2 Review: The logic of here-and-there with constraints

The logic of Here-and-There with constraints ( $\textrm {HT}_{\!c}$ ) along with its equilibrium models provides logical foundations for constraint satisfaction problems (CSPs) in the setting of ASP. Its approach follows the one of the traditional Boolean Logic of Here-and-There (Heyting Reference Heyting1930) and its nonmonotonic extension of Equilibrium Logic (Pearce Reference Pearce1997, Reference Pearce2006). In $\textrm {HT}_{\!c}$ , a CSP is expressed as a triple $\langle \mathcal{X},{\mathcal{D}},{\mathcal{C}} \rangle$ , also called signature, where $\mathcal{X}$ is a set of variables over some non-empty domain $\mathcal{D}$ and $\mathcal{C}$ is a set of constraint atoms. A constraint atom provides an abstract way to relate values of variables and constants according to the atom’s semantics. Most useful constraint atoms have a structured syntax, but in the general case, we may simply consider them as strings.Footnote ¹ For instance, linear equations are expressions of the form “ $x+y=4$ ”, where $x$ and $y$ are variables from $\mathcal{X}$ and $4$ is a constant representing some element from $\mathcal{D}$ .

Variables can be assigned some value from $\mathcal{D}$ or left undefined. For the latter, we use the special symbol $\mathbf{u} \notin {\mathcal{D}}$ and the extended domain $\mathcal{D}_{\mathbf{u}} \stackrel {\mathrm{def}}{=} {\mathcal{D}} \cup \{\mathbf{u}\}$ . The set $\mathit{vars}(c) \subseteq \mathcal{X}$ collects all variables occurring in constraint atom $c$ .

A valuation $v$ over $\mathcal{X},{\mathcal{D}}$ is a set of pairs $x \mapsto d$ where $x \in \mathcal{X}$ and $d \in {\mathcal{D}}$ such that for each $x \in \mathcal{X}$ there is at most one pair $x \mapsto d$ in $v$ . The restriction of $v$ to a set of variables $Z \subseteq \mathcal{X}$ , denoted ${v|}_{\hspace {-1pt}Z}$ , is the set of pairs $x \mapsto d$ in $v$ such that $x \in Z$ . We define the domain of a valuation $v$ as the set of variables that have a defined value, namely,

\begin{equation*}\mathit{dom}(v) \stackrel {\mathrm{def}}{=} \{x \in \mathcal{X} \mid x \mapsto d \in v \text{ for some } d \in {\mathcal{D}}\}.\end{equation*}

We sometimes use function notation for writing valuations and write $v(x)=d$ to denote that $x \mapsto d$ is in $v$ , and $v(x)=\mathbf{u}$ to denote that there is no pair $x \mapsto d$ in $v$ for any $d \in {\mathcal{D}}$ . Using this notation, $v\subseteq v'$ means that $v(x) \in {\mathcal{D}}$ implies $v'(x)=v(x)$ for all $x \in \mathcal{X}$ . We also allow for applying a valuation $v$ to fixed domain values, and so extend their type to $v:\mathcal{X}\cup \mathcal{D}_{\mathbf{u}}\rightarrow \mathcal{D}_{\mathbf{u}}$ by fixing $v(d) = d$ for any $d \in {\mathcal{D}}_{\mathbf{u}}$ . The semantics of constraint atoms is defined in $\textrm {HT}_{\!c}$ via denotations, which are functions of form $[\![ \, \cdot \, ]\!]:{\mathcal{C}}\rightarrow 2^{\mathcal{V}}$ , mapping each constraint atom to a set of valuations. Following Cabalar et al. (Reference Cabalar, Fandinno, Schaub and Wanko2020a), we assume in the sequel that they satisfy the following properties for all $c\in {\mathcal{C}}$ , $x\in \mathcal{X}$ , and $v,v' \in \mathcal{V}$ :

1. 1. $v \in [\![ \, c \, ]\!]$ and $v \subseteq v'$ imply $v' \in [\![ \, c \, ]\!]$ ,

2. 2. $v \in [\![ \, c \, ]\!]$ implies $v \in [\![ \, c[x/v(x)] \, ]\!]$ ,

3. 3. if $v(x)=v'(x)$ for all $x \in \mathit{vars}(c)$ then $v \in [\![ \, c \, ]\!]$ iff $v' \in [\![ \, c \, ]\!]$ .

where $c[s/s']$ is the syntactic replacement in $c$ of subexpression $s$ by $s'$ . We also assume that $c[x/d] \in {\mathcal{C}}$ for any constraint atom $c \in {\mathcal{C}}$ containing $x$ , for variable $x \in \mathcal{X}$ and $d \in \mathcal{D}_{\mathbf{u}}$ . That is, replacing a variable by any element of the extended domain results in a syntactically valid constraint atom. Intuitively, Condition 1 makes constraint atoms behave monotonically with respect to definedness. Condition 2 stipulates that denotations respect the role of variables as placeholders for values, that is, replacing variables by their assigned values does not change how an expression is evaluated. Condition 3 asserts that the denotation of $c$ is fixed by combinations of values for $\mathit{vars}(c)$ ; other variables may freely vary.

A formula $\varphi$ over signature $\langle \mathcal{X},{\mathcal{D}},{\mathcal{C}} \rangle$ is defined as

\begin{align*} \varphi ::= \bot \mid c\mid \varphi \land \varphi \mid \varphi \lor \varphi \mid \varphi \rightarrow \varphi \quad \text{ where }c\in {\mathcal{C}}. \end{align*}

We define $\top$ as $\bot \rightarrow \bot$ and $\neg \varphi$ as $\varphi \rightarrow \bot$ for any formula $\varphi$ . We let $\mathit{vars}(\varphi )$ stand for the set of variables in $\mathcal{X}$ occurring in all constraint atoms in formula $\varphi$ . An HT_c-theory is a set of formulas.

In $\textrm {HT}_{\!c}$ , an interpretation over $\mathcal{X},{\mathcal{D}}$ is a pair $\langle h,t \rangle$ of valuations over $\mathcal{X},{\mathcal{D}}$ such that $h\subseteq t$ . The interpretation is total if $h=t$ . Given a denotation $[\![ \, \cdot \, ]\!]$ , an interpretation $\langle h,t \rangle$ satisfies a formula $\varphi$ , written $\langle h,t \rangle \models \varphi$ , if

1. 1. $\langle h,t \rangle \models c \text{ if } h\in [\![ \, c \, ]\!]$

2. 2. $\langle h,t\rangle \models \varphi \land \psi \text{ if } \langle h,t\rangle \models \varphi \text{ and } \langle h,t\rangle \models \psi$

3. 3. $\langle h,t\rangle \models \varphi \lor \psi \text{ if } \langle h,t\rangle \models \varphi \text{ or } \langle h,t\rangle \models \psi$

4. 4. $\langle h,t\rangle \models \varphi \rightarrow \psi \text{ if }\langle w,t\rangle \not \models \varphi \text{ or }\langle w,t\rangle \models \psi \text{ for every }w\in \{h,t\}$

We say that an interpretation $\langle h,t \rangle$ is a model of a theory $\Gamma$ , written $\langle h,t \rangle \models \Gamma$ , when $\langle h,t \rangle \models \varphi$ for every $\varphi \in \Gamma$ . We write $\Gamma \models \Gamma '$ when every model of $\Gamma$ is also a model of $\Gamma '$ . We write $\Gamma \equiv \Gamma '$ if $\Gamma$ and $\Gamma '$ have the same models. We omit braces whenever a theory is a singleton.

The first item reflects the well-known persistence property in constructive logics. The second one tells us that negation is only evaluated in the there world. The third item allows us to derive the strong equivalence of two expressions in $\textrm {HT}_{\!c}$ from their equivalence in $\textrm {HT}$ when treating constraint atoms monolithically, though more equivalences can usually be derived using $\textrm {HT}_{\!c}$ .

$\textrm {HT}_{\!c}$ makes few assumptions about the syntactic form or the semantics of constraint atoms. In the current paper, however, we introduce a specific kind of constraint atom that is useful later on for some of the formal results. Given a subset ${\mathcal{D}}'\subseteq {\mathcal{D}}$ , we define the associatedFootnote ² constraint atom $x:{\mathcal{D}}'$ with denotation:

\begin{eqnarray*} [\![ \, x:{\mathcal{D}}' \, ]\!] \ \stackrel {\mathrm{def}}{=} \ \{ v \in \mathcal{V}^{\mathcal{X}, {\mathcal{D}}} \mid v(x) \in {\mathcal{D}}'\} \end{eqnarray*}

and $\mathit{vars}(x:{\mathcal{D}}')=\{x\}$ , that is, $x:{\mathcal{D}}'$ asserts that $x$ has some value in subdomain ${\mathcal{D}}'$ . Note that $v(x) \in {\mathcal{D}}'$ implies that $x$ is defined in $v$ , since $\mathbf{u} \not \in {\mathcal{D}} \supseteq {\mathcal{D}}'$ . We use the abbreviation $\mathit{def}(x)$ to stand for $x:{\mathcal{D}}$ , so that this atom holds iff $x$ has some value, that is it is not undefined, or $v(x) \neq \mathbf{u}$ .

