formula

(PI) $$ \begin{align} \bigwedge_{I\subseteq n}\{x_i \mid i\notin I\}\perp_{\{x_i \mid i\in I\}\vec z}\{y_i \mid i\in I\}. \end{align} $$

Note that as with no-signalling, the version of parameter independence presented in [Reference Abramsky1] would correspond to the formula $\bigwedge _{i<n}\{ x_j \mid j\neq i \}\perp _{x_i\vec {z}}y_i$ .

A team X is said to support outcome-independence if the following holds: Suppose the team X has two measurement-outcome combinations s and $s'$ with the same total input data $\vec {x}$ and the same hidden variables $\vec {z}$ , i.e., $s(\vec {x})=s'(\vec {x})$ and $s(\vec {z})=s'(\vec {z})$ . We demand that outcome $s(y_i)$ should occur as an outcome also if the outcomes $s(\{y_j \mid j\ne i\})$ are changed to $s'(\{y_j \mid j\ne i\})$ . In other words, the variables $y_i$ , $i<n$ , are mutually independent whenever $\vec {x}\vec {z}$ is fixed.

Definition 3.11 (Outcome Independence)

A hidden-variable team X supports outcome independence if it satisfies the formula

(OI) $$ \begin{align} \bigwedge_{i<n} y_i\perp_{\vec{x}\vec{z}}\{y_j \mid j\neq i\}. \end{align} $$

All the previous examples were, from the point of view of independence logic, atoms or conjunctions of atoms of the same kind with a certain organization of the variables. We shall now consider a property which is slightly more complicated.

This is the crucial notion of locality, which expresses the idea that the possible outcomes of a party can only depend on the input to that party, together with the values of the hidden variables, and not on the outcomes of any other party. This strengthens the no-signalling condition, which only requires independence from the inputs of the other parties. Whereas quantum mechanics satisfies no-signalling, it violates locality—hence allowing for non-local correlations of outcomes. This condition is formalized in team semantics as follows.

Definition 3.12 (Locality)

A hidden-variable team X satisfies locality if

$$ \begin{align*} \forall s_0,&\dots,s_{n-1}\in X \left[ \exists s\in X\bigwedge_{i<n} s(x_i \vec{z}) = s_i(x_i \vec{z}) \right. \\ & \left. \implies \exists s'\in X\bigwedge_{i<n}s'(x_i y_i \vec{z}) = s_i(x_i y_i \vec{z}) \right]. \end{align*} $$

The definition of locality is not per se an expression of independence logic. However, in Lemma 3.16 below, we prove that locality can be defined, after all, by a conjunction of independence atoms.

3.4 Relationships between the properties

We present several logical consequences of independence logic and demonstrate how they can be interpreted in the context of empirical and hidden-variable teams. In many cases, we can derive the logical consequence relation from the axioms of Definition 2.5. Semantic proofs are due to [Reference Abramsky1],

It should be noted that logical consequence in independence logic is, in principle, a highly complex concept. For example, it cannot be axiomatized because the set of Gödel numbers of valid sentences (of even dependence logicFootnote ⁵ ) is non-arithmetical. Even the implication problem for the independence atoms is undecidable [Reference Herrmann29], while for dependence atoms, it is decidable [Reference Armstrong and Rosenfeld7]. Logical implication between finite conjunctions of independence atoms is, however, recursively axiomatizable, as it can be reduced to logical consequence in first-order logic by introducing a new predicate symbol.

Because of the complexity of logical consequence, it is important to accumulate good examples. We claim that the below examples arising from quantum mechanics are illustrative examples and may guide us in finding a more systematic approach.

In words, if a hidden-variable team supports weak determinism, then it supports outcome independence.

In words, if a hidden-variable team supports strong determinism, then it supports parameter independence

Proof Fix $I\subseteq n$ . Using Armstrong’s axioms, one can obtain

$$\begin{align*}\{\mathop{=}\hspace{-0.7pt}(x_i\vec{z},y_i) \mid i\in I\}\vdash\mathop{=}\hspace{-0.7pt}(\{x_i \mid i\in I\}{\vec{z}},\{y_i \mid i\in I\}). \end{align*}$$

Note that $\mathop {=}\hspace {-0.7pt}(\{x_i \mid i\in I\}{\vec {z}},\{y_i \mid i\in I\})$ means $\{y_i \mid i\in I\}\perp _{\{x_i \mid i\in I\}{\vec {z}}}\{y_i \mid i\in I\}$ . Now the Constancy Rule of independence atoms gives $\{y_i \mid i\in I\}\perp _{\{x_i \mid i\in I\}{\vec {z}}}\vec {w}$ for any variable tuple $\vec {w}$ , in particular, when $\vec {w} = \{ x_i \mid i\notin I \}$ . Finally, we obtain $\{x_i \mid i\notin I\}\perp _{\{x_i \mid i\in I\}\vec z}\{y_i \mid i\in I\}$ by using the Symmetry Rule.

In words, if a hidden-variable team supports parameter independence and weak determinism, then it supports strong determinism.

Proof Fix $i<n$ . $\mathop {=}\hspace {-0.7pt}(\vec {x}\vec {z},\vec {y})$ means $\vec {y}\perp _{\vec {x}\vec {z}}\vec {y}$ , from which we get $y_i\perp _{\vec {x}\vec {z}}y_i$ using the Weakening Rule. From parameter independence, we obtain $\{ x_j \mid j\neq i \}\perp _{x_i\vec {z}}y_i$ by choosing the conjunct with $I = n\setminus \{i\}$ . Then we have

$$\begin{align*}\{ x_j \mid j\neq i \}\perp_{x_i\vec{z}}y_i \land y_i\perp_{\vec{x}\vec{z}}y_i. \end{align*}$$

Finally, the First Transitivity Rule yields $y_i\perp _{x_i\vec {z}}y_i$ , which means $\mathop {=}\hspace {-0.7pt}(x_i\vec {z}, y_i)$ .

Lemma 3.16. Locality is equivalent to the formula

$$\begin{align*}\left(\bigwedge_{I\subseteq n} \{ x_i \mid i\notin I \}\perp_{\{x_i \mid i\in I\}\vec{z}}\{y_i \mid i\in I\}\right) \land \left(\bigwedge_{i<n} y_i\perp_{\vec{x}\vec{z}}\{y_j \mid j\neq i\} \right). \end{align*}$$

In words, a hidden-variable team X supports locality if and only if it supports both parameter independence and outcome independence.

Proof It is essentially proved in [Reference Abramsky1] that the weaker version of parameter independence

$$\begin{align*}\bigwedge_{i<n}\{x_j \mid j\neq i\}\perp_{x_i{\vec{z}}}y_i \end{align*}$$

together with outcome independence is equivalent to locality. What is left is to show that the stronger version of parameter independence still follows from locality. So suppose that X supports locality and fix $I\subseteq n$ . We show that

$$\begin{align*}X\models\{ x_i \mid i\notin I \}\perp_{\{x_i \mid i\in I\}\vec{z}}\{y_i \mid i\in I\}. \end{align*}$$

Let $s,s'\in X$ be such that $s(x_i\vec {z})=s'(x_i\vec {z})$ for all $i\in I$ . We wish to find $s"\in X$ with $s"({\vec {x}}{\vec {z}})=s({\vec {x}}{\vec {z}})$ and $s"(\{y_i \mid i\in I\})=s'(\{y_i \mid i\in I\})$ . Let $s_i = s'$ for $i\in I$ and $s_i = s$ for $i\notin I$ . Now s is such that for $i\in I$ , $s(x_i\vec {z}) = s'(x_i\vec {z}) = s_i(x_i\vec {z})$ , but also for $i\notin I$ we have $s(x_i\vec {z}) = s_i(x_i\vec {z})$ (as $s = s_i$ ). Hence we have $s(x_i\vec {z})=s_i(x_i\vec {z})$ for all $i<n$ , so by locality there exists $s"\in X$ with $s"(x_iy_i\vec {z}) = s_i(x_iy_i\vec {z})$ for all $i<n$ . But then $s"(x_i{\vec {z}}) = s_i(x_i\vec {z}) = s(x_i\vec {z})$ for all $i<n$ and $s"(y_i) = s_i(y_i) = s'(y_i)$ for $i\in I$ . Thus $s"$ is as desired.

