2 Metric logic of here-and-there

We begin by introducing the logic of Metric Here-and-There ( $\textrm {MHT}$ ), a metric extension of $\textrm {HT}$ which serves as the monotonic foundation for Metric Equilibrium Logic ( $\textrm {MEL}$ ).

Given $m \in \mathbb{N}$ and $n \in \mathbb{N} \cup \{\omega \}$ , we let $[m..n]$ stand for the set $\{i \in \mathbb{N} \mid m \leq i \leq n\}$ , $[m..n)$ for $\{i \in \mathbb{N} \mid m \leq i \lt n\}$ , $(m..n]$ for $\{i \in \mathbb{N} \mid m \lt i \leq n\}$ and $(m..n)$ stand for $\{i \in \mathbb{N} \mid m \lt i \lt n\}$ . We use letters $I, J$ to denote intervals and, since they stand for sets, we assume standard set relations like inclusion $I \subseteq J$ or membership $i \in I$ .

Given a set $\mathcal{A}$ of propositional variables (called alphabet), a metric formula $\varphi$ is defined by the grammar:

where $p \in \mathcal{A}$ is an atom, $\otimes$ is any binary Boolean connective $\otimes \in \{\to ,\wedge ,\vee \}$ . As usual in intuitionistic logic, we define $\neg \varphi \stackrel {\mathit{def}}{=} \varphi \to \bot$ , $\varphi \leftrightarrow \psi \stackrel {\mathit{def}}{=} \left (\varphi \to \psi \right )\wedge \left (\psi \to \varphi \right )$ and $\top \stackrel {\mathit{def}}{=} \neg \bot$ .

The binary temporal operators $\varphi _1 \mathbin {\boldsymbol{\mathsf{S}}}_{I} \varphi _2$ and $\varphi _1 \mathbin {\boldsymbol{\mathsf{T}}}_{I} \varphi _2$ denote that $\varphi _1$ holds since or is triggered by $\varphi _2$ , respectively. Similarly, represents the until operator, while signifies that $\varphi _2$ is released by $\varphi _1$ . The non-metric counterparts of these operators share the same intuitive semantics but omit explicit reference to the interval $I$ . The unary operators $\unicode{x25EF}_{I} \varphi$ and $\unicode{x2B24}_{I}\varphi$ (along with their non-metric versions $\unicode{x25EF} \varphi$ and $\unicode{x2B24} \varphi$ ) indicate that $\varphi$ holds in the next or previous state, respectively. We distinguish between metric and non-metric variants based on the presence of interval constraints: in the metric case, satisfaction is contingent upon conditions being met within the intervals augmenting the operators, whereas non-metric operators impose no such quantitative bounds. Additionally, the following derived operators can be defined:

2.2 Strict traces

Next, we consider a group of results that hold under the assumption of strict traces, namely, that $\tau (i) \lt \tau (i+1)$ for any pair of consecutive time points. We can enforce metric models to be traces with a strict timing function $\tau$ . This can be achieved with the addition of the formula $\square \neg \unicode{x25EF}_{0} \top$ to the context, where $0$ abbreviates $[0..0]$ .

Proposition 1. For any $\textrm {MHT}$ trace $(\langle \mathbf{H},\mathbf{T} \rangle ,\tau )$ of length $\lambda \in \mathbb{N}\cup \lbrace \omega \rbrace$ , we have that $(\langle \mathbf{H},\mathbf{T} \rangle ,\tau ) \models \square \neg \unicode{x25EF}_{0} \top$ iff $\tau$ is a strict function.

