2 Background

In this section we present the necessary background that will be used throughout the paper. We start by defining the syntax and the semantics of the four-valued logic introduced by Charalambidis et al. (Reference Charalambidis, Rondogiannis and Troumpoukis2021) and discuss how this logic can be used to redefine the semantics of LPODs.

Similarly to the paper by Faber et al. (Reference Faber, Tompits and Woltran2008), we do not consider strong negation, for reasons of simplicity.

• Every element of $\Sigma$ is a well-formed formula,

• If $\phi_1$ and $\phi_2$ are well-formed formulas, then $(\phi_1 \wedge \phi_2)$ , $(\phi_1 \vee \phi_2)$ , $(\mathit{not}{\,} \phi_1)$ , $(\phi_1 \leftarrow \phi_2)$ , and $(\phi_1 \times \phi_2)$ are well-formed formulas.

We will use capital variables, like A, B, C, D, and their subscripted versions, to denote atoms; we will use L, and its subscripted versions, to denote literals (namely, atoms or negated atoms).

In order to define the semantics of well-formed formulas, we use the set $V = \{F, F^*, T^*, T\}$ of truth values, which are ordered as follows:

\[ F < F^* < T^* < T.\]

Definition 2 An interpretation I is a function from $\Sigma$ to V. We can extend I to apply to formulas, as follows:

\[\begin{array}{lll} I(\mathit{not}{\,}\phi) & = & \begin{cases} T & \mbox{if $I(\phi)\leq F^*$}\\ F & \mbox{otherwise} \end{cases} \\ I(\phi \leftarrow \psi) & = & \begin{cases} T & \mbox{if $I(\phi) \geq I(\psi)$}\\ F & \mbox{otherwise} \end{cases} \\I(\phi_1 \wedge \phi_2) & = & \min\{I(\phi_1),I(\phi_2)\}\\I(\phi_1 \vee \phi_2) & = & \max\{I(\phi_1),I(\phi_2)\}\\I(\phi_1 \times \phi_2) & = & \begin{cases} I(\phi_2) & \mbox{if $I(\phi_1) = F^*$}\\ I(\phi_1) & \mbox{otherwise} \end{cases}. \\\end{array}\]

It is straightforward to see that the meanings of “ $\vee$ ”, “ $\wedge$ ”, and “ $\times$ ” are associative, and therefore we can write $I(\phi_1 \vee \cdots\vee \phi_n)$ , $I(\phi_1 \wedge \cdots \wedge \phi_n)$ , and $I(\phi_1 \times\cdots \times \phi_n)$ unambiguously (without the need of extra parentheses). Moreover, given literals $L_1,\ldots,L_n$ , we will often write $L_1,\ldots,L_n$ instead of $L_1\wedge \cdots \wedge L_n$ .

LPODs are sets of formulas of a special kind, specified by the following definition.

Definition 3 An LPOD is a finite set of rules of the form:

\[ C_1 \times \cdots \times C_n \leftarrow A_1,\ldots,A_m,{\mathit{not}{\,} B_1},\ldots,{\mathit{not}{\,} B_k}.\]

where $n\geq 1$ , $m,k\geq 0$ , and the $C_i, A_j$ , and $B_l$ are atoms.

We will use capital letters like P, Q, and their subscripted versions, to denote LPODs.

Charalambidis et al. (Reference Charalambidis, Rondogiannis and Troumpoukis2021) defined the semantics of LPODs, namely the precise characterization of their most-preferred answer sets, based on the above four-valued logic. More specifically, the most-preferred answer sets of an LPOD are generated using a two-step procedure. In the first step, a subset of the models of the program is selected using a minimization procedure according to an ordering relation $\preceq$ defined below. These models are called answer sets of the given LPOD, because they can also be produced using a reduct-based approach similar to the one defined in the paper by Brewka et al. (Reference Brewka, Niemelä and Syrjänen2004b). In the second step, a subset of the answer sets is selected using a minimization procedure that examines the set of atoms that have the value $F^*$ in each answer set. These two steps are formally defined below.

It is easy to verify that $\preceq$ is a partial order.

