2.1 Syntax overview

The only fluent in our example scenario is ${{\textit{Attention}}}$ . It is boolean, that is, it may take values $\top$ and $\bot$ . In EPEC, this is written

(1) \begin{equation} {{{Attention}}}\, {{ \mathbf{takes-values} }}\, \{ \top, \bot \}. \end{equation}

In the example, we consider the agent to be an automated system that is responsible for triggering a re-engagement strategy when necessary. On the other hand, the child is considered part of the environment. Therefore, the actions are PlaySound and TrackObject, where PlaySound is the agent action of playing a sound and TrackObject is the environmental action which corresponds to the child tracking the object with her eyes. The effects of not tracking the object are defined by the following proposition:

(2) \begin{equation} \neg {{{TrackObject}}}\, {{ \mathbf{causes-one-of} }}\, \{ (\{\neg {{{Attention}}} \}, 0.9), (\emptyset,0.1) \}. \end{equation}

Looking at the object is never considered to be due to chance, therefore we also include the following proposition:

(3) \begin{equation} {{{TrackObject}}}\, {{ \mathbf{causes-one-of} }}\, \{ (\{{{{Attention}}} \}, 1) \}. \end{equation}

For what regards action PlaySound under the control of the agent, we want to formalize the conditional plan stating that this action must be performed at instant 2 if the child is believed to be distracted, that is, if the probability of Attention has fallen below a predefined threshold, say $0.1$ . This can be written in EPEC as:

(4) \begin{equation} {{{PlaySound}}}\, {{ \mathbf{performed-at} }}\, 2 \,{{ \mathbf{if-believes} }}\, ({{{Attention}}}, [0, 0.1]). \end{equation}

We consider the child to be initially paying full attention to the task. This translates to:

(5) \begin{equation} {{\mathbf{initially-one-of} }}\, \{ (\{{{{Attention}}}\},1) \}. \end{equation}

Finally, the action TrackObject occurs twice at instants 0 and 1 with confidence $0.25$ and $0.13$ respectively. This translates to the following propositions:

(6) \begin{gather} {{{TrackObject}}}\, {{ \mathbf{occurs-at} }}\, 0 {{ \mathbf{with-prob} }}\, 0.25\end{gather}

(7) \begin{gather} {{{TrackObject}}}\, {{ \mathbf{occurs-at} }}\, 1\, {{ \mathbf{with-prob} }} 0.13. \end{gather}

In EPEC, Proposition (1) is known as a v-proposition (v for “value”), Propositions (2) and (3) are known as c-propositions (c for “causes”), Proposition (4) is known as a p-proposition (p for “performs”), Proposition (5) is known as an i-proposition (i for “initially”), Propositions (6) and (7) are known as o-propositions (o for occurs). Propositions (4), (6) and (7) are known as the narrative part of the domain.

The set of propositions {(1),….,(7)} constitutes what in EPEC is called a domain description and is usually denoted by $\mathcal{D}$ (with appropriate subscripts and/or supercripts). The narrative part of a domain is denoted by ${narr}(\mathcal{D})$ . A domain description is given meaning using a bespoke semantics.

2.2 Semantics overview

The semantics of EPEC enumerates all the possible evolutions (worlds in the terminology of EPEC) of the environment being modeled, starting from the initial state. It then applies a series of standard reasoning about actions principles (e.g., persistence of fluents, closed world assumption for actions) to filter out those evolutions of the environment that do not make intuitive sense with respect to the given domain. The remaining worlds, dubbed well-behaved worlds to indicate that they are meaningful with respect to the given domain description, are assigned a probability value. Some well-behaved worlds for our example domain can be depicted as follows:

World $W_1$ represents a possible evolution of the environment starting from a state in which the child is initially (at instant 0) fully attentive and tracking the object with her eyes. At instant 1, the child is still paying attention to the task, but he/she is no longer tracking the object. At instant 2, the child is distracted and again not tracking the object. The sound is never played. EPEC’s semantics assigns a probability of $0.19575$ to this world, as this results from the product of $0.25$ (due to proposition (6) and TrackObject occurring at 0 in $W_1$ ), $1-0.13$ (due to proposition (7) and TrackObject not occurring at 1 in $W_1$ ) and $0.9$ (due to proposition $2$ and since not tracking the object at instant 1 caused Attention not to hold at instant 2). The intuitive meaning of world $W_2$ and its probability can be figured similarly. In $W_2$ the child is always attentive but never tracking the object, the sound is never played, and its probability is $ 0.006525 $ .

