2.1 Timelines and temporal intervals

A timeline $\mathbb{T}$ is either the set $\mathbb{Q}$ of rationals or the set $\mathbb{Z}$ of integers. A $\mathbb{T}$ -time point is an element of $\mathbb{T}$ . A $\mathbb{T}$ -interval $\varrho$ is a subset of $\mathbb{T}$ satisfying both of the following properties:

• $t \in\varrho$ for all $t_1,t_2 \in \varrho$ and $t \in \mathbb{T}$ such that $t_1<t<t_2$ ; and

• the greatest lower bound $\varrho^-$ and the least upper bound $\varrho^+$ of $\varrho$ are both in $\mathbb{T}\, \cup \{-\infty ,\infty\}$ .

The bounds $\varrho^-$ and $\varrho^+$ are called the left and right endpoints of $\varrho$ , respectively. A $\mathbb{T}$ -interval is punctual if it contains exactly one $\mathbb{T}$ -time point, it is non-negative if it contains no negative $\mathbb{T}$ -time points, it is bounded if both its endpoints are in $\mathbb{T}$ , and it is closed if it includes both of its endpoints. In these and similar notions, we often omit the reference to $\mathbb{T}$ if it is clear from the context. We consider binary representations of integers and fractional representations of rationals, with an integer numerator and a positive integer denominator, encoded in binary. We use standard representations of the form $\langle \varrho^-,\varrho^+ \rangle$ for a non-empty interval $\varrho$ (i.e., $\varrho \cap \mathbb{T} \neq \emptyset$ ), where the left bracket $\langle$ is either [ or (, the right bracket $\rangle$ is either ] or), and, if numeric, the endpoints $\varrho^-$ and $\varrho^{+}$ are represented as explained above. Brackets [ and ] indicate that the endpoints are included in the interval, whereas (and) indicate that they are not included; note that, by this convention, [ and ] cannot be used with endpoints $-\infty$ and $\infty$ . We often abbreviate a punctual interval [t,t] as t. If it is clear from the context, we abuse notation and identify each interval representation with the interval it represents.

2.2 Syntax and semantics of metric temporal expressions

Assume a function-free first-order vocabulary and a timeline $\mathbb{T}$ . A relational atom is an expression of the form $P(\mathbf{s})$ , where P is a predicate and $\mathbf{s}$ is a tuple of constants and variables of the same arity as P. A metric atom is an expression given by the following grammar, where $P(\mathbf{s})$ ranges over relational atoms and $\varrho$ over non-empty, non-negative intervals:

A metric atom is ground if it mentions no variables. A metric fact is an expression $M @ \varrho$ , with M a ground metric atom and $\varrho$ a non-empty $\mathbb{T}$ -interval; it is relational if so is M. A dataset is a finite set of relational facts; it is bounded if so are all intervals it mentions. For a dataset $\mathcal{D}$ , we denote with $t^{\min}_{\mathcal{D}}$ and $t^{\max}_{\mathcal{D}}$ the smallest and the largest numbers mentioned in $\mathcal{D}$ ; if $\mathcal{D}$ mentions no numbers, we let $t^{\min}_{\mathcal{D}}=t^{\max}_{\mathcal{D}}=0$ .

An interpretation $\mathfrak{I}$ is a function which assigns to each time point $t \in \mathbb{T}$ a set of ground relational atoms; if an atom $P(\mathbf{c})$ belongs to this set, we say that $P(\mathbf{c})$ is satisfied at t in $\mathfrak{I}$ and we write $\mathfrak{I},t \models_\mathbb{T} P(\mathbf{c})$ . This notion extends to other ground metric atoms as given in Table 1. Interpretation $\mathfrak{I}$ is a model of a metric fact $M @ \varrho$ , written $\mathfrak{I} \models_\mathbb{T} M @ \varrho$ , if $\mathfrak{I},t \models M$ for all $t \in \varrho$ ; it is a model of a set $\mathcal{M}$ of metric facts (e.g., a dataset) if it is a model of all facts in $\mathcal{M}$ . Set $\mathcal{M}$ entails a set $\mathcal{M}'$ of metric facts, written $\mathcal{M} \models \mathcal{M}'$ , if every model of $\mathcal{M}$ is a model of $\mathcal{M}'$ . An interpretation $\mathfrak{I}$ contains an interpretation $\mathfrak{I}'$ , written $\mathfrak{I}' \subseteq \mathfrak{I}$ , if for each ground relational atom $P(\mathbf{c})$ and each time point $t \in \mathbb{T}$ , having $\mathfrak{I}',t \models_\mathbb{T} P(\mathbf{c})$ implies $\mathfrak{I},t \models_\mathbb{T} P(\mathbf{c})$ . Furthermore, $\mathfrak{I}$ is the least interpretation in a set X of interpretations, if $\mathfrak{I} \subseteq \mathfrak{I}'$ for every $\mathfrak{I}' \in X$ .

Table 1. Semantics of ground metric atoms

5.1 General programs

It will be convenient for our presentation to assume that programs are in a normal form analogous to that by Tena Cucala et al. (Reference Tena Cucala, Wałęga, Cuenca Grau and Kostylev2021) for stratifiable programs: in each normalised rule the head is a relational atom or $\bot$ , there is neither nesting of metric operators nor occurrences of or in rule bodies, and the only unbounded interval allowed is $[0, \infty)$ .