Next we indicate connections between properties of empirical teams and properties of hidden-variable teams, again following [Reference Abramsky1].

In words, every empirical team is realized by a hidden-variable team supporting single-valuedness.

Proof By the Reflexivity Rule, we have $\vec {z}\perp _{\vec {z}}\vec {z}$ . Using introduction of existential quantifier l times, we obtain $\exists z_0\dots \exists z_{l-1}\ \vec {z}\perp _{\vec {z}}\vec {z}$ . Using elimination of existential quantifier and introduction of dependence l times, we obtain

$$\begin{align*}\exists z_0\dots\exists z_{l-1}\left(\bigwedge_{k<l}\mathop{=}\hspace{-0.7pt}(z_k)\land\vec{z}\perp_{\vec{z}}\vec{z}\right). \end{align*}$$

As, from Armstrong’s axioms, one can infer $\bigwedge _{k<l}\mathop {=}\hspace {-0.7pt}(z_k)\vdash \mathop {=}\hspace {-0.7pt}(\vec {z})$ , we then easily obtain $\exists \vec {z}\mathop {=}\hspace {-0.7pt}(\vec {z})$ . Then, assuming $\varphi $ , by using elimination and introduction of existential quantifier l times, we obtain $\exists \vec {z}(\mathop {=}\hspace {-0.7pt}(\vec {z})\land \varphi )$ . Finally, the first introduction rule of $\tilde \exists $ gives $\tilde \exists z_0\exists z_1\dots \exists z_{l-1} (\mathop {=}\hspace {-0.7pt}(\vec {z})\wedge \varphi )$ .

Proposition 3.18. Let

$$\begin{align*}\varphi = \bigwedge_{I\subseteq n}\{x_i \mid i\notin I\}\perp_{\{x_i\mid i\in I\}}\{y_i \mid i\in I\} \end{align*}$$

and

$$\begin{align*}\psi = \tilde{\exists}z_0\exists z_1\dots\exists z_{l-1} \left( \vec{z}\perp\vec{x} \land \bigwedge_{I\subseteq n} \{x_i \mid i\notin I\}\perp_{\{x_i\mid i\in I\}\vec{z}}\{y_i\mid i\in I\} \right). \end{align*}$$

Then $\varphi \dashv \vdash \psi $ .

In words, an empirical team supports no-signalling if and only if it can be realized by a hidden-variable team supporting ${\vec {z}}$ -independence and parameter independence.

Proof We first show that $\varphi \vdash \psi $ . First of all, assume $\mathop {=}\hspace {-0.7pt}({\vec {z}})$ , which is harmless in view of Proposition 3.17. Note that $\mathop {=}\hspace {-0.7pt}({\vec {z}})$ means $\vec z\perp \vec z$ . By the Constancy Rule, $\vec z\perp \vec z$ entails $\vec z\perp \vec x$ . Then fix $I\subseteq n$ . Note that $\mathop {=}\hspace {-0.7pt}({\vec {z}})$ also means $\mathop {=}\hspace {-0.7pt}(\emptyset ,{\vec {z}})$ . Now from Armstrong’s axioms, one can obtain $\mathop {=}\hspace {-0.7pt}(\vec w,{\vec {z}})$ for any variable tuple $\vec w$ , in particular, when $\vec w = \{x_i \mid i\in I\}\cup \{y_i \mid i\in I\}$ . Now $\mathop {=}\hspace {-0.7pt}(\{x_i \mid i\in I\}\{y_i \mid i\in I\},\vec z)$ means $\vec z\perp _{\{x_i \mid i\in I\}\{y_i \mid i\in I\}}\vec z$ . By the Constancy Rule, $\vec z\perp _{\{x_i \mid i\in I\}\{y_i \mid i\in I\}}\vec z$ entails $\{x_i \mid i\notin I\}\perp _{\{x_i \mid i\in I\}\{y_i \mid i\in I\}}\vec z$ . Then by Contraction, $\{x_i \mid i\notin I\}\perp _{\{x_i \mid i\in I\}}\{y_i \mid i\in I\}$ and $\{x_i \mid i\notin I\}\perp _{\{x_i \mid i\in I\}\{y_i \mid i\in I\}}\vec z$ together entail $\{ x_i \mid i\notin I \}\perp _{\{x_i \mid i\in I\}}\{y_i \mid i\in I\}\vec {z}$ , which by Weak Union entails $\{ x_i \mid i\notin I \}\perp _{\{x_i \mid i\in I\}\vec {z}}\{y_i \mid i\in I\}$ .

Hence from the assumptions $\mathop {=}\hspace {-0.7pt}(\vec z)$ and ${\{x_i \mid i\notin I\}\perp _{\{x_i \mid i\in I\}}\{y_i \mid i\in I\}}$ , we can deduce $\vec z\perp \vec x \land \{ x_i \mid i\notin I \}\perp _{\{x_i \mid i\in I\}\vec {z}}\{y_i \mid i\in I\}$ . Introducing conjunctions and existential quantifiers, we obtain that $\mathop {=}\hspace {-0.7pt}({\vec {z}})$ and $\varphi $ entail

$$\begin{align*}\exists z_0\dots\exists z_{l-1} \left( \vec{z}\perp\vec{x} \land \bigwedge_{I\subseteq n} \{x_i \mid i\notin I\}\perp_{\{x_i\mid i\in I\}\vec{z}}\{y_i\mid i\in I\} \right), \end{align*}$$

and hence by eliminating the existential quantifiers of the formula $\exists z_0\dots \exists z_{l-1}\mathop {=}\hspace {-0.7pt}(\vec z)$ , we obtain that $\exists z_0\dots \exists z_{l-1}\mathop {=}\hspace {-0.7pt}(\vec z)$ and $\varphi $ together entail

$$\begin{align*}\exists z_0\dots\exists z_{l-1} \left( \vec{z}\perp\vec{x} \land \bigwedge_{I\subseteq n}\{x_i \mid i\notin I\}\perp_{\{x_i\mid i\in I\}{\vec{z}}}\{y_i \mid i\in I\} \right). \end{align*}$$

Then the elimination rule of $\tilde \exists $ gives that $\tilde \exists z_0\exists z_1\dots \exists z_{l-1}\mathop {=}\hspace {-0.7pt}(\vec z)$ and $\varphi $ entail  $\psi $ . As $\tilde \exists z_0\exists z_1\dots \exists z_{l-1}\mathop {=}\hspace {-0.7pt}(\vec z)$ can be deduced with no assumptions, we obtain the desired deduction.

Next we show that $\psi \vdash \varphi $ . Given $I\subseteq n$ , consider the assumptions ${\vec {z}\perp \vec {x}}$ and $\{ x_i \mid i\notin I \}\perp _{\{x_i\mid i\in I\}\vec {z}}\{y_i \mid i\in I\}$ . By Weak Union, Permutation, and Symmetry, from $\vec z\perp \vec x$ , we obtain $\{ x_i \mid i\notin I \}\perp _{\{ x_i \mid i\in I \}}\vec z$ . From $\{ x_i \mid i\notin I \}\perp _{\{ x_i \mid i\in I \}}\vec z$ and $\{x_i \mid i\notin I\}\perp _{\{x_i\mid i\in I\}\vec {z}}\{y_i\mid i\in I\}$ , Contraction gives $\{x_i \mid i\notin I\}\perp _{\{x_i\mid i\in I\}}\{y_i\mid i\in I\}{\vec {z}}$ , whence the Weakening Rule yields $\{x_i \mid i\notin I\}\perp _{\{x_i\mid i\in I\}}\{y_i\mid i\in I\}$ . Introducing conjunctions, $\vec {z}\perp \vec {x}$ together with $\bigwedge _{I\subseteq n} \{x_i \mid i\notin I\}\perp _{\{x_i\mid i\in I\}\vec {z}}\{y_i\mid i\in I\}$ entails $\bigwedge _{I\subseteq n} \{x_i \mid i\notin I\}\perp _{\{x_i\mid i\in I\}}\{y_i\mid i\in I\}$ , so by elimination of existential quantifiers in $\exists z_0\dots \exists z_{l-1} ( \vec {z}\perp \vec {x} \land \bigwedge _{I\subseteq n} \{x_i \mid i\notin I\}\perp _{\{x_i\mid i\in I\}\vec {z}}\{y_i\mid i\in I\} )$ , we obtain the deduction $\exists z_0\dots \exists z_{l-1} ( \vec {z}\perp \vec {x} \land \bigwedge _{I\subseteq n} \{x_i \mid i\notin I\}\perp _{\{x_i\mid i\in I\}\vec {z}}\{y_i\mid i\in I\} )$ $\vdash \varphi $ . By the elimination rule of $\tilde \exists $ , $\psi \vdash \varphi $ finally follows.

