A declarative approach to data narration

The atomic components of the model are characters and measures. For instance, the DN of Figure 1 (bottom) includes character ‘black women’ and measure ‘stroke death’. To keep things simple in this preliminary version of the model, measure values and units (e.g., 57/100000) are not part of the present preliminary model. The semantics of the message is given by a predicate connecting characters and measures, in the spirit of semantic triples.

We also assume binary relations between characters as the bases for the relations between messages. Relations between characters are at the core of relations between findings (see Figure 2). In data narration, it is common to use these relations as transitions between episodes of the narrative’s plot, and it has been seen that most transitions are of the following nature: specialisation, temporal or spatial (cf. e.g., (Hullman et al., 2017)). More precisely,

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  a specialization relation, noted $c\prec c^{\prime}$, allows to classify characters in a hierarchy, for instance to indicate that black women is more specific than women,

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  a spatial relation, noted $c\vdash c^{\prime}$ to indicate that $c$ is in a spatial relation with $c^{\prime}$. For instance Greece $\vdash$ France.

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  a temporal relation, noted $c\dashv c^{\prime}$ to indicate that $c$ is in temporal relation with $c^{\prime}$. For instance, in Europe, Spring $\dashv$ 2nd quarter.

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  we also assume a general similarity relation noted $c\approx c^{\prime}$ to indicate that character $c$ is similar to character $c^{\prime}$, for instance birth control pills is similar to abortion pills.

A message is a tuple associating characters with measures. Formally, a message $m$ is a tuple $\langle C,V,P\rangle$ where $C$ is a set of characters, $V$ is a set of measures and $P$ is a predicate³³3For the sake of consistency, $P$ is a singleton.. The simplest message is the empty message, $\langle\emptyset,\emptyset,\emptyset\rangle$.

Consider the messages of Figure 3. Messages $m_{1}$ to $m_{5}$ and $m_{6}$ to $m_{8}$ are inspired from those of the DNs of Figure 1, restricting to a subset of messages and simplifying many characters. Message $m_{9}$ is inspired from a DN about covid. $\Box$

Since findings can be related to one another (e.g. findings about black women are more specific than those concerning all women), we consider that relations over characters also applies to messages. For example, among messages of Figure 3, $m_{3}$ is more general than $m_{1}$ regarding characters ‘black women’ and ‘women’. Given two messages $m,m^{\prime}$ we consider the transition relation between them which can be one of: spatial, temporal, generalization, similarity.

Formally, for $m=\langle C,V,P\rangle$ and $m^{\prime}=\langle C^{\prime},V^{\prime},P^{\prime}\rangle$ it is $mRm^{\prime}$ if $\exists c\in C,c^{\prime}\in C^{\prime},cRc^{\prime}$ and $R$ is one of $\prec,\approx,\vdash,\dashv$.

The schema of a DN of length $k$ (i.e., with $k$ messages) is a couple $\langle h,k\rangle$ where $h\in\mathcal{H}$ is the DN name. A DN instance is a tuple of messages.

For the sake of readability, in what follows we will consider a DN of length $k$ as an injective function from $\mathcal{M}$ to $2^{\mathbb{N}}$. For instance, the DN $n=\langle m,m,m^{\prime}\rangle$ can be seen as the function $n$ where $n(m)=\{1,2\}$ and $n(m^{\prime})=\{3\}$. We abuse notations and note $m\in n$ if a message $m$ appears in the DN $n$, and $messages(n)$ the set of messages of DN $n$.

A DNDB schema is a set of DN schemas and a DNDB instance is a set of DN instances.

Continuing the running example, $I=\{n_{1},n_{2},n_{3}\}$, is a DNDB instance that organizes the messages of Figure 3:

$n_{1}=\langle m_{1},m_{2},m_{3},m_{4},m5\rangle$; $n_{2}=\langle m_{6},m_{7},m_{8}\rangle$; $n_{3}=\langle m_{9}\rangle$. $\Box$

Selects the DNs in instance $I$ that satisfy a given condition.

$\sigma_{\varphi}(I)=\{n\in I|\varphi\}$

where $\varphi$ is a logical formula to express selection conditions, for instance:

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  $\exists m\in n,m=\langle C,V,P\rangle,c\in C$ (has character $c$)

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  $\exists m\in n,m=\langle C,V,P\rangle,v\in V$ (has measure $v$)

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  $\exists m\in n,m=\langle C,V,P\rangle,p\in P$ (has predicate $p$)

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  $\exists m\in n,m=\langle C,V,P\rangle,\exists c^{\prime}\in C,c^{\prime}Rc$ (has character in relation $R$ with $c$, $R$ being one of the relations over characters)

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  $\exists m,m^{\prime}\in n,mRm^{\prime}$ (has messages that are in relation $R$ where $R$ is one of the relations over messages)

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  $\forall m\in n,m=\langle\emptyset,\emptyset,\emptyset\rangle$ (has only empty messages)

The operation $\sigma_{\exists m\in n,m=\langle C,V,P\rangle,`strokeDeaths^{\prime}\in V}(I)$
looks for DNs about stroke deaths in instance $I$ of the running example. Its output is $\{n_{1},n_{2}\}$. $\Box$

For each DN in I, keeps only the messages satisfying a condition.