Second, we iteratively eliminate nested temporal operators from rule bodies. To this end, consider a rule r whose body has an occurrence M of a metric atom that mentions only one temporal operator and which is in the scope of some other temporal operator. If M is of one of the forms , , $\boxminus_\varrho P(\mathbf{s})$ , or $\boxplus_\varrho P(\mathbf{s})$ , then we replace it with $P'(\mathbf{s})$ and add a rule $P'(\mathbf{s}) \gets M$ , where P’ is a fresh predicate of the same arity as P. If M mentions $\mathcal{S}$ or $\mathcal{U}$ , we need to proceed in a special way to ensure safety of the new rules. If M is of the form $P_1(\mathbf{s}_1) \mathcal{S}_{\varrho} P_2(\mathbf{s}_2)$ , we remove r and proceed as follows (note that the conditions below are not exclusive):

• if $0 \in \varrho$ , we add the rule obtained by replacing M in r with $P_2(\mathbf{s}_2)$ ,

• if $1 \in \varrho$ , we add the rule obtained by replacing M in r with $\boxminus_1 P_2(\mathbf{s}_2)$ ,

• we add the rule and the rule obtained by replacing M in r with $P'(\mathbf{s}_1, \mathbf{s}_2)$ , where P’ is a fresh predicate whose arity equals the sum of arities of $P_1$ and $P_2$ .

If M is of the form $P_1(\mathbf{s}_1) \mathcal{U}_{\varrho} P_2(\mathbf{s}_2)$ , we proceed analogously to the case when M is of the form $P_1(\mathbf{s}_1) \mathcal{S}_{\varrho} P_2(\mathbf{s}_2)$ , but instead of $\boxminus$ and we use, respectively, $\boxplus$ and in the new rules. Moreover, if M mentions $\top$ or $\bot$ , we treat these symbols as nullary predicates and proceed as before.

Next, weliminate diamond operators by replacing and with, respectively, $\top \mathcal{S}_\varrho M$ and $\top \mathcal{U}_\varrho M$ . Finally, we eliminate unbounded intervals $\varrho$ different from $[0,\infty)$ as follows. If a rule r mentions $\Box_{\varrho} P(\mathbf{s})$ , we replace this $\Box_{\varrho} P(\mathbf{s})$ with $\Box_{t} P(\mathbf{s})$ and add a rule ${P'(\mathbf{s}) \gets \Box_{[0,\infty) }P(\mathbf{s})}$ , where $\Box$ is either $\boxminus$ or $\boxplus$ , t is the least natural number in $\varrho$ (so $t \geq 1$ ), and P’ is a fresh predicate of the same arity as P. In the case of operators $\mathcal{S}$ and $\mathcal{U}$ we need to pay special attention to ensure that the new rules are safe. Assume that a rule r mentions ${M = P_1(\mathbf{s}_1) \mathcal{S}_{\varrho} P_2(\mathbf{s}_2) }$ with an unbounded $\varrho \neq [0,\infty)$ . We remove r and proceed as follows, for t the least natural number belonging to $\varrho$ :

• if $1 \in \varrho$ , we add the rule obtained by replacing M in r with $\boxminus_1 P_2(\mathbf{s}_2)$ ,

• if $1 \not\in \varrho$ , we add the rule obtained by replacing M in r with $ \boxminus_{ t} P_2(\mathbf{s}_2) \land \boxminus_{(0,t)} P_1(\mathbf{s}_1)$

• we add the rule , and the rule obtained by replacing M in r with $\boxminus_{ t } P'(\mathbf{s}_1, \mathbf{s}_2) \land \boxminus_{(0,t]} P_1(\mathbf{s}_1)$ , where P’ is a fresh predicate whose arity equals the sum of arities of $P_1$ and $P_2$ .

In the case of atoms mentioning $\mathcal{U}$ , as well as $\top$ or $\bot$ , we proceed analogously.

In the remainder of this section, we fix a normalised program $\Pi$ and a dataset $\mathcal{D}$ , and let $\mathsf{ground}(\Pi,\mathcal{D})$ be the subset of $\mathsf{ground}(\Pi)$ mentioning only constants from $\Pi$ and $\mathcal{D}$ . Then, $\mathsf{at}(\Pi, \mathcal{D})$ is the set consisting of all relational atoms in $\mathcal{D}$ , all metric atoms in $\mathsf{ground}(\Pi,\mathcal{D})$ , and all metric atoms of the forms $\boxminus_{ [ 0,\infty)} M$ and $\boxplus_{ [ 0,\infty)} M$ , with M a relational atom mentioned in $\mathsf{ground}(\Pi,\mathcal{D})$ .

We next define the notion of a window – a fragment of an HT-interpretation over a particular interval; such windows will serve as states of our automata.

The window’s length is the length of $\varrho$ . Finally, we say that a window is initial if either $H=T$ and $b=0$ , or $H \neq T$ and $b=1$ .