In Lemma 1 below, the use of strict traces enables a recursive definition for operators of the form $Q_{\leq n}$ , with . This novel characterization, similar to the recursive definition of binary temporal operators in $\textrm {LTL}$ (Pnueli Reference Pnueli1977), is more suitable for the Tseitin-style translation developed in this paper. For this reason, we henceforth restrict our focus to strict traces, allowing us to assume the implicit inclusion of the aforementioned axiom. The following equivalences state that the interval $[0..0]$ makes all binary metric operators collapse into their right-hand argument formula, whereas unary operators collapse to a truth constant. For metric formulas $\psi$ and $\varphi$ and for strict traces, we have:

(2)

(3) \begin{align} \unicode{x25EF}_{0} \, \varphi &\equiv \unicode{x2B24}_{0} \, \varphi \equiv \bot\\[-20pt]\nonumber \end{align}

(4) \begin{align} \widehat {\unicode{x25EF}}_{0} \, \varphi &\equiv \widehat {\unicode{x2B24}}_{0} \, \varphi \equiv \top \end{align}

The last two lines are precisely an effect of dealing with strict traces. For instance, $\unicode{x25EF}_{0} \, \varphi \equiv \bot$ tells us that it is always impossible to have a successor state with the same time as the current one, regardless of the formula $\varphi$ at hand. Lemma 1 below allows unfolding until, release, since, and trigger for intervals of the form $[0..n]$ .

Lemma 1 (Becker et al. Reference Becker, Cabalar, Diéguez, Schaub and Schuhmann2024b).

For metric formulas $\psi$ and $\varphi$ and for $n \ge 0$ , we have:

(5)

(6)

(7) \begin{align} \varphi \mathbin {\boldsymbol{\mathsf{S}}}_{\leq n} \psi & {} \equiv \textstyle \psi \vee (\varphi \wedge \bigvee _{x=1}^{n} \unicode{x2B24}_{x} (\varphi \mathbin {\boldsymbol{\mathsf{S}}}_{\leq (n-x)} \psi ) )\\[-8pt]\nonumber \end{align}

(8) \begin{align} \varphi \mathbin {\boldsymbol{\mathsf{T}}}_{\leq n} \psi & {} \equiv \textstyle \psi \wedge ( \varphi \vee \bigwedge _{x=1}^{n} \widehat {\unicode{x2B24}}_{x} (\varphi \mathbin {\boldsymbol{\mathsf{T}}}_{\leq (n-x)} \psi ) ), \end{align}

where intervals of the form $x$ stand for $[x..x]$ .

The equivalences presented above hold under the assumption of strict traces. To show it, let us consider the formula and let us define the timed $\textrm {HT}$ -trace $\mathbf{M}=(\langle \mathbf{T},\mathbf{T} \rangle ,\tau )$ of length $\lambda =2$ as $T_0 = \lbrace \mathit{p} \rbrace$ , $T_1 = \lbrace \mathit{a} \rbrace$ , and $\tau (0)=\tau (1)=0$ . Since $\tau (0) \not \lt \tau (1)$ , $\mathbf{M}$ is not a strict trace. Hence, but , which falsifies (5). Formulas of the form , where $[a..b)$ is a general interval, accept more complicated recursive definitions (see (Becker et al. Reference Becker, Cabalar, Diéguez, Schaub and Schuhmann2024b) for details) and they are left out of the scope of this paper.

3 Three-valued semantics

It is well known that $\textrm {HT}$ semantics is equivalent to three-valued Gödel logic (Gödel Reference Gödel1932). The use of the latter semantics in the context of equilibrium logic and its temporal extensions facilitates correctness proofs of Tseitin reductions to logic programs and temporal logic programs, since it allows equivalences to be evaluated more conveniently. Three-valued Gödel logic was extended to the temporal setting in Cabalar (Reference Cabalar, Janhunen and Niemelä2010), within the context of reducing temporal theories to temporal logic programs. Since, in this paper, we extend Cabalar’s results to the metric case, we also provide a three-valued characterization of $\textrm {MHT}$ . In our setting, a three-valued interpretation is a mapping

\begin{equation*}m: \left ([0..\lambda ) \to \mathbb{N}\right ) \to \left (\left ([0..\lambda ) \times \mathcal{A}\right ) \to \lbrace 0,1,2\rbrace \right )\end{equation*}

where, broadly speaking, the first input function $\left ( [0..\lambda ) \to \mathbb{N}\right )$ represents a timing function that gives the different time points associated with each state of the trace; we denote it by $\tau$ in the rest of this section. Given $\tau$ , we define

\begin{equation*} m(\tau ): [0..\lambda ) \times \mathcal{A} \to \lbrace 0,1,2\rbrace \end{equation*}

as the associated evaluation for propositional variables, that is, a function that provides the truth value for each propositional variable $p \in \mathcal{A}$ at each time point $k \in [0..\lambda )$ . Such $m(\tau )$ Footnote ² can be extended to $\boldsymbol{m}$ to cope with arbitrary formulas as shown below.