In words, every empirical team is realized by a hidden-variable team supporting strong determinism.

In words, every empirical team is realized by a hidden-variable team supporting weak determinism and ${\vec {z}}$ -independence.

Proposition 3.22. Let

$$ \begin{align*} \varphi &= \bigwedge_{I\subseteq n}\left( \{x_i \mid i\notin I\}\perp_{\{x_i\mid i\in I\}\vec{z}}\{y_i\mid i\in I\} \right), \\ \psi &= \bigwedge_{i<n}\left( y_i\perp_{\vec{x}\vec{z}}\{y_j \mid j\neq i \} \right) \text{ and} \\ \theta &= \bigwedge_{i<n} \mathop{=}\hspace{-0.7pt}(x_i\vec{z},y_i). \end{align*} $$

Then $\tilde {\exists }z_0\exists z_1\dots \exists z_{l-1}({\vec {z}}\perp {\vec {x}}\land \varphi \land \psi ) \models \tilde {\exists }z_0\exists z_1\dots \exists z_{l-1}({\vec {z}}\perp {\vec {x}}\land \theta )$ .

In words, any hidden-variable team supporting ${\vec {z}}$ -independence and locality is equivalent (in the sense of Definition 3.2) to a hidden-variable team supporting ${\vec {z}}$ -independence and strong determinism.

We do not know whether the logical consequences of Propositions 3.19–3.22 are provable from our axioms. This is a subject of further study. Remark 3.20 would suggest that stronger axioms for $\tilde \exists $ be required, as the semantic proofs of these propositions make use of the possibility of acquiring values for the hidden variables ${\vec {z}}$ outside of the values that are possible for ${\vec {x}}$ and ${\vec {y}}$ .

3.5 Representation of no-go theorems in team semantics

We now turn to the representation of no-go theorems in the foundations of quantum mechanics in terms of team semantics. These results have fundamental significance, both foundationally, and also for their implications for quantum information and computation. They rule out the possibility, even in principle, of accounting for quantum behaviour by means of local hidden-variable theories. This shows that the behaviour of quantum mechanics is essentially and unavoidably non-local. This non-locality is, on the one hand, highly challenging in terms of understanding what quantum mechanics is telling us about the nature of physical reality. On the other hand, this non-classicality opens up the possibility of performing information processing tasks using quantum resources which provably exceed what can be done classically.

How can these no-go theorems be represented in terms of team semantics? All the results so far are examples of logical consequences and equivalences between team properties (or formulas). To prove the no-go theorems, we shall exhibit some counter-example teams that demonstrate certain failures of logical consequence. These failures will imply the impossibility of describing these teams in terms of local hidden variables. As we shall see later, these teams do arise as behaviours of certain quantum mechanical systems.

The original result in this line is the celebrated Bell’s Theorem [Reference Bell9]. However, that result in its original form is probabilistic in character, and hence not amenable to formalization in terms of team semantics.Footnote ⁶ Two later constructions, due to Greenberger–Horne–Zeilinger (GHZ) [Reference Greenberger, Horne, Shimony and Zeilinger22, Reference Liu, Zhou, Meng, Yang, Li, Meng, Su, Chen, Sun, Xu, Li and Guo37] and Hardy [Reference Hardy27], strengthen Bell’s result by constructions which work at the purely possibilistic level, and hence can be formalized directly in team semantics.

In Tarski semantics of first-order logic, some existential formulas, such as $\exists z (x=z \land \neg y=z)$ , are not valid while others, such as $\exists z (x=z \lor x=y)$ , are. To decide which are valid and which are not is particularly simple, especially in the empty vocabulary because first-order logic has in that case elimination of quantifiers. In team semantics where such quantifier elimination is not known to be possible, non-valid existential-conjunctive formulas can be quite complicated, as the examples below show. As we shall see, the no-go results of quantum mechanics give rise to very interesting teams.

We turn firstly to the GHZ construction.

Definition 3.23. Assume that $n=3$ . Let X be an empirical team with $\operatorname {\mathrm {rng}}(X) = \{0,1\}$ . Denote

$$ \begin{align*} P &= \{(0,1,1), (1,0,1), (1,1,0)\}, \\ Q &= \{(0,0,0), (0,1,1), (1,0,1), (1,1,0)\} \text{ and}\\ R &= \{(0,0,1), (0,1,0), (1,0,0), (1,1,1)\}. \end{align*} $$

We say that X is a GHZ team if it satisfies the following conditions.

1. (i) $Q = \{s(\vec {y}) \mid s\in X, s(\vec {x})\in P\}$ and $P \subseteq \{s(\vec {x}) \mid s\in X, s(\vec {y})\in Q\}$ .

2. (ii) $R = \{s(\vec {y}) \mid s\in X, s(\vec {x}) = (0,0,0)\}$ .

The following is a minimal example of a GHZ team:

The following is like [Reference Abramsky1, Proposition 6.2].

Proposition 3.24. The formula $\tilde {\exists }z_0\exists z_1\dots \exists z_{l-1}({\vec {z}}\perp {\vec {x}}\land \varphi \land \psi )$ , where

$$ \begin{align*} \varphi &= \bigwedge_{I\subseteq n}\left( \{x_i \mid i\notin I\}\perp_{\{x_i\mid i\in I\}\vec{z}}\{y_i\mid i\in I\} \right) \text{ and} \\ \psi &= \bigwedge_{i<n}\left( y_i\perp_{\vec{x}\vec{z}}\{y_j \mid j\neq i \} \right), \end{align*} $$

is not valid, as demonstrated by any GHZ team.

In words, no GHZ team can be realized by a hidden-variable team supporting ${\vec {z}}$ -independence and locality.

Next, we consider the Hardy construction.

Definition 3.25. Assume that $n=2$ . Let X be an empirical team with $\operatorname {\mathrm {rng}}(X) = \{0,1\}$ . Let $s_0,\dots ,s_3$ be as in the following table.

We say that X is a Hardy team if the following hold:

1. (i) $s_0\in X$ but $s_1,s_2,s_3\notin X$ , and

2. (ii) for every pair $\vec {a}\in \{0,1\}^2$ , there is some $s\in X$ with $s(\vec {x})=\vec {a}$ .Footnote ⁷

A minimal example of a Hardy team would be the following:

The following is like [Reference Abramsky1, Proposition 6.3].

Theorem 3.26 gives an alternative proof that the formula in Theorem 3.24 is not valid.

4.1 Probabilistic teams

So far, we have only been looking at possibilistic (i.e., two-valued relational) versions of the independence notions of quantum physics while these notions are usually taken to be probabilistic. To be able to discuss the probabilistic notions from the point of view of team semantics, we need a suitable framework. For this, we consider probabilistic team semantics.

The study of a probabilistic variant of independence logic was first done in a multiteam setting in [Reference Durand, Hannula, Kontinen, Meier and Virtema14]. Prior to that, multiteams were studied in [Reference Hyttinen, Paolini and Väänänen30, Reference Hyttinen, Paolini and Väänänen31, Reference Väänänen47]. Probabilistic teams were then introduced by Durand et al. in [Reference Durand, Hannula, Kontinen, Meier, Virtema, Ferrarotti and Woltran15] as a way to generalize multiteams, and further investigated in [Reference Hannula, Hirvonen, Kontinen, Kulikov and Virtema25]. They can be thought of as a special case of measure teams, another approach to probabilities in team semantics given in [Reference Hyttinen, Paolini and Väänänen31].

It should be observed that we are not introducing probabilistic logic in the sense of formulas having probabilities. In our approach, only the teams are probabilistic and the logic is two-valued.