The nonmonotonic extension of $\textrm {HT}_{\!c}$ is defined in terms of equilibrium models, being minimal models in $\textrm {HT}_{\!c}$ in the following sense.

As detailed by Cabalar et al. (Reference Cabalar, Kaminski, Ostrowski and Schaub2016), the original logic of Here-and-There can be obtained as a special case of $\textrm {HT}_{\!c}$ with a signature $\langle \mathcal{X},{\mathcal{D}},{\mathcal{C}} \rangle$ , where $\mathcal{X}=\mathcal{A}$ represents logical propositions, the domain ${\mathcal{D}}=\{\mathbf{t}\}$ contains a unique value (standing for “true”) and the set of constraint atoms ${\mathcal{C}}=\mathcal{A}$ coincides with the set of propositions. In this way, each logical proposition $\mathtt {a}$ becomes both a constraint atom $\mathtt {a} \in {\mathcal{C}}$ and a homonymous variable $a \in \mathcal{X}$ so that $\mathit{vars}(\mathtt {a})=\{a\}$ (we use different fonts for the same name to stress the different role). The denotation of a propositional atom is fixed to

\begin{gather*} [\![ \, \mathtt {a} \, ]\!] \stackrel {\mathrm{def}}{=} [\![ \, a:\{\mathbf{t}\} \, ]\!] = \{v \in \mathcal{V}^{\mathcal{X},{\mathcal{D}}} \mid v(a)=\mathbf{t} \} \end{gather*}

In the following, we refer to constraint atoms with this denotation as propositional atoms. Moreover, we can establish a one-to-one correspondence between any propositional interpretation, represented as a set $X$ of propositional atoms, and the valuation $v$ that assigns $\mathbf{t}$ to all members of $X$ , that is, $X=\{\mathtt {a} \mid v(a)=\mathbf{t} \}$ . Note that a false atom, viz. $\mathtt {a} \not \in X$ , is actually undefined in the valuation, viz. $v(a)=\mathbf{u}$ , since having no value is the default situation.Footnote ³ Once this correspondence is established, it is easy to see that the definition of equilibrium and stable models in Definition1 collapses to their standard definition for propositional theories (and also, logic programs in ASP).

Treating logical propositions as constraint atoms allows us not only to capture standard ASP but also to combine propositional atoms with other constraint atoms in a homogeneous way, even when we deal with a larger domain ${\mathcal{D}} \supset \{\mathbf{t}\}$ such as, for instance, ${\mathcal{D}} = \mathbb{Z} \cup \{\mathbf{t}\}$ . In this setting, we may observe that while $[\![ \, \mathtt {a} \, ]\!] \subseteq [\![ \, \mathit{def}(a) \, ]\!]$ (i.e., if the atom has value $\mathbf{t}$ , it is obviously defined), the opposite $[\![ \, \mathit{def}(a) \, ]\!] \subseteq [\![ \, \mathtt {a} \, ]\!]$ does not necessarily hold. For instance, we may have now some valuation $v(a)=7$ , and so $v \in [\![ \, \mathit{def}(a) \, ]\!]$ , but $v \not \in [\![ \, \mathtt {a} \, ]\!]$ . For this reason, we assume the inclusion of the following axiom

(1) \begin{eqnarray} \mathit{def}(a) \to \mathtt {a} \end{eqnarray}

to enforce $[\![ \, \mathit{def}(a) \, ]\!]=[\![ \, \mathtt {a}\, ]\!]$ for each propositional atom $\mathtt {a}$ . This forces propositional atoms to be either true or undefined. It does not constrain non-propositional atoms from taking any value in the domain.

Finally, one more possibility we may consider in $\textrm {HT}_{\!c}$ is treating all constraint atoms as propositional atoms, so that we do not inspect their meaning in terms of an external theory but only consider their truth as propositions.

In other words, regular stable models are the result of considering an $\textrm {HT}_{\!c}$ -theory $\Gamma$ as a propositional $\textrm {HT}$ -theory where constraint atoms are treated as logical propositions. This definition is useful in defining the semantics of logic programs with constraints according to clingo (see Section 4).

3 Strong equivalence in ${\textrm{HT}}_{c}$

One of the most important applications of $\textrm {HT}$ is its use in ASP for equivalent transformations among different programs or fragments of programs. In general, if we want to safely replace program $P$ by $Q$ , it does not suffice to check that their sets of stable models coincide, because the semantics of ASP programs cannot be figured out by looking at single rules (or groups of rules) in isolation. We can easily extrapolate this concept to $\textrm {HT}_{\!c}$ as follows:

Theory $\Delta$ is sometimes called a context, so that strong equivalence guarantees that $\Gamma$ can be replaced by $\Gamma '$ (and vice versa) in any context. An important property of $\textrm {HT}$ proved by Lifschitz et al. (Reference Lifschitz, Pearce and Valverde2001) is that two logic programs $P$ and $Q$ are strongly equivalent if and only if they are equivalent in $\textrm {HT}$ .

Accordingly, we may wonder whether a similar result holds for $\textrm {HT}_{\!c}$ . The following theorem is an immediate result: the equivalence $\Gamma \equiv \Gamma '$ in $\textrm {HT}_{\!c}$ implies that $\textrm {HT}_{\!c}$ -theories $\Gamma$ and $\Gamma '$ are strongly equivalent.

For illustration, let us use Theorem1 to prove the strong equivalence of two simple $\textrm {HT}_{\!c}$ -theories. Consider the following two formulas, similar to the ones from the introductory section:

(2) \begin{align} &\bot \leftarrow \neg \mathtt {a} \wedge s \geq 120 \\[-10pt] \nonumber \end{align}

(3) \begin{align} &\mathtt {a} \leftarrow s \gt 100 \\[8pt] \nonumber \end{align}

In fact, we show in Section 7 that these two formulas are an essential part of the $\textrm {HT}_{\!c}$ -based translation of the logic program mentioned in the introduction. Let us show that $\Gamma = \{ (2), (3)\}$ is strongly equivalent to $\Gamma ' = \{ (3)\}$ . By Theorem1, a sufficient condition is showing that $\Gamma \equiv \Gamma '$ and, since $\Gamma ' \subseteq \Gamma$ , it suffices to show that (3) $\to$ (2) is an $\textrm {HT}_{\!c}$ -tautology. The following property is useful to prove this result.

Let $\varphi [c/\psi ]$ be the uniform replacement of a constraint atom $c$ occurring in formula $\varphi$ by a formula $\psi$ , and $\varphi [c]$ be the formula $\varphi$ with a distinguished occurrence of constraint atom $c$ . We show below that $\textrm {HT}_{\!c}$ satisfies the rule of uniform substitution.

Let us now resume our example. The $\textrm {HT}$ -tautology

\begin{gather*} (\alpha \to \beta ) \to ((\beta \to \gamma ) \to \neg (\neg \gamma \wedge \alpha )) \end{gather*}

is also an $\textrm {HT}_{\!c}$ -tautology by Proposition1. By applying Proposition2 to this $\textrm {HT}_{\!c}$ -tautology with substitution $\alpha \mapsto s\geq 120$ , $\beta \mapsto s \gt 100$ , and $\gamma \mapsto \mathtt {a}$ , we obtain the following $\textrm {HT}_{\!c}$ -tautology:

\begin{align*} (s\geq 120 \to s\gt 100) \to ((3) \to (2)) \end{align*}

To show that $(3)\to (2)$ is an $\textrm {HT}_{\!c}$ -tautology and, thus, $\Gamma$ and $\Gamma '$ are strongly equivalent, it suffices to show that the antecedent of this implication is an $\textrm {HT}_{\!c}$ -tautology. This immediately follows once we assume the usual semantics of linear inequalities and, thus, that our denotation satisfies

\begin{align*} [\![ \, s \geq 120 \, ]\!] \subseteq [\![ \, s\gt 100 \, ]\!] . \end{align*}

As expected, the result may not hold if we assume that these two constraint atoms have a different meaning than the one they have for linear inequalities.