\begin{align*} \boldsymbol{m}(k,\bot ) & = 0 \\ \boldsymbol{m}(k,a) &= m(\tau )(k,a) \hspace {20pt} \text{for any atom } a\\ \boldsymbol{m}(k,\varphi \wedge \psi ) & = \min (\boldsymbol{m}(k,\varphi ),\boldsymbol{m}(k,\psi ))\\ \boldsymbol{m}(k,\varphi \vee \psi ) & = \max (\boldsymbol{m}(k,\varphi ),\boldsymbol{m}(k,\psi ))\\ \boldsymbol{m}(k,\varphi \to \psi ) & = \begin{cases} 2 & \text{ if } \boldsymbol{m}(k,\varphi ) \le \boldsymbol{m}(k,\psi )\\ \boldsymbol{m}(k,\psi ) & \text{ otherwise } \end{cases}\\ \boldsymbol{m}(k,\unicode{x2B24}_{I} \varphi ) & = \begin{cases} 0 & \text{if } k=0 \text{ or } \tau (k)-\tau (k-1) \not \in I\\ \boldsymbol{m}(k-1,\varphi ) & \text{otherwise } \end{cases}\\ \boldsymbol{m}(k,\widehat {\unicode{x2B24}}_{I} \varphi ) & = \begin{cases} 2 & \text{if } k=0 \text{ or } \tau (k)-\tau (k-1) \not \in I\\ \boldsymbol{m}(k-1,\varphi ) & \text{otherwise } \end{cases}\kern95pt\end{align*}

$ \boldsymbol{m}(k,\varphi \mathbin {\boldsymbol{\mathsf{S}}}_{I} \psi ) = \sup \{\min \{\boldsymbol{m}(i,\psi ),\boldsymbol{m}(j,\varphi )\mid j \in (i..k]\} \mid i \in [0..k] \text{ and } \tau (k)-\tau (i) \in I\}$ Footnote ³

\begin{align*} \boldsymbol{m}(k,\varphi \mathbin {\boldsymbol{\mathsf{T}}}_{I} \psi ) & = \inf \{\max \{\boldsymbol{m}(i,\psi ),\boldsymbol{m}(j,\varphi )\mid j \in (i..k]\} \mid i \in [0..k] \text{ and } \tau (k)-\tau (i)\in I\}\\ \boldsymbol{m}(k,\unicode{x25EF}_{I} \varphi ) & = \begin{cases} 0 & \text{if } k+1=\lambda \ (\neq \omega ) \text{ or } \tau (k+1)-\tau (k)\not \in I\\ \boldsymbol{m}(k+1,\varphi ) & \text{otherwise } \end{cases}\\ \boldsymbol{m}(k,\widehat {\unicode{x25EF}}_{I} \varphi ) & = \begin{cases} 2 & \text{if } k+1=\lambda \ (\neq \omega ) \text{ or } \tau (k+1)-\tau (k)\not \in I\\ \boldsymbol{m}(k+1,\varphi ) & \text{otherwise } \end{cases} \end{align*}

The three-valued semantics defined above offers a significant advantage when establishing equivalences. For instance, demonstrating that $\boldsymbol{m}(k,\varphi \leftrightarrow \psi )=2$ is equivalent to showing that $\boldsymbol{m}(k,\varphi ) = \boldsymbol{m}(k,\psi )$ for all formulas $\varphi$ and $\psi$ . Similarly, proving that $\boldsymbol{m}(0, \square (\varphi \leftrightarrow \psi ))$ is equivalent to establishing that $\boldsymbol{m}(k,\varphi )=\boldsymbol{m}(k,\psi )$ for all $k \in [0..\lambda )$ . These properties streamline the formal verification process, as they allow complex logical equivalences to be treated as identities within our three-valued framework.