Ordinary teams of size k can be seen as probabilistic teams by giving each assignment in the team probability $1/k$ and assignments not in the team probability zero. This idea of treating ordinary teams as uniformly distributed probabilistic teams generalizes to multiteams, i.e., teams in which assignments can have several occurrences. Then an assignment which occurs m times is given the probability $m/k$ . In fact, it is not difficult to see that any probabilistic team with rational probabilities corresponds to a multiteam.

We will call a probabilistic team with variable domain $V_{\text {m}}\cup V_{\text {o}}$ a probabilistic empirical team and a probabilistic team with variable domain $V_{\text {m}}\cup V_{\text {o}}\cup V_{\text {h}}$ a probabilistic hidden-variable team.

Note that if $\mathbb {X}$ is a probabilistic team of $\mathfrak {A}$ , then the possibilistic collapse X is a team of $\mathfrak {A}$ .

We may consider a probabilistic team a “probabilistic realization” of its collapse. Of course, an ordinary team has a multitude of such probabilistic realizations.

The possibilistic collapse of a probabilistic team $\mathbb {X}$ is also called the support of $\mathbb {X}$ and denoted by $\operatorname {\mathrm {supp}}\mathbb {X}$ .

We denote by $\left | \mathbb {X}_{\vec {u}=\vec {a}} \right |$ the number

$$\begin{align*}\sum_{\substack{s(\vec{u})=\vec{a} \\ s\in\operatorname{\mathrm{supp}}\mathbb{X}}}\mathbb{X}(s), \end{align*}$$

i.e., the marginal probability of the variable tuple $\vec {u}$ having the value $\vec {a}$ in $\mathbb {X}$ .

Next we define the probabilistic analogue for an empirical team being realized by a hidden-variable team.

The intuition behind the definition is the following: $\mathbb {Y}$ realizes $\mathbb {X}$ if the probability of the event “ $s(\vec {x})=\vec {a}\,$ ” is non-zero in both teams exactly the same time, and in the case that the probability indeed is non-zero, the probability of the event “ $s(\vec {y})=\vec {b}\,$ ”, conditional to “ $s(\vec {x})=\vec {a}\,$ ”, is the same in both teams. Uniform realizability appears to be a stronger concept than realizability, but we do not have an example for that yet.

Proposition 4.4 says that one obtains the same team by first projecting away hidden variables and then taking the possibilistic collapse as one gets by first taking the possibilistic collapse and then projecting away the hidden variables, i.e., the diagram in Figure 1 commutes.

Figure 1 Probabilistic realization implies possibilistic realization.

Proof of Proposition 4.4

Suppose that $\mathbb {Y}$ realizes $\mathbb {X}$ , and denote by Y and X the respective possibilistic collapse. In order to prove that Y realizes X, we need to show that for all assignments s,

$$\begin{align*}s\in X \iff \exists s'\in Y \bigwedge_{i<n}(s'(x_i)=s(x_i) \land s'(y_i)=s(y_i)). \end{align*}$$

We show only one direction, the other one is similar.

Suppose that $s\in X$ . Denote $\vec {a}=s(\vec {x})$ and $\vec {b}=s(\vec {y})$ . The aim is to show that there is some $s'\in Y$ with $s'(\vec {x})=\vec {a}$ and $s'(\vec {y})=\vec {b}$ . Since X is the possibilistic collapse of $\mathbb {X}$ and $s\in X$ , we have $\mathbb {X}(s)>0$ , and as in addition $s(\vec {x})=\vec {a}$ , we obtain $\left | \mathbb {X}_{\vec {x}=\vec {a}} \right |> 0$ . Thus, as $\mathbb {Y}$ realizes $\mathbb {X}$ , $\left | \mathbb {Y}_{\vec {x}=\vec {a}} \right |> 0$ . Similarly, $\left | \mathbb {X}_{\vec {x}\vec {y}=\vec {a}\vec {b}} \right |> 0$ , and as

$$\begin{align*}\left| \mathbb{Y}_{\vec{x}\vec{y}=\vec{a}\vec{b}} \right| = \frac{\left| \mathbb{X}_{\vec{x}\vec{y}=\vec{a}\vec{b}} \right|\cdot\left| \mathbb{Y}_{\vec{x}=\vec{a}} \right|}{\left| \mathbb{X}_{\vec{x}=\vec{a}} \right|}, \end{align*}$$

also $\left | \mathbb {Y}_{\vec {x}\vec {y}=\vec {a}\vec {b}} \right |> 0$ . This means that there is some $s'$ with $\mathbb {Y}(s')> 0$ and $s'(\vec {x}\vec {y})=\vec {a}\vec {b}$ . Since Y is the possibilistic collapse of $\mathbb {Y}$ , this means that $s'\in Y$ , as desired.

4.2 Probabilistic independence logic

We now present the semantics of the probabilistic (conditional) independence atom $\vec {u}\mathbin {\perp \!\!\!\perp }_{\vec {v}}\vec {w}$ , as defined in [Reference Durand, Hannula, Kontinen, Meier, Virtema, Ferrarotti and Woltran15].

Definition 4.5. Let $\mathfrak {A}$ be a structure and $\mathbb {X}$ a probabilistic team of $\mathfrak {A}$ , and let $\vec {u}$ , $\vec {v}$ , and $\vec {w}$ be tuples of variables. Then $\mathbb {X}$ satisfies the formula $\vec {u}\mathbin {\perp \!\!\!\perp }_{\vec {v}}\vec {w}$ in $\mathfrak {A}$ , in symbols $\mathfrak {A}\models _{\mathbb {X}}\vec {u}\mathbin {\perp \!\!\!\perp }_{\vec {v}}\vec {w}$ , if for all $\vec {a}$ , $\vec {b}$ , and $\vec {c}$ ,

$$\begin{align*}\left| \mathbb{X}_{\vec{u}\vec{v}=\vec{a}\vec{b}} \right| \cdot \left| \mathbb{X}_{\vec{v}\vec{w}=\vec{b}\vec{c}} \right| = \left| \mathbb{X}_{\vec{u}\vec{v}\vec{w}=\vec{a}\vec{b}\vec{c}} \right| \cdot \left| \mathbb{X}_{\vec{v}=\vec{b}} \right|. \end{align*}$$

The intention behind the atom is to capture the notion of conditional independence in probability theory: denoting a probability measure by p, two events A and B are conditionally independent over an event C (assuming $p(C)>0$ ) if

$$\begin{align*}p(A\mid C)p(B\mid C) = p(A\cap B\mid C). \end{align*}$$

Recalling that $p(D\mid C) = p(D\cap C)/p(C)$ , we can multiply both sides of the equation by $p(C)^2$ and obtain

$$\begin{align*}p(A\cap C)p(B\cap C) = p(A\cap B\cap C)p(C), \end{align*}$$

which is exactly what the probabilistic dependence atom expresses. In the case when $p(C)=0$ , both sides of the new equation are $0$ , so in that case, the independence atom is vacuously true.

Exactly the same way as in ordinary independence logic, we can define the probabilistic dependence atom $\mathop {=}\hspace {-0.7pt}(\vec {v},\vec {w})$ via the independence atom.

The syntax of probabilistic independence logic is the same as the syntax of ordinary independence logic, except that we use the symbol $\mathbin {\perp \!\!\!\perp }$ instead of $\perp $ . Next we present the semantics of more complex formulas of probabilistic independence logic, as defined in [Reference Durand, Hannula, Kontinen, Meier, Virtema, Ferrarotti and Woltran15]. First we define the r-scaled union $\mathbb {X}\sqcup _r\mathbb {Y}$ of two probabilistic teams $\mathbb {X}$ and $\mathbb {Y}$ with the same variable and value domain, for $r\in [0,1]$ , by setting

We define the (“duplicated”) team $\mathbb {X}[A_{{\mathfrak {s}}(v)}/v]$ by setting

for all $a\in A_{{\mathfrak {s}}(v)}$ . If $v$ is a fresh variable, i.e., not in the variable domain of $\mathbb {X}$ , then $\mathbb {X}[A/v](s(a/v))=\mathbb {X}(s)/\left | A_{{\mathfrak {s}}(v)} \right |$ for all $s\in \operatorname {\mathrm {supp}}\mathbb {X}$ . Finally, given a function F from the set $\operatorname {\mathrm {supp}}\mathbb {X}$ to the set of all probability distributions on $A_{{\mathfrak {s}}(v)}$ , we define the (“supplemented”) team $\mathbb {X}[F/v]$ by setting

for all $a\in A_{{\mathfrak {s}}(v)}$ . Again, if $v$ is fresh, then $\mathbb {X}[F/v](s(a/v)) = \mathbb {X}(s)F(s)(a)$ . It is easy to see that both duplication and supplementation give rise to well-defined probabilistic teams.