Intuitively, a window $(\varrho, H,T,b)$ is a fragment of an HT-interpretation $(\mathfrak{H},\mathfrak{T})$ restricted to $\varrho$ , where H and T describe facts holding within $\varrho$ in $\mathfrak{H}$ and $\mathfrak{T}$ , respectively. Windows will serve as states of the automata recognising word representations of specific HT-interpretations, and in this process the flag b is used to distinguish between stable and non-stable models; in particular, our automata will ensure that flag b is set to 1 in each state $\mathcal{W}$ of a run such that $H \neq T$ in either $\mathcal{W}$ or in some previous state of this run.

By definition, a window can be extended to an HT-interpretation. This HT-interpretation can be an HT-model of $\Pi$ only if the window locally satisfies $\Pi$ , which we define next.

Next, given an initial window $\mathcal{W}_0$ , we define automata $\mathcal{A}_{\mathcal{W}_0}^\leftarrow$ and $\mathcal{A}_{\mathcal{W}_0}^\rightarrow$ , which will allow us to recognise HT-models of $\Pi$ that extend $\mathcal{W}_0$ . In particular, if $\mathcal{A}_{\mathcal{W}_0}^\leftarrow$ and $\mathcal{A}_{\mathcal{W}_0}^\rightarrow$ accept words ${{{\overleftarrow{w}}}}$ and ${{{\overrightarrow{w}}}}$ respectively, then we will be able to construct an HT-model extending $\mathcal{W}_0$ , for which the part located to the left of $\mathcal{W}_0$ is described by ${{{\overleftarrow{w}}}}$ , and the part to the right of $\mathcal{W}_0$ by ${{{\overrightarrow{w}}}}$ .

The automaton $\mathcal{A}_{\mathcal{W}_0}^\to$ is defined analogously, except that we let ${\varrho'= [\varrho^- +1, \varrho^+ +1] }$ , in the definition of T’ we replace $\varrho^-{-}1$ with $\varrho^+ {+} 1$ , and in the definition of F we use $\boxplus$ and $\mathcal{U}$ instead of $\boxminus$ and $\mathcal{S}$ , respectively.

Accepting runs of these automata will correspond to HT-interpretations. Indeed, as we will show next, each HT-interpretation can be decomposed into an infinite sequence of windows $ \dots , \mathcal{W}_{-1}, \mathcal{W}_0, \mathcal{W}_1,\dots$ such that $\mathcal{W}_{0},\mathcal{W}_{-1}, \dots$ and $\mathcal{W}_0, \mathcal{W}_1,\dots$ are accepting runs of $\mathcal{A}_{\mathcal{W}_0}^\gets$ and $\mathcal{A}_{\mathcal{W}_0}^\to$ , respectively. We define the decomposition of an HT-interpretation as follows.

To prove that Statement 2, we observe that $\mathcal{W}_0$ is an initial window by construction, so the automaton $\mathcal{A}_{\mathcal{W}_0}^\leftarrow = (\mathcal{Q}, \Sigma, \delta, q_0,F)$ is well-defined. By the definition of the transition function $\delta$ of $\mathcal{A}_{\mathcal{W}_0}^\gets$ (see Definition 5.4) as well as by the construction of $\mathcal{W}_0, \mathcal{W}_{-1}, \dots$ and ${{{\overleftarrow{w}}}} = \sigma_{-1} \sigma_{-2} \dots$ , we get that ${\mathcal{W}_{i-1} \in \delta(\mathcal{W}_i, \sigma_{i-1})}$ , for each integer $i \leq 0$ . Thus, $\mathcal{W}_0, \mathcal{W}_{-1}, \dots$ is a run of $\mathcal{A}_{\mathcal{W}_0}^\gets$ on ${{{\overleftarrow{w}}}}$ . It remains to show that this run is accepting, that is, for every set S in the accepting condition F, there are infinitely many integers $i < 0$ such that $\mathcal{W}_i \in S$ . Towards a contradiction suppose that there exists $S \in F$ which does not satisfy this property. Thus, there exists $i \leq 0$ such that $\mathcal{W}_j \notin S$ , for all $j \leq i$ . Assume first that S is of one of the following forms

\begin{align*}\{(\varrho',H, T,b) \in \mathcal{Q}& \mid \text{there exists } t \in \varrho' \text{ such that }\boxminus_{[0,\infty)} M @ t \in H \text{ or } M @ t \notin H \}, \text{ or}\\\{(\varrho',H, T,b) \in \mathcal{Q}& \mid \text{there exists } t \in \varrho' \text{ such that }\boxminus_{[0,\infty)} M @ t \in T \text{ or } M @ t \notin T \},\end{align*}

for some ${\boxminus_{[0,\infty)} M \in \mathsf{at}(\Pi,\mathcal{D})}$ . As a result, $\mathfrak{I} \not\models \boxminus_{[0,\infty)} M @ t $ and $\mathfrak{I} \models M @ t $ , for each $t \leq \varrho_i^-$ , where the interpretation $\mathfrak{I}$ is either $\mathfrak{H}$ or $\mathfrak{T}$ , depending whether S is of the first or the second form presented above. This, however, contradicts the semantics of $\boxminus_{[0,\infty)}$ . It remains to consider the case where S is of one of the following forms

