3.2. Basic notions

We will define the concept of missing information which is a formula containing the pieces of information the agent needs in order to be able to achieve a goal. The main idea is that with the arrival of new (maybe relevant) information, the amount of missing information should reduce and never increase. Therefore, the more a piece of information allows to reduce the missing information the more useful it is for the user. We will then be led to compare formulas before and after receiving some information in order to measure the usefulness of the new piece of information.

The extension we are proposing here will consider different ways to achieve a goal. We represent those different ways as a disjunction of conjunctions, one conjunction corresponding to one way to achieve a goal. We need then to define a notion of inclusion for disjunctions of conjunctions. Two different types of inclusion will be considered. The first is related to the literals contained in the formulas and the second to the number of literals contained in a formula.

We would like to stress that our formalism is not supposed to consider what happens after the goal has been achieved. We made the hypothesis that as soon as the agent has the necessary information to reach its goal it will not look for that information anymore—we do not consider the forgetting aspect.

In the following definitions, the term ‘literal’ means ‘literal of a propositional language’ (Buss, Reference Buss1998).

Equipped with this notion of formula inclusion, we can now define a preference relation over disjunctions of conjunctions based on it.

In some settings, the above preference relation might be too strong. For this reason, we now define a weaker notion of formula inclusion, which only cares about the number of literals they contain, and which will be used to define a weaker preference relation.

Obviously, $D \subseteq D'$ implies $D \sqsubseteq D'$ . But the contrary is not true. Indeed, $p \vee q \sqsubseteq (r \wedge s)$ but $p \vee q \not\subseteq (r \wedge s)$ . Moreover, $D \subset D'$ implies $D \sqsubset D'$ but the contrary is not true. Indeed, we have $p \vee q \sqsubset (r \wedge s)$ but $p \vee q \not\subset (r \wedge s)$ .

We are now ready to define a weaker preference relation, based on $\sqsubseteq$ , which we will name $\preceq^2$ .

Thus, $S \preceq^1 S'$ (resp., $S \preceq^2 S'$ ) if and only if S does not have more elements than S ^′; if S and S ^′ have the same number of elements, then the disjunctions in S are included in the disjunctions of S ^′ (resp., are less complex than those of S ^′).

Let us notice that:

• $\preceq^1$ is a preorder but not total. Some sets of disjunctions of conjunctions are incomparable, such as $\{p, q \wedge r\} \not\preceq^1 \{r, s\}$ and $\{r, s\} \not\preceq^1 \{p, q \wedge r\}$ .

• $\preceq^2$ is a preorder but not total. For instance, $\{p \vee (q \wedge r), q \wedge r\} \not\preceq^2$ $\{p \vee q, q \vee (r \wedge s)\}$ and $ \{p \vee q, q \vee (r \wedge s)\} \not\preceq^2 \{p \vee (q \wedge r), q \wedge r\}$ .

• $S \preceq^1 S'$ implies $S \preceq^2 S'$ but the contrary is not true in general.

• $S \prec^1 S'$ if and only if (i) $ |S| \lt |S'|$ or (ii) $|S|=|S'|$ and there is a bijection $f\;:\; S \rightarrow S'$ such that: $\forall D \in S\;D \subseteq f(D)$ and $\exists D_0 \in S$ such that $D_0 \subset f(D_0)$ .

• $S \prec^2 S'$ if and only if (i) $ |S| \lt |S'|$ or (ii) $|S|=|S'|$ and there is a bijection $f\;:\; S \rightarrow S'$ such that: $\forall D \in S\;D \sqsubseteq f(D)$ and $\exists D_0 \in S$ such that $D_0 \sqsubset f(D_0)$ .

• $S \prec^1 S'$ implies $S \prec^2 S'$ but the contrary is not true in general.

3.3. An extended defnition of missing information

In this section we propose an extension of the definition of missing information we have proposed in Cholvy and da Costa Pereira (Reference Cholvy and da Costa Pereira2019):

Definition 6 Let a be an agent with its belief base $B_a$ and its goal set $G_a$ . Let $g \in G_a$ be such that $B_a \not\models g$ . $Missing(B_a,g)$ , is defined as follows:

\begin{align*}Missing(B_a,g) = \bigwedge_{l: l \in premise(g)\;and\;B_a \not\models l} l\end{align*}

$Missing(B_a,g)$ is the conjunction of all the literals in the premise of g which cannot be deduced from $B_a$ (i.e., which are not yet believed by the agent). Therefore, in the particular case in which $B_a^m = \emptyset$ , $Missing(B_a,g) = premise(g)$ , that is, the missing piece of information to achieve g is premise(g).