Again, when it is clear what is meant, we write $\mathbb {X}\models \varphi $ instead of $\mathfrak {A}\models _{\mathbb {X}}\varphi $ .

By definition, first-order atomic formulas are satisfied by a probabilistic team if and only if the underlying possibilistic collapse satisfies them. This property is in the multiteam setting of [Reference Durand, Hannula, Kontinen, Meier and Virtema14] called weak flatness.

Definition 4.8. We say that a formula $\varphi $ of probabilistic independence logic is weakly flat if for all probabilistic teams $\mathbb {X}$ , we have

$$\begin{align*}\mathbb{X}\models\varphi \iff \operatorname{\mathrm{supp}}\mathbb{X}\models\varphi^{*}, \end{align*}$$

where $\varphi ^{*}$ is the formula of independence logic obtained from $\varphi $ by replacing each occurrence of the symbol $\mathbin {\perp \!\!\!\perp }$ by the symbol $\perp $ . A sublogic of probabilistic independence logic is weakly flat if every formula of the logic is.

Later on, we simply write $\varphi $ instead of $\varphi ^{*}$ whenever it is obvious what is meant.

It turns out that one direction of the equivalence in the definition of weak flatness always holds.

Proposition 4.10. Let $\mathbb {X}$ be a probabilistic team and $\varphi $ a formula of independence logic. Denote by X the possibilistic collapse of $\mathbb {X}$ . Then

$$\begin{align*}\mathbb{X}\models\varphi \implies X\models\varphi. \end{align*}$$

We omit the proof, as it is available in [Reference Albert and Grädel6].

Proof Suppose that $\varphi $ and $\psi $ are weakly flat. We then show that the formulas $\varphi \land \psi $ , $\varphi \lor \psi $ , $\exists v\varphi $ , $\forall v\varphi $ , $\tilde {\exists }v\varphi $ , and $\tilde {\forall }v\varphi $ are weakly flat. Let $\mathfrak {A}$ be a structure and $\mathbb {X}$ a probabilistic team of $\mathfrak {A}$ , and let X be the possibilistic collapse of $\mathbb {X}$ . Note that to show that a formula $\theta $ is weakly flat, we only need to show that

$$\begin{align*}\mathfrak{A}\models_X\theta \implies \mathfrak{A}\models_{\mathbb{X}}\theta, \end{align*}$$

as the other direction follows from Proposition 4.10.

1. (i) The case for conjunction is trivial.

2. (ii) Suppose that $X\models \varphi \lor \psi $ . Then there are $X_0$ and $X_1$ such that $X_0\models \varphi $ and $X_1\models \psi $ and $X=X_0\cup X_1$ . Let $\mathbb {X}_i$ be a probabilistic team with collapse $X_i$ such that

   $$\begin{align*}\mathbb{X}_i(s) = \begin{cases} \mathbb{X}(s)/(2p_i+q) & \text{if } s\in X_0\cap X_1, \\ \mathbb{X}(s)/(p_i+q/2) & \text{otherwise,} \end{cases} \end{align*}$$
   where $p_i=\sum _{s\in X_i\setminus X_{1-i}}\mathbb {X}(s)$ and $q = \sum _{s\in X_0\cap X_1}\mathbb {X}(s)$ . As $\varphi $ and $\psi $ are weakly flat, $\mathbb {X}_0\models \varphi $ and $\mathbb {X}_1\models \psi $ . Now, if $s\in X_0\setminus X_1$ , then
   $$ \begin{align*} \mathbb{X}(s) &= \frac{(p_0+q/2)\mathbb{X}(s)}{p_0+q/2} = (p_0+q/2)\mathbb{X}_0(s) \\ &= (\mathbb{X}_0\sqcup_{p_0 + q/2}\mathbb{X}_1)(s), \end{align*} $$
   and if $s\in X_1\setminus X_0$ , then
   $$ \begin{align*} \mathbb{X}(s) &= \frac{(p_1+q/2)\mathbb{X}(s)}{p_1+q/2} = (p_1+q/2)\mathbb{X}_1(s)\\ & = (1 - (p_0 + q/2))\mathbb{X}_1(s) = (\mathbb{X}_0\sqcup_{p_0 + q/2}\mathbb{X}_1)(s), \end{align*} $$
   and if $s\in X_0\cap X_1$ , then
   $$ \begin{align*} \mathbb{X}(s) &= \frac{\mathbb{X}(s)}{2} + \frac{\mathbb{X}(s)}{2} = \frac{(p_0 + q/2)\mathbb{X}(s)}{2(p_0 + q/2)} + \frac{(p_1 + q/2)\mathbb{X}(s)}{2(p_1 + q/2)} \\ &= \frac{(p_0 + q/2)\mathbb{X}(s)}{2p_0 + q} + \frac{(p_1 + q/2)\mathbb{X}(s)}{2p_1 + q} \\ &= (p_0 + q/2)\mathbb{X}_0(s) + (p_1 + q/2)\mathbb{X}_1(s) \\ &= (p_0 + q/2)\mathbb{X}_0(s) + (1 - (p_0 + q/2))\mathbb{X}_1(s) \\ &= (\mathbb{X}_0\sqcup_{p_0 + q/2}\mathbb{X}_1)(s). \end{align*} $$

   Hence $\mathbb {X}=\mathbb {X}_0\sqcup _{p_0 + q/2}\mathbb {X}_1$ and thus $\mathbb {X}\models \varphi \lor \psi $ .

3. (iii) Suppose that $X\models \exists v\varphi $ . Then there is a function $F\colon X\to A_{{\mathfrak {s}}(v)}$ such that $X[F/v]\models \varphi $ . Define a function $G\colon X\to \{p\in [0,1]^{A_{{\mathfrak {s}}(v)}} \mid \,p \text { is a distribution}\}$ by setting

   $$\begin{align*}G(s)(a) = \begin{cases} 1/\left| F(s) \right| & \text{if } a\in F(s), \\ 0 & \text{otherwise.} \end{cases} \end{align*}$$

   Then $X[F/v]$ is the possibilistic collapse of $\mathbb {X}[G/v]$ , as

   $$ \begin{align*} & \mathbb{X}[G/v](s(a/v))> 0 \iff \sum_{\substack{t\in X \\ t(a/v)=s(a/v)}}\mathbb{X}(t)G(t)(a) > 0 \\ &\quad \iff \exists t\in X\ (G(t)(a) > 0\ \text{and}\ t(a/v)=s(a/v)) \\ &\quad \iff \exists t\in X\ (a\in F(t)\ \text{and}\ t(a/v)=s(a/v)) \\ &\quad \iff s(a/v)\in X[F/v]. \end{align*} $$

   Then as $\varphi $ is weakly flat, $\mathbb {X}[G/v]\models \varphi $ , so $\mathbb {X}\models \varphi $ .

4. (iv) The universal quantifier case is similar.

5. (v) Suppose that $\mathfrak {A}\models _{X}\tilde {\exists }v\varphi $ . Then there is an expansion $\mathfrak {B}$ of $\mathfrak {A}$ of the new sort ${\mathfrak {s}}(v)$ such that $\mathfrak {B}\models _{X}\exists v\varphi $ . We already showed that $\exists v\varphi $ is weakly flat, so thus $\mathfrak {B}\models _{\mathbb {X}}\exists v\varphi $ . Thus $\mathfrak {A}\models _{\mathbb {X}}\tilde {\exists }v\varphi $ .

6. (vi) The universal sort quantifier case is similar.