\begin{align*}\{ (\varrho',H, T,b) \in \mathcal{Q}& \mid \text{there exists } t \in \varrho' \text{ such that } M_1 \mathcal{S}_{[0,\infty)} M_2 @ t \notin H \text{ or } M_2 @ t \in H \}, \text{ or}\\\{ (\varrho',H, T,b) \in \mathcal{Q}& \mid \text{there exists } t \in \varrho' \text{ such that } M_1 \mathcal{S}_{[0,\infty)} M_2 @ t \notin T \text{ or } M_2 @ t \in T \},\end{align*}

for some $M_1 \mathcal{S}_{[0,\infty)} M_2 \in \mathsf{at}(\Pi,\mathcal{D})$ . Then, $\mathfrak{I} \models M_1 \mathcal{S}_{[0,\infty)} M_2 @ t $ and ${\mathfrak{I} \not\models M_2 @ t }$ , for each $t \leq \varrho_i^-$ , where $\mathfrak{I}$ is either $\mathfrak{H}$ or $\mathfrak{T}$ , depending whether S is of the first or the second form presented above. This directly contradicts the semantics of $\mathcal{S}_{[0,\infty)}$ . Consequently, $\mathcal{W}_0, \mathcal{W}_{-1}, \dots$ needs to be an accepting run of $\mathcal{A}_{\mathcal{W}_0}^\gets$ on ${{{\overleftarrow{w}}}}$ .

The proof of Statement 3 is analogous to the proof of Statement 2, due to the symmetry between the automata $\mathcal{A}_{\mathcal{W}_0}^\gets$ and $\mathcal{A}_{\mathcal{W}_0}^\to$ .

To check existence of a stable model, however, we require automata that recognise HT-models $(\mathfrak{H},\mathfrak{T})$ with $\mathfrak{H}=\mathfrak{T}$ , and automata that recognise HT-models $(\mathfrak{H},\mathfrak{T})$ with $\mathfrak{H} \subsetneq \mathfrak{T}$ . The intersection of the former with the complement of the latter allows us to recognise stable models – that is, essentially, HT-models $(\mathfrak{T},\mathfrak{T})$ for which there are no models $(\mathfrak{H},\mathfrak{T})$ with $\mathfrak{H} \subsetneq \mathfrak{T}$ . To this end, we define two more types of automata as follows:

Intuitively, if $\mathcal{W}_0$ satisfies $\mathcal{D}$ , then the automata $\mathcal{B}_{\mathcal{W}_0}^\gets$ and $\mathcal{B}_{\mathcal{W}_0}^\to$ recognise interpretations $\mathfrak{T}$ such that $(\mathfrak{T},\mathfrak{T})$ is an HT-model of $\Pi$ and $\mathcal{D}$ . Furthermore, interpretations $\mathfrak{T}$ accepted by $\mathcal{A}_{\mathcal{W}_0}^\gets$ and $\mathcal{C}_{\mathcal{W}_0}^\to$ , or by $\mathcal{C}_{\mathcal{W}_0}^\gets$ and $\mathcal{A}_{\mathcal{W}_0}^\to$ are such that $(\mathfrak{H},\mathfrak{T})$ is an HT-model of $\Pi$ and $\mathcal{D}$ , for some $\mathfrak{H} \subsetneq \mathfrak{T}$ . Hence, as we show next, we can use these automata to recognise stable models. This, however, requires the additional assumption that the windows in the automata are long enough to allow for capturing the semantics of metric operators occurring in a program. To this end, we will use windows of the same length as the interval ${\varrho_{(\Pi,\mathcal{D})} = [t^{\min}_{\mathcal{D}},t^{\max}_{\mathcal{D}}+t_{\Pi}]}$ . If the length of an initial window $\mathcal{W}_0$ is as required, then we can reduce checking existence of a stable model to checking particular properties of our automata, as stated next.

To show that Condition 1 holds, we observe that, by Lemma 5.6, $\mathcal{W}_0, \mathcal{W}_{-1}, \dots$ is an accepting run of $\mathcal{A}_{\mathcal{W}_0}^\gets$ on ${{{\overleftarrow{w}}}}$ , and $\mathcal{W}_0, \mathcal{W}_{1}, \dots$ is an accepting run of $\mathcal{A}_{\mathcal{W}_0}^\to$ on ${{{\overrightarrow{w}}}}$ . Moreover, since $H_i=T_i$ and $b_i = 0$ for all $i \in \mathbb{Z}$ , we obtain that $\mathcal{W}_0, \mathcal{W}_{-1}, \dots$ is an accepting run of $\mathcal{B}_{\mathcal{W}_0}^\gets$ on ${{{\overleftarrow{w}}}}$ , and $\mathcal{W}_0, \mathcal{W}_{1}, \dots$ is an accepting run of $\mathcal{B}_{\mathcal{W}_0}^\to$ on ${{{\overrightarrow{w}}}}$ .