Example 1 (taken from the paper by Reference Charalambidis, Rondogiannis and Troumpoukis Charalambidis et al. (2021) ) Consider the following program whose declarative reading is I prefer to buy a Mercedes than a BMW. In case a Mercedes is available, I prefer a gas model to a diesel one. A gas model of Mercedes is not available.

$$\begin{array}{l} {\tt mercedes \times bmw \leftarrow}\\ {\tt gas\_mercedes \times diesel\_mercedes \leftarrow mercedes}\\ {\tt false \leftarrow gas\_mercedes, not \ false}\end{array}$$

The last clause is a standard technique in ASP in order to state that an atom (gas_mercedes in our case) is not true. The above program has two answer sets, namely:

\[\begin{array}{l}\{({\tt mercedes},T),({\tt bmw},F),({\tt gas\_mercedes},F^*), ({\tt diesel\_mercedes},T),({\tt false},F^*)\}\\\{({\tt mercedes},F^*),({\tt bmw},T),({\tt gas\_mercedes},F^*), ({\tt diesel\_mercedes},F^*),({\tt false},F^*)\}\end{array}\]

According to the $\sqsubset$ ordering, the most-preferred answer set is the first one because it minimizes the $F^*$ values. It is worth noting that under the original semantics of LPODs (Brewka Reference Brewka2002; Brewka et al. Reference Brewka, Niemelä and Syrjänen2004b) two answer sets are produced that are incomparable (and therefore they are both considered as “most-preferred”).

3 A logical characterization of strong equivalence for LPODs

In this section we establish a new, purely logical characterization of strong equivalence for LPODs. Our investigation has as a starting point the work of Faber et al. (Reference Faber, Tompits and Woltran2008), in which an exhaustive study of different forms of strong equivalence for LPODs was performed. Not all forms of strong equivalence studied by Faber et al. (Reference Faber, Tompits and Woltran2008) are applicable in our case. An explanation of this state of affairs and a detailed comparison of our technique with that of Faber et al. (Reference Faber, Tompits and Woltran2008) is given in Section 5. In our work, we examine two notions of strong equivalence, namely strong equivalence under the most-preferred answer sets, and strong equivalence under all the answer sets ^{Footnote 1} . We demonstrate that these notions can be captured by establishing logical equivalence in the four-valued logic of Section 2 of the programs involved.

( $\Rightarrow$ ) Assume that $P_1$ and $P_2$ are strongly equivalent. Suppose that $P_1$ has a model M which is not a model of $P_2$ . Without loss of generality, we may assume that $M(A)=F$ , for every atom A in $\Sigma$ that does not occur in $P_1 \cup P_2$ .

We will show that we can construct an interpretation M’, and a program P such that M’ is a most-preferred answer set of one of $P_1 \cup P$ and $P_2 \cup P$ but not of the other, contradicting our assumption of strong equivalence.

First, we construct two sets of atoms that will help us define P. In particular, we construct two sets of atoms $\mathcal{T}$ and $\mathcal{F}$ each one containing a new atom for every A in P such that $M(A) = F^*$ . More formally, let $\mathcal{T} = \{ t_A \mid M(A) = F^* \}$ and $\mathcal{F} = \{ f_A \mid M(A) = F^* \}$ , where all $t_A$ and $f_A$ do not appear in $P_1$ and $P_2$ . We define M’ as:

\[ M'(A) = \begin{cases} T & M(A) = T^* \\ T & A \in \mathcal{T} \\ F^* & A \in \mathcal{F} \\ M(A) & \text{otherwise} \end{cases}.\]

We claim that M’ is a model of $P_1$ . To verify this, take any rule in $P_1$ of the form

\[ C_1 \times \cdots \times C_n \leftarrow A_1,\ldots,A_m,{\mathit{not}{\,} B_1},\ldots,{\mathit{not}{\,} B_k}.\]

If $M(A_1, \ldots, A_m, \mathit{not}{\,} B_1, \ldots, \mathit{not}{\,} B_k) \geq T^*$ , then it is also $M(C_1 \times \cdots \times C_n) \geq T^*$ , since M is a model of $P_1$ . Then, there exists $j \leq n$ such that $M(C_i) = F^*$ for all $i<j$ , and $M(C_j) \geq T^*$ . It follows that $M'(C_i) = F^*$ for all $i<j$ , and $M'(C_j) = T$ , which implies $M'(C_1 \times \cdots \times C_n) = T$ . Therefore, M’ satisfies the rule in this case.