To establish the equivalence between the standard $\textrm {MHT}$ semantics and our three-valued characterization, we define a function $f$ from the set of timed $\textrm {HT}$ -traces to the set of three-valued interpretations. For any timed $\textrm {HT}$ -trace $\mathbf{M} = (\langle \mathbf{H},\mathbf{T} \rangle ,\tau )$ of length $\lambda$ , let $f(\mathbf{M}) = m(\tau )$ where $m(\tau ): [0 \cdot \cdot \lambda ) \times \mathcal{A} \to \lbrace 0,1,2\rbrace$ is defined for any $k \in [0..\lambda )$ and $a \in \mathcal{A}$ as follows:

\begin{equation*} m(\tau )(k,a) = \begin{cases} 0 & \text{if} \ a \not \in T_k \\ 1 & \text{if} \ a \in T_k\setminus H_k \\ 2 & \text{if} \ a \in H_k \end{cases}\\ \end{equation*}

This function provides us with a 1-1 correspondence between timed $\textrm {HT}$ -traces and three-valued interpretations. It can be checked that $f$ is bijective. The following lemma shows that the satisfaction of arbitrary metric temporal formulas is preserved among the semantics.

For our Tseitin translation, we impose the following restriction: the intervals associated to binary temporal operators $\mathbin {\boldsymbol{\mathsf{S}}}_{I}$ , $\mathbin {\boldsymbol{\mathsf{T}}}_{I}$ , and are of the form $I=[0..n]$ , with $n\lt \omega$ . By abbreviating the interval $[0..n]$ by ‘ ${\le }\,n$ ’, the metric temporal operators are of the form $\mathbin {\boldsymbol{\mathsf{S}}}_{\leq n}$ , $\mathbin {\boldsymbol{\mathsf{T}}}_{\leq n}$ , , . The following proposition shows how equivalences (5)–(8) lead to an alternative three-valued interpretation for the metric operators.

Proposition 3. For any interpretation $\boldsymbol{m}$ of length $\lambda$ , any $k \in [0..\lambda )$ and any $n \in [0..\omega )$ , we have:

Note that, by applying Proposition 3, we can conclude that . In what follows, we use those semantics when dealing with metric modalities of the form $Q_{\leq n}$ with .

4 Translation into metric temporal logic programs

In the propositional setting, the Tseitin transformation is typically performed by substituting subformulas with auxiliary atoms that represent their respective truth values. In the metric temporal case, the recursive nature of the metric temporal operators necessitates a more complex approach. For instance, if we follow Proposition 1, the satisfaction of the formula requires the evaluation of , for all $x \in [1..3]$ , which are not subformulas of .

Inspired by the Fisher-Ladner closure used in dynamic logics (Fischer and Ladner Reference Fischer and Ladner1979), the set of subformulas must be closed under the recursive definitions of the binary metric operators. This closure ensures that the auxiliary atoms correctly capture the temporal expansion of formulas across the discrete time points of the timed trace.

For instance, the closure of the formula (1), , corresponds to the set $\mathit{cl}$ ((1)) below. For brevity, we abbreviate $\mathit{press}$ , $\mathit{moving}$ and $\mathit{arrives}$ as $\mathit{p}$ , $\mathit{m}$ and $\mathit{a}$ , respectively.

Each formula in has been labeled in order to be referenced when displaying the translation of (1) into a metric temporal logic program shown in Table 4. Note that the formula $\;\mathbf{(F1)}$ corresponds to (1). The next proposition shows that the size of the closure of a formula $\varphi$ , in symbols $\lvert \mathit{cl}(\varphi )\rvert$ can be bounded by a polynomial that depends on two parameters: the size of $\varphi$ (denoted by $\lvert \varphi \rvert$ ) and the maximum number $n$ used in a binary temporal operator occurring in $\varphi$ .