The extension proposed here is based on the fact that there may be more than one way to achieve a goal instead of just one as we proposed in Cholvy and da Costa Pereira (Reference Cholvy and da Costa Pereira2019). For example, suppose you have a goal of knowing the mail address of Mr John. The different ways to achieve your goal could be to have access to his homepage or to have access to the directory of his department. This will imply to account for the usefulness of a piece of information with respect to the two possibilities to achieve the goal.

It is well known that a rule containing a disjunction, like $p \land (q \lor r) \to g$ can be decomposed into two distinct rules, like $p \land q \to g$ and $p \land r \to g$ , to be considered as being connected by a disjunction (i.e., $(p \land q \to g) \lor (p \land r \to g)$ . However, computing the usefulness of incoming information in this case cannot be just trivially performed by computing its usefulness separately for each rule having the same goal and then combining them, for example by the maximum operator. Indeed, with the above example and assuming to already know p, if the incoming information is $q \lor r$ , this is not enough to deduce g by either decomposed rule separately; however, it surely is enough if we consider the original (complete) rule.

The extension of Definition 6 can now be stated as follows.

Definition 7 Let a be an agent with its goal base $G_a$ and its belief base $B_a$ . Let $g \in G_a$ .

\begin{align*}Missing(B_a,g) = \bigvee_{k = 1..n_g\;} \bigwedge_{l:\; l \in premise^k(g) \;and\; B_a \not\models l} l \end{align*}

Thus, $Missing(B_a,g)$ is a disjunction of conjunctions of literals. There are many conjunctions as premises of g. The $k\mathrm{th}$ conjunction is the conjunction of the literals in $premise^k(g)$ which are not yet believed by a if there are some. If $B_a \models premise^k(g)$ Footnote 1 then this $k\mathrm{th}$ element is any tautology, thus $Missing(B_a,g)$ is also any tautology and is denoted $\top$ . This means that g is achieved, that is, $B_a \models g$ . Intuitively, the $n_g$ premisses (conjunctions of literals) associated to goal g represent the possible ways to achieve the goal. If at least one of them is believed by the agent, it means that the goal is achieved.

Definition 8 Let a be an agent whose belief base is $B_a$ and goal set is $G_a = \{g_1,\ldots,g_n\}$ . $Missing(B_a,G_a)$ is a multiset defined by:

\begin{align*}Missing(B_a,G_a) = \{Missing(B_a,g_1), \ldots, Missing(B_a,g_m)\}\end{align*}

with $g_i \in G_a$ , $B_a \not\models g_i$ and $i \in \{1 \ldots m\}$ .

Thus, the cardinality of $Missing(B_a,G_a)$ is the number of goals that are not yet achieved.

The following proposition ensures that the usefulness measures we are proposing satisfy some common-sense properties, namely that (i) if some information is received, the missing information to achieve a goal cannot increase, (ii) more general information will not diminish missing information to achieve a given goal less than more specific information, and (iii) receiving a conjunction of two information items has the same effect as receiving the two items one after the other.

The following proposition shows that adding a belief to the belief base $B_a$ does not increase the number of goals that remain to be achieved. Finally, if adding a belief to the belief base $B_a$ reduces the number of missing conjunctions, then this means that such new belief allows to achieve one or more goals.

Notice however that, even if $|Missing(B_a \cup \varphi,G_a)| \leq |Missing(B_a, G_a)|$ , it may happen that $Missing(B_a \cup \varphi,G_a) \not\subseteq Missing(B_a, G_a)$ . See example below.

As we have pointed out in Section 2, the aim of traditional IR systems is to provide to the user a list of documents that best fit with the user query. The proposed documents are then ranked with respect to the aboutness measure, which, roughly speaking, represents the degree of matching between the document and the user query. However, such a measure does not consider what the user already knows, nor their informative goals; as a consequence, it does not necessarily help providing documents that make the user knowledge advance toward their goals.