If $M(A_1, \ldots, A_m, \mathit{not}{\,} B_1, \ldots, \mathit{not}{\,} B_k) = F^*$ , then there exists $A_i$ such that $M(A_i) = F^*$ . By the definition of M’, $M'(A_i) = F^*$ , and thus $M'(A_1, \ldots, A_m, \mathit{not}{\,} B_1, \ldots, \mathit{not}{\,} B_k) \leq F^*$ . Since M is a model of $P_1$ it satisfies the given rule and thus $M(C_1 \times \cdots \times C_n) \geq F^*$ . If $M(C_1 \times \cdots \times C_n) = F^*$ , then for all $C_i$ , $M(C_i)=F^*$ , which implies that $M'(C_i)=F^*$ and therefore $M'(C_1 \times \cdots \times C_n)=F^*$ . If $M(C_1 \times \cdots \times C_n) > F^*$ , then there exists $j\leq n$ such that for all $i<j$ , $M(C_i)=F^*$ and $M(C_j)>F^*$ , which implies that for all $i<j$ , $M'(C_i)=F^*$ and $M'(C_j)>F^*$ , and therefore $M'(C_1 \times \cdots \times C_n)>F^*$ . In both cases, M’ satisfies the given rule.

If $M(A_1, \ldots, A_m, \mathit{not}{\,} B_1, \ldots, \mathit{not}{\,} B_k) = F$ , then either there exists $A_i$ such that $M(A_i) = F$ or there exists $B_j$ such that $M(B_j) \geq T^*$ . It follows, by the definition of M’, that $M'(A_i) = F$ or $M'(B_j) = T$ and as a result $M'(A_1, \ldots, A_m, \mathit{not}{\,} B_1, \ldots, \mathit{not}{\,} B_k) = F$ . Therefore M’ satisfies the rule in this case. Thus, M’ is a model of $P_1$ .

We proceed by distinguishing two cases that depend on whether M’ is a model of $P_2$ or not.

Case 1: M’ is not a model of $P_2$ . We take:

\[P = \{ A\leftarrow \mid M'(A) = T \} \cup \{ A \times t_A\leftarrow \mid t_A \in {\cal T} \} \cup \{ f_A \leftarrow \mathit{not}{\,} f_A, A \mid f_A \in {\cal F} \} \]

We claim that every model N of $\{ A \times t_A \leftarrow \mid t_A \in {\cal T} \} \cup \{ f_A \leftarrow \mathit{not}{\,} f_A, A \mid f_A \in {\cal F} \} $ has the following property:

(1) $$\begin{equation}\text{for every atom } A, \text{ if } M'(A) = F^*, \text{ then } N(A) \neq F.\end{equation}$$

In order to prove our claim we distinguish two cases for atoms such that $M'(A) = F^*$ : the atoms where $M(A) = F^*$ and the atoms in $\mathcal{F}$ . For the first case, assume that for some A it is $M(A) = F^*$ and $N(A) = F$ . But then there exists a rule $A \times t_A\leftarrow$ in P which is not satisfied by N, which is a contradiction. So, for all such atoms A it should be $N(A) \geq F^*$ . For the second case, assume that for some $f_A$ , it is $N(f_A) = F$ . Then, the rule $f_A \leftarrow \mathit{not}{\,} f_A, A$ is not satisfied by N (since $N(A) \geq F^*$ ), which is also a contradiction. Therefore, our claim holds.

Now, it is easy to see that M’ is a model of P and therefore a model of $P_1 \cup P$ . Moreover, it is a most-preferred answer set of $P_1 \cup P$ . Indeed, let N be a model of $P_1 \cup P$ and $N \prec M'$ . Since M’ does not assign any $T^*$ , there exists A such that either $M'(A) = T$ and $N(A) \prec T$ or $M'(A) = F^*$ and $N(A) = F$ . In the first case, P contains a fact $A \leftarrow$ , which is not satisfied by N. In the second case, N does not satisfy property P1. In both cases, N is not a model of P, which is a contradiction. It follows that M’ is $\preceq$ -minimal model of $P_1 \cup P$ .

Assume now that there exists some N which is a most-preferred answer set of $P_1 \cup P$ and $N \sqsubset M'$ . There must exist some atom A such that $M'(A) = F^*$ and $N(A) \neq F^*$ . We will show in the following that it should be $N(A) = T$ for those atoms. First, notice that $N(A) \neq T^*$ because since N is most-preferred it is also solid. Also, it must be $N(A) \neq F$ , since N is a model of P and thus it satisfies property P1. Therefore, $N(A) = T$ for the atoms such that $M'(A) = F^*$ and $N(A) \neq F^*$ .