In ordinary team semantics, the dependence atom is downwards closed, meaning that if $X\models \mathop {=}\hspace {-0.7pt}(\vec {v},\vec {w})$ , then for any $Y\subseteq X$ also $Y\models \mathop {=}\hspace {-0.7pt}(\vec {v},\vec {w})$ . We define an analogous concept of downwards closedness and show that dependence logic is downwards closed also in probabilistic team semantics.

The concept of a weak subteam is the weakest notion of subteam that one would think of. Still, due to weak flatness, it seems to be enough.

Notice that all atomic formulas of probabilistic dependence logic are weakly flat, as proved in [Reference Durand, Hannula, Kontinen, Meier and Virtema14]. Hence we obtain the following corollary.

Next we show that logical operations preserve downwards closedness. To make it easier, we first show that one may change the name of a bound variable without affecting the truth of the formula.

Lemma 4.16 (Locality, [Reference Durand, Hannula, Kontinen, Meier, Virtema, Ferrarotti and Woltran15])

Let $\varphi $ be a formula of probabilistic independence logic, with its free variables among $v_0,\dots ,v_{m-1}$ . Then for all probabilistic teams $\mathbb {X}$ whose variable domain D includes the variables $v_i$ and any set V such that $\{v_0,\dots ,v_{m-1}\}\subseteq V\subseteq D$ , we have

$$\begin{align*}\mathbb{X}\models\varphi \iff \mathbb{X}\restriction V\models\varphi, \end{align*}$$

where $\mathbb {X}\restriction V$ is the probabilistic team $\mathbb {Y}$ with variable domain V defined by $\mathbb {Y}(s)=\sum _{t\restriction V = s}\mathbb {X}(t)$ .

Proof The statement of the lemma easily follows from the following claim.

Let $\mathbb {X}$ and $\mathbb {Y}$ be probabilistic teams with variable domain $D_{\mathbb {X}} = \{v_0,\dots ,v_{n-1}\}$ and $D_{\mathbb {Y}} = \{w_0,\dots ,w_{n-1}\}$ , respectively, such that

1. (i) $\operatorname {\mathrm {supp}}\mathbb {Y} = \{s^{*} \mid s\in \operatorname {\mathrm {supp}}\mathbb {X}\}$ , where $s^{*}$ is the assignment with domain $D_{\mathbb {Y}}$ such that $s^{*}(w_i) = s(v_i)$ for all $i<n$ , and

2. (ii) $\mathbb {X}(s) = \mathbb {Y}(s^{*})$ for all $s\in \operatorname {\mathrm {supp}}\mathbb {X}$ .

Then for any $\varphi $ with free variables in $D_{\mathbb {X}}$ ,

$$\begin{align*}\mathbb{X}\models\varphi \iff \mathbb{Y}\models\varphi(w_0/v_0,\dots,w_{n-1}/v_{n-1}). \end{align*}$$

The claim can be proved with a straightforward induction on $\varphi $ .

One might wonder whether ordinary independence logic is a result of “collapsing” probabilistic independence logic in the sense that a team X satisfies a formula $\varphi $ if and only if it is the collapse of some probabilistic team $\mathbb {X}$ such that $\mathbb {X}$ satisfies the probabilistic version of $\varphi $ . It turns out, in Proposition 4.10, that, indeed, given a probabilistic team that satisfies a formula, also the collapse will satisfy the (possibilistic version of the) formula. But given an ordinary team that satisfies a formula, there may not be any probabilistic realization of that team that would satisfy the (probabilistic version of the) formula. We will see in Proposition 4.30 that such a formula and a team can be quite simple.

We add a new operation ${\mathsf {P}\hspace {-0.5pt}\mathsf {R}}$ to ordinary independence logic, defined by

One can then ask whether this operation is downwards closed, closed under unions, $\Sigma _1^1$ -definable, etc.

By weak flatness, for any formula $\varphi $ of dependence logic, we have $\varphi \equiv {\mathsf {P}\hspace {-0.5pt}\mathsf {R}}\varphi $ .

A similar discussion can be applied to logical consequence. Logical consequence in ordinary team semantics is different from logical consequence in probabilistic team semantics: as was shown by Studený [Reference Studeny44] and also pointed out in the team semantics context by Albert and Grädel [Reference Albert and Grädel6], the following is an example of a rule that is sound in team semantics but not in probabilistic team semantics:

$$\begin{align*}\{{\vec{x}}\perp_{{\vec{z}}}{\vec{y}},{\vec{x}}\perp_{{\vec{u}}}{\vec{y}}, {\vec{z}}\perp_{{\vec{x}}{\vec{y}}}{\vec{u}}\} \quad\text{entails}\quad {\vec{x}}\perp_{{\vec{z}}{\vec{u}}}{\vec{y}}, \end{align*}$$

while the following is an example of a rule that is sound in probabilistic team semantics but not in ordinary team semantics:

$$\begin{align*}\{{\vec{x}}\mathbin{\perp\!\!\!\perp}_{{\vec{y}}{\vec{z}}}{\vec{y}}, {\vec{z}}\mathbin{\perp\!\!\!\perp}_{{\vec{x}}}{\vec{u}}, {\vec{z}}\mathbin{\perp\!\!\!\perp}_{{\vec{y}}}{\vec{u}}, {\vec{x}}\mathbin{\perp\!\!\!\perp}{\vec{y}}\} \quad\text{entails}\quad {\vec{z}}\mathbin{\perp\!\!\!\perp}{\vec{u}}. \end{align*}$$

Certainly this means that such rules cannot be derived from the axioms of Definition 2.5 (or even the semigraphoid axioms), since the axioms are satisfied by both ordinary and probabilistic independence logic, as will be shown in Proposition 4.25.

It was recently proved that the implication problem for probabilistic independence atoms is undecidable [Reference Li36].

Earlier we noted that in ordinary team semantics, being a realization of a hidden-variable team is expressible by means of existential quantifiers. We show that this is also the case in probabilistic team semantics.

Lemma 4.19. Let $\mathfrak {A}$ be a structure and $\mathfrak {B}$ an expansion of $\mathfrak {A}$ by the hidden-variable sort. Let $\mathbb {X}$ be a probabilistic empirical team of $\mathfrak {A}$ and $\mathbb {Y}$ be a probabilistic hidden-variable team of $\mathfrak {B}$ . Then $\mathbb {X}$ is uniformly realized by $\mathbb {Y}$ if and only if

$$\begin{align*}\mathbb{Y}=\mathbb{X}[F_0/z_0]\dots[F_0/z_{l-1}] \end{align*}$$

for some functions $F_i$ .

Proof For simplicity, we assume that $l=1$ and $\vec {z}=z$ .

Suppose that $\mathbb {Y}$ uniformly realizes $\mathbb {X}$ . Then for all $\vec {a}$ and $\vec {b}$ ,

1. (i) $\left | \mathbb {X}_{\vec {x}=\vec {a}} \right |=\left | \mathbb {Y}_{\vec {x}=\vec {a}} \right |$ , and

2. (ii) $\left | \mathbb {X}_{\vec {x}\vec {y}=\vec {a}\vec {b}} \right |\left | \mathbb {Y}_{\vec {x}=\vec {a}} \right | = \left | \mathbb {Y}_{\vec {x}\vec {y}=\vec {a}\vec {b}} \right |\left | \mathbb {X}_{\vec {x}=\vec {a}} \right |$ .