There are 8 well-behaved worlds with respect to our example domain, and they are such that their probabilities sum up to 1. Formal results about the semantics of EPEC guarantee that this is always the case for any domain description and the corresponding set of well-behaved worlds.

We now want to address the question “What is the level of attention of the child at instant 2?”. EPEC first needs to query the domain description about the probability of fluent Attention at instant 2. In EPEC, queries are expressed in the form of an i-formula (i.e., a formula with instants attached). In our case, the query has the form $Q_1={[{{{Attention}}}]@2}$ , but more complicated queries are also possible, for example $Q_2=({[\neg Attention]@1} \vee {[\neg {{{TrackObject}}}]@1}) \wedge {[{{{Attention}}}]@2}$ . We say that i-formula $Q_1$ has instant 2 (as this is the only instant appearing in the formula) and that the i-formula $Q_2$ has instants $>0$ (as all the instants appearing in the formula are strictly greater than 0). Answering query ${[{{{Attention}}}]@2}$ amounts to summing the probabilities of those well-behaved worlds in which the i-formula is true, that is, such that Attention is true at instant 2. This results in $0.158275$ , and can also be written as:

Note that this value falls outside the interval $[0, 0.1]$ in the precondition of Proposition (4). Therefore, EPEC deduces that the answer to our last question in Scenario 2.1 (“Should the sound be played at time 2?”) is no. Then, Proposition (4) does not fire and the action PlaySound, under the control of the agent, is not played in any of the well-behaved worlds.

2.3 Simulating epistemic actions via c-propositions and o-propositions

The reader may have noticed that our target application domain mainly deals with sensors. In its original formulation (D’Asaro et al. Reference D’Asaro, Bikakis, Dickens and Miller2020), epistemic modeling is reserved to an additional, type of propositions, called s-propositions (s for “senses”), of the following form:

(8) \begin{equation} S {{ \mathbf{senses} }} F {{ \mathbf{with-accuracies} }} M \end{equation}

for an action S, a fluent F and some matrix M representing the accuracy of A when sensing F. Given their role within EPEC, we would need s-propositions to model sensors in our domain. However, s-propositions add a layer of complexity that in this work we would like to avoid due to the necessity of taking decisions at runtime. In this section, we show a possible translation procedure of s-propositions into pairs composed by a c-proposition and an o-proposition. This aims to demonstrate that one can make “improper” use of c-propositions and o-propositions (that are typically used for effectors) as a means to model epistemic aspects of a domain. Whether a sound and complete translation is available remains however a subject for future work.

To illustrate, consider a simple domain description in which an agent (imperfectly) tests twice for Flu:

(9) \begin{gather} {{{Flu}}} {{ \mathbf{takes-values} }} \{ \top, \bot \}\end{gather}

(10) \begin{gather} {{\mathbf{initially-one-of} }} \{ ( \{ {{{Flu}}} \}, 0.7 ), ( \{ \neg {{{Flu}}} \}, 0.3 ) \}\end{gather}

(11) \begin{gather} {{{Test}}} {{ \mathbf{senses} }} {{{Flu}}} {{ \mathbf{with-accuracies} }} \begin{pmatrix}0.8 & 0.2\\0.4 & 0.6\end{pmatrix}\end{gather}

(12) \begin{gather} {{{Test}}} {{ \mathbf{performed-at} }} 0\end{gather}

(13) \begin{gather} {{{Test}}} {{ \mathbf{performed-at} }} 1 \end{gather}

which means that the Test has the effect of producing knowledge about whether the agent has Flu with associated confusion matrix $\begin{pmatrix} 0.8 & 0.2\\0.4 & 0.6\end{pmatrix}$ , where $0.8$ is the true positives rate and $0.6$ is the true negatives rate. The domain description above entails the following b-proposition (b for “believes”):

(14) \begin{align} { { \bf{at} }} 2 &{{ \bf{believes} }} {[{{{Flu}}}]@0} {{ \mathbf{with-probs} }}\end{align}

(15)

(16)

(17)

(18)

This proposition may be interpreted as saying that an agent sitting at instant 2, having tested for Flu twice, is in one of the four following knowledge states:

• • The test came up negative twice (as in the case of (15)). Given the initial knowledge about the distribution of Flu, this is likely to happen with probability $0.136$ . In this case, the agent will believe that the probability of ${[{{{Flu}}}]@0}$ decreased from the initial value $0.7$ to $0.206$ .