Next, let us suppose towards a contradiction that there exists ${\mathcal{W}_0' =(\varrho_0',H_0', T_0', b_0') }$ witnessing a violation of Condition 2. Hence, $\mathcal{W}_0'$ is an initial window locally satisfying $\Pi$ such that $\varrho_0'=\varrho_0$ , $T_0' = T_0$ , and $H_0' \models \mathcal{D}$ . Moreover, we assume that ${{{\overleftarrow{w}}}}$ and ${{{\overrightarrow{w}}}}$ are accepted by $\mathcal{C}_{\mathcal{W}'_0}^\gets$ and $\mathcal{A}_{\mathcal{W}'_0}^\to$ , respectively. Hence, $\mathcal{C}_{\mathcal{W}'_0}^\gets$ has an accepting run $\mathcal{W}'_0, \mathcal{W}'_{-1}, \dots$ on ${{{\overleftarrow{w}}}}$ and $\mathcal{A}_{\mathcal{W}'_0}^\to$ has an accepting run $\mathcal{W}'_0, \mathcal{W}'_{1}, \dots$ on ${{{\overrightarrow{w}}}}$ , where we let ${\mathcal{W}'_i= (\varrho_i', H_i', T_i', b_i')}$ , for all $i \in \mathbb{Z}$ . Clearly, $\varrho_i' = \varrho_i$ , for all $i \in \mathbb{Z}$ . Moreover, by the definition of the transition functions of the automata and the construction of ${{{\overleftarrow{w}}}}$ and ${{{\overrightarrow{w}}}}$ , we obtain that $T_i' = T_i$ , for all $i \in \mathbb{Z}$ . Therefore, $\mathfrak{T}$ is the least model of all relational facts in $\bigcup_{i\in \mathbb{Z}}T'_i$ . We let $\mathfrak{H}$ be the least model of all relational facts in $\bigcup_{i\in \mathbb{Z}}H_i$ ; we will show that $(\mathfrak{H}, \mathfrak{T})$ is an HT-model of $\Pi$ and $\mathcal{D}$ .

Since $H_i' \subseteq T_i'$ , for all $i \in \mathbb{Z}$ , we obtain that $\mathfrak{H} \subseteq \mathfrak{T}$ , and so $(\mathfrak{H}, \mathfrak{T})$ is an HT-interpretation. Moreover, $(\mathfrak{H}, \mathfrak{T})$ is an HT-model of $\mathcal{D}$ , as $H_0' \models \mathcal{D}$ . Next we will show that $(\mathfrak{H}, \mathfrak{T})$ is an HT-model of $\Pi$ . Since each $\mathcal{W}_i'$ is a window of length $[t^{\min}_{\mathcal{D}},t^{\max}_{\mathcal{D}}+t_{\Pi}]$ (which is the length of $\varrho_{(\Pi,\mathcal{D})}$ ), it holds that $\mathfrak{H} \models M@t $ if and only if $M@t \in H_i'$ ; as well as $\mathfrak{T} \models M@t $ if and only if $M@t \in T_i'$ , for any $M \in \mathsf{at}(\Pi,\mathcal{D})$ and $t \in \varrho_i'$ . Indeed, we have shown an analogous statement for positive (Wałęga et al. Reference Wałęga, Cuenca Grau, Kaminski and Kostylev2019, Lemma 9) and stratifiable programs (Tena Cucala et al. Reference Tena Cucala, Wałęga, Cuenca Grau and Kostylev2021), and the same argument applies here. Now, to show that $(\mathfrak{H},\mathfrak{T})$ is a model of $\Pi$ , we fix a ground rule from $\mathsf{ground}(\Pi,\mathcal{D})$ of Form (1) and a time point t. If $\mathfrak{H} ,t \models M_i$ for all ${i \in \{1, \ldots, k\} }$ and ${\mathfrak{T} ,t \not\models M_j}$ for all ${j \in \{k+1, \ldots, m\} }$ , then $M_i@t \in H_n'$ for all $i \in \{1, \ldots, k\}$ and $M_j@t \notin T_n$ for all $j \in \{k+1, \ldots, m\}$ , for each n such that $t \in \varrho_n'$ . Since $\mathcal{W}_n'$ is a window locally satisfying $\Pi$ , we obtain that $M@t \in H_n$ (where M is the head of r), which implies $\mathfrak{H} ,t \models M$ by definition of $\mathfrak{H}$ and the fact that $\Pi$ is in normal form so M is a relational atom. Similarly, if $\mathfrak{T} ,t \models M_i$ for all $i \in \{1, \ldots, k\}$ and ${\mathfrak{T} ,t \not\models M_j}$ for all $j \in \{k+1, \ldots, m\}$ , then we obtain that $\mathfrak{T} ,t \models M$ . Hence, $(\mathfrak{H},\mathfrak{T})$ is indeed an HT-model of $\Pi$ .

However, the accepting condition of $\mathcal{C}_{\mathcal{W}'_0}^\gets$ guarantees that $b'_i=1$ , for some $i \leq 0$ , and so ${H_i' \subsetneq T_i' }$ . Therefore, $\mathfrak{H} \subsetneq \mathfrak{T}$ , and so $\mathfrak{T}$ is not a stable model, which rises a contradiction. If ${{{\overleftarrow{w}}}}$ and ${{{\overrightarrow{w}}}}$ are accepted by $\mathcal{A}_{\mathcal{W}'_0}^\gets$ and $\mathcal{C}_{\mathcal{W}'_0}^\to$ , respectively, then we construct in an analogous way a run $\mathcal{W}'_0, \mathcal{W}'_{-1}, \dots$ of $\mathcal{A}_{\mathcal{W}'_0}^\gets$ and a run $\mathcal{W}'_0, \mathcal{W}'_{1}, \dots$ of $\mathcal{C}_{\mathcal{W}'_0}^\to$ . Repeating the argumentation above, we construct an HT-model $(\mathfrak{H},\mathfrak{T})$ of $\Pi$ and $\mathcal{D}$ , and then, we show that there exists $i\geq 0$ such that $b'_i=1$ . Thus, $H_i' \subsetneq T_i'$ , so $\mathfrak{H} \subsetneq \mathfrak{T}$ , and consequently, $\mathfrak{T}$ is not a stable model, raising again a contradiction. Thus, Condition 2 holds.