Note that the closure of a formula contains other formulas than subformulas, which are also translated into logical rules. To prove the correctness, we cannot only rely on structural induction since our translation depends on the notion of closure, which contains formulas that are not subformulas. For instance, let us consider the formula . Following Proposition 3, the formulas and are $\textrm {MHT}_{\!f}$ -equivalent. Our translation replaces , , and by fresh atoms and then adds extra formulas that ensure the relationship between those and the formulas they replace. Although the proof strategy is to proceed by structural induction, it is not directly possible here since, for instance, is not a proper subformula of . We aim to define a well-founded strict partial order that takes into account not only the complexity of the input formula but also the unfolding of the , , $\mathbin {\boldsymbol{\mathsf{S}}}_{\leq n}$ and $\mathbin {\boldsymbol{\mathsf{T}}}_{\leq n}$ operators. We associate with each metric temporal formula $\varphi$ a natural number $\mathit{u}(\varphi )$ that is used to define our well-founded strict partial order.

Given two metric formulas, we say that $\varphi \prec _{\mathit{u}} \psi$ if and only if $\mathit{u}(\varphi ) \lt \mathit{u}(\psi )$ . Since the previous definition is given as a mapping into the natural numbers, we can readily prove the following result.

Regarding our running example, we get , , and . Consequently, , and , which allows us to apply the induction on the unfolding of binary metric temporal operators. Thanks to Proposition 5, we can use $\mathit{u}$ to provide an induction hypothesis, which is not based on the typical structural induction.

The following theorem states that an arbitrary metric temporal formula $\varphi$ Footnote ⁴ over an alphabet $\mathcal{A}$ can be compiled into a metric temporal logic program $\Gamma$ over an extended alphabet $\mathcal{A}_{{\ell _{}}}\supseteq \mathcal{A}$ in such a way that there exists a bijection between the timed $\textrm {HT}$ -traces of length $\lambda \in [1..\omega )$ satisfying $\varphi$ and the traces of length $\lambda$ satisfying $\Gamma$ projected onto $\mathcal{A}$ . That is, we have equivalence modulo auxiliary atoms.

Theorem 1 (Metric temporal logic program reduction).

For every metric temporal formula $\varphi$ ⁴ over $\mathcal{A}$ and for any $\lambda \in [1..\omega )$ , we have that

(9) \begin{equation} \textrm {MHT}(\varphi ,\lambda ) = \lbrace \mathbf{M}|_{\mathcal{A}} \mid \mathbf{M} \in \textrm {MHT}(\lbrace {\ell _{\varphi }} \rbrace \cup \eta ^*(\varphi ), \lambda ) \rbrace . \end{equation}

For transforming arbitrary temporal formulas into metric temporal logic programs, we use a Tseitin-style reduction (Tseitin Reference Tseitin1968) that relies on an alphabet extended by new atoms for each formula in the original language. The equivalence result in Theorem 1 is then obtained after removing auxiliary atoms and, in fact, is still preserved inside the context of a larger theory for the original vocabulary. The results presented in Theorem 1 are preserved when selecting the minimal models. This means that there exists a bijection between the metric equilibrium models of $\varphi$ and $\Gamma$ projected onto $\mathcal{A}$ . That is, we have strong equivalence modulo auxiliary atoms.

Corollary 2. For every metric temporal formula $\varphi$ ⁴ over $\mathcal{A}$ , for any $\lambda \in [1..\omega )$ and any theory $\Gamma$ over $\mathcal{A}$ ,

(10) \begin{equation} \textrm {MEL}(\lbrace \varphi \rbrace \cup \Gamma ,\lambda ) = \{\langle (\mathbf{H},\mathbf{T}),\tau \rangle |_{\mathcal{A}} \mid \langle (\mathbf{H},\mathbf{T}),\tau \rangle \in \textrm {MEL}(\lbrace {\ell _{\varphi }} \rbrace \cup \eta ^*(\varphi ) \cup \Gamma ,\lambda )\}. \end{equation}