However, we now claim that N is not $\preceq$ -minimal. Indeed, we can construct N’ from N that is also a model of $P_1 \cup P$ and $N' \prec N$ . Define N’ as:

\[ N'(A) = \begin{cases} T^* & \text{if $A \in {\cal F}$ and $N(A) = T$} \\ N(A) & \text{otherwise} \end{cases}.\]

First, we need to establish that $N' \prec N$ , that is, there exists atom A such that $N'(A) \prec N(A)$ . By the assumption $N \sqsubset M'$ we know that there exists atom A such that $N(A) \neq M'(A)$ and we have established that $M'(A) = F^*$ and $N(A) = T$ . The first case is for the atom A to be $A=f_B \in {\cal F}$ for some B, and it is straightforward that $N(f_B) = T$ and $N'(f_B) = T^*$ . The second case is for A to be an atom such that $M(A) = F^*$ ; then there exists a rule $f_A \leftarrow \mathit{not}{\,} f_A, A$ in P which must be satisfied by N. But since $N(A) = T$ , and N is solid, the only way to satisfy this rule is when $N(f_A) = T$ . By definition of N’, $N'(f_A) = T^*$ . Therefore, $N' \prec N$ . It is also easy to see that N’ is a model of $P_1 \cup P$ because it satisfies all rules $f_A \leftarrow \mathit{not}{\,} f_A, A$ and $f_A$ does not occur in any other rule of $P_1 \cup P$ .

Therefore, M’ is a most-preferred answer set of $P_1 \cup P$ . This contradicts the assumption of strong equivalence because M’ is not even a model for $P_2 \cup P$ .

Case 2: M’ is a model of $P_2$ . Let D be an atom in $\Sigma - ({\cal T} \cup {\cal F})$ that does not occur in $P_1 \cup P_2$ . Such an atom always exists, since $\Sigma$ is a countably infinite set and ${\cal T}$ , ${\cal F}$ , $P_1$ , and $P_2$ are finite; moreover, $M(D) = F$ , by our assumption about M. We take

\[ \begin{split} P = & \{ A\leftarrow\ \mid M(A) = T \} \, \cup\\ & \{ A \times t_A\leftarrow\ \mid t_A \in {\cal T} \} \cup \{ f_A \leftarrow \mathit{not}{\,} f_A, A \mid f_A \in {\cal F} \}\, \cup \\ & \{ B \leftarrow A \mid A \neq B \text{ and } M(A) = M(B) = T^* \}\, \cup \\ & \{ D \leftarrow \mathit{not}{\,} A \mid M(A) = T^* \} \end{split}.\]

It is easy to see that M’ satisfies every formula in P, and therefore it is a model of both $P_1 \cup P$ and $P_2 \cup P$ . We will show that M’ is a most-preferred answer set of $P_2 \cup P$ but not a most-preferred answer set of $P_1 \cup P$ .

We proceed by showing that M’ is a $\preceq$ -minimal model of $P_2 \cup P$ . Assume there exists a model N of $P_2 \cup P$ such that $N \prec M'$ .

We first show that there exists an atom A such that $M(A) = T^*$ and $N(A) = T$ . Consider an arbitrary atom C. If $M(C)=T$ , then it is also $N(C)=T$ , because P contains $C \leftarrow$ and N is a model of P. If $M(C)=F^*$ , then by the construction of M’ it is $M'(C)=F^*$ . Since N is a model of P, by property P1 we obtain $N(C) \neq F$ . This implies $N(C) = F^*$ , because $N \prec M'$ . If $M(C) = F$ , then by the construction of M’ it is $M'(C) = F$ , and since $N \prec M'$ we get $N(C) = F$ . Therefore, if $M(C) \neq T^*$ , then $M(C)=N(C)$ . There should be, however, an atom A that occurs in $P_2$ such that $N(A) \neq M(A)$ because N is a model of $P_2$ and M is not. Obviously, for that atom it must be $M(A) = T^*$ and $N(A) \neq T^*$ . Now, notice that there exists a rule $D \leftarrow \mathit{not}{\,} A$ in P where $M(D) = F$ and must be satisfied by N since it is also a model of P. Since $M(D) = F$ implies $N(D) = F$ , the only remaining possibility is $N(A) = T$ .