For the converse implication, assume that there exist $\mathcal{W}_0$ , ${{{\overleftarrow{w}}}}$ , and ${{{\overrightarrow{w}}}}$ as described in the statement of the theorem. By Condition 1, there is an accepting run $\mathcal{W}_0, \mathcal{W}_{-1}, \dots$ of $\mathcal{B}_{\mathcal{W}_0}^\gets$ on ${{{\overleftarrow{w}}}}$ , and an accepting run $\mathcal{W}_0, \mathcal{W}_1, \dots$ of $\mathcal{B}_{\mathcal{W}_0}^\to$ on ${{{\overrightarrow{w}}}}$ , where ${\mathcal{W}_i= (\varrho_i, T_i, T_i, 0)}$ . We argue that the least model $\mathfrak{T}$ of relational facts in $\bigcup_{i\in\mathbb{Z}} T_i$ is a stable model of $\Pi$ and $\mathcal{D}$ . By an argument analogous to the second to last paragraph above, $(\mathfrak{T},\mathfrak{T})$ is an HT-model of $\Pi$ and $\mathcal{D}$ . Suppose for contradiction that $\mathfrak{T}$ is not stable, so there exists $\mathfrak{H} \subsetneq \mathfrak{T}$ such that $(\mathfrak{H}, \mathfrak{T})$ is an HT-model of $\Pi$ and $\mathcal{D}$ . Let $\dots, \mathcal{W}'_{-1}, \mathcal{W}'_0, \mathcal{W}'_1, \dots$ be the $\varrho_{(\Pi,\mathcal{D})}$ -decomposition of $(\mathfrak{H},\mathfrak{T})$ , with ${\mathcal{W}_i'= (\varrho_i', H_i', T_i', b_i')}$ , and observe that $\mathcal{W}'_0$ is an initial window. By Lemma 5.6, $\mathcal{W}'_0, \mathcal{W}'_{-1}, \dots$ is an accepting run of $\mathcal{A}_{\mathcal{W}'_0}^\gets$ on ${{{\overleftarrow{w}}}}$ , and $\mathcal{W}'_0, \mathcal{W}'_{1}, \dots$ is an accepting run of $\mathcal{A}_{\mathcal{W}'_0}^\to$ on ${{{\overrightarrow{w}}}}$ . Moreover, since $\mathfrak{H} \subsetneq \mathfrak{T}$ , there is $i \in \mathbb{Z}$ such that $H_i' \neq T_i'$ , and so $b_i'=1$ . If $i \leq 0$ , then $b_j' =1$ for all $j \leq i$ , and so $\mathcal{C}_{\mathcal{W}'_0}^\gets$ accepts ${{{\overleftarrow{w}}}}$ ; analogously, if $i \geq 0$ , then $\mathcal{C}_{\mathcal{W}'_0}^\to$ accepts ${{{\overrightarrow{w}}}}$ . Thus, Condition 2 does not hold, leading to a contradiction.

Theorem 5.8 reduces checking existence of a stable model to checking specific properties of our automata. We aim at showing that the latter is feasible in ${{\rm E}{\small\rm XP}{\rm S}{\small\rm PACE}}$ . The main obstacle, however, is the size of states in the automata: $\mathcal{W}_0$ is exponential in size with respect to $\mathcal{D}$ , and states of the automata from Theorem 5.8 can be arbitrarily large since time points in windows can be arbitrary integers. To remedy the first issue, we let $t_\Pi$ be the largest positive number mentioned in $\Pi$ , and we let $t_{\Pi}=1$ if $\Pi$ mentions no positive numbers – this choice of value is arbitrary since in this case we only need $t_\Pi$ to be a positive number in the timeline. Then, we show in Lemma 5.10 that it suffices to consider automata with states (i.e., windows) of length $t_\Pi$ , which does not depend on $\mathcal{D}$ . The second issue is addressed by Lemma 5.13 which tells us that, rather than considering automata with states of unbounded size (each of length ${t_\Pi}$ ), we can construct equivalent automata with polynomial-size states instead.

To state Lemma 5.10, we define the left-most and right-most fragments of length $t_\Pi$ of a window as follows.

Analogously, we let $\mathcal{W}^{R}=( \varrho^R, H^R,T^R, b^R )$ , where

• ${\varrho^R=[\varrho^+ - t_\Pi, \varrho^+] }$ ,

• ${H^R = \{M@t \in H \mid t \in \varrho^R \} }$ ,

• ${T^R = \{M@t \in T \mid t \in \varrho^R \} }$ ,

• $b^R = 1$ if ${H^R \neq T^R}$ ; otherwise, $b^R = 0$ .