• • The test came up negative at instant 0 and positive at instant 1 (as in the case of (16)). This is likely to happen with probability $0.184$ , and in this case the agent will believe that the probability of ${[{{{Flu}}}]@0}$ is $0.608$ .

• • The test came up positive at instant 0 and negative at instant 1 (as in the case of (17)). This is likely to happen with probability $0.184$ , and in this case the agent will believe that the probability of ${[{{{Flu}}}]@0}$ is $0.608$ .

• • The test came up positive twice (as inthe case of (18)). This is likely to happen with probability $0.496$ , and in this case the agent will believe that the probability of ${[{{{Flu}}}]@0}$ is $0.903$ .

This syntax shows the epistemic core of EPEC, which allows one to reasoning about past, present and future: for example, note that in (14) we reason about the perspective of an agent at instant 2 who is reasoning about whether it had Flu at instant 0 in the past. This powerful syntax comes at expenses of computational efficiency, and therefore in this work we adopt a number of simplifications due to the fact that we only need to reason about the present. The first of such simplifications is that we drop s-propositions and simulate them via o-propositions and c-propositions, in a way that we now aim to make clearer.

Assume that the agent modeled in the domain description above tests positive twice. Then, we may note that substituting the s-proposition (11) and the two p-propositions (12) and (13) with

(19) \begin{gather} {{{Test}}} {{ \mathbf{causes} }} \{ ( \{ {{{Flu}}} \}, 1 ) \}\end{gather}

(20) \begin{gather} {{{Test}}} {{ \mathbf{occurs-at} }} 0 {{ \mathbf{with-prob} }} 0.412\end{gather}

(21) \begin{gather} {{{Test}}} {{ \mathbf{occurs-at} }} 1 {{ \mathbf{with-prob} }} 0.452 \end{gather}

yields

(22)

which matches with the result in (18), and therefore we may say that simulates it. Although a formal characterization of the translation from s-proposition to c-propositions and o-propositions is beyond the scope of this work, note that an s-proposition for a boolean fluent F of the form

(23) \begin{equation} S {{ \mathbf{senses} }} F {{ \mathbf{with-accuracies} }} \begin{pmatrix}a & 1-a\\b & 1-b\end{pmatrix} \end{equation}

is simulated either by a c-proposition

(24) \begin{equation} S^+ {{ \mathbf{causes-one-of} }} \{ ( \{ F \}, 1 ) \} \end{equation}

and an o-proposition

(25) \begin{equation} S^+ {{ \mathbf{occurs-at} }} I {{ \mathbf{with-prob} }} \frac{p(a-b)}{p(a-b)+b} \end{equation}

if S is performed at I and produces a positive result, or by a c-proposition

(26) \begin{equation} S^- {{ \mathbf{causes-one-of} }} \{ ( \{ \neg F \}, 1 ) \} \end{equation}

and an o-proposition

(27) \begin{equation} S^- {{ \mathbf{occurs-at} }} I {{ \mathbf{with-prob} }} \frac{(p-1)(a-b)}{p(a-b)+b-1} \end{equation}

if S is performed at I and produces a negative result, where p is the probability such that

in the domain $\mathcal{D}$ under consideration. This informally shows that it is possible to meaningfully interpret c-propositions and o-propositions as epistemic actions in place of s-propositions, and justifies their employment to perform sensing.

5.1 Scalability

Experiment (a). To demonstrate that PEC-RUNTIME scales favorably compared with PEC-ASP and PEC-ANGLICAN we tested the three implementations on a simple domain description in which an action repeatedly occurs and causes the probability of a fluent to slowly decay:

\begin{gather*} F {{ \mathbf{takes-values} }} \{ \top, \bot \}\\ {{\mathbf{initially-one-of} }} \{ (\{ F \}, 1 ) \}\\ A {{ \mathbf{causes-one-of} }} \{ (\{ \neg F \},0.2), (\emptyset,0.8) \}\\ \forall I,\ A {{ \mathbf{performed-at} }} I {{ \mathbf{with-prob} }} 0.5 \end{gather*}

where the set of instants is $\{ 0, 1, \dots, 15 \}$ . The probability of $\neg F$ as calculated by the three frameworks is shown in Figure 3. Table 1 summarizes the performances of PEC-RUNTIME, PEC-ASP and PEC-ANGLICAN.