Define a function F by setting

$$\begin{align*}F(s)(\gamma) = \frac{\mathbb{Y}(s(\gamma/z))}{\left| \mathbb{Y}_{\vec{x}\vec{y}=s(\vec{x}\vec{y})} \right|} \end{align*}$$

for all $s\in \operatorname {\mathrm {supp}}\mathbb {X}$ and $\gamma $ . Now $F(s)$ is a distribution, as

$$\begin{align*}\sum_\gamma F(s)(\gamma) = \sum_\gamma \frac{\mathbb{Y}(s(\gamma/z))}{\left| \mathbb{Y}_{\vec{x}\vec{y}=s(\vec{x}\vec{y})} \right|} = \frac{\left| \mathbb{Y}_{\vec{x}\vec{y}=s(\vec{x}\vec{y})} \right|}{\left| \mathbb{Y}_{\vec{x}\vec{y}=s(\vec{x}\vec{y})} \right|} = 1. \end{align*}$$

Then

$$ \begin{align*} \mathbb{X}[F/z](\vec{x}\vec{y}z\mapsto\vec{a}\vec{b}\gamma) &= \mathbb{X}(\vec{x}\vec{y}\mapsto\vec{a}\vec{b})F(\vec{x}\vec{y}\mapsto\vec{a}\vec{b})(\gamma) \\ &= \left| \mathbb{X}_{\vec{x}\vec{y}=\vec{a}\vec{b}} \right|\cdot\frac{\mathbb{Y}(\vec{x}\vec{y}z\mapsto\vec{a}\vec{b}\gamma)}{\left| \mathbb{Y}_{\vec{x}\vec{y}=\vec{a}\vec{b}} \right|} \\ &= \frac{\left| \mathbb{X}_{\vec{x}=\vec{a}} \right|}{\left| \mathbb{Y}_{\vec{x}=\vec{a}} \right|}\mathbb{Y}(\vec{x}\vec{y}z\mapsto\vec{a}\vec{b}\gamma) \\ &= \mathbb{Y}(\vec{x}\vec{y}z\mapsto\vec{a}\vec{b}\gamma), \end{align*} $$

whence $\mathbb {Y}=\mathbb {X}[F/z]$ . Thus $\mathbb {X}[F/z]\models \varphi $ , so $\mathbb {X}\models \exists z\varphi $ .

Conversely, suppose that $\mathbb {Y}=\mathbb {X}[F/z]$ for some F. Then it is clear that $\left | \mathbb {X}_{\vec {x}=\vec {a}} \right |=0$ if and only if $\left | \mathbb {Y}_{\vec {x}=\vec {a}} \right |=0$ for any $\vec {a}$ . Now

$$ \begin{align*} \left| \mathbb{Y}_{\vec{x}\vec{y}=\vec{a}\vec{b}} \right|\left| \mathbb{X}_{\vec{x}=\vec{a}} \right| &= \left| \mathbb{X}_{\vec{x}=\vec{a}} \right|\sum_\gamma\mathbb{Y}(\vec{x}\vec{y}z\mapsto\vec{a}\vec{b}\gamma) \\ &= \left| \mathbb{X}_{\vec{x}=\vec{a}} \right|\sum_\gamma\mathbb{X}(\vec{x}\vec{y}\mapsto\vec{a}\vec{b})F(\vec{x}\vec{y}\mapsto\vec{a}\vec{b})(\gamma) \\ &= \left| \mathbb{X}_{\vec{x}=\vec{a}} \right|\mathbb{X}(\vec{x}\vec{y}\mapsto\vec{a}\vec{b}) \\ &= \left| \mathbb{X}_{\vec{x}=\vec{a}} \right|\left| \mathbb{X}_{\vec{x}\vec{y}=\vec{a}\vec{b}} \right| \\ &= \left| \mathbb{X}_{\vec{x}\vec{y}=\vec{a}\vec{b}} \right|\left| \mathbb{Y}_{\vec{x}=\vec{a}} \right|. \end{align*} $$

Thus $\mathbb {Y}$ uniformly realizes $\mathbb {X}$ .

A consequence of Lemma 4.19 is that if $\varphi (\vec {x},\vec {y},\vec {z})$ is a formula of probabilistic independence logic and defines a property of probabilistic hidden-variable teams, then $\tilde {\exists } z_0\exists z_1\dots \exists z_{l-1}\varphi $ defines the class of probabilistic empirical teams that are uniformly realized by hidden-variable teams satisfying $\varphi $ .

4.3 K-teams

In this section, we study a version of team semantics using semirings. It can be viewed as a generalization of both ordinary and probabilistic team semantics.

Semiring relations, which we can consider semiring teams, were introduced in [Reference Green, Karvounarakis and Tannen21] to study the provenance of relational database queries. In [Reference Grädel and Tannen19], provenance of first-order formulas was studied by defining a semiring semantics for first-order logic. A similar approach to team semantics was taken in [Reference Barlag, Hannula, Kontinen, Pardal, Virtema, Marquis, Son and Kern-Isberner8]. K-relations are studied in [Reference Hannula24] as a unifying framework for conditional independence and other similar notions.

In the sheaf-theoretic approach to contextuality and non-locality introduced in [Reference Abramsky and Brandenburger3], semiring-valued distributions are used to give a unified account of probabilistic and possibilistic forms of contextuality and non-locality, as well as signed measures (“negative probabilities”).

We say that a semiring K is commutative if also multiplication is commutative. We say that K is multiplicatively cancellative if for all ${a,b,c\in K}$ ,

$$\begin{align*}ab = ac \text{ and } a \neq 0 \implies b = c. \end{align*}$$

We say that K is positive if it is plus-positive, i.e.,

$$\begin{align*}a+b = 0 \implies a=b=0, \end{align*}$$

and has no zero-divisors, i.e.,

$$\begin{align*}ab = 0 \implies a = 0 \text{ or } b = 0. \end{align*}$$

Canonical semirings include

• • the natural numbers $(\mathbb {N},+,\cdot ,0,1)$ ,

• • multivariate polynomials $(\mathbb {N}[X_0,\dots ,X_{n-1}],+,\cdot ,0,1)$ over $\mathbb {N}$ ,

• • the Boolean semiring $\mathbb {B} = (\{0,1\},\lor ,\land ,0,1)$ , and

• • the non-negative reals $\mathbb {R}_{\geq 0} = ([0,\infty ),0,\cdot ,0,1)$ ,

as well as all rings.

K-teams are the natural generalization of probabilistic teams: all the relevant definitions are the same but with the interval $[0,1]$ of real numbers replaced with K.

Definition 4.21. Let K be a semiring and X the full team of a finite structure $\mathfrak {A}$ . A K-team of $\mathfrak {A}$ is a function $\mathbb {X}\colon X\to K$ . We denote by $\operatorname {\mathrm {supp}}\mathbb {X}$ the set $\{s\in X \mid \mathbb {X}(s) \neq 0\}$ . By $|\mathbb {X}_{{\vec {u}} = {\vec {a}}}|$ , we denote the sum

$$\begin{align*}\sum_{\substack{s(\vec{u})=\vec{a} \\ s\in\operatorname{\mathrm{supp}}\mathbb{X}}}\mathbb{X}(s). \end{align*}$$

We can view probabilistic teams as $\mathbb {R}_{\geq 0}$ -teams, multiteams as $\mathbb {N}$ -teams, and ordinary teams as $\mathbb {B}$ -teams. We say that a K-team $\mathbb {X}$ is total if $\operatorname {\mathrm {supp}}\mathbb {X}$ is the full team, i.e., $\mathbb {X}(s)> 0$ for all assignments s.

Definition 4.22. Let $\mathfrak {A}$ be a finite structure, $\mathbb {X}$ a K-team of $\mathfrak {A}$ , and ${\vec {u}}$ , ${\vec {v}}$ , and ${\vec {w}}$ tuples of variables. Then $\mathfrak {A}\models _{\mathbb {X}} {\vec {u}}\mathbin {\perp \!\!\!\perp }_{{\vec {v}}}{\vec {w}}$ if for all ${\vec {a}}$ , ${\vec {b}}$ , and ${\vec {c}}$ ,

$$\begin{align*}\left| \mathbb{X}_{\vec{u}\vec{v}=\vec{a}\vec{b}} \right| \cdot \left| \mathbb{X}_{\vec{v}\vec{w}=\vec{b}\vec{c}} \right| = \left| \mathbb{X}_{\vec{u}\vec{v}\vec{w}=\vec{a}\vec{b}\vec{c}} \right| \cdot \left| \mathbb{X}_{\vec{v}=\vec{b}} \right|. \end{align*}$$

It is easy to see that the definition of independence is invariant under scaling, i.e., if we denote by $a\mathbb {X}$ the K-team $s\mapsto a\mathbb {X}(s)$ , then

$$\begin{align*}\mathbb{X}\models{\vec{u}}\mathbin{\perp\!\!\!\perp}_{{\vec{v}}}{\vec{w}} \iff a\mathbb{X}\models{\vec{u}}\mathbin{\perp\!\!\!\perp}_{{\vec{v}}}{\vec{w}}. \end{align*}$$

Proof Let $\mathbb {X}$ be a K-team. Let ${\vec {a}}$ and ${\vec {b}}$ be arbitrary. Then

$$ \begin{align*} \left| \mathbb{X}_{{\vec{x}}{\vec{x}} = {\vec{a}}{\vec{a}}} \right|\cdot\left| \mathbb{X}_{{\vec{x}}{\vec{y}} = {\vec{a}}{\vec{b}}} \right| = \left| \mathbb{X}_{{\vec{x}} = {\vec{a}}} \right|\cdot\left| \mathbb{X}_{{\vec{x}}{\vec{x}}{\vec{y}} = {\vec{a}}{\vec{a}}{\vec{b}}} \right| =\left| \mathbb{X}_{{\vec{x}}{\vec{x}}{\vec{y}} = {\vec{a}}{\vec{a}}{\vec{b}}} \right|\cdot\left| \mathbb{X}_{{\vec{x}} = {\vec{a}}} \right|. \end{align*} $$

Hence $\mathbb {X}\models {\vec {x}}\mathbin {\perp \!\!\!\perp }_{\vec {x}}{\vec {y}}$ .