This illustrates how we can use Theorem1 to prove the strong equivalence of two $\textrm {HT}_{\!c}$ -theories. To prove that $\textrm {HT}_{\!c}$ -equivalence is also a necessary condition for strong equivalence, we depend on the form of the context theory $\Delta$ . For instance, in the case in which $\Gamma$ and $\Gamma '$ are regular ASP programs, it is well-known that if we restrict the form of $\Delta$ to sets of facts, we obtain a weaker concept called uniform equivalence, that may hold even if $\Gamma$ and $\Gamma '$ do not have the same $\textrm {HT}$ models. In the case of $\textrm {HT}_{\!c}$ , the variability in the possible context theory $\Delta$ is much higher since the syntax and semantics of constraint atoms are very general, and few assumptions can be made on them. Yet, if our theory accepts at least a constraint atom $\mathit{def}(x)$ in $\mathcal{C}$ for each $x \in \mathcal{X}$ , then we can use this construct (in the context theory $\Delta$ ) to prove the other direction of the strong equivalence characterization.

Proof. We proceed by contraposition, that is, proving that $\Gamma \not \equiv \Gamma '$ implies that $\Gamma$ and $\Gamma '$ are not strongly equivalent. To this aim, we build some theory $\Delta$ such that $\Gamma \cup \Delta$ and $\Gamma ' \cup \Delta$ have different stable models. Without loss of generality, suppose there exists some model $\langle h,t \rangle \models \Gamma$ but $\langle h,t \rangle \not \models \Gamma '$ . Note that, by persistence, $\langle t,t \rangle \models \Gamma$ , but we do not know whether $\langle t,t \rangle \models \Gamma '$ or not, so we separate the proof in two cases.

Case 1. Suppose first that $\langle t,t \rangle \not \models \Gamma '$ . Let us build the theory

\begin{eqnarray*} \Delta =\{\mathit{def}(x) \mid x \in \mathcal{X}, t(x)\neq \mathbf{u}\} \end{eqnarray*}

that exclusively consists of constraint atoms. We can easily see that $\langle t,t \rangle \models \Delta$ because $\Delta$ collects precisely those $\mathit{def}(x)$ for which $t(x) \neq \mathbf{u}$ and so, $t \in [\![ \, \mathit{def}(x) \, ]\!]$ trivially. As a result, $\langle t,t \rangle \models \Gamma \cup \Delta$ and, to prove that $\langle t,t \rangle$ is in equilibrium, suppose we have a smaller $v \subset t$ satisfying $\Gamma \cup \Delta$ . Then there is some variable $x \in \mathcal{X}$ for which $v(x)=\mathbf{u}$ whereas $t(x) \neq \mathbf{u}$ . The former implies $\langle v,t \rangle \not \models \mathit{def}(x)$ while the latter implies $\mathit{def}(x) \in \Delta$ , so we conclude $\langle v,t \rangle \not \models \Delta$ reaching a contradiction.

Case 2. Suppose that $\langle t,t \rangle \models \Gamma '$ . Take the theory $\Delta = \Delta _1 \cup \Delta _2$ with:

\begin{eqnarray*} \Delta _1 & = & \{\mathit{def}(x) \mid x \in \mathcal{X}, \, h(x) \neq \mathbf{u} \} \end{eqnarray*}

and $\Delta _2$ consisting of all rules $\mathit{def}(x) \leftarrow \mathit{def}(y)$ for all pair variables $x,y$ in the set:

\begin{equation*} \{z \in \mathcal{X} \mid h(z)= \mathbf{u}, t(z)\neq \mathbf{u} \} \end{equation*}

We prove first that $\langle t,t \rangle $ is not an equilibrium model for $\Gamma \cup \Delta$ because $\langle h,t \rangle$ is, indeed, a model of this theory. To show this, it suffices to see that $\langle h,t \rangle \models \Delta$ . First, it follows that $\langle h,t \rangle \models \Delta _1$ because $\Delta _1$ only contains facts of the form $\mathit{def}(x)$ per each variable $x$ satisfying $h(x) \neq \mathbf{u}$ . Second, $\langle h,t \rangle \models \Delta _2$ also follows because for all the implications of the form of $\mathit{def}(x) \leftarrow \mathit{def}(y)$ in $\Delta _2$ , both $\langle h,t \rangle \not \models \mathit{def}(y)$ – because $h(y)=\mathbf{u}$ – and $\langle t,t \rangle \models \mathit{def}(x)$ – because $t(x) \neq \mathbf{u}$ . It remains to be proven that $\langle t,t \rangle$ is an equilibrium model of $\Gamma ' \cup \Delta$ . We can see that $\langle t,t \rangle \models \Delta$ follows by persistence because $\langle h,t \rangle \models \Delta$ , and thus, $\langle t,t \rangle \models \Gamma ' \cup \Delta$ . Suppose, for the sake of contradiction, that there existed some $v \subset t$ such that $\langle v,t \rangle \models \Gamma ' \cup \Delta$ . Since $\langle v,t \rangle \models \Delta _1$ , any variable $x$ defined in $h$ must be defined in $v$ as well, but as $v \subset t$ , $v(x)=t(x)=h(x)$ and so, we conclude $h \subseteq v$ . However, we also know $\langle h,t \rangle \not \models \Gamma ' \subseteq \Gamma ' \cup \Delta$ , and so $v \neq h$ , that is, $h \subset v \subset t$ . Consider any variable $y$ defined in $v$ but not in $h$ , that is, $h(y)=\mathbf{u}$ and $\mathbf{u} \neq v(y) = t(y)$ by persistence. Now, $y$ is undefined in $h$ and defined in $t$ , and suppose we take any other variable $x$ in the same situation. Then, we have an implication $\mathit{def}(x) \leftarrow \mathit{def}(y)$ in $\Delta _2$ that must be satisfied by $\langle v,t \rangle$ whereas, on the other hand, $\langle v,t \rangle \models \mathit{def}(y)$ because $v(y) \neq \mathbf{u}$ . As a result, $\langle v,t \rangle \models \mathit{def}(x)$ for all variables $x$ undefined in $h$ but not in $t$ . But as $\langle v,t \rangle \models \Delta _1$ the same happens for variables defined in $h$ . As a result, $\langle v,t \rangle \models \mathit{def}(x)$ iff $\langle t,t \rangle \models \mathit{def}(x)$ and this implies $v(x)=t(x)$ for all variables, namely, $v=t$ reaching a contradiction. Therefore, $\langle t,t \rangle$ is an equilibrium model of $\Gamma ' \cup \Delta$ .

This proof is analogous to the original one for $\textrm {HT}$ by Lifschitz et al. (Reference Lifschitz, Pearce and Valverde2001), using here constraint atoms $\mathit{def}(x)$ to play the role of propositional atoms in the original proof.

4 Review: Logic programs with abstract theories as $\textrm {HT}_{c}$ -theories

The main distinctive feature of clingo 5 is the introduction of theory atoms in its syntax (Gebser et al. Reference Gebser, Kaminski, Kaufmann, Ostrowski, Schaub and Wanko2016). We review its most recent semantic characterization based on the concept of abstract theories (Cabalar et al. Reference Cabalar, Fandinno, Schaub and Wanko2023). We consider an alphabet consisting of two disjoint sets, namely, a set $\mathcal{A}$ of propositional (or regular) atoms and a set $\mathcal{T}$ of theory atoms, whose truth is governed by some external theory. We use letters $\mathtt {a}$ , $\mathtt {s}$ , and $b$ and variants of them for atoms in $\mathcal{A}$ , $\mathcal{T}$ , and $\mathcal{A} \cup \mathcal{T}$ , respectively. In clingo 5, theory atoms are expressions preceded by ‘ $\mathtt {\&}$ ’, but their internal syntax is not predetermined: it can be defined by the user to build new extensions. As an example, the system clingcon extends the input language of clingo with linear equations, represented as theory atoms of the form

(4) \begin{align} \mathtt {\&sum\{} \mathit{k_1*x_1};\dots ;\mathit{k_n*x_n} \mathtt {\}} \prec k_0 \end{align}

where each $x_i$ is an integer variable and each $k_i\in \mathbb{Z}$ an integer constant for $0\leq i\leq n$ , whereas $\prec$ is a comparison symbol such as ${\mathtt{ \lt} =}, \texttt { = }, \texttt { !=}, \mathtt{ \lt }, \mathtt{ \gt }, \mathtt{ \gt =}$ . Several theory atoms may represent the same theory entity. For instance, $\mathtt {\&sum\{} x \mathtt {\}} \mathtt { \gt } 0$ and $\mathtt {\&sum\{x\}}{ \gt =} 1$ actually represent the same condition (as linear equations).