We next show that there exists an atom B such that $M(B) = N(B) = T^*$ . Since $N \prec M'$ , there exists B such that $N(B) \prec M'(B)$ . The last relation immediately implies $M'(B) \neq F$ . Notice also that, by the construction of M’, it is $M'(B)\neq T^*$ . Moreover, it cannot be $M'(B) = F^*$ , since in that case from $N(B) \prec M'(B)$ we would obtain $N(B) = F$ , which contradicts property P1. Therefore, the only remaining value is $M'(B) = T$ . For that atom, it cannot be $M(B) = T$ because then it would also be $N(B) = T$ (since $B \leftarrow$ is a rule in P and N is a model of P), which would contradict $N(B) \prec M'(B)$ . It follows by the construction of M’ that $M(B) = T^*$ . We claim that $N(B) = T^*$ , that is, it cannot be $N(B) = F$ . Since $M(B) = T^*$ there exists a rule $D \leftarrow \mathit{not}{\,} B$ where $M(D) = F$ . Since, $M(D) = F$ , we get $N(D) = F$ . If we assume that also $N(B) = F$ then N does not satisfy this rule which is a contradiction. Therefore, $N(B) = T^*$ .

Since $M(A) = M(B) = T^*$ there exists rule $B \leftarrow A$ in P that is not satisfied by N because we have showed that $N(B) = T^*$ and $N(A) = T$ . Therefore, N is not a model of $P_2 \cup P$ , which is a contradiction.

We conclude that M’ is $\preceq$ -minimal model of $P_2 \cup P$ . Following an identical reasoning as in the final paragraph of the proof of Case 1, we can show that M’ is a most-preferred answer set of $P_2 \cup P$ . In order to conclude the proof, it suffices to show that M’ is not a most-preferred answer set of $P_1 \cup P$ . We define M” as:

\[ M''(A) = \begin{cases} T & A \in \mathcal{T} \\ F^* & A \in \mathcal{F} \\ M(A) & \text{otherwise} \end{cases}.\]

M” is not a model of $P_2$ because M is not a model of $P_2$ . By definition, $M'' \preceq M'$ . But $M'' \neq M'$ because M’ is a model of $P_2$ and M” is not. Therefore, $M'' \prec M'$ . Observe that M” agrees with M for all A that appear in $P_1$ and since M is a model of $P_1$ , M” is also a model of $P_1$ . M” also satisfies the rules of P, and therefore it is a model of $P_1 \cup P$ . Therefore, M’ is not a most-preferred answer set of $P_1 \cup P$ .

We now consider the second notion of strong equivalence that is applicable in our setting.

The proof of the above theorem follows the same steps as that of the proof of Theorem 1, omitting the parts of the proof related to $\sqsubset$ -minimization.

We feel that the above corollary highlights an interesting fact: it states that assessing the observable behaviour of two programs with respect to the most-preferred answer sets, suffices to determine strong equivalence of the programs.

Due to the above corollary, in the following we will often talk about “strong equivalence of LPODs” without specifying the exact type of equivalence (since they coincide).

Example 2 One can easily verify (using a four-valued truth table or a case analysis) that the programs:

$$\begin{array}{l} {\tt a \times b \leftarrow}\\ {\tt a \leftarrow} \end{array},$$

and the program that consists of just the following fact:

$$\begin{array}{l} {\tt a \leftarrow},\end{array}$$

are strongly equivalent. Similarly, one can verify that the programs given in Example 3 of the paper by Faber et al. (Reference Faber, Tompits and Woltran2008), namely:

$$\begin{array}{l} {\tt c \times a \times b \leftarrow}\\ {\tt a \leftarrow c}\\ {\tt b \leftarrow c}\\ {\tt c \leftarrow a,b}\end{array},$$

and:

$$\begin{array}{l} {\tt c \times a \times b \leftarrow}\\ {\tt c \times c \times b \times a \leftarrow}\\ {\tt a \leftarrow c}\\ {\tt b \leftarrow c}\\ {\tt c \leftarrow a,b}\end{array},$$

are also strongly equivalent. Notice that the above two programs are also strongly equivalent under the relations $\equiv^{i}_{s,\times}$ and $\equiv_{s,\times}$ defined in the paper by Faber et al. (Reference Faber, Tompits and Woltran2008) (see the discussion in Example 3, page 441, of the aforementioned paper).

We now demonstrate that our characterization of strong equivalence, when restricted to normal logic programs, retains the spirit of the initial characterization of strong equivalence for such programs ^{Footnote 2} (Lifschitz et al. Reference Lifschitz, Pearce and Valverde2001). More specifically, we show that in order to characterize strong equivalence for normal logic programs, it suffices to look at their models that contain only the truth values F, $T^*$ , and T.

We define strong equivalence for normal programs in the standard way (Lifschitz et al. Reference Lifschitz, Pearce and Valverde2001). The “standard answer set semantics” is the usual stable model semantics (Gelfond and Lifschitz Reference Gelfond and Lifschitz1988) of normal logic programs.