Fig. 3. Probability of F as a function of time for the example discussed in Section 5.1, Experiment (a).

Table 1. Time (in seconds) to execute the query ${[\neg F]@I}$ in the example discussed in Section 5.1. The numbers in bracket in the case of PEC-ANGLICAN refer to the number of sampled well-behaved worlds used to approximate the result of the query. For every implementation, reported times include grounding and processing of the domain description

As it can be seen from these results, PEC-RUNTIME outperforms both PEC-ASP and PEC-ANGLICAN on all queries. In terms of performance, PEC-ANGLICAN is close to PEC-RUNTIME when only 100 well-behaved worlds are sampled. However, sampling 100 worlds causes precision issues as it is clear from Figure 3.

Experiment (b). In this experiment we consider the effect of varying the number of actions in the domain description. To isolate the effect of actions from that of other side-effects we consider domain descriptions of the simplest possible form:

\begin{gather*} F {{ \mathbf{takes-values} }} \{ \top, \bot \}\\ {{\mathbf{initially-one-of} }} \{ ( \{ F \}, 1 ) \}\\ \forall I,\ A_1 {{ \mathbf{performed-at} }} I {{ \mathbf{with-prob} }} 0.5\\ \vdots\\ \forall I,\ A_n {{ \mathbf{performed-at} }} I {{ \mathbf{with-prob} }} 0.5 \end{gather*}

where $\mathcal{I} = \{ 1, \dots, 15 \}$ and we consider different values of n ranging from 0 to 15. We evaluate the computation time for query ${[F]@I}$ at all instants. Results are plotted in Figure 4 and confirm the intuition that, at an instant I, computation time scales exponentially with the number of action occurrences at that instant.

Fig. 4. Time (in seconds) to query the domain in Section 5.1, Experiments (b) and (c), expressed as a function of the number of actions and fluents. The results show averages over 15 runs – however, standard error was not plotted as it it significantly small ( $<0.05$ at all data points).

Experiment (c). To empirically evaluate the effect of the number of fluents in the domain, we performed a similar experiment with domain descriptions of the following forms:

\begin{gather*} F_1 {{ \mathbf{takes-values} }} \{ \top, \bot \}\\ \vdots\\ F_n {{ \mathbf{takes-values} }} \{ \top, \bot \}\\ {{\mathbf{initially-one-of} }} \{ ( \{ F_1, \dots, F_n \}, 1 ) \} \end{gather*}

where $\mathcal{I}=\{ 1, \dots, 15 \}$ and n ranges from 1 to 15. Results are shown in Figure 4, and confirm the intuition that computational time scales exponentially as a function of the number of fluents in the domain.

Experiment (d). This experiment combines (b) and (c) in that it shows how computation time scales with both fluents and actions. The considered domain descriptions are:

\begin{gather*} F_1 {{ \mathbf{takes-values} }} \{ \top, \bot \}\\ \vdots\\ F_n {{ \mathbf{takes-values} }} \{ \top, \bot \}\\ {{\mathbf{initially-one-of} }} \{ ( \{ F_1,\dots,F_n \}, 1 ) \}\\ \forall I,\ A_1 {{ \mathbf{performed-at} }} I {{ \mathbf{with-prob} }} 0.5\\ \vdots\\ \forall I,\ A_m {{ \mathbf{performed-at} }} I {{ \mathbf{with-prob} }} 0.5 \end{gather*}

where $\mathcal{I}=\{ 1,\dots,15 \}$ , the number of fluents varies from 1 to 15 and the number of actions varies from 0 to 15. Result are shown in Figure 5 and extend those of Figure 4.

Fig. 5. Contour plot showing time (in seconds) to query the domain in Section 5.1, Experiment (d), expressed as a function of the number of actions and fluents.

Experiment (e). In this experiment we empirically evaluate the effect of initial conditions, and show that computation time scales linearly with their number. To this aim, we consider the following domain description:

\begin{gather*} F_1 {{ \mathbf{takes-values} }} \{ \top, \bot \}\\ \vdots\\ F_{10} {{ \mathbf{takes-values} }} \{ \top, \bot \}\\ {{\mathbf{initially-one-of} }} \{ ( \{ F_1, \dots, F_{10} \}, 1/n ), \dots, ( \{ \neg F_1, \dots, F_{10} \}, 1/n ),\\ ( \{ \neg F_1, \dots, \neg F_{10} \}, 1/n ) \} \end{gather*}

where $\mathcal{I}=\{ 1, \dots, 15 \}$ and n ranges from 1 to $2^{10}$ , that is, the maximum number of possible initial conditions with 10 fluents. Note that n is also the number of well-behaved worlds considered by PEC-RUNTIME at each progression step. Results are shown in Figure 6 and confirm the intuitive hypothesis that computational time scales linearly with the number of initial conditions/well-behaved worlds.