A literal is any atom $b \in \mathcal{A} \cup \mathcal{T}$ or its default negation $\neg b$ . A $\mathcal{T}$ -program over $\langle \mathcal{A}$ , $\mathcal{T}\rangle$ is a set of rules of the form

(5) \begin{align} b_0 \leftarrow b_1,\dots ,b_n,\neg b_{n+1},\dots ,\neg b_m \end{align}

where $b_i\in \mathcal{A}\cup \mathcal{T}$ for $1 \leq i \leq m$ and $b_0\in \mathcal{A}\cup \mathcal{T} \cup \{ \bot \}$ with $\bot \not \in \mathcal{A} \cup \mathcal{T}$ denoting the falsum constant. We sometimes identify (5) with the formula

\begin{equation*}b_1 \wedge \dots \wedge b_n \wedge \neg b_{n+1} \wedge \dots \wedge \neg b_m \to b_0.\end{equation*}

We let notations $\mathit{h}(r)\stackrel {\mathrm{def}}{=} b_0$ , ${\mathit{B}(r)}^+\stackrel {\mathrm{def}}{=} \{b_1,\dots ,b_n\}$ and ${\mathit{B}(r)}^-\stackrel {\mathrm{def}}{=}\{b_{n+1},\dots ,b_m\}$ stand for the head, the positive and the negative body atoms of a rule $r$ as in (5). The set of body atoms of $r$ is just $\mathit{B}(r) \stackrel {\mathrm{def}}{=} {\mathit{B}(r)}^+ \cup {\mathit{B}(r)}^-$ . Finally, the sets of body and head atoms of a program $P$ are defined as $\mathit{B}(P) \stackrel {\mathrm{def}}{=} \bigcup _{r \in P} \mathit{B}(r)$ and $\mathit{H}(P)\stackrel {\mathrm{def}}{=} \{\mathit{h}(r)\mid r\in P\}\setminus \{\bot \}$ , respectively.

The semantics of $\mathcal{T}$ -programs in clingo (Gebser et al. Reference Gebser, Kaminski, Kaufmann, Ostrowski, Schaub and Wanko2016), relies on a two-step process: (1) generate regular stable models (as in Definition2) and (2) select the ones passing a theory certification. We present next this semantics following the formalization recently introduced by Cabalar et al. (Reference Cabalar, Fandinno, Schaub and Wanko2023).

An abstract theory $\mathfrak{T}$ is a triple $ \langle \mathcal{T}, \mathit{Sat}, \overparen{\cdot } \,\rangle$ where $\mathcal{T}$ is the set of theory atoms constituting the language of the abstract theory. The set $\mathit{Sat} \subseteq 2^{\mathcal{T}}$ contains all possible sets of theory atoms that are considered satisfiable by the theory: each one of these sets $S \in \mathit{Sat}$ is said to be $\mathfrak{T}$ -satisfiable. Finally, $\overparen{\cdot }:\mathcal{T} \rightarrow \mathcal{T}$ is a function mapping theory atoms to their complement such that $\overparen{\overparen{ \mathtt{s}}}= \mathtt {s}$ for any $\mathtt {s} \in \mathcal{T}$ . We define $\overparen{{S}} = \{ \overparen{ \mathtt{s}} \mid \mathtt {s} \in S\}$ for any set $S\subseteq \mathcal{T}$ .

We partition the set of theory atoms into two disjoint sets, namely, a set $\mathcal{E}$ of external theory atoms and a set $\mathcal{F}$ of founded theory atoms. Intuitively, the truth of each external atom in $\mathcal{E}$ requires no justification. Founded atoms on the other hand must be derived by the $\mathcal{T}$ -program. We assume that founded atoms do not occur in the body of rules. We refer to the work by Janhunen et al. (Reference Janhunen, Kaminski, Ostrowski, Schellhorn, Wanko and Schaub2017) for a justification for this assumption.

Given any set $S$ of theory atoms, we define its (complemented) completion with respect to external atoms $\mathcal{E}$ , denoted by $\mathit{Comp}_{\mathcal{E}}(S)$ , as:

\begin{eqnarray*} \mathit{Comp}_{\mathcal{E}}(S) \stackrel {\mathrm{def}}{=} S \ \cup \ (\overparen{ \mathcal{E}\setminus S}) \end{eqnarray*}

In other words, we add the complement atom $\overparen{ \mathtt{s}}$ for every external atom $\mathtt {s}$ that does not occur explicitly in $S$ .

That is, $S$ is a $\langle \mathfrak{T},\mathcal{E}\rangle$ -solution whenever both $S$ and $\mathit{Comp}_{\mathcal{E}}(S)$ are $\mathfrak{T}$ -satisfiable.

Definition 5 (Theory stable modelFootnote ⁴; Cabalar et al. Reference Cabalar, Fandinno, Schaub and Wanko2023). Given a theory $\mathfrak{T}=\langle \mathcal{T}, \mathit{Sat}, \overparen{\cdot } \,\rangle$ and a set $\mathcal{E} \subseteq \mathcal{T}$ of external theory atoms, a set $X\subseteq \mathcal{A}\cup \mathcal{T}$ of atoms is a $\langle \mathfrak{T}, \mathcal{E} \rangle$ - $\textit{stable model}$ of a $\mathcal{T}$ -program $P$ , if there is some $\langle \mathfrak{T},\mathcal{E}\rangle$ -solution $S$ such that $X$ is a regular stable model of the program

(6) \begin{align} P \cup \{ {\mathtt {s}\leftarrow } \mid \mathtt {s}\in (S\cap \mathcal{E}) \} \cup \{ {\bot \leftarrow \mathtt {s}} \mid \mathtt {s}\in ((\mathcal{T}\cap \mathit{H}(P)) \setminus S) \} \ . \end{align}

As an example of an abstract theory, consider the case of linear equations $\mathfrak{L}$ which can be used to capture clingcon-programs. This theory is formally defined as $\mathfrak{L}=\langle \mathcal{T}, \mathit{Sat},\overparen{\cdot} \, \rangle$ , where

• • $\mathcal{T}$ is the set of all expressions of form (4),

• • $\mathit{Sat}$ is the set of all subsets $S\subseteq \mathcal{T}$ of expressions of form (4) for which there exists an assignment of integer values to their variables that satisfies all linear equations in $S$ according to their usual meaning, and

• • the complement function $\overparen{\mathtt {\&sum\{\cdot \}} \prec c} $ is defined as $\mathtt {\&sum\{\cdot \}}\mathrel {\overparen{\prec }}c$ with ${\overparen{\prec }}$ defined according to the following table:

$\prec$	$\mathtt {\lt =}$	$\mathtt {=}$	$\mathtt {!=}$	$\mathtt {\lt }$	$\mathtt {\gt }$	$\mathtt {\gt =}$
${\overparen{\prec }}$	$\mathtt {\gt }$	$\mathtt {!=}$	$\mathtt {=}$	$\mathtt {\gt =}$	$\mathtt {\lt =}$	$\mathtt {\lt }$

Using the theory atoms of theory $\mathfrak{L}$ , we can write the running example from the introduction as the $\mathcal{T}$ -program:

(7) \begin{align} \bot & \leftarrow \neg \mathtt {a},\, \mathtt {\&sum\{} s \mathtt {\}}\!\gt =\!120 \\[-10pt] \nonumber \end{align}

(8) \begin{align} \mathtt {a} & \leftarrow \mathtt {\&sum\{} s \mathtt {\}}\!\gt \!100 \\[10pt] \nonumber \end{align}

5 A new and simpler definition of theory stable model

In this section, we introduce a new, simplified version of the definition of $\langle \mathfrak{T}, \mathcal{E} \rangle$ -stable model that, under certain conditions introduced below, is equivalent to Definition5. This streamlined definition not only enhances the clarity of the technical development presented in the subsequent sections but also holds the potential to facilitate future advancements in hybrid solver design.

Definition 6 (Theory stable model simplified). Given a theory $\mathfrak{T}=\langle \mathcal{T}, \mathit{Sat}, \overparen{\cdot } \,\rangle$ and a set $\mathcal{E} \subseteq \mathcal{T}$ of external theory atoms, a set $X\subseteq \mathcal{A} \cup \mathcal{T}$ of atoms is a $\langle \mathfrak{T}, \mathcal{E} \rangle$ - ${\textit stable\,model}$ of a $\mathcal{T}$ -program $P$ , if $(X \cap \mathcal{T}) \in \mathit{Sat}$ and $X$ is a regular stable model of the theory

(9) \begin{align} P \cup \{ \mathtt {s} \vee \overparen{\mathtt {s}}\, \mid \mathtt {s} \in \mathcal{E}\} . \end{align}

This definition drops the existential quantifier used in Definition5 for identifying a $\langle \mathfrak{T}, \mathcal{E} \rangle$ -solution $S$ .