Theorem 1 and Corollary 2 can be extended to arbitrary theories by defining the translation $\sigma$ as the metric temporal logic program:

\begin{align*} \sigma (\Gamma ) = \left \lbrace {\ell _{\gamma }} \mid \gamma \in \Gamma \right \rbrace \cup \left \lbrace \eta ^*(\mu ) \mid \mu \in cl(\Gamma )\right \rbrace \end{align*}

and then replacing $\lbrace {\ell _{\varphi }} \rbrace \cup \eta ^*(\varphi )$ by $\sigma (\Gamma )$ in expressions (9) and (10). Table 4 shows the translation of the expression in (1), which is done by applying the translation $\eta (\mu )$ for each $\mu \in \mathit{cl}$ ((1)). A metric temporal logic program can be obtained by using $\eta ^*(\mu )$ instead. The rest of this section is devoted to proving Theorem 1, for which we extend Cabalar’s reduction for the case $\textrm {THT}_{\!\omega }$ (Cabalar Reference Cabalar, Janhunen and Niemelä2010) to the case of $\textrm {MHT}_{\!f}$ . The reduction uses an extended alphabet $\mathcal{A}_{{\ell _{}}}\supseteq \mathcal{A}$ that additionally contains a new atom $\ell _{\varphi }$ (aka label) for each formula $\varphi$ in the original language over $\mathcal{A}$ . For convenience, we use ${\ell _{\varphi }} \overset {\mathit{def}}{=} \varphi$ if $\varphi$ is $\top , \bot$ or an atom $a \in \mathcal{A}$ . For any non-atomic formula $\gamma$ over $\mathcal{A}$ , we introduce the translation $\eta$ given along Tables 1, 2, and 3, and call $\eta (\gamma )$ the definition of $\gamma$ (the translation rules for Boolean and non-metric operators are provided in Tables A1 and A2). This definition is not a logic program yet, but it contains some double implication that can be easily transformed into a metric temporal logic program format by simple, non-modal transformations in the propositional logic of $\textrm {HT}$ . This translation of $\eta (\mu )$ into a metric temporal logic program is shown in the tables as $\eta ^*(\mu )$ . In the rest of the section, we indistinctly use $\eta (\mu )$ and $\eta ^*(\mu )$ , as the former is more suitable for developing the proofs. As said before, their equivalence can be easily tested inside logic HT, and so, is not provided in this paper.

Table 1.

Translation for metric next, weak next, previous and weak previous

Table 1. Long description

The table has four rows and three columns. The columns are labeled with mu, eta(μ), and η*(μ). The rows are labeled with φ, φ subscript I, Iφ, and Iφ subscript I. Each cell contains specific logical expressions and their translations. Row 1: μ: φ, η(μ): hat O box (bullet T arrow right bullet (lμ arrow left lφ)), hat O diamond (bullet T arrow right bullet lμ), box (F arrow right lμ), η*(μ): hat O box (bullet lμ arrow right lφ), hat O diamond (bullet T land lφ arrow right bullet lμ), hat O diamond (bullet T arrow right bullet lμ), box (F arrow right lμ). Row 2: μ: φ subscript I, η(μ): hat O box (bullet T arrow right bullet (lμ arrow left lφ)), hat O diamond (bullet T arrow right bullet lμ), box (F arrow right lμ), η*(μ): hat O box (bullet lμ arrow right lφ), hat O diamond (bullet T land lφ arrow right bullet lμ), hat O diamond (bullet T arrow right bullet lμ), box (F arrow right lμ). Row 3: μ: Iφ, η(μ): hat O diamond (lμ arrow right I lφ), arrow right lμ, η*(μ): hat O box (lμ arrow right I lφ), hat O diamond (I lφ arrow right lμ), arrow right lμ. Row 4: μ: Iφ subscript I, η(μ): hat O box (lμ arrow right hat I lφ), lμ, η*(μ): hat O box (lμ land T arrow right lφ), hat O diamond (I lφ arrow right lμ), lμ.