Fig. 6. Time (in seconds) to query the domain in Section 5.1, Experiment (e), expressed as a function of the number of initial conditions.

Experiment (f). This experiment aims to study how PEC-RUNTIME behaves in a more realistic domain where every sensor has a fixed probability p of producing a reading at every instant. We consider the following domain descriptions:

\begin{gather*} F_1 {{ \mathbf{takes-values} }} \{ \top, \bot \}\\ F_2 {{ \mathbf{takes-values} }} \{ \top, \bot \}\\ F_3 {{ \mathbf{takes-values} }} \{ \top, \bot \}\\ F_4 {{ \mathbf{takes-values} }} \{ \top, \bot \}\\ F_5 {{ \mathbf{takes-values} }} \{ \top, \bot \}\\ {{\mathbf{initially-one-of} }} \{ ( \{ \neg F_1, \neg F_2, \neg F_3, \neg F_4, \neg F_5 \}, 1 ) \}\\ \neg A_1 \wedge \neg A_2\wedge \neg A_3 \wedge \neg A_4 \wedge A_5 {{ \mathbf{causes-one-of} }} \{ (\{ \neg F_1, \neg F_2, \neg F_3, \neg F_4, F_5 \},4/5), (\emptyset,1/5) \}\\ \vdots\\ A_1 \wedge A_2\wedge A_3 \wedge A_4 \wedge \neg A_5 {{ \mathbf{causes-one-of} }} \{ (\{ F_1, F_2, F_3, F_4, \neg F_5 \},4/5), (\emptyset,1/5) \}\\ A_1 \wedge A_2\wedge A_3 \wedge A_4 \wedge A_5 {{ \mathbf{causes-one-of} }} \{ (\{ F_1, F_2, F_3, F_4, F_5 \},4/5), (\emptyset,1/5) \}\\ \end{gather*}

and, for every instant I and action A, the proposition

\begin{equation*} A_i {{ \mathbf{performed-at} }} I {{ \mathbf{with-prob} }} 0.5 \end{equation*}

has a some fixed probability p of being included in the domain description, where p varies in the set $\{ 0.1, 0.2, \dots, 1 \}$ . Results are shown in Figure 7.

Fig. 7. Time (in seconds) to query the example discussed in Section 5.1, Experiment (f). The results show averages over 30 runs.

5.2 Effect of noise

One of the characteristics of EPEC (and its dialects) that makes it highly suitable as an interface with machine learning algorithms is that it supports events annotated with probabilities. If a classifier provides a confidence score for its output, it is possible to feed such score into EPEC in the form of a probability. This may be useful to account for possible artifacts and flickering of a given classifier. For instance, consider a classifier C that alternatively produces readings true and false for some characteristic F. This may be represented in EPEC as the following stream of events:

\begin{gather*} {{{Ctrue}}} {{ \mathbf{occurs-at} }} 0 {{ \mathbf{with-prob} }} P_{true}\\ {{{Cfalse}}} {{ \mathbf{occurs-at} }} 1 {{ \mathbf{with-prob} }} P_{false}\\ {{{Ctrue}}} {{ \mathbf{occurs-at} }} 2 {{ \mathbf{with-prob} }} P_{true}\\ {{{Cfalse}}} {{ \mathbf{occurs-at} }} 3 {{ \mathbf{with-prob} }} P_{false}\\ \vdots\\ {{{Ctrue}}} {{ \mathbf{occurs-at} }} 18 {{ \mathbf{with-prob} }} P_{true}\\ {{{Cfalse}}} {{ \mathbf{occurs-at} }} 19 {{ \mathbf{with-prob} }} P_{false} \end{gather*}

where Ctrue (resp. Cfalse) is an event that causes the value of fluent F to be $\top$ (resp. $\bot$ ), that is:

\begin{gather*} {{{Ctrue}}} {{ \mathbf{causes-one-of} }} \{ (\{ F \}, 1) \}\\ {{{Cfalse}}} {{ \mathbf{causes-one-of} }} \{ (\{ \neg F \}, 1) \} \end{gather*}

and the truth value of f is initially unknown, that is:

\begin{gather*} {{\mathbf{initially-one-of} }} \{ (\{F\},0.5), (\{\neg F\},0.5) \} \end{gather*}