To state the conditions under which Definitions5 and 6 are equivalent, we need the following concepts (Cabalar et al. Reference Cabalar, Fandinno, Schaub and Wanko2023). An abstract theory $\mathfrak{T} = \langle \mathcal{T}, \mathit{Sat}, \overparen{\cdot } \,\rangle$ is consistent if none of its satisfiable sets contains complementary theory atoms, that is, there is no $S \in \mathit{Sat}$ such that $\mathtt {s} \in S$ and $\overparen{ \mathtt{s}}\in S$ for some atom $\mathtt {s}\in \mathcal{T}$ . A set $S$ of theory atoms is closed if $\mathtt {s} \in S$ implies $\overparen{ \mathtt{s}} \in S$ . A set $S$ of theory atoms is called $\mathcal{E}$ -complete, if for all $\mathtt {s} \in \mathcal{E}$ , either $\mathtt {s}\in S$ or $\overparen{ \mathtt{s}}\in S$ . A theory $\mathfrak{T}=\langle \mathcal{T}, \mathit{Sat}, \overparen{\cdot } \,\rangle$ is monotonic if $S \subseteq S'$ and $S' \in \mathit{Sat}$ implies $S \in \mathit{Sat}$ . Programs over a consistent theory with a closed set of external theory atoms have the following interesting properties:

The preconditions of Theorem3 cover many hybrid extensions of clingo such as clingcon, clingo[dl], and clingo[lp]. Therefore, in the rest of the paper, we assume these conditions hold and use Definition6 as the definition of a $\langle \mathfrak{T},\mathcal{E}\rangle$ -stable model.

6 Review: Structured and compositional theories

The approach presented in Sections 4 and 5 is intentionally generic in its formal definitions. No assumption is made on the syntax or inner structure of theory atoms. An abstract theory is only required to specify when a set of theory atoms is satisfiable and provide a complement function for each theory atom. Definition6 is a bit more specific, and further assumes consistent and monotonic theories (with a closed set of external atoms), something we may expect in most cases while still being very generic. The advantage of this generality is that it allows us to accommodate external theories without requiring much knowledge about their behavior. However, this generality comes at a price: ignoring the structure of the external theory may prevent in depth formal elaborations, such as, for instance, the study of strong equivalence for logic programs with constraint atoms.

Also, in many practical applications of hybrid systems, we are interested in the assignment of values to variables rather than the theory atoms that are satisfied, something not reflected in Definitions5 or 6. This is what happens, for instance, in most hybrid extensions of answer set solvers including the hybrid versions of clingo, namely, clingcon, clingo[dl], and clingo[lp]. Since the presence of variables in theory atoms can be exploited to describe their semantics in more detail, we resort to refined types of theories that are enriched with a specific structure.

An assignment over $\mathcal{X},{\mathcal{D}}$ is a valuation over $\mathcal{X},{\mathcal{D}}$ such that $v(x)\in {\mathcal{D}}$ for all $x \in \mathcal{X}$ .

Whenever an abstract theory $\mathfrak{T}$ is associated with such a structure, we call it structured (rather than abstract). Observe that structured theories are based on the same concepts as $\textrm {HT}_{\!c}$ , viz. a domain, a set of variables, valuation functions, and denotations for atoms. In fact, we assume that the previously introduced definitions and notation for these concepts still apply here. This similarity is intended, since $\textrm {HT}_{\!c}$ was originally thought of as a generalization of logic programs with theory atoms. One important difference between an $\textrm {HT}_{\!c}$ valuation and a structured theory valuation $v \in \mathcal{V}_{\mathfrak{T}}$ is that the latter cannot leave a variable undefined in $\textrm {HT}_{\!c}$ terms, that is, for a structured theory, we assume that the “undefined” value is not included in the domain $\mathbf{u} \not \in {\mathcal{D}}_{\mathfrak{T}}$ , and so, $v(x) \neq \mathbf{u}$ for all $x \in \mathcal{X}_{\mathfrak{T}}$ and $v \in \mathcal{V}_{\mathfrak{T}}$ . We also emphasize the use of the theory name $\mathfrak{T}$ subindex in all the components of a structure because, as we see below, an $\textrm {HT}_{\!c}$ characterization allows us to accommodate multiple theories in the same formalization.

Given a set $S$ of theory atoms, we define its denotation as $[\![ \, S \, ]\!]_{\mathfrak{T}} \stackrel {\mathrm{def}}{=} \bigcap _{\mathtt {s}\in S}[\![ \, \mathtt {s} \, ]\!]_{\mathfrak{T}}$ . For any structured theory $\mathfrak{T} = \langle \mathcal{T}, \mathit{Sat}, \overparen{\cdot } \,\rangle$ with structure $(\mathcal{X}_{\mathfrak{T}}, {\mathcal{D}}_{\mathfrak{T}}, \mathit{vars}_{\mathfrak{T}}, [\![ \, \cdot \, ]\!]_{\mathfrak{T}})$ , if $X \subseteq \mathcal{A} \cup \mathcal{T}$ is a set of atoms, we define $\mathit{ans}(X)$ as the set

\begin{equation*} \{ (Y,{v|}_{\hspace {-1pt}\Sigma }) \mid v \in [\![ \, X \cap \mathcal{T} \, ]\!]_{\mathfrak{T}} \} \quad \text{ with } Y = X \cap \mathcal{A} \text{ and } \Sigma = \mathit{vars}_{\mathfrak{T}}(X\cap \mathcal{T}) . \end{equation*}

Intuitively, $\mathit{ans}(X)$ collects all pairs $(Y,v')$ where $Y$ is fixed to the propositional atoms in $X$ and $v'$ varies among all valuations in $[\![ \, X \cap \mathcal{T} \, ]\!]_{\mathfrak{T}}$ restricted to the variables occurring in the theory atoms in $X$ . Note that $v$ is only defined for a subset of variables, namely, $\Sigma =\mathit{vars}_{\mathfrak{T}}(X\cap \mathcal{T})\subseteq \mathcal{X}$ .

Let $\mathfrak{T} = \langle \mathcal{T}, \mathit{Sat}, \overparen{\cdot } \rangle$ be a theory with structure $(\mathcal{X}_{\mathfrak{T}}, {\mathcal{D}}_{\mathfrak{T}}, \mathit{vars}_{\mathfrak{T}}, [\![ \, \cdot \, ]\!]_{\mathfrak{T}})$ . We say that $\mathfrak{T}$ has an absolute complement whenever the denotation of $\overparen{ \mathtt{s}}$ is precisely the set complement of $[\![ \, \mathtt {s} \, ]\!]_{\mathfrak{T}}$ , that is, $[\![ \, \overparen{ \mathtt{s}} \, ]\!]_{\mathfrak{T}} = \mathcal{V}_{\mathfrak{T}} \setminus [\![ \, \mathtt {s} \, ]\!]_{\mathfrak{T}}$ . We also say that $\mathfrak{T} $ is compositional, if it satisfies $\mathit{Sat} = \{S \subseteq \mathcal{T} \mid [\![ \, S \, ]\!]_{\mathfrak{T}} \neq \emptyset \}$ , that is, a set $S$ is $\mathfrak{T}$ -satisfiable iff its denotation is not empty. This means that, for compositional theories, the set $\mathit{Sat}$ of $\mathfrak{T}$ -satisfiable sets does not need to be explicitly stated as it can be derived from the denotation. Hence, we can write $\mathfrak{T}=\langle \mathcal{T}, \overparen{\cdot } \,, \mathcal{X}_{\mathfrak{T}},{\mathcal{D}}_{\mathfrak{T}}, \mathit{vars}_{\mathfrak{T}}, [\![ \, \cdot \, ]\!]_{\mathfrak{T}}\rangle$ to denote a compositional theory.