In the following three sections, we propose three different measures, of increasing refinement, allowing a system to evaluate the usefulness of a piece of information to the user (given the beliefs about the user goals and about the current user knowledge), in various ways: the first one is binary, in that it just tells whether a piece of information is useful or not; the second one allows to compare two pieces of information and decide if one of them is more useful than the other; finally, the third measure assigns a numerical value of information usefulness to a given piece of information, which, in addition to telling if one piece of information is more useful that another, also quantifies by how much.

6.1. Postulates for a usefulness measure

In this section, we follow a numerical approach by associating each piece of information with a usefulness degree. To begin with, we state some rationality postulates, which are based on the results of our survey, such a measure must satisfy. The general case will not be treated, and we will limit ourselves to calculating the degree of usefulness of conjunctions of positive literals.

Again, consider an agent a with its belief base $B_a$ and its goal base $G_a$ . Let $\varphi$ be a conjunction of positive literals. We define:

• $ E_{B_a}(\varphi) = \{g \in G_a, B_a \not\models g \;and\; B_a \cup \varphi \models g\}$

• $Cons(B_a,\varphi) = \{l\; positive \; literal\;of\;L\;:\; B_a \cup \varphi \models l\}$

• $N_m(\varphi)= \Sigma_{g \in G_a} Min_{C \in Missing(B_a,g)} |C \setminus Cons(B_a,\varphi)|$

• $N_u(\varphi) = |\varphi \setminus \cup_{g \in G_a} \cup_{C \in Missing(B_a,g)} C|$

$ E_{B_a}(\varphi)$ is the set of goals that a can achieve after adding $\varphi$ to its beliefs. Thus, $\mid E_{B_a}(\varphi) \mid $ is the number of goals that a can achieve after adding $\varphi$ to its beliefs.

$Cons(B_a,\varphi)$ is the set of positive literals of L that are deducible from $B_a \cup \varphi$ .

$N_m(\varphi)$ sums, for all goals in $G_a$ , the smallest number of literals that are missing to achieve the goals and that are not deductible from $B_a \cup \varphi$ . The smaller $N_m(\varphi)$ , the more $\varphi$ reduces information which are missing to achieve the goals.

$N_u(\varphi)$ counts the positive literals deducible from $B_a \cup \varphi$ that are not literals of missing information. Adding them is therefore not useful to achieve the goals. Thus, the greater $N_u(\varphi)$ , the more $\varphi$ brings non useful information.

Let $U(\varphi)$ be a real number representing how much $\varphi$ is useful for a. We have based our definition on the following postulates.

Monotonicity on the number of goals

(P1) $|E(\varphi_1)| \lt |E(\varphi_2)| \;\Longrightarrow\; $ $ U(\varphi_1) \lt U(\varphi_2)$ .

Table 1. The values of $N_m$ and $N_u$ measurements

The number of goals that a piece of information allows an agent to achieve should influence the degree of usefulness of such a piece of information for the agent. Intuitively, a piece of information which allows to achieve a higher number of goals (with respect to another piece of information) should be more useful.

6.2. The usefulness measure U

In this section we will present an extension of the numerical measure of usefulness proposed in Cholvy and da Costa Pereira (Reference Cholvy and da Costa Pereira2019):

Definition 13 Let a be an agent whose goals are in $G_a$ and let $\varphi$ be a conjunction of positive literals. The usefulness degreeFootnote 2 has been defined as follows:

\begin{align*} U(\varphi) = \frac{1}{|G_a| + 1}\left[|E(\varphi)| + \frac{N_1(\varphi)}{N_1(\varphi) + N_2(\varphi) + \frac{N_3(\varphi)}{N_3(\varphi) + 1}}\right]\end{align*}

$N_1(\varphi)$ quantifies the useful part of $\varphi$ for the agent, while $N_2(\varphi)$ quantifies the agent’s disappointment (lack of needed information) toward $\varphi$ and, finally, $N_3(\varphi)$ quantifies the disturbance caused to the agent by the unexpected and unnecessary content of $\varphi$ . The above definition of usefulness takes these three aspects into consideration.