We analyze the behavior of EPEC in different scenarios (corresponding to different values for $P_{true}$ and $P_{false}$ ):

Scenario (a). The classifier C produces true with high confidence and false with low confidence, for example, $P_{true}=0.99$ and $P_{false}=0.01$ . This domain description entails “ ${[F]@20} {{ \mathbf{holds-with-prob }}} P$ ” where $P\approx 0.9899$ , which makes intuitive sense as the false events have low confidence and are discarded as background noise.

Scenario (b). The classifier C produces true and false with a high degree of uncertainty. However, it assigns true a slightly higher degree of confidence, for example $P_{true}=0.501$ and $P_{false}=0.499$ . This domain description entails “ ${[F]@20} {{ \mathbf{holds-with-prob }}} P$ ” where $P\approx 0.653246$ , which reflects that such a sequence of events does not carry as much information as in Scenario (a), due to the the small difference between $P_{true}$ and $P_{false}$ .

Scenario (c). The classifier C produces both true and false with low confidence, with $P_{true}\gg P_{false}$ . For example, let $P_{true} = 0.49$ and $P_{false}=0.01$ . This domain description entails “ ${[F]@20} {{ \mathbf{holds-with-prob }}} P$ ” where $P\approx 0.979286$ , which again makes intuitive sense as false readings are again discarded as background noise, with the repeated true readings making the probability of F steadily grow throughout the timeline.

These scenarios show that EPEC is sensibly more accurate than any logical framework that is not capable of handling probabilities in the presence of noisy sensors. In fact, these frameworks would have to set a threshold, and only accept events with probabilities higher than that threshold. In our example, setting a threshold for example of $0.5$ would lead to significant errors especially in Scenarios (b) and (c). In Scenario (b), all Cfalse events would be discarded, implying that F is detected to hold true at instant 20 with certainty (compare this to the probability $0.653246$ assigned to the same query by EPEC). In Scenario (c), all events would be discarded and it would not be possible neither that F holds at 20 nor that it does not (compare again to the probability $0.979286$ assigned by EPEC).

5.3 An example from AVATEA

As discussed in Section 4, a layer of classifiers directly injects events into PEC-RUNTIME, which merges them together and decides what action to take according to a predefined conditional plan. We show how PEC-RUNTIME accounts for these classifiers in the following example.

We use the Head Pose Recognition module to decide whether the child is looking at the screen or not, and therefore this module can trigger an EyesNotFollowingTarget event (and possibly an associated probability). Similarly, the Shoulders Alignment Module triggers a BadPosture event if it detects that the child’s shoulders are not correctly aligned. Finally, the Emotion Recognition Module triggers a LowValence event when it thinks the child is experiencing negative valence. On the basis of these events, we aim to evaluate the probabilities of TaskCorrect, that is, that the child is performing the therapeutic task correctly, and Engagement, that is, that the child and engaged to the task. Consider the following narrative of events:

\begin{gather*} {{{EyesNotFollowingTarget}}} {{ \mathbf{occurs-at} }} 0 {{ \mathbf{with-prob} }} 1 \\ {{{BadPosture}}} {{ \mathbf{occurs-at} }} 0 {{ \mathbf{with-prob} }} 1 \\ {{{LowValence}}} {{ \mathbf{occurs-at} }} 0 {{ \mathbf{with-prob} }} 1 \\ {{{EyesNotFollowingTarget}}} {{ \mathbf{occurs-at} }} 1 {{ \mathbf{with-prob} }} 1 \\ {{{BadPosture}}} {{ \mathbf{occurs-at} }} 1 {{ \mathbf{with-prob} }} 1 \\ {{{LowValence}}} {{ \mathbf{occurs-at} }} 1 {{ \mathbf{with-prob} }} 1 \\ {{{EyesNotFollowingTarget}}} {{ \mathbf{occurs-at} }} 2 {{ \mathbf{with-prob} }} 76/100 \\ {{{LowValence}}} {{ \mathbf{occurs-at} }} 2 {{ \mathbf{with-prob} }} 87/100 \\ {{{EyesNotFollowingTarget}}} {{ \mathbf{occurs-at} }} 3 {{ \mathbf{with-prob} }} 1 \\ {{{BadPosture}}} {{ \mathbf{occurs-at} }} 3 {{ \mathbf{with-prob} }} 1 \\ {{{LowValence}}} {{ \mathbf{occurs-at} }} 3 {{ \mathbf{with-prob} }} 1 \\ {{{EyesNotFollowingTarget}}} {{ \mathbf{occurs-at} }} 7 {{ \mathbf{with-prob} }} 7/100 \\ {{{EyesNotFollowingTarget}}} {{ \mathbf{occurs-at} }} 8 {{ \mathbf{with-prob} }} 89/100 \\ {{{EyesNotFollowingTarget}}} {{ \mathbf{occurs-at} }} 9 {{ \mathbf{with-prob} }} 74/100 \\ {{{EyesNotFollowingTarget}}} {{ \mathbf{occurs-at} }} 11 {{ \mathbf{with-prob} }} 1 \end{gather*}