As an example for a compositional, structured theory with an absolute complement, let us associate the theory of linear equations $\mathfrak{L}$ with the structure $(\mathcal{X}_{\mathfrak{L}},{\mathcal{D}}_{\mathfrak{L}},\mathit{vars}_{\mathfrak{L}},[\![ \, \cdot \, ]\!]_{\mathfrak{L}})$ , where

• • $\mathcal{X}_{\mathfrak{L}}$ is an infinite set of integer variables,

• • ${\mathcal{D}}_{\mathfrak{L}}=\mathbb{Z}$ ,

• • $\mathit{vars}_{\mathfrak{L}}(\mathtt {\&sum\{} k_1\!*\!x_1;\!{\small \dots }\!;k_n\!*\!x_n \mathtt {\}} \prec k_0)=\{x_1,\!{\small \dots }\!,x_n\}$ , and

• • $[\![ \, \mathtt {\&sum\{} k_1*x_1;\dots ;k_n*x_n \mathtt {\}} \prec k_0 \, ]\!]_{\mathfrak{L}} =$

  \begin{equation*}\big \{ v\in \mathcal{V}_{\mathfrak{L}} \mid \{k_1,v(x_1),\dots k_n,v(x_n)\}\subseteq \mathbb{Z} \text{ and } \sum\nolimits_{1\leq i \leq n}k_i*v(x_i)\prec k_0 \big \} \ . \end{equation*}

With $\mathfrak{L}$ , a set $S$ of theory atoms capturing linear equations is $\mathfrak{L}$ -satisfiable whenever $[\![ \, S \, ]\!]_{\mathfrak{L}}$ is non-empty.

In general, we have a one-to-many correspondence between $\langle \mathfrak{T},\mathcal{E}\rangle$ -stable models and their associated $\langle \mathfrak{T},\mathcal{E}\rangle$ -answer sets. That is, the cardinality of $\mathit{ans}(X)$ is generally greater than one for a single stable model $X$ . However, if we focus on consistent, compositional theories that have an absolute complement, then we can establish a one-to-one correspondence between any $\langle \mathfrak{T},\mathcal{E}\rangle$ -stable model of a program and a kind of equivalence classes among its associated $\langle \mathfrak{T},\mathcal{E}\rangle$ -answer sets. Each of the equivalence classes may contain many $\langle \mathfrak{T},\mathcal{E}\rangle$ -answer sets, but any of them has enough information to reconstruct the corresponding $\langle \mathfrak{T},\mathcal{E}\rangle$ -stable model. An answer set $(Y,v)$ satisfies an atom $b \in \mathcal{A} \cup \mathcal{T}$ , written $(Y,v) \models b$ , if

• • $b\in Y$ for propositional atoms $b \in \mathcal{A}$ and

• • $v\mathbin {\hat {\in }}[\![ \, b \, ]\!]_{\mathfrak{T}}$ for theory atoms $b\in \mathcal{T}$ ,

where $v \mathbin {\hat {\in }} [\![ \, b \, ]\!]_{\mathfrak{T}}$ holds iff there exists some valuation $w \in [\![ \, b \, ]\!]_{\mathfrak{T}}$ such that $v$ and $w$ agree on the variables in $\mathit{dom}(v)$ . Note that we use $v\mathbin {\hat {\in }} [\![ \, b \, ]\!]_{\mathfrak{T}}$ instead of $v \in [\![ \, b \, ]\!]_{\mathfrak{T}}$ because $v$ can be partial and, if so, we just require that there exists some complete valuation in $[\![ \, b \, ]\!]_{\mathfrak{T}}$ that agrees with the values assigned by $v$ to its defined variables $\mathit{dom}(v)$ . For any negative literal $\neg b$ , we say that $(Y,v) \models \neg b$ simply when $(Y,v) \not \models b$ . If $B$ is a rule body, we write $(Y,v) \models B$ to stand for $(Y,v) \models \ell$ for every literal $\ell$ in $B$ . For a program $P$ and an $\langle \mathfrak{T},\mathcal{E}\rangle$ -answer set $(Y,v)$ , we define

\begin{align*} \mathit{stb}_{P}(Y,v) & = Y \cup \big \{ \mathtt {s} \in \mathcal{E} \mid v \mathbin {\hat {\in }} [\![ \, \mathtt {s} \, ]\!]_{\mathfrak{T}} \big \} \cup \big \{ \mathtt {s} \in \mathcal{F} \mid (\mathtt {s} \leftarrow B) \in P, (Y,v) \models B \big \} \\[-15pt] \end{align*}

We also write $(Y_1,v_1) \sim (Y_2,v_2)$ if $\mathit{stb}_{P}(Y_1,v_1) = \mathit{stb}_{P}(Y_2,v_2)$ and say that $(Y_1,v_1)$ and $(Y_2,v_2)$ belong to the same equivalence class with respect to $P$ .

7 $\textrm {HT}_{\!c}$ -characterization of logic programs with structured and compositional theories

We now present a direct encoding of $\mathcal{T}$ -programs as $\textrm {HT}_{\!c}$ theories. This encoding is “direct” in the sense that it preserves the structure of each program rule by rule and atom by atom, only requiring the addition of a fixed set of axioms.

In the following, we restrict ourselves to consistent compositional theories with an absolute complement and refer to them just as theories.

We start embodying compositional theories in $\textrm {HT}_{\!c}$ by mapping their respective structures (domain, variables, valuations, and denotations), while having in mind that $\textrm {HT}_{\!c}$ may tolerate multiple abstract theories in the same formalization. For this reason, when we encode a theory $\mathfrak{T}= \langle \mathcal{T}_{\mathfrak{T}}, \overparen{\cdot }\, , \mathcal{X}_{\mathfrak{T}},{\mathcal{D}}_{\mathfrak{T}},\mathit{vars}_{\mathfrak{T}}, [\![ \, \cdot \, ]\!]_{\mathfrak{T}} \rangle$ into an $\textrm {HT}_{\!c}$ theory over a signature $\langle \mathcal{X},{\mathcal{D}},{\mathcal{C}} \rangle$ , we only require $\mathcal{X}_{\mathfrak{T}} \subseteq \mathcal{X}$ and ${\mathcal{D}}_{\mathfrak{T}} \subseteq {\mathcal{D}}$ , so that the $\textrm {HT}_{\!c}$ signature may also include variables and domain values other than the ones in $\mathfrak{T}$ . We then map each abstract theory atom $\mathtt {s} \in \mathcal{T}$ into a corresponding $\textrm {HT}_{\!c}$ -constraint atom $\tau(\mathtt {s}) \in {\mathcal{C}}$ with the same variables $\mathit{vars}(\tau(\mathtt {s})) = \mathit{vars}_{\mathfrak{T}}(\mathtt {s})$ . We also require the following relation between the theory denotation of a theory atom and the $\textrm {HT}_{\!c}$ denotation of its corresponding constraint atom:

(10) \begin{align} [\![ \, \tau(\mathtt {s}) \, ]\!] & \ \stackrel {\mathrm{def}}{=} \ \{v\in \mathcal{V}^{\mathcal{X},{\mathcal{D}}} \mid \exists w\in [\![ \, \mathtt {s} \, ]\!]_{\mathfrak{T}}, \ {v|}_{\hspace {-1pt}\mathcal{X}_{\mathfrak{T}}}=w \} \end{align}

Note that $\textrm {HT}_{\!c}$ valuations $v\in \mathcal{V}^{\mathcal{X},{\mathcal{D}}}$ apply to a (possibly) larger set of variables $\mathcal{X} \supseteq \mathcal{X}_{\mathfrak{T}}$ and on larger domains $\mathcal{D}_{\mathbf{u}} \supset {\mathcal{D}}_{\mathfrak{T}}$ , which include the element $\mathbf{u} \not \in {\mathcal{D}}_{\mathfrak{T}}$ to represent undefined variables in $\textrm {HT}_{\!c}$ . The denotation $[\![ \, \tau(\mathtt {s}) \, ]\!]$ collects all possible $\textrm {HT}_{\!c}$ -valuations that coincide with some $\mathfrak{T}$ -valuation $w \in [\![ \, \mathtt {s} \, ]\!]_{\mathfrak{T}}$ on the variables of theory $\mathfrak{T}$ , letting everything else vary vary freely.