$\pi_{\varphi}(I)=\{n|_{\{m\in n|\varphi\}}|n\in I\}$

where $\varphi$ is a logical formula, similar to those used for the selection and $n|_{\{m\in n|\varphi\}}$ is the restriction of DN $n$ to the set of messages satisfying $\varphi$.

The operation $\pi_{\exists m\in n,m=\langle C,V,P\rangle,`BlackWomen^{\prime}\in C}(I)$ produce DNs $n_{4},n_{5}$ and $n_{6}$, projecting a subset of messages of DNs in instance $I$, i.e., those containing Black women as characters.

$n_{4}=\langle m_{3}\rangle$; $n_{5}=\langle m_{6},m_{7}\rangle$; $n_{6}=\langle\rangle$. $\Box$

For each DN in $I$, keeps only one occurrence of each message.

$\delta(I)=\{n^{\prime}|n\in I,\forall m\in n,n^{\prime}(m)=min(n(m))\}$

For each DN in $I$, groups the messages using a grouping condition and aggregates each group using a specific aggregation function.

$\gamma_{\varphi_{1},\ldots,\varphi_{j},agg_{1},\ldots,agg_{j}}(I)=$

$\{\langle agg_{1}(\{m\in n|\varphi_{1}\}),\ldots,agg_{j}(\{m\in n|\varphi_{j}\})\rangle|n\in I\}$

where the $\varphi_{i}$ are grouping conditions on the set of messages of the DN. It is not requested that the grouping conditions partition the set of messages, which allows one message to appear in several groups. The $agg_{i}$ are aggregation functions for merging a set of messages into one message.

The operation $\gamma_{BW,WW,A,A}(I)$, where $BW$ and $WW$ are conditions about characters ‘Black women’ and ‘White women’ and $A$ is an aggregation function computing the union of characters and measures in the input messages. The output DNs contain two messages, the former concerning Black women and the latter White women: $n_{7}=\langle m_{3},\langle\emptyset,\emptyset,\emptyset\rangle\rangle$; $n_{8}=\langle A(m_{6},m_{7}),A(m_{6},m_{7})\rangle$;
$n_{9}=\langle\langle\emptyset,\emptyset,\emptyset\rangle,\langle\emptyset,\emptyset,\emptyset\rangle\rangle$. $\Box$

Another group-aggregate operation allows to merge messages across DNs into one DN.

$\gamma^{across}_{\varphi_{1},\ldots,\varphi_{j},agg_{1},\ldots,agg_{j}}(I)=$

$\{\langle agg_{1}(\{m|\varphi_{1}\}),\ldots,agg_{j}(\{m|\varphi_{j}\})\rangle|m\in\bigcup_{n\in I}messsages(n)\}$

The operation $\gamma^{across}_{BW,WW,A,A}(I)$, with $BW$, $WW$ and $A$ as in previous example, outputs

$n_{10}=\langle A(m_{3},m_{6},m_{7}),A(m_{6},m_{7})\rangle$ $\Box$

Allows to change the order of messages in the DNs.

$\tau_{\varphi_{1},\ldots,\varphi_{J},sort_{1},\ldots,sort_{J}}(I)=$

$\{\langle sort_{1}(\{m\in n|\varphi_{1}\}),\ldots,sort_{J}(\{m\in n|\varphi_{J}\})\rangle|n\in I\}$

where the $\varphi_{i}$ are selection conditions over messages (like the ones used e.g., for the selection operation) and the $sort_{i}$ are sorting functions, mapping a set of messages to a tuple of messages.

Flattens all DNs in I into one DN, by concatenating all their messages.

$\chi(I)=\{\langle n_{1},...n_{k}\rangle|n_{i}\in I\}$

: By definition, each operator defines a set of DNs from a set of DNs.

From a given set of messages, all narratives can be obtained using the constant and cross-product operators, which together form the core of the algebra.

The minimal set of operators includes constant, cross product, both group-aggregates, order by, union and difference. All other operations can be expressed from this set. We can anticipate that some expressions will be popular and deserve to be promoted as operators, like the join in the relational algebra that is a shortcut from cross product followed by selection.

Like in the case of the relational algebra, the cross product is non commutative, admits one absorbing element (the empty set) and one neutral element (the empty DN, i.e., $\langle\rangle$). Set operations keep their usual properties. Note that, because DNDB instances are sets of DNs of different length, intersection cannot be expressed by combining cross product, selection and projection, while it is the case for the relational algebra.