The following definition and theorem characterize strong equivalence of normal programs in our setting.

4 The complexity of strong equivalence for LPODs

In this section we examine the complexity of strong equivalence under our new characterization. Since the two versions of strong equivalence that we have examined have an identical characterization (see Theorems 1 and 2), the same complexity applies in both cases.

Our proof establishes $\mathbf{coNP}$ -hardness by a direct (and quite simple) reduction from 3SAT, which uses the special structure of LPODs in a crucial way. The corresponding proof by Faber et al. (Reference Faber, Tompits and Woltran2008) utilizes the more involved $\mathbf{coNP}$ -hardness result of Lin (Reference Lin2002) for strong equivalence of normal logic programs ^{Footnote 3} . In this respect, we feel that the proof that follows, apart from the fact that it applies to our new characterization of strong equivalence, also has an interest in its own right due to its different approach.

We next prove that strong equivalence of LPODs is also a $\mathbf{coNP}$ -hard problem, using a polynomial time reduction of 3SAT to the complement of this problem.

Let $\phi= \bigwedge_{i=1}^n c_i$ be a propositional formula in conjunctive normal form, where $c_i = L_{i,1} \vee L_{i,2} \vee L_{i,3}$ and $L_{i,j}$ is a literal (i.e. either a variable or the negation of a variable). For convenience, we may assume that the variables that occur in $\phi$ are elements of $\Sigma$ .

We will construct two programs $P_1$ , $P_2$ , such that $\phi$ is satisfiable if and only if $P_1$ and $P_2$ are not strongly equivalent.

For every literal L we define $\widetilde{L}$ as follows:

$$\widetilde{L} = \left\{\begin{array}{ll} L & \mbox{if $L = C$, for some $C \in \Sigma$}\\ not\ C & \mbox{if $L = \neg C$, for some $C \in \Sigma$}\\ \end{array}\ \ .\right.$$

Let A, B be two propositional variables in $\Sigma$ that do not occur in $\phi$ and let Q be

\[Q = \{ A \leftarrow \widetilde{L_{i,1}},\widetilde{L_{i,2}},\widetilde{L_{i,3}} \mid 1 \leq i \leq n\}.\]

The LPODs $P_1$ and $P_2$ are defined as follows:

\[ P_1 = Q \cup \{ A \times B \leftarrow \}, \]

\[ P_2 = Q \cup \{ A \times B \leftarrow \} \cup \{ A \leftarrow \}.\]

Assume that $\phi$ is satisfiable and let J be a two-valued interpretation such that $J(\phi) = T$ . We define the four-valued interpretation I as follows:

$$\begin{align*}\quad I(A) & = F^* \\\quad I(B) & = T {}\\I(C) & = F, \text{ if } C \text{ occurs in } \phi \text{ and } J(C)=T \\I(C) & = T, \text{ if } C \text{ occurs in } \phi \text{ and } J(C)=F.\end{align*}$$

Consider an arbitrary rule $A \leftarrow \widetilde{L_{i,1}},\widetilde{L_{i,2}},\widetilde{L_{i,3}}$ in Q. Then, $L_{i,1} \vee L_{i,2} \vee L_{i,3}$ is a clause in $\phi$ ; since J satisfies $\phi$ , it holds $J(L_{i,j})=T$ , for some $j\ \in \{1,2,3\}$ . Therefore, $I(\widetilde{L_{i,j}})=F$ , which implies that I satisfies the rule $A \leftarrow \widetilde{L_{i,1}},\widetilde{L_{i,2}},\widetilde{L_{i,3}}$ . Moreover, $I(A \times B) = T$ . We conclude that I is a model of $P_1$ ; however, I is not a model of $P_2$ , since $I(A)=F^*$ . Therefore, $P_1$ and $P_2$ are not logically equivalent in the four-valued logic, which implies that they are not strongly equivalent.

Conversely, assume that $P_1$ and $P_2$ are not strongly equivalent. Then, $P_1$ and $P_2$ are not logically equivalent in our four-valued logic. Since $P_1 \subset P_2$ , there exists a four-valued interpretation I that is a model of $P_1$ , but not a model of $P_2$ . We define the following two-valued interpretation for the variables in $\phi$ :

$$J(C) = \left\{\begin{array}{ll} T & \mbox{if $I(C) \leq F^*$}\\ F & \mbox{if $I(C) \geq T^*$}\\ \end{array}\\\ .\right.$$

We will show that J satisfies $\phi$ . We first prove some properties of I.