We write $\tau(S)$ for $\{\tau(\mathtt {s}) \mid \mathtt {s} \in S \}$ and $S \subseteq \mathcal{T}$ . The following result shows that this mapping of denotations preserves $\mathfrak{T}$ -satisfiability:

Let us now consider $\mathcal{T}$ -programs $P$ over $\langle \mathcal{A},\mathcal{T}\rangle$ , that are associated with a structured theory $\mathfrak{T}= \langle \mathcal{T}_{\mathfrak{T}}, \overparen{\cdot }\, , \mathcal{X}_{\mathfrak{T}},{\mathcal{D}}_{\mathfrak{T}},\mathit{vars}_{\mathfrak{T}}, [\![ \, \cdot \, ]\!]_{\mathfrak{T}} \rangle$ . As explained in Section 2, we can capture regular programs by identifying propositional atoms $\mathcal{A}$ with both variables and constraint atoms and having the truth constant $\mathbf{t}$ in the domain. We can capture this by adding propositional atoms both as variables and as constraint atoms to the signature of the $\textrm {HT}_{\!c}$ theory, so we have

(11) \begin{align} \mathcal{A} \cup \mathcal{X}_{\mathfrak{T}} &\subseteq \mathcal{X}\\[-10pt] \nonumber \end{align}

(12) \begin{align} \mathcal{A} \cup \{ \tau (\mathtt {s}) \mid \mathtt {s} \in \mathcal{T}_{\mathfrak{T}} \} &\subseteq {\mathcal{C}} \\[10pt] \nonumber \end{align}

and the truth constant $\mathbf{t}$ in the domain, so we have ${\mathcal{D}}_{\mathfrak{T}} \cup \{ \mathbf{t} \} \subseteq {\mathcal{D}}$ . For each propositional atom $\mathtt {a} \in \mathcal{A}$ , its variables and denotation are defined as follows:

• • $\mathit{vars}(\mathtt {a})=\{a\}$ , and

• • $[\![ \, \mathtt {a} \, ]\!] = [\![ \, a:\{\mathbf{t}\} \, ]\!] = \{v \in \mathcal{V}^{\mathcal{X},{\mathcal{D}}} \mid v(a)=\mathbf{t}\}$ .

With this encoding, each propositional atom $\mathtt {a} \in \mathcal{A}$ is mapped into itself in the $\textrm {HT}_{\!c}$ -theory, viz. $\tau(\mathtt {a}) \stackrel {\mathrm{def}}{=} \mathtt {a}$ .

Since the resulting $\textrm {HT}_{\!c}$ -theory contains variables from the abstract theory and the propositional atoms, we are interested in forcing variables to range only over their corresponding subdomain, one from the external theory and one for Boolean values. This is achieved by including an axiom of form

(13) \begin{eqnarray} \mathit{def}(x) \to x:{\mathcal{D}}_{\mathfrak{T}} \end{eqnarray}

for each $x \in \mathcal{X}_{\mathfrak{T}}$ and each abstract theory $\mathfrak{T}$ encoded in our $\textrm {HT}_{\!c}$ formalization. The application of any valuation $v \in \mathcal{V}^{\mathcal{X}_{\mathfrak{T}},{\mathcal{D}}}$ satisfying (13) to a variable $x \in \mathcal{X}_{\mathfrak{T}}$ from the abstract theory returns some element from the theory domain $v(x) \in {\mathcal{D}}_{\mathfrak{T}}$ or is undefined, viz. $v(x) = \mathbf{u}$ . Let us denote by $\delta (\mathfrak{T},\mathcal{A})$ the set of all the axioms of the form of (13) for each $x \in \mathcal{X}_{\mathfrak{T}}$ plus all the axioms of the form (1) for every $\mathtt {a} \in \mathcal{A}$ .

We are now ready to introduce the direct translation of a $\mathcal{T}$ -program $P$ over $\langle \mathcal{A},\mathcal{T} \rangle$ with external theory $\mathfrak{T}$ into an $\textrm {HT}_{\!c}$ -theory $\tau(P,\mathfrak{T},\mathcal{E})$ where $\mathcal{E}$ is a set of theory atoms considered external in $P$ . Given any rule $r$ of the form of (5), we let $\tau^B(r)$ stand for the formula

(14) \begin{align} \tau(b_1)\land \dots \land \tau(b_n)\land \neg \tau(b_{n+1})\land \dots \land \neg \tau(b_{m}) \end{align}

representing the body of $r$ , where each occurrence of an atom $b_i$ in the program is replaced by $\tau(b_i)$ . We extend the application of $\tau$ to the whole rule $r$ and let $\tau(r)$ stand for

(15) \begin{align} \tau^B(r) \rightarrow \tau(b_0) \end{align}

assuming that, when the head is $b_0=\bot$ , its translation is simply $\tau(\bot ) \stackrel {\mathrm{def}}{=} \bot$ . We further write $\tau(P) \stackrel {\mathrm{def}}{=} \{\tau(r) \mid r \in P \}$ , applying our transformation to all rules in program $P$ . The complete translation of $\mathcal{T}$ -program $P$ is defined as

(16) \begin{align} \tau(P,\mathfrak{T},\mathcal{E}) \stackrel {\mathrm{def}}{=} \tau(P) \cup \sigma (\mathfrak{T},\mathcal{E}) \cup \delta (\mathfrak{T},\mathcal{A}) \end{align}

where $\sigma (\mathfrak{T},\mathcal{E})$ consists of

(17) \begin{align} \mathit{def}(x) \ \text{ for every } x \in \mathit{vars}(\mathtt {s}) \text{ and }\mathtt {s} \in \mathcal{E}. \end{align}

Formula (17) asserts that each variable in any external atom can be arbitrarily assigned some (defined) value.

Proposition 6. Let $P$ be a $\mathcal{T}$ -program over $\langle \mathcal{A},\mathcal{T} \rangle$ with external theory $\mathfrak{T}$ and external atoms $\mathcal{E}$ . For every $\mathtt {s} \in \mathcal{E}$ ,

\begin{gather*} \tau(P,\mathfrak{T},\mathcal{E}) \models \tau(\mathtt {s}) \vee \tau(\overparen{ \mathtt{s}}). \end{gather*}

This proposition means that either $\mathtt {s}$ or its complement $\overparen{ \mathtt{s}}$ must hold. In other words, axioms of form (17) entail a kind of strong excluded middle for external atoms similarly to the simplified definition of a $\langle \mathfrak{T},\mathcal{E}\rangle$ -stable model (in Definition6). This is a stronger version of the usual choice construct $\tau(\mathtt {s})\vee \neg \tau(\mathtt {s})$ in $\textrm {HT}$ : we can freely add $\tau(\mathtt {s})$ or not but, when the latter happens, we further provide evidence for the complement $\tau(\overparen{ \mathtt{s}})$ .

Following our running example, consider program $P_0$ consisting of rules (7) and (8), plus the fact

(18) \begin{align} \mathtt {\&sum\{} s \mathtt {\}} = 130. \end{align}

This program has a unique answer set where $\mathtt {a}$ is true and $s$ is assigned the value $130$ .

We assume the following translation of constraint atoms:

\begin{align*} \tau(\mathtt {\&sum\{} s \mathtt {\}} \gt =120) \ &\stackrel {\mathrm{def}}{=} \ s\geq 120 \\ \tau(\mathtt {\&sum\{} s \mathtt {\}} \gt 100) \ &\stackrel {\mathrm{def}}{=} \ s\gt 100 \\ \tau(\mathtt {\&sum\{} s \mathtt {\}} = 130) \ &\stackrel {\mathrm{def}}{=} \ s=130 \end{align*}

Hence, $\tau(P_0)$ is the $\textrm {HT}_{\!c}$ theory $\{(2),\,(3),\, {s=130}\}$ . Given that constraint atoms $\mathtt {\&sum\{} s \mathtt {\}}\geq 120$ and $\mathtt {\&sum\{} s \mathtt {\}}\gt 100$ occur in the body, they must be external. We assume that constraint atom (18) is not external (see discussion below). Thus, $\tau(P_0,\mathfrak{L},\mathcal{E})$ is the result of adding to $\tau(P_0)$ the following axioms:

\begin{align*} \qquad \qquad \qquad \qquad\qquad \qquad \qquad \qquad &\mathit{def}(a) \rightarrow \mathtt {a}\qquad \qquad \qquad \qquad\qquad \qquad \qquad \qquad \text{(1)} \\[-30pt] \nonumber \end{align*}

(19) \begin{align} &\mathit{def}(s) \rightarrow s:{\mathcal{D}}_{\mathfrak{L}} \\[-10pt] \nonumber \end{align}

(20) \begin{align} &\mathit{def}(s)\\[8pt] \nonumber \end{align}

We can replace the last two axioms simply by $s:{\mathcal{D}}_{\mathfrak{L}}$ , which ensures that the variable $s$ is assigned a value from the domain ${\mathcal{D}}_{\mathfrak{L}}$ of linear constraints, namely, an integer. Recall that (1) ensures that $a$ is assigned a Boolean value, either $\mathtt{t}$ or $\mathbf{u}$ . This $\textrm {HT}_{\!c}$ -theory has a unique stable model

\begin{equation*}\{ a \mapsto \mathbf{t}, \, s \mapsto 130 \}.\end{equation*}

This stable model corresponds to the unique answer set of program $P_0$ .

The following theorem states the relation between the answer sets of a $\mathcal{T}$ -program $P$ and the equilibrium models of its translation $\tau(P,\mathfrak{T},\mathcal{E})$ , generalizing the correspondence illustrated in the previous example.

