## 3 Symbolic Temporal Regression
On top of [PERSON]'s relations, the operator \(\langle=\rangle\) is added to model decisions that are taken on the same interval. For each \(X\in\{A,L,B,E,D,O\}\), the modality \(\langle\overline{X}\rangle\), corresponding to the inverse relation \(R_{\overline{X}}\) of \(R_{X}\), is said to be the transpose of the modality \(\langle X\rangle\), and vice versa. Intuitively, the formulas of HS can express the properties of a time series such as if there exists an interval in which \(A_{1}\) is high, during which \(A_{2}\) is low, then \(\ldots\), as an example of using existential operators, or as if during a certain interval, \(A_{1}\) is always low, then \(\ldots\), as an example of using universal ones. Formally, HS formulas are interpreted on time series. We define:  
\[T=\langle\mathbb{I}([l]),V\rangle,\]  
where \([l]=\{1,\ldots,l\}\) is the domain of the time series, \(\mathbb{I}([l])\) is the set of all strict intervals over \([l]\) having cardinality \(l(l-1)/2\), and:  
\[V:\mathcal{AP}\to 2^{\mathbb{I}([l])}\]  
is a valuation function, which assigns to each proposition \(p\in\mathcal{AP}\) the set of intervals \(V(p)\) on which \(p\) holds. Following the presentation, note that we deliberately use \(l\) for the domain of \(T\), which is also the maximum fixed lag. The truth of formula \(\varphi\) on a given interval \([x,y]\) in a time series \(T\) is defined by structural induction on formulas as follows:  
\[ \begin{array}{llll}T_{r}\left[x,y\right]\Vdash p&\text{iff}&\left[x,y \right]\in V(p),\text{ for all }p\in\mathcal{AP};\\ T_{r}\left[x,y\right]\Vdash\
eg\psi&\text{iff}&T_{r}\left[x,y\right]\
ot \models\psi\text{ (i.e., it is not the case that }T,\left[x,y\right]\Vdash\psi);\\ T_{r}\left[x,y\right]\Vdash\psi_{1}\vee\varphi_{2}&\text{iff}&T_{r}\left[x,y \right]\Vdash\psi_{1}\text{ or }T,\left[x,y\right]\Vdash\varphi_{2};\\ T_{r}\left[x,y\right]\Vdash\langle=\rangle\psi&\text{iff}&T_{r}\left[x,y \right]\Vdash\psi;\\ T_{r}\left[x,y\right]\Vdash\langle X\rangle\psi&\text{iff}&\text{there is }[w,z]\text{ s.t. }\left[x,y\right]R_{X}\left[w,z\right]\text{ and }T,\left[w,z\right]\Vdash\psi;\\ T_{r}\left[x,y\right]\Vdash\langle\overline{X}\rangle\psi&\text{iff}&\text{there is }[w,z]\text{ s.t. }\left[x,y\right]R_{\overline{X}}\left[w,z\right]\text{ and }T,\left[w,z\right]\Vdash\psi;\end{array} \]  
where \(X\in\{A,L,B,E,D,O\}\). It is important to point out, however, the we use logic as a tool; through it, we describe the time series that predict a certain value, so that the expert is able to understand the underlying phenomenon. The semantics of the relations \(R_{X}\) allows us to ease such an interpretation: \begin{tabular}{c c c} \(R_{A}\) & (meets) & an interval that meets the current one; \\ \(R_{L}\) & (later than) & an interval that is later than the current one; \\ \(R_{E}\) & (ends) & an interval that ends the current one; \\ \(R_{B}\) & (starts) & an interval that starts the current one; \\ \(R_{D}\) & (during) & an interval that is during the current one; \\ \(R_{O}\) & (overlaps) & an interval that overlaps the current one. \\ \end{tabular}   
Thus, a formula of the type \(p\wedge\langle A\rangle q\) is interpreted as \(p\) holds now (in the current interval), and there is an interval that starts when the current one ends in which \(q\) holds. From the syntax, we can easily generalize the concept of decision and define a set of temporal and atemporal decisions \(\mathcal{S}=\mathcal{S}_{\circ}\cup\mathcal{S}_{=}\), where:  
\[ \begin{array}{rcl}\mathcal{S}_{\circ}&=&\{\langle X\rangle(A\bowtie_{ \gamma}a),\langle\overline{X}\rangle(A\bowtie_{\gamma}a)\mid A\text{ is an attribute and }a\in dom(A)\},\\ \mathcal{S}_{=}&=&\{A\bowtie_{\gamma}a\mid A\text{ is an attribute and }a\in dom(A)\}, \end{array}  \tag{13}\]  
where \(\bowtie\in\{\leq,=,\
eq,>\}\), \(\gamma\in(0.0,1.0]\), and \(\langle X\rangle\) is an interval operator of the language of HS.