Since I is a model of $P_1$ , it must be either $I(A)=T$ , or $I(A)=F^*$ and $I(B)=T$ , so that the rule $A \times B\leftarrow$ is satisfied. However, in the former case, I should also be a model of $P_2$ (since $P_2-P_1 = \{ A \leftarrow \}$ ), which is a contradiction. Therefore, only the latter case is possible, that is, $I(A)=F^*$ .

Consider an arbitrary clause $c_i = L_{i,1} \vee L_{i,2} \vee L_{i,3}$ in $\phi$ . Since I is a model of $P_1$ , I satisfies the rule $A \leftarrow \widetilde{L_{i,1}},\widetilde{L_{i,2}},\widetilde{L_{i,3}}$ in $Q \subset P_1$ . Therefore, $\min\{I(\widetilde{L_{i,1}}),I(\widetilde{L_{i,2}}),I(\widetilde{L_{i,3}})\} \leq I(A) = F^*$ , which implies that there exists a $j \in \{1,2,3\}$ such that $I(\widetilde{L_{i,j}}) \leq F^*$ . But then, $J(L_{i,j})=T$ . We conclude that $\phi$ is satisfiable.

5 Related and future work

The work on strong equivalence, started with the pioneering results of Lifschitz et al. (Reference Lifschitz, Pearce and Valverde2001), but has since been extended to various formal systems. In particular, strong equivalence has been abstractly studied as a property across a variety of preferential formalisms (Faber et al. Reference Faber, Truszczynski and Woltran2013). To our knowledge however, the only existing work on the strong equivalence of LPODS is the paper by Faber et al. (Reference Faber, Tompits and Woltran2008). In that work the authors present an exhaustive study of several notions of strong equivalence for LPODs. More specifically, given LPODs P, Q, they consider the following notions of strong equivalence:

1. 1. $P \equiv_s Q$ holds iff the standard answer sets of P and Q coincide under any extension by ordinary (namely, normal) programs.

2. 2. $P \equiv_{s,\times} Q$ holds iff the standard answer sets of P and Q coincide under any extension by LPODs.

3. 3. $P \equiv_s^{\sigma} Q$ holds iff the $\sigma$ -preferred answer sets of P and Q coincide under any extension by ordinary programs, where $\sigma \in \{i,p,c\}$ and the indices i, c, and p correspond to the inclusion, Pareto, and cardinality orderings, respectively (see the paper by Brewka et al. (Reference Brewka, Niemelä and Syrjänen2004b) for formal definitions of these orderings).

4. 4. $P \equiv_{s,\times}^{\sigma} Q$ holds iff the $\sigma$ -preferred answer sets of P and Q coincide under any extension by LPODs, where $\sigma \in \{i,p,c\}$ .

Considering the above notions, the study of Faber et al. (Reference Faber, Tompits and Woltran2008) is certainly broader than the present work. We have not considered cases (1) and (3) above because in the standard definition of strong equivalence (Lifschitz et al. Reference Lifschitz, Pearce and Valverde2001) both the programs under comparison and the context programs, all belong to the same source language (in our case, LPODs). Of course, there may exist application domains where relations like $\equiv_s$ and $\equiv_s^{\sigma}$ might be of interest. In such a case, it might prove interesting to extend the present work in this direction. Case (2) above is covered by our Theorem 2. Finally, from case (4) above, we cover only the subcase where $\sigma$ is the inclusion preference. The subcases of Pareto and cardinality preferences are not covered because the semantics of Charalambidis et al. (Reference Charalambidis, Rondogiannis and Troumpoukis2021) on which the present work is based, is defined using the relation $\sqsubset$ , which is the model-theoretic version of the inclusion preference of Brewka (Reference Brewka2002) and Brewka et al. (Reference Brewka, Niemelä and Syrjänen2004b). It is important, however, to stress that the inclusion preference is probably the most fundamental among the three orderings and the initial paper introducing LPODs (Brewka Reference Brewka2002), used only this one. The Pareto and cardinality preferences were proposed subsequently in order to remedy the shortcomings of the initial semantics of LPODs (Brewka et al. Reference Brewka, Niemelä and Syrjänen2004b, see the discussion in page 342). Notice also that the cardinality preference can not be generalized in a direct way to first-order programs whose ground instantiation consists of an infinite number of rules.