Table 2.

Translation of metric since and metric trigger

Table 2. Long description

The table presents translations of metric temporal logic expressions. It includes two main columns labeled eta(mu) and eta*(mu), each containing various logical expressions. The expressions involve logical operators such as implication, conjunction, disjunction, and negation, along with temporal operators like next, previous, since, and trigger. The table is divided into two sections, each with multiple rows of logical formulas. The first section starts with the expression phi subscript S less than or equal to n psi and translates it into a series of implications involving temporal operators. The second section starts with the expression phi subscript T less than or equal to n psi and similarly translates it into a series of implications involving temporal operators. Each row in the table shows a different logical expression and its corresponding translation.

Table 3.

Translation of metric until and metric release

Table 3. Long description

The table presents translations of metric temporal logic expressions. It includes two main sections, each with a formula and its corresponding translations. The first section shows translations for metric next, weak next, previous, and weak previous. The second section shows translations for metric since and metric trigger. Each section includes multiple rows with different translations, indicated by symbols such as ‘ell’, ‘ell subscript psi’, ‘ell subscript phi’, and various logical operators. The translations are presented in a structured format with clear labeling and logical expressions.

Table 4.

Translation of (1) into a metric temporal logic program. Equivalences of the form ${\ell _{\varphi }}\leftrightarrow p$ , with $p\in \mathcal{A}$ , are not shown and $p$ is used instead of $\ell _{p}$

Table 4. Long description

An equation with multiple lines labeled from F1 to F11. Each line represents a different formula. The formulas include various logical operators such as diamonds, boxes, implications, and conjunctions. Variables and constants are used throughout the equations, with some lines containing nested logical expressions. The equations are structured to show the translation of a specific formula into a metric temporal logic program.

Given a trace $\mathbf{M}= \langle (\mathbf{H},\mathbf{T}),\tau \rangle$ of length $\lambda \in \mathbb{N}\cup \lbrace \omega \rbrace$ , we define its projection onto the alphabet $\mathcal{A}$ as $\mathbf{M}\mid _{\mathcal{A}} = \langle (\mathbf{H}',\mathbf{T}'),\tau \rangle$ where $H'_i = H_i \cap \mathcal{A}$ and $T'_i = T_i \cap \mathcal{A}$ , for all $i \in [0..\lambda )$ .

Let us discuss now the proof of Theorem 1. We show that for any metric temporal formula $\varphi$ over $\mathcal{A}$ and any $\lambda \in [1..\omega )$ ,

\begin{equation*} \textrm {MHT}(\varphi ,\lambda ) = \lbrace \mathbf{M}|_{\mathcal{A}} \mid \mathbf{M} \in \textrm {MHT}(\lbrace {\ell _{\varphi }} \rbrace \cup \eta (\varphi ), \lambda ) \rbrace . \end{equation*}

The ‘ $\subseteq$ ’ direction follows from Lemma 2. For the ‘ $\supseteq$ ’ direction, take some $\textrm {MHT}$ model $\mathbf{M}$ of $\eta (\varphi )$ of length $\lambda \in [1..\omega )$ . This implies that its associated $3$ -valuation satisfies ${\boldsymbol{m'}}(0,{\ell _{\varphi }})=2$ , which is included in $\eta (\varphi )$ . Since $\varphi \in cl(\varphi )$ , we can apply Lemma 3 to conclude ${\boldsymbol{m'}}(k,{\ell _{\varphi }})={\boldsymbol{m'}}(k,\varphi )$ for all $k \in [0..\lambda )$ but then, in particular, $2={\boldsymbol{m'}}(0,{\ell _{\varphi }})={\boldsymbol{m'}}(0,\varphi )$ and so, $\mathbf{M}$ is also a model of $\varphi$ . Finally, $\mathbf{M}|_{\mathcal{A}}$ is still a model of $\varphi$ because the latter is a theory over vocabulary $\mathcal{A}$ . The theorem below provides a parametric polynomial bound of the size of $\eta ^*(\varphi )$ .