These events are given a meaning by means of a domain description (see Appendix A for the full domain) that specify what bearing each event has on fluents TaskCorrect and Engagement.

Furthermore, consider a conditional plan of the following form:

\begin{gather*} \forall I,\ {{{PlaySound}}} {{ \mathbf{performed-at} }} I {{ \mathbf{if-believes} }} ({{{Attention}}},[0,0.3])\\ \forall I,\ {{{LowerDifficultyLevel}}} {{ \mathbf{performed-at} }} I {{ \mathbf{if-believes} }} ({{{TaskCorrect}}},[0,0.3]) \end{gather*}

which states that a sound must be played by the system whenever the Attention of the child is low, and that the difficulty level must be lowered if the child does not manage to perform it correctly.

PEC-RUNTIME works out the probabilities in Figure 8, and triggers decisions when they fall below the given threshold. In particular, PlaySound is triggered at instant 4, and LowerDifficultyLevel is triggered at instants 2, 3 and 4. It is worth noting here that thresholds may induce the so-called rubberband effect, where abrupt probability changes (due e.g., to noise) may lead to decisions being taken too frequently. Unfortunately, EPEC (and therefore also PEC-RUNTIME) does not allow to control this effect, therefore we implemented a bespoke semaphore in the underlying Python script that only allows decision to be taken every 20 s.

Fig. 8. Engagement and TaskCorrect as a function of time in the example from Section 5.3.

These decisions are forwarded to the game. At the end of each therapeutic session, this narrative is saved to a file so that it can be used to provide feedback to the therapists and re-consulted if needed. In this scenario, PEC-RUNTIME provides feedback such as “The child performed the task correctly for a total of 10 instants out of 13 ( $76.9\%$ ). The game difficulty was lowered at instants 2, 3 and 4 as s/he was unable to perform the exercise. His attention was lowest at instant 4.”. A graph similar to that in Figure 8 is also shown to the therapists, so that they can have a quick overview of the session.

6.1 Gamification

Gamification strategies consist in using game-like elements (e.g., points, rewards, performance graphs, etc.) in serious contexts such as ours. These have proven to be extremely successful to engage young children in diagnostic and therapeutic exercises, even before the advent of digital gaming. While games to test cognitive capabilities (Belpaeme et al.Belpaeme, Baxter, Read, Wood, Cuayáhuitl, Kiefer, Racioppa, Kruijff-Korbayová, Athanasopoulos, Enescu, et al Reference Belpaeme, Baxter, Read, Wood, Cuayáhuitl, Kiefer, Racioppa, Kruijff-Korbayová, Athanasopoulos, Enescu, Looije, Neerincx, Demiris, Ros-Espinoza, Beck, Cañamero, Hiolle, Lewis, Baroni, Nalin, Cosi, Paci, Tesser, Sommavilla and Humbert2012) do not form a sharply defined class, games designed to test and improve motor skills are usually referred to as exergames. The effects of exergames have been found to be generally positive (Vernadakis et al. Reference Vernadakis, Papastergiou, Zetou and Antoniou2015). Gamification systems are usually effective in engaging young users in playful activities, while adapting the current challenge according to level of user competence. Sessions are typically logged in order to provide detailed feedback to therapists. On the cognitive side, these adaptive systems have been designed to evaluate subjective well-being (Wu et al. Reference Wu, Cai and Tu2019) and phonological acquisition (Origlia et al. Reference Origlia, Cosi, Rodà and Zmarich2017) among others. Adaptive exergames have been used for example in the context of children with spinal impairments (Mulcahey et al. Reference Mulcahey, Haley, Duffy, Ni and Betz2008), and to test gross motor skills (Huang et al. Reference Huang, Tung, Chou, Wu, Chen and Hsieh2018).