4.4 Properties of probabilistic teams

By simply replacing the symbol $\perp $ by the symbol $\mathbin {\perp \!\!\!\perp }$ , we get the probabilistic versions of the previously introduced possibilistic team properties of empirical and hidden-variable teams. These are in line with the definitions in [Reference Brandenburger and Yanofsky10], with the exception of no-signalling and parameter independence which suffer from the same weakness as their possibilistic counterparts and which we have generalized here.

Lemma 3.16, stating that locality is equivalent to the conjunction of parameter and outcome independence, remains true in the probabilistic world, at least with the simpler definition of parameter independence from [Reference Brandenburger and Yanofsky10].

Lemma 4.27. Probabilistic locality is equivalent to the formula

$$\begin{align*}\bigwedge_{i<n}\left(\{ x_j \mid j\neq i \}\mathbin{\perp\!\!\!\perp}_{x_i\vec{z}}y_i \land y_i\mathbin{\perp\!\!\!\perp}_{\vec{x}\vec{z}}\{y_j \mid j\neq i\}\right)\!. \end{align*}$$

We conjecture that even with the more general definition of parameter independence, probabilistic locality is equivalent to the conjunction of probabilistic parameter independence and probabilistic outcome independence, as it is in the possibilistic case.

As by Proposition 4.25, the axioms presented in Section 2.2 are valid also in the probabilistic setting, all the results from Section 3.4 that were proved from the axioms are true also for probabilistic teams.

The above (i)–(vi) may seem like somewhat arbitrary observations. However, let us recall that they arise from examples motivated by quantum mechanics and each one of them has an intuitive interpretation in physics. It would seem more satisfactory to present a systematic study of such logical consequences and equivalences but we have already observed that it would be a formidable task bordering the impossible.

4.5 Building probabilistic teams

As we saw in Section 4.4, properties of probabilistic teams are inherited by their possibilistic collapses. Here we prove results concerning the question to what extent the converse holds: when can one construct a probabilistic team out of a possibilistic one, with the same properties?

Following [Reference Abramsky1], we proceed to show that some no-signalling teams have no probabilistic realization that would also support probabilistic no-signalling.

This also shows that $\varphi \models {\mathsf {P}\hspace {-0.5pt}\mathsf {R}}\varphi $ is not true for all formulas $\varphi $ of independence logic.

Proposition 4.30. Suppose that $n=2$ . There is an empirical team X supporting no-signalling such that there is no probabilistic team $\mathbb {X}$ that supports probabilistic no-signalling and whose possibilistic collapse is X, i.e.,

$$\begin{align*}\bigwedge_{i<n}\{x_j \mid j\neq i\}\perp_{x_i}y_i\ \not\models\ {\mathsf{P}\hspace{-0.5pt}\mathsf{R}}\bigwedge_{i<n}\{x_j \mid j\neq i\}\perp_{x_i}y_i. \end{align*}$$

Proof We let $X = \{s_0,\dots ,s_{11}\}$ , where the assignments $s_i$ are as follows:

It is straightforward to check that X supports no-signalling. Suppose for a contradiction that $\mathbb {X}$ is a probabilistic team that supports probabilistic no-signalling and whose possibilistic collapse is X. Then there are positive numbers $p_0,\dots ,p_{11}$ with $\sum _{i<12}p_i = 1$ such that $\mathbb {X}(s_i)=p_i$ for all $i<12$ . By probabilistic no-signalling,

$$\begin{align*}\mathbb{X}\models x_{1-i}\mathbin{\perp\!\!\!\perp}_{x_i} y_i, \end{align*}$$

for all $i\in \{0,1\}$ , so we have, for all $a,b,c,i\in \{0,1\}$ ,

$$\begin{align*}\left| \mathbb{X}_{x_{1-i} x_i = ab} \right|\cdot\left| \mathbb{X}_{x_i y_i = bc} \right| = \left| \mathbb{X}_{x_{1-i} x_i y_i = abc} \right|\cdot\left| \mathbb{X}_{x_i=b} \right|. \end{align*}$$

Calculating the marginal probabilities and applying the above condition, we get the following four equations:

1. (i) $p_2 p_3 = (p_0 + p_1)(p_4 + p_5)$ ,

2. (ii) $p_0 p_8 = (p_1 + p_2)(p_6 + p_7)$ ,

3. (iii) $p_6 p_{11} = (p_7 + p_8)(p_9 + p_{10})$ , and

4. (iv) $p_5 p_9 = (p_3 + p_4)(p_{10} + p_{11})$ .

From this, using the third and the fourth equations, we get

$$\begin{align*}p_6 p_{11} p_5 p_9 = (p_7 + p_8)(p_9 + p_{10})(p_3 + p_4)(p_{10} + p_{11})> p_8 p_9 p_3 p_{11}, \end{align*}$$

whence $p_5 p_6> p_3 p_8$ . Then by multiplying by $p_2$ and using the first equation, we get

$$\begin{align*}p_2 p_5 p_6> p_2 p_3 p_8 = (p_0 + p_1)(p_4 + p_5)p_8 > p_0 p_5 p_8, \end{align*}$$

whence $p_2 p_6> p_0 p_8$ . Then finally, using the second equation, we get

$$\begin{align*}p_2 p_6> p_0 p_8 = (p_1 + p_2)(p_6 + p_7) > p_2 p_6, \end{align*}$$

which is a contradiction.

A minimal example of a possibilistic no-signalling team which is not a collapse of any probabilistic no-signalling team can be obtained by translating an example of [Reference Abramsky, Barbosa, Kishida, Lal and Mansfield2]—which occurs as part of a discussion about the question whether there exists an intrinsic characterization of the class of no-signalling teams that are collapses of probabilistic no-signalling teams—into our team-semantic framework.

The next property of empirical and hidden-variable teams, measurement locality, was introduced by the first author in [Reference Abramsky1]. Measurement locality states that the measurement variables are mutually independent of each other.

Definition 4.31 (Measurement Locality)

An empirical team X supports measurement locality if it satisfies the formula

(ML) $$ \begin{align} \bigwedge_{i<n} x_i\perp\{x_j \mid j\neq i\}. \end{align} $$

A hidden-variable team X supports measurement locality if it satisfies the formula

(ML) $$ \begin{align} \bigwedge_{i<n} x_i\perp_{\vec{z}}\{x_j \mid j\neq i\}. \end{align} $$

Definition 4.32 (Probabilistic Measurement Locality)

A probabilistic empirical team $\mathbb {X}$ supports probabilistic measurement locality if it satisfies the formula

(PML) $$ \begin{align} \bigwedge_{i<n} x_i\mathbin{\perp\!\!\!\perp}\{x_j \mid j\neq i\}. \end{align} $$

A probabilistic hidden-variable team $\mathbb {X}$ supports probabilistic measurement locality if it satisfies the formula

(PML) $$ \begin{align} \bigwedge_{i<n} x_i\mathbin{\perp\!\!\!\perp}_{\vec{z}}\{x_j \mid j\neq i\}. \end{align} $$