The intuitive idea behind this definition of usefulness is as follows. The usefulness of information can be seen as a calculation of the similarity between the information the agent needs to achieve its goals and the piece of information that arrives. The more direct or indirect elements (that can be deduced) there are in common between the two, the more useful the information will be. We would like to stress that this fact allows to account for the serendipity factor (Toms, Reference Toms2000) in the definition of usefulness. Indeed, an agent gets (asks for) a piece of information to achieve a given goal, but if the received piece of information also enables other goals to be achieved, this fact is taken into considered in the computation of the usefulness. However, the number of common elements is not always enough to distinguish the degrees of usefulness between two pieces of information. Indeed, in some cases it would also be necessary to take into account their differences. We have been inspired by Tversky’s idea (Tversky, 1977), according to which, in order to calculate the similarity between two objects A and B, we should consider, in addition to what they have in common, what distinguishes them, that is, the features of A which are not features of B and vice versa. This is the reason why these three values, $N_1(\varphi)$ , $N_2(\varphi)$ , and $N_3(\varphi)$ , we have been considered.

Here, $N_1(\varphi)$ is no longer needed; for the sake of intuitiveness, the extension of $N_2(\varphi)$ will be renamed $N_m(\varphi)$ (for ‘number of missing [literals]’) and the extension of $N_3(\varphi)$ will be renamed $N_u(\varphi)$ (for ‘number of useless or unexpected [literals]’).

Table 2. The values of $N_m$ , $N_u$ , and U measurements

This gives the following extension of Definition 13.

Definition 14 We denote $\alpha_a = \Sigma_{g \in G_a} Min_{C \in Missing(B_a,g)} \mid C \mid$ . Then the usefulness measure U is defined by:

\begin{align*} U(\varphi) = \frac{1}{|G_a| + 1}\left[|E(\varphi)| + \frac{\alpha_a - N_m(\varphi)}{\alpha_a + \frac{N_u(\varphi)}{N_u(\varphi) + 1}}\right].\end{align*}

$N_m(\varphi)$ quantifies how far the agent is from achieving its goals after acquiring $\varphi$ , while $N_u(\varphi)$ quantifies the disturbance caused to the agent by the unexpected and unnecessary content of $\varphi$ .

Proposition 7 We can notice that for any conjunction of positive literals $\varphi$ we have:

\begin{align*} 0 \leq \frac{\alpha_a - N_m(\varphi)}{\alpha_a + \frac{N_u(\varphi)}{N_u(\varphi) + 1}} \leq 1\end{align*}

\begin{align*} \frac{\alpha_a - N_m(\varphi)}{\alpha_a + \frac{N_u(\varphi)}{N_u(\varphi) + 1}} = 1\;\Longrightarrow\;E(\varphi) = G_a\end{align*}

Proof

• First we notice that $N_m(\varphi) \leq \alpha_a$ . Indeed $\forall g \in G_a\; \forall C \in Missing(B_a,g)\;\mid C \setminus Cons(\varphi) \mid \leq \mid C \mid$ . Thus, $\forall g \in G_a\; Min_{C \in Missing(B_a,g)} \mid C \setminus Cons(\varphi) \mid \leq Min_{C \in G_a} \mid C \mid$ . Finally, $\Sigma_{g \in G_a} Min_{C \in Missing(B_a,g)} \leq \Sigma_{g \in G_a} \mid C \mid$ Thus, $\alpha_a - N_m(\varphi)$ is positive.

  Moreover, $\alpha_a + \frac{N_u(\varphi)}{N_u(\varphi) + 1}$ is positive. Thus, $\frac{\alpha_a - N_m(\varphi)}{\alpha_a + \frac{N_u(\varphi)}{N_u(\varphi) + 1}}$ is positive.

  Besides $\alpha_a - N_m(\varphi) \leq \alpha_a + \frac{N_u(\varphi)}{N_u(\varphi) + 1}$ . Thus, $\frac{\alpha_a - N_m(\varphi)}{\alpha_a + \frac{N_u(\varphi)}{N_u(\varphi) + 1}} \leq 1$

• $\frac{\alpha_a - N_m(\varphi)}{\alpha_a + \frac{N_u(\varphi)}{N_u(\varphi) + 1}} = 1$ implies $\alpha_a - N_m(\varphi) = \alpha_a + \frac{N_u(\varphi)}{N_u(\varphi) + 1}$ . Thus, $\frac{N_u(\varphi)}{N_u(\varphi) + 1} + N_m(\varphi) = 0$ . Since these two numbers are positive, this implies $N_m(\varphi) = N3(\varphi) = 0$ . Finally, $N_m(\varphi)=0$ implies that any goal is achieved that is $E(\varphi) = G_a$ .