(Deep) Machine Learning techniques are not advantageous in the case of exergames if they are used alone, as they must be trained on big amounts of data even when expert knowledge can be (relatively) easily extracted from experts and encoded in symbolic form (e.g., using logic programming), with the further advantage that symbolic knowledge can be used to automatically produce explanations and reports. As a centralized reasoning system, we use probabilistic logic programming techniques instead. These systems are a convenient option as they allow both for the representation of formal constraints needed to implement a clinically effective exercise, and for the statistical modeling of intrinsically noisy data sources.

6.2 Probabilistic reasoning about actions

Recently, logic-based techniques have been successfully applied to several fields of Artificial Intelligence, including among others event recognition from security cameras (Skarlatidis et al. Reference Skarlatidis, Artikis, Filippou and Paliourasz2015a; Skarlatidis et al. Reference Skarlatidis, Paliouras, Artikis and Vouros2015b), robot location estimation (Belle and Levesque Reference Belle and Levesque2018), understanding of tenses (Van Lambalgen and Hamm Reference Van Lambalgen and Hamm2008), natural language processing (Nadkarni et al. Reference Nadkarni, Ohno-Machado and Chapman2011), probabilistic diagnosis (Lee and Wang Reference Lee and Wang2018) and intention recogntion (Acciaro et al. Reference Acciaro, D’Asaro and Rossi2021). Given the importance of Machine Learning and Probability Theory in AI, these frameworks and languages have gradually started employing probabilistic semantics (Sato Reference Sato1995) to incorporate and deal with uncertainty. This has given birth to the field of Probabilistic Logic Programming (Riguzzi Reference Riguzzi2018), and we are particularly concerned with Probabilistic Reasoning about Action frameworks as they can deal with agents interacting with some (partially known) environment. For example Bacchus et al. (Reference Bacchus, Halpern and Levesque1999), Belle and Levesque (Reference Belle and Levesque2018), that are based on the Situation Calculus ontology (Reiter Reference Reiter2001), can model imperfect sensors and effectors. The Situation Calculus’ branching structure makes these frameworks mostly suitable for planning under partial states of information. A recent extension of language $\mathcal{C}+$ (Giunchiglia et al. Reference Giunchiglia, Lee, Lifschitz, McCain and Turner2004), dubbed $p\mathcal{BC}+$ , is presented in Lee and Wang (Reference Lee and Wang2018). The authors show how $p\mathcal{BC}+$ can be implemented in $\text{LP}^{\text{MLN}}$ , a probabilistic extension of ASP that can be readily implemented using tools such as LPMLN2ASP (Lee et al. Reference Lee, Talsania and Wang2017). Unlike our work, which focuses on runtime reasoning, $p\mathcal{BC}+$ is mostly suited for probabilistic diagnosis and parameter learning. The two languages MLN-EC (Skarlatidis et al. Reference Skarlatidis, Paliouras, Artikis and Vouros2015b) and ProbEC (Skarlatidis et al. Reference Skarlatidis, Artikis, Filippou and Paliourasz2015a) extend the semantics of the Event Calculus (Kowalski and Sergot Reference Kowalski and Sergot1986; Miller and Shanahan Reference Miller and Shanahan2002) ontology, using Markov Logic Networks (Richardson and Domingos Reference Richardson and Domingos2006) and ProbLog (De Raedt et al. Reference De Raedt, Kimmig and Toivonen2007) to perform event recognition from security cameras. In their proposed case study, the logical part of the architecture receives time-stamped events as inputs and processes them in order to detect complex long-term activities (e.g., infer that two people are fighting from the fact that they have been close to each other and moving abruptly during the last few seconds). Given their semi-probabilistic nature, these frameworks are able to handle uncertainty in the input events (ProbEC) or in the causal rules linking events and fluents (MLN-EC) but they do not deal with any form of epistemic knowledge. On the other hand, EPEC (D’Asaro et al. Reference D’Asaro, Bikakis, Dickens and Miller2020) has the advantage of being based on the Event Calculus (making it particularly well suited for Event Recognition tasks) and being able to deal with epistemic aspects. Its semantics abstracts from any specific programming language and therefore it lends itself to task-specific optimizations, such as the runtime adaptation presented in this work.