Recapitulating, the two notions of strong equivalence that we cover in the present paper (Theorems 1 and 2), correspond to the relations $\equiv^{i}_{s,\times}$ and $\equiv_{s,\times}$ defined in the paper by Faber et al. (Reference Faber, Tompits and Woltran2008). In our case, both notions of strong equivalence coincide, because they have a unique characterization as logical equivalence in our four-valued logic. On the other hand, the relations $\equiv^{i}_{s,\times}$ and $\equiv_{s,\times}$ do not coincide (Faber et al. Reference Faber, Tompits and Woltran2008, see Theorem 21). This means that our approach and that of Faber et al. (Reference Faber, Tompits and Woltran2008) are different: there exist programs that are strongly equivalent with respect to one of the approaches and not strongly equivalent with respect to the other approach. This was expected since the two approaches are based on markedly different semantics. Although it does not seem straightforward to establish a formal relation between our framework and that of Faber et al. (Reference Faber, Tompits and Woltran2008), we can find examples where the two approaches give different results.

Example 3 Consider the following two programs given in Example 2 of the paper by Faber et al. (Reference Faber, Tompits and Woltran2008):

$$\begin{array}{l} {\tt c \times a \times b \leftarrow}\\ {\tt c \leftarrow a,b}\\ {\tt d \leftarrow c,not \ d} \end{array},$$

and:

$$\begin{array}{l} {\tt c \times b \times a \leftarrow}\\ {\tt c \leftarrow a,b}\\ {\tt d \leftarrow c,not \ d} \end{array}.$$

It is intuitively clear that in the first program a is preferred over b, while in the second program b is preferred over a. Despite this difference, the two programs are strongly equivalent under the $\equiv_{s,\times}$ semantics of Faber et al. (Reference Faber, Tompits and Woltran2008). Under our characterization the two programs are not strongly equivalent. To see this, consider the interpretation $I=\{({\tt a},T),({\tt b},F),({\tt c},F^*),({\tt d},F^*)\}$ , which is a model of the first program but not a model of the second. Therefore, the two programs are not logically equivalent in our four-valued logic, and consequently they are not strongly equivalent in our setting.

Although our study does not cover all the notions of strong equivalence examined in Faber et al. (Reference Faber, Tompits and Woltran2008), we believe that it has important advantages. Our work characterizes strong equivalence as logical equivalence in the four-valued logic of Charalambidis et al. (Reference Charalambidis, Rondogiannis and Troumpoukis2021). This result extends in a smooth way the well-known characterization of strong equivalence for normal logic programs (Lifschitz et al. Reference Lifschitz, Pearce and Valverde2001). Notice that the corresponding characterization of the inclusion preferred strong equivalence in Faber et al. (Reference Faber, Tompits and Woltran2008) is much more involved and uses certain binary functions over the sets of models of the programs that rely on the syntax of the given programs (see Faber et al. (Reference Faber, Tompits and Woltran2008), Definition 8 and Theorem 19). We believe that this is not an inherent shortcoming of the work of Faber et al. (Reference Faber, Tompits and Woltran2008), but instead a possibly unavoidable consequence of the fact that the original semantics of LPODs (Brewka Reference Brewka2002; Brewka et al. Reference Brewka, Niemelä and Syrjänen2004b) is not purely model-theoretic. The simplicity of our characterization makes us believe that it can be extended to broader classes of programs, such as for example to LPODs with strong negation and to disjunctive LPODs (Charalambidis et al. Reference Charalambidis, Rondogiannis and Troumpoukis2021).

One important aspect that we have not examined in this paper, is the possible practical use of the proposed strong equivalence characterization. To our knowledge, all major ASP systems are two-valued, and it is therefore a legitimate question of how our four-valued framework can be embedded in such systems. We believe that a promising direction for future work would be to define a notion of collapsed strong equivalence for LPODs:

Notice that in the above definition we do not need to collapse $T^*$ to T because, by Definition 7, answer sets do not contain the $T^*$ value. The logical characterization of collapsed strongly equivalent LPODs is probably an interesting question that deserves further investigation.

Finally, a very interesting question raised by one of the reviewers is whether the techniques developed in the paper by Charalambidis et al. (Reference Charalambidis, Rondogiannis and Troumpoukis2021) can also be used to derive a novel and simpler semantics for Qualitative Choice Logic (QCL) (Brewka et al. Reference Brewka, Benferhat and Berre2004a). Notice that QCL has also recently been investigated with respect to strong equivalence (Bernreiter et al. Reference Bernreiter, Maly, Woltran and Zhou2021), so the work developed in the present paper may be also relevant in this more general context.

Supplementary material

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