• (P1). Suppose $|E(\varphi_1)| \lt |E(\varphi_2)| $ . Then $|E(\varphi_1)| + 1 \leq |E(\varphi_2)|.$ Suppose now $ \frac{\alpha_a - N_m(\varphi_1)}{\alpha_a + \frac{N_u(\varphi_1)}{N_u(\varphi_1) + 1}} = 1. $ This would imply (by Proposition 7) $|E(\varphi_1)| = |G_a|$ . This is impossible because $|E(\varphi_1)| \lt |E(\varphi_2)| $ . Thus, $ \frac{\alpha_a - N_m(\varphi_1)}{\alpha_a + \frac{N_u(\varphi_1)}{N_u(\varphi_1) + 1}} \lt 1. $ Thus, $|E(\varphi_1)| + \frac{\alpha_a - N_m(\varphi_1)}{\alpha_a + \frac{N_u(\varphi_1)}{N_u(\varphi_1) + 1}} \lt |E(\varphi_2)|$ . Finally, $U(\varphi_1) \lt U(\varphi_2)$ .

• (P2) Suppose $|E(\varphi_1)| = |E(\varphi_2)|$ and $N_m(\varphi_1) \lt N_m(\varphi_2)$

  We denote $U = \frac{N_u(\varphi_2)}{N_u(\varphi_2) + 1}$ and $V = \frac{N_u(\varphi_1)}{N_u(\varphi_1) + 1}$ .

  Thus, $ U(\varphi_2) - U(\varphi_1) = \frac{1}{|G_a| + 1}\left[\frac{\alpha_a - N_m(\varphi_2)}{\alpha_a + U} - \frac{\alpha_a - N_m(\varphi_1)}{\alpha_a + V}\right] $

  We distinguish 3 cases:

• – $U = V$ . We denote $\alpha_a + U = \alpha_a + V = D'$ . Then

  $ U(\varphi_2) - U(\varphi_1) = \frac{- N_m(\varphi_2) + N_m(\varphi_1)}{D'} \lt 0$ , because $N_m(\varphi_1) \lt N_m(\varphi_2)$ .

• – $U \lt V \Rightarrow \frac{1}{\alpha_a + V} \lt \frac{1}{\alpha_a + U} \Rightarrow \frac{- N_m(\varphi_2)}{\alpha_a + U} \lt \frac{- N_m(\varphi_1)}{\alpha_a + V} \\[5pt] \Rightarrow \frac{\alpha_a - N_m(\varphi_2)}{\alpha_a + U} \lt \frac{\alpha_a - N_m(\varphi_1)}{\alpha_a + V} \Rightarrow U(\varphi_2) - U(\varphi_1) \lt 0$

• – $V \lt U \Rightarrow \frac{1}{\alpha_a + U} \lt \frac{1}{\alpha_a + V} \Rightarrow \frac{N_m(\varphi_1)}{\alpha_a + U} \lt \frac{N_m(\varphi_2)}{\alpha_a + V} \Rightarrow \frac{- N_m(\varphi_2)}{\alpha_a + V} \lt \frac{- N_m(\varphi_1)}{\alpha_a + U}\\[5pt] \Rightarrow \frac{\alpha_a - N_m(\varphi_2)}{\alpha_a + V} \lt \frac{\alpha_a - N_m(\varphi_1)}{\alpha_a + U} \\[5pt] \Rightarrow \frac{\alpha_a - N_m(\varphi_2)}{\alpha_a + U} \lt \frac{\alpha_a - N_m(\varphi_2)}{\alpha_a + V} \lt \frac{\alpha_a - N_m(\varphi_1)}{\alpha_a + U} \lt \frac{\alpha_a - N_m(\varphi_1)}{\alpha_a + V} \\[5pt] \Rightarrow U(\varphi_2) - U(\varphi_1) \lt 0$

• (P3) Suppose $|E(\varphi_1)| = |E(\varphi_2)|$ , $N_m(\varphi_1)=N_m(\varphi_2)$ and $N_u(\varphi_1)\lt N_u(\varphi_2)$ . Since $N_u(\varphi_1)\lt N_u(\varphi_2)$ we also have $ \frac{N_u(\varphi_1)}{N_u(\varphi_1) + 1} \lt \frac{N_u(\varphi_2)}{N_u(\varphi_2) + 1}$ . Thus, $\frac{1}{\alpha_a + \frac{N_u(\varphi_2)}{N_u(\varphi_2) + 1}} \lt \frac{1}{\alpha_a + \frac{N_u(\varphi_1)}{N_u(\varphi_1) + 1}}$ .

