2 Explanations as support graphs

We start from a finite ^{Footnote 1} signature $\mathit{At}$ , a non-empty set of propositional atoms. A (labelled) rule is an implication of the form:

(1) \begin{eqnarray}\ell : p_1 \vee \dots \vee p_m \leftarrow q_1 \wedge \dots \wedge q_n \wedge \neg s_1 \wedge \dots \wedge \neg s_j \wedge \neg\neg t_1 \wedge \dots \wedge \neg\neg t_k.\end{eqnarray}

Given a rule r like (1), we denote its label as $\mathit{Lb}(r)\stackrel{\text{df}}{=} \ell$ . We also call the disjunction in the consequent $p_1 \vee \dots \vee p_m$ the head of r, written $\mathit{Head}(r)$ and denote the set of head atoms as $\mathit{H}(r) \stackrel{\text{df}}{=} \{p_1, \dots, p_m\}$ ; the conjunction in the antecedent is called the body of r and denoted as $\mathit{Body}(r)$ . We also define the positive and negative parts of the body, respectively, as the conjunctions $\mathit{Body}^+(r) \stackrel{\text{df}}{=} q_1 \wedge \dots \wedge q_n$ and $\mathit{Body}^-(r) \stackrel{\text{df}}{=} \neg s_1 \wedge \dots \wedge \neg s_j \wedge \neg\neg t_1 \wedge \dots \wedge \neg\neg t_k$ . The atoms in the positive body are represented as $\mathit{B}^+(r) \stackrel{\text{df}}{=} \{q_1,\dots,q_n\}$ . As usual, an empty disjunction (resp. conjunction) stands for $\bot$ (resp. $\top$ ). A rule r with empty head $\mathit{H}(r)=\emptyset$ is called a constraint. On the other hand, when $\mathit{H}(r)=\{p\}$ is a singleton, $\mathit{B}^+(r)=\emptyset$ and $\mathit{Body}^-(r)=\top$ the rule has the form $\ell: p \leftarrow \top$ and is said to be a fact, simply written as $\ell:p$ . The use of double negation in the body allows representing elementary choice rules. For instance, we will sometimes use the abbreviation $\ell : \{p\} \leftarrow B$ to stand for $\ell : p \leftarrow B \wedge \neg \neg p$ . A (labelled) logic program P is a set of labelled rules where no label is repeated. Note that P may still contain two rules r, r’ with same body and head $\mathit{Body}(r)=\mathit{Body}(r')$ and $\mathit{H}(r)=\mathit{H}(r')$ , but different labels $\mathit{Lb}(r) \neq \mathit{Lb}(r')$ . A program P is positive if $\mathit{Body}^-(r)=\top$ for all rules $r \in P$ . A program P is non-disjunctive if $|\mathit{H}(r)|\leq 1$ for every rule $r \in P$ . Finally, P is Horn if it is both positive and non-disjunctive: note that this may include (positive) constraints $\bot \leftarrow B$ .

A propositional interpretation I is any subset of atoms $I \subseteq \mathit{At}$ . We say that a propositional interpretation is a (classical) model of a labelled program P if $I \models \mathit{Body}(r) \to \