We observe that if $\mathcal{W}$ is a window locally satisfying $\Pi$ , then $\mathcal{W}^L$ and $\mathcal{W}^R$ (which are “fragments” of $\mathcal{W}$ ) also are windows locally satisfying $\Pi$ . Furthermore, both $\mathcal{W}^L$ and $\mathcal{W}^R$ are initial windows by definition. Thus, if $\mathcal{W}_0$ is such that $X_{\mathcal{W}_0}^\gets$ and $X_{\mathcal{W}_0}^\to$ are well-defined automata, for some $X \in \{ \mathcal{A},\mathcal{B},\mathcal{C} \}$ , then $X_{\mathcal{W}_0^{L}}^\gets$ and $X_{\mathcal{W}_0^{R}}^\to$ are also well-defined automata. Moreover, we can show several equivalences between these automata.

For the opposite direction of Statement 1, let us assume that $\mathcal{A}_{\mathcal{W}_{0}^L}^\gets$ has an accepting run $\mathcal{W}_{0}^L, \mathcal{W}_{-1}', \mathcal{W}_{-2}',\dots$ on a word $w = \sigma_{-1},\sigma_{-2},\dots$ ; we will show that $\mathcal{A}_{\mathcal{W}_0}^\gets$ has an accepting run on w. To facilitate the definition of this run, we divide $\mathcal{W}_0 = (\varrho_0,H_0,T_0,b_0)$ into tuples $\mathcal{W}_{0}', \dots, \mathcal{W}_{k}'$ , where k is the difference between the lengths of $\mathcal{W}_0$ and $\mathcal{W}_0^L$ . In particular, for each $i \in \{0, \dots, k \}$ , we define ${\mathcal{W}_{i}' = (\varrho_i',H_i',T_i',b_i') }$ , such that ${\varrho_i'=[\varrho_0^- + k , \varrho_0^- +t_{\Pi} + k ]}$ , ${H_i' = \{M@t \in H_0 \mid t \in \varrho_i' \} }$ , ${T_i' = \{M@t \in T_0 \mid t \in \varrho_i' \} }$ , whereas $b_i' = 1$ if $H'_i \neq T'_i$ and $b'_i=0$ if $H'_i = T'_i$ . Recall that since $\mathcal{A}_{\mathcal{W}_0}^\gets$ is well-defined, $\mathcal{W}_0$ is an initial window locally satisfying $\Pi$ , and as we already mentioned, by fragmenting windows locally satisfying $\Pi$ we obtain windows which also locally satisfy $\Pi$ . Hence all of $\mathcal{W}_{0}', \dots, \mathcal{W}_k'$ locally satisfy $\Pi$ . Furthermore, observe that $\mathcal{W}_{0}' = \mathcal{W}_{0}^L$ . Next, we show how to merge the windows $\mathcal{W}_k', \dots, \mathcal{W}_0', \mathcal{W}_{-1}',\dots$ to obtain an accepting run of $\mathcal{A}_{\mathcal{W}_{0}}^\gets$ on w. To this end, for arbitrary windows $\mathcal{W} = ( \varrho, H,T, b )$ and ${\mathcal{W}' =( \varrho', H',T', b' )}$ such that $\varrho \cap \varrho' \neq \emptyset$ , we define their union, written as $\mathcal{W} \cup \mathcal{W}'$ , as the tuple ${(\varrho \cup \varrho', H \cup H', T \cup T', \max(b,b'))}$ . We can then establish the following claim:

Proof of Claim 5.11

Let $\mathcal{W}'' = \mathcal{W} \cup \mathcal{W}' = (\varrho'', H'', T'', b'')$ . If $\mathcal{W}$ and $\mathcal{W}'$ locally satisfy $\Pi$ , then clearly $\mathcal{W}''$ also locally satisfies $\Pi$ . It remains to show that $\mathcal{W}''$ is a window. To this end, for $\mathfrak{I}$ an interpretation, $\varrho_b$ a closed interval, and J a set of metric facts $M@t$ with $M \in \mathsf{at}(\Pi,\mathcal{D})$ and $t \in \varrho_b$ , we say that $\mathfrak{I}$ corresponds to J over $\varrho_b$ if, for every $t \in \varrho_b$ and $M \in \mathsf{at}(\Pi,\mathcal{D})$ , it holds that $\mathfrak{I} \models M@t$ if and only if $M@t \in J$ . To show that $\mathcal{W}''$ is a window, the only non-trivial condition that needs to be verified is that there exist two interpretations that correspond, respectively, to H” and T” over $\varrho''$ . We show this for H” only, as the argument for T” is analogous. Let $\mathfrak{H}$ (resp. $\mathfrak{H}'$ ) be an interpretation corresponding to H over $\varrho$ (resp. H’ over $\varrho'$ ); such an interpretation exists, since $\mathcal{W}$ (resp. $\mathcal{W}'$ ) is a window. Let $\mathfrak{H}''$ be an arbitrary interpretation that coincides with $\mathfrak{H}$ over $\varrho$ and all time points to its left, and with $\mathfrak{H}'$ over $\varrho'$ and all time points to its right. Note that $\mathfrak{H}$ and $\mathfrak{H}'$ agree over $\varrho \cap \varrho'$ since, by assumption, $M@t \in H$ if and only if $M@t \in H'$ for all $t \in \varrho \cap \varrho'$ ; thus, $\mathfrak{H}''$ is well-defined. It remains to show that $\mathfrak{H}''$ corresponds to H” over $\varrho''$ – that is, for each $M@t$ with $M \in \mathsf{at}(\Pi,\mathcal{D})$ and $t \in \varrho''$ , we must show that $\mathfrak{H}'' \models M @ t$ if and only if $M @ t \in H''$ . We show this explicitly for the case where M is of the form $\boxminus_{\varrho_b} P$ , for $\varrho_b$ a bounded interval and P a relational atom; the remaining cases can be proved in a similar way; see Wałęga et al. (Reference Wałęga, Cuenca Grau, Kaminski and Kostylev2019), Lemma 7. Suppose that $t \in \varrho$ . Since $\mathfrak{H}''$ and $\mathfrak{H}$ coincide over $\varrho$ and all time points to its left, we have that $\mathfrak{H}'' \models \boxminus_{\varrho_b} P @t$ if and only if $\mathfrak{H} \models \boxminus_{\varrho_b} P@t$ . Then, since $\mathfrak{H}$ corresponds to H over $\varrho$ , it holds that $\mathfrak{H} \models \boxminus_{\varrho_b} P @ t$ if and only if $\boxminus_{\varrho_b} P @t \in H$ . Finally, since H and H” coincide over $\varrho$ , $\boxminus_{\varrho_b} P @t\in H$ if and only if $\boxminus_{\varrho_b} P @ t \in H''$ . Thus, by chaining together these double implications, we have that $\mathfrak{H}'' \models \boxminus_{\varrho_b} P @t$ if and only if $\boxminus_{\varrho_b} P @ t \in H''$ , which is the target equivalence. Suppose now that $t \notin \varrho$ , so t is the right endpoint of $\varrho'$ . By the semantics of metric atoms, $\mathfrak{H}'' \models \boxminus_{\varrho_b} P @ t$ if and only if $\mathfrak{H}'' \models P @ t-t'$ for each $t' \in \varrho_b$ . Since the length of $\varrho'$ is at least $t_{\Pi}$ , we have $t - t' \in \varrho'$ for each $t' \in \varrho_b$ . Hence, since $\mathfrak{H}''$ and $\mathfrak{H}'$ coincide over $\varrho'$ , $\mathfrak{H}'' \models\boxminus_{\varrho_b} P$ if and only if $\mathfrak{H}' \models\boxminus_{\varrho_b} P$ , and the rest follows by an argument analogous to the previous case, finishing the proof of Claim 5.11.

We resume the proof of Lemma 5.10. Now, we construct an accepting run of $\mathcal{A}_{\mathcal{W}_0}^\gets$ on w as follows. For each integer $ i < 0$ , we let $\mathcal{W}_i = (\varrho_i,H_i,T_i,b_i)$ be $\mathcal{W}_i' \cup \dots \cup \mathcal{W}_{i+k}'$ with the exception that $b_i =1$ if there exists $j \in \{i, \dots, k \}$ such that $H'_i \neq T'_i$ and $b_i =0$ otherwise.

Now, we will show that $\mathcal{W}_0, \mathcal{W}_{-1}, \dots$ is indeed an accepting run of $\mathcal{A}_{\mathcal{W}_0}^\gets$ on w. To this end, we observe that each $\mathcal{W}_{i}$ is a window locally satisfying $\Pi$ , since so is ${ \mathcal{W}_i' \cup \dots \cup \mathcal{W}_{i+k}' }$ by Claim 5.11; furthermore, each $\mathcal{W}_i$ it is of the same length as $\mathcal{W}_0$ . Therefore, each $\mathcal{W}_i$ is a state of the automaton $\mathcal{A}_{\mathcal{W}_0}^\gets$ . Moreover, the definition of $\mathcal{W}_i$ ensures that if there exists a transition on $\sigma_{i-1}$ from $\mathcal{W}_i'$ to $\mathcal{W}_{i-1}'$ in $\mathcal{A}_{\mathcal{W}_0^L}^\gets$ , then there exists a transition on $\sigma_{i-1}$ from $\mathcal{W}_i$ to $\mathcal{W}_{i-1}$ in $\mathcal{A}_{\mathcal{W}_0}^\gets$ , so $\mathcal{W}_0, \mathcal{W}_{-1}, \dots$ is a run of $\mathcal{A}_{\mathcal{W}_0}^\gets$ on w. Furthermore, if $\mathcal{W}_i'$ belongs to some set S in the accepting condition of $\mathcal{A}_{\mathcal{W}_0^L}^\gets$ , then the bigger window $\mathcal{W}_i$ belongs to the set corresponding to S in the accepting condition of $\mathcal{A}_{\mathcal{W}_0}^\gets$ . Hence, $\mathcal{A}_{\mathcal{W}_0}^\gets$ accepts the run $\mathcal{W}_0, \mathcal{W}_{-1}, \dots$ . The case of $\mathcal{A}_{\mathcal{W}_0}^\to$ is symmetric, so we obtain that if $\mathcal{A}_{\mathcal{W}_0^R}^\to$ accepts a word w, then so does $\mathcal{A}_{\mathcal{W}_0}^\to$ .