  We denote $\alpha_a - N_m(\varphi_1) = \alpha_a - N_m(\varphi_2) = k$ .

  Then $\frac{k}{\alpha_a + \frac{N_u(\varphi_2)}{N_u(\varphi_2) + 1}} \lt \frac{k}{\alpha_a + \frac{N_u(\varphi_1)}{N_u(\varphi_1) + 1}}$ . Finally, $U(\varphi_2) \lt U(\varphi_1)$

• (P4) Straightforward.

6.3. Particular case

In this section, we consider the particular case in which there is only one way to achieve each goal that is, $\forall g \in G_a$ $n_g = 1$ , which was the case considered in Cholvy and da Costa Pereira (Reference Cholvy and da Costa Pereira2019).

• First, we prove that the definition of U given in Definition 14 is an extension of the definition of U given in Cholvy and da Costa Pereira (Reference Cholvy and da Costa Pereira2019). Indeed, under this assumption, we have:

  • – $N_m(\varphi)= \Sigma_{g \in G_a} |Missing(B_a,g) \setminus Cons(B_a,\varphi)|$

  • – $N_u(\varphi) = |\varphi \setminus \cup_{g \in G_a} Missing(B_a,g) |$

  • – $\alpha_a = \Sigma_{g \in G_a} \mid Missing(B_a,g)\mid$

  Now consider the two terms $\alpha_a - N_m(\varphi)$ and $\alpha_a + \frac{N_u(\varphi)}{N_u(\varphi) + 1}$ in Definition 14.

• – $\alpha_a - N_m(\varphi) = \Sigma_{g \in G_a} \mid Missing(B_a,g)\mid -\Sigma_{g \in G_a} |Missing(B_a,g) \setminus Cons(B_a,\varphi)|$ .Thus, $\alpha_a - N_m(\varphi) = \Sigma_{g \in G_a} \mid Missing(B_a,g) \cap Cons(B_a,\varphi) \mid$ .

• – $\alpha_a + \frac{N_u(\varphi)}{N_u(\varphi) + 1} = \Sigma_{g \in G_a} \mid Missing(B_a,g)\mid + \frac{|\varphi \setminus \cup_{g \in G_a} Missing(B_a,g) |}{|\varphi \setminus \cup_{g \in G_a} Missing(B_a,g) | +1}$

  Thus, U as defined in Definition 14 becomes

  \begin{align*} U(\varphi) = \frac{1}{|G_a| + 1}\left[|E(\varphi)| +\frac{ \Sigma_{g \in G_a} \mid Missing(B_a,g) \cap Cons(B_a,\varphi) \mid }{\Sigma_{g \in G_a} \mid Missing(B_a,g)\mid +\frac{|\varphi \setminus \cup_{g \in G_a} Missing(B_a,g) |}{|\varphi \setminus \cup_{g \in G_a} Missing(B_a,g) | +1}}\right]\end{align*}
  This is exactly the one which was given in Cholvy and da Costa Pereira (Reference Cholvy and da Costa Pereira2019).

• Moreover, now assume that $B_a^W = \emptyset$ . In this case, the only beliefs of the agent concern the agent’s needs in terms of information about the ways to achieve its goals and we have: $Missing(B_a,g) = premise(g)$ and $Cons(B_a,\varphi) = \varphi$ . $U(\varphi)$ can then be written as:

  \begin{align*} U(\varphi) = \frac{1}{| G_a + 1|} \cdot \left[|E(\varphi)| + \frac{ \Sigma_{g \in G_a} |\varphi \cap Premise(g)|}{\Sigma_{g \in G_a} |premise(g)| +\frac{|\varphi \setminus \cup_{g \in G_a} Premise(g)| }{ |\varphi \setminus \cup_{g \in G_a} Premise(g)| + 1}}\right]\end{align*}
  with $E(\varphi) = \{g \in G_a, premise(g) \subseteq \varphi\}$ ,