This result shows that the semantics of a $\mathcal{T}$ -program $P$ can alternatively be described as the equilibrium models of the $\textrm {HT}_{\!c}$ -theory $\tau(P,\mathfrak{T},\mathcal{E})$ . Intuitively, formulas of form (15) capture the rules in the $\mathcal{T}$ -program and are used for the same purpose, that is, to decide which founded atoms from $\mathcal{T} \setminus \mathcal{E}$ can be eventually derived. Furthermore, due to the minimization imposed to obtain an equilibrium model $\langle t,t \rangle$ , if a founded atom $\mathtt {s}$ is not derived (i.e., $\langle t,t \rangle \not \models \tau(\mathtt {s})$ ), then all its variables $x \in \mathit{vars}_{\mathfrak{T}}(\mathtt {s})$ not occurring in external atoms or other derived atoms are left undefined, viz. $t(x)=\mathbf{u}$ .

An interesting consequence of the characterization provided by Theorem4 is that all atoms whose variables occur in external atoms are implicitly external, even if we do not declare them as that. This result is trivial when looked at the definition of $\tau(P,\mathfrak{T},\mathcal{E})$ , but it is far from obvious when considering the definition of $\langle \mathfrak{T},\mathcal{E}\rangle$ -answer sets based on Definitions5 or 6.

In other words, extending the set of external atoms by adding new ones whose variables are already occurring in other external atoms does not change the answer sets of the program. For instance, continuing with our running example, we have assumed that constraint atom (18) is not external, but because its only variable $s$ occurs in external atoms, it is implicitly external. As stated above, assuming that it is external instead, does not change the answer sets of the program.

8 Strong equivalence of $\mathcal{T}$ -programs

So far, the notion of strong equivalence applies to $\textrm {HT}_{\!c}$ -theories, but not to $\mathcal{T}$ -programs. We now leverage the translation introduced in Section 7 to lift strong equivalence to $\mathcal{T}$ -programs.

We can easily observe that, if $\mathcal{T}$ -programs $P$ and $Q$ are strongly equivalent, then $P \cup R$ and $Q \cup R$ have the same answer sets for any program $R$ . Our main result is an immediate consequence of Theorems1 and 2 and lifts the characterization of strong equivalence from $\textrm {HT}_{\!c}$ to $\mathcal{T}$ -programs.

We now revisit the question posed in the introduction: can we remove rule (7) from any program containing rules (7) and (8) without changing its semantics? Continuing with our running example, recall that

(21) \begin{align} \tau(P_1,\mathfrak{L},\mathcal{E}) \ = \ \{{(1), (2), (3), (19), (20)}\} \end{align}

where $P_1 = \{(7),(8)\}$ . We saw in Section 3 that $\{(2),(3)\}$ and (3) are equivalent in $\textrm {HT}_{\!c}$ . Therefore, (21) is equivalent in $\textrm {HT}_{\!c}$ to

\begin{equation*} \tau(P_2,\mathfrak{L},\mathcal{E}) = \{{(1), (3), (19), (20)}\} \end{equation*}

where $P_2 = \{ (8) \}$ . By Theorem5, this means that programs $P_1$ and $P_2$ are strongly equivalent, and, consequently, rule (7) can be safely removed from any program containing both rules without changing its semantics.

Table 1.

Summary of complexity results

Table 1. Long description

The table has three columns and two rows. The columns are labeled Unrestricted, Oracle, and Polynomial. The rows are labeled Satisfiability and Strong equivalence. Row 1: Unrestricted, Undecidable; Oracle, N P superscript 0; Polynomial, N P. Row 2: Unrestricted, Undecidable; Oracle, co N P superscript 0; Polynomial, co N P.

This table summarizes the complexity results for $\mathcal{T}$ -programs. The first column considers unrestricted $\mathcal{T}$ -programs, the second column considers $\mathcal{T}$ -programs where the complexity of the satisfiability problem of the external theory is in some complexity class $\textsf{O}$ , and the third column considers $\mathcal{T}$ -programs where the satisfiability problem of the external theory is decidable in polynomial time.

9 Complexity

In this section, we address the computational complexity of the satisfiability and strong equivalence problems for $\mathcal{T}$ -programs. Table 1 summarizes our results, which we discuss in detail below. The complexity of $\mathcal{T}$ -programs highly depends on the associated external theory $\mathfrak{T}$ , and is in general undecidable as illustrated by the following example.

Example 1. Let $\mathfrak{DE}= \langle \mathcal{T}, \overparen{\cdot }, \mathcal{X}_{\mathfrak{T}},{\mathcal{D}}_{\mathfrak{T}}, \mathit{vars}_{\mathfrak{T}}, [\![ \, \cdot \, ]\!]_{\mathfrak{T}} \rangle$ be a structured theory with variables $\mathcal{X}_{\mathfrak{T}} = \{ x_1, x_2, \dotsc \}$ , whose domain is the integers and whose atoms are of the form

\begin{align*} \mathtt {\&sum\{} c_1 \cdot x_1^{e_1},\dots ,c_n \cdot x_n^{e_n} \mathtt {\}} = 0 \end{align*}

where each $c_i\in \mathbb{Z}$ , $e_i\in \mathbb{N}$ and $x_i$ is a variable. The denotation $[\![ \, \cdot \, ]\!]_{\mathfrak{T}}$ of these atoms is the set of all assignments of integer values to the variables that satisfy the corresponding Diophantine equation.

Since the satisfiability problem for Diophantine equations is undecidable (Matiyasevich Reference Matiyasevich1972), the satisfiability problem for $\mathcal{T}$ -programs with theory $\mathfrak{DE}$ is also undecidable. The satisfiability problem for $\mathcal{T}$ -program s consists of deciding whether there is a set of atoms that is a $\langle \mathfrak{T},\mathcal{E} \rangle$ -stable model of some program.

We can provide a more detailed analysis of the complexity in terms of oracles that can solve the satisfiability problem for the external theory $\mathfrak{T}$ . In what follows, we assume that the reader is familiar with the basic concepts of complexity theory.Footnote ⁵ For convenience, we briefly recapitulate the definitions and some elementary properties of the complexity classes considered in our analysis. The class $\textsf{NP}$ consists of all decision problems that can be solved by a non-deterministic Turing machine in polynomial time. As usual, for any complexity class $\textsf{C}$ , by $\textsf{co-C}$ we understand the class of all problems which are complementary to the problems in $\textsf{C}$ . Thus, $\textsf{co-NP}$ is the class of all problems whose complements are in $\textsf{NP}$ . A non-deterministic Turing machine with an oracle is a non-deterministic Turing machine that has a special query tape and three special states: query, yes, and no. The computations of a Turing machine with oracle $\textsf{O}$ proceed as usual, except when the machine enters the query state. In such case, the next state is either the yes state or the no state, depending on whether the string written on the query tape is in the language decided by the oracle $\textsf{O}$ . For complexity classes $\textsf{C}$ and $\textsf{O}$ , we denote by $\textsf{C}^{\textsf{O}}$ the class of languages which can be decided by a Turing machine with oracle $\textsf{O}$ of the same sort and time bound as $\textsf{C}$ (Papadimitriou Reference Papadimitriou1994, pp. 339–340). For example, $\textsf{NP}^{\textsf{O}}$ is the class of all decision problems that can be solved by a non-deterministic Turing machine with oracle $\textsf{O}$ in polynomial time.

Since regular ASP programs are a particular case of $\mathcal{T}$ -programs, the following result immediately follows from the $\textsf{NP}$ -completeness of the satisfiability problem for programs without disjunctions in ASP (Dantsin et al. Reference Dantsin, Eiter, Gottlob and Voronkov2001). All completeness results in this section hold under logarithmic space reductions. Recall that a logarithmic space reduction from a language $A$ to a language $B$ is a function $f$ that can be computed by a Turing machine using logarithmic space such that every string $x$ satisfies $x\in A$ if and only if $f(x)\in B$ . A problem $A$ is hard for a complexity class $\textsf{C}$ if every problem in $\textsf{C}$ can be reduced to $A$ by a logarithmic space reduction. A problem $A$ is complete for a complexity class $\textsf{C}$ if it is hard for $\textsf{C}$ and it is in $\textsf{C}$ .

Finally, we obtain the following results for the complexity of checking whether two $\mathcal{T}$ -program s are strongly equivalent.

Since regular ASP programs are a particular case of $\mathcal{T}$ -programs, the following result immediately follows from the $\textsf{co-NP}$ -completeness of the strongly equivalence problem for ASP problems (Lin Reference Lin2002, Corollary 2.1).Footnote ⁶