• Now assume that $B_a^m = \emptyset$ and $G_a$ is a singleton. In this case, the agent has a single goal, $g_0$ , and its only beliefs is the formula which expresses the information need for achieving that single goal. $U(\varphi)$ can then be written as follows:

  \begin{align*} U(\varphi) = \frac{1}{2} \cdot \left(n(\varphi) + \frac{ |\varphi \cap premise(g_0)| }{ |premise(g_0)| + \frac{|\varphi \setminus premise(g_0)|}{|\varphi \setminus premise(g_0)| + 1}}\right)\end{align*}
  with $n(\varphi) = 1$ if $premise(g_0) \subseteq \varphi$ and $n(\varphi) = 0$ otherwise.

7.1. A refresher on relevance dimensions

The aboutness (Cooper, Reference Cooper1971) dimension is a core notion in IR. It is used to compute the topical matching between a document and a user query. However, its modeling gave raise to several distinct interpretations, which characterize a variety of IR models, of which the vector space model is an example. Formally, in the vector space model, a piece of information or, more generally, a document d, can be represented as a vector of T elements, $d = [w_{1d}, \ldots, w_{T d}]$ . The user interests are represented by a vector $q = [w_{1q}, \ldots, w_{T q}]$ , T being the size of the term vocabulary used. Different choices have been made in the literature regarding the values of $w_{id}$ , for example: simply based on the presence or absence of a word in the document, in this case the vector contains values in $\{0,1\}$ , or based on the frequency of the word in the document and in the whole repository (TF-IDF) (Baeza-Yates & Ribeiro-Neto, Reference Baeza-Yates and Ribeiro-Neto2011). Here, we will use the vector space model interpretation, and, like in da Costa Pereira et al. (Reference da Costa Pereira, Dragoni, Pasi, Boughanem, Berrut, Mothe and Soulé-Dupuy2009), in addition to the aboutness measure, we will consider the appropriateness dimension (proposed in da Costa Pereira et al., Reference da Costa Pereira, Dragoni, Pasi, Boughanem, Berrut, Mothe and Soulé-Dupuy2009) and the coverage dimension (proposed in Pasi et al., Reference Pasi, Bordogna and Villa2007). We have considered those three relevance dimensions because they explicitly account for the user query/goals. This is not the case for the popularity relevance dimension for example.

7.1.1. Aboutness

The term aboutness (topical relevance) is formally defined as follows. Let $d = [w_{1d} \ldots w_{T d}]$ and $q = [w_{1q} \ldots w_{T q}]$ representing document d and query q, respectively, with T representing the size of the term vocabulary used. The measure of aboutness (topical relevance) is calculated by the standard cosine-similarity (Salton & McGill, Reference Salton and McGill1984):

(1) \begin{equation}{\mathrm{Aboutness}}_{IR}(d,q) = \frac{\sum_{i=1}^{T}(w_{iq}.w_{id})}{\sqrt{\sum_{i=1}^{T} w_{iq}^2 \;.\; \sum_{i=1}^{T} w_{id}^2}}.\end{equation}

7.1.2. Coverage

The coverage criterion is assessed on the document representation and on the user profile representation. It measures how strongly the user interests are included in a document.

(2) \begin{equation}{\mathrm{Coverage}}_{IR}(d,q) = \frac{\sum_{i=1}^{T}{\min(w_{iq}, w_{id})}}{\sum_{i=1}^{T} w_{iq}}.\end{equation}

This function produces the maximum value 1 when the non null elements in q ^′s vector also belong to d ^′s vector. It produces the value zero when the two vectors have no common element. Moreover, the value of the function increases with the increase of the number of common elements.

7.1.3. Appropriateness

This dimension allows to measure how appropriate or how seemly a document is with respect to the user interests.

(3) \begin{equation}{\mathrm{Appropriateness}}_{IR}(d, q) = 1 - \frac{\sum_{i=1}^{T}|w_{iq} - w_{id}|}{T}.\end{equation}

According to this definition, a piece of information is considered fully appropriate if it covers all the user interests. However, if in addition it covers other subjects, it is considered less appropriate.

7.2. Reformulation in logic

We can consider a user query in IR as the information needed to achieve a goal.

This way, the premise of the goal can be represented by a formula that corresponds to the agent’s information need. Let $\varphi$ and $\psi$ be two conjunctions of positive literals of a propositional language. We have:

\begin{eqnarray*} {\mathrm{Aboutness}}_{Logic}(\varphi,\psi)) &=& \frac{|\varphi \cap \psi|}{\sqrt{|\varphi| \;.\; |\psi|}},\\[5pt] {\mathrm{Coverage}}_{Logic}(\varphi,\psi) &=& \frac{|\varphi \cap \psi|}{|\psi|},\\[5pt] {\mathrm{Appropriateness}}_{Logic}(\varphi,\psi) &=& 1 - \frac{|\varphi \setminus \psi| + |\psi \setminus \varphi|}{|L|}.\end{eqnarray*}

After replacing the premises of the agent’s goal by the formula $\psi$ , the measure defined in Definition 14 is then re-written as follows:

\begin{align*} U(\varphi,\psi) = \frac{1}{2} \cdot \left(n(\varphi) + \frac{ |\varphi \cap \psi|}{ |\psi| + \frac{|\varphi \setminus \psi|}{|\varphi \setminus \psi | + 1}}\right)\end{align*}

with $n(\varphi) = 1$ if $\psi \subseteq \varphi$ and $n(\varphi) = 0$ otherwise.

More precisely, we consider a propositional language L that has T propositional letters $p_1 \ldots p_{T}$ and a letter $g_0$ representing the goal of the user. A document d can then be represented by a formula noted $\varphi_d$ defined as: $\varphi_d = \bigwedge_{i=1, \ldots, T\;and\;w_{i,d}=1} p_i$ . A query q can also be represented by a formula noted $premise(g_0)$ defined by $\psi_q = \bigwedge_{i=1, \ldots, T\;and\;w_{i,q}=1} p_i$ .

The following two propositions allows us to, respectively, reformulate in logic the three IR relevance dimensions we have considered from the literature, and to provide some comparisons between the U measure and the IR ones.

Proposition 9

\begin{align*}{\mathrm{Aboutness}}_{I\; R}(d,q) &= {\mathrm{Aboutness}}_{Logic}(\varphi_d,\psi_q) \\[5pt]{\mathrm{Coverage}}_{I\; R}(d,q) &= {\mathrm{Coverage}}_{Logic}(\varphi_d,\psi_q) \\[5pt]{\mathrm{Appropriateness}}_{I\; R}(d,q) &= {\mathrm{Appropriateness}}_{Logic}(\varphi_d,\psi_q) \end{align*}

Example 10 Let us consider again Example 9, with the propositional language whose letters are p, q, r and $g_0$ . $\psi = p \wedge q$ and let us consider the five formulas: $\varphi_1 = r$ , $\varphi_2 = p \wedge r$ , $\varphi_3 = p$ , $\varphi_4 = p \wedge q \wedge r$ , and $\varphi_5 = p \wedge q$ . The following table summarizes the values of the four measurements.

A number of observations emerge from these results. First of all, we notice that two formulas can have identical degrees of coverage without their degrees of usefulness being identical. Thus, $Coverage(\varphi_2, \psi) = Coverage(\varphi_3, \psi)$ but $U(\varphi_2, \psi) \not= U(\varphi_3, \psi)$ . Similarly, two formulas may have identical degrees of appropriateness without their degrees of usefulness being identical. Thus, $Appropriatemess(\varphi_3, \psi) = Appropriateness(\varphi_4, \psi)$ but $U(\varphi_3, \psi) \not= U(\varphi_4, \psi)$ . We also notice that p and $p \wedge q \wedge r$ have identical appropriateness values although for different reasons: $appriopriateness(p, p \wedge q) = 2/3$ because p says nothing about q, whereas this is part of the user’s information need, and $appriopriateness(p \wedge q \wedge r, p \wedge q) = 2/3$ because $p \wedge q \wedge r$ , although providing all the information the user need to achieve his/her goal, it provides unnecessary information, r. On the other hand, these different reasons lead to different degrees of usefulness and, in particular, $U(p, p \wedge q)$ is much lower than $U(p \wedge q \wedge r, p \wedge q)$ . Indeed, by definition, U favors information that allows the user need to be satisfied (this is fully the case with $p \wedge q \wedge r$ whereas it is partially the case with p). Even if $p \wedge q \wedge r$ provides unnecessary information, namely r, the user will be able to achieve his/her goal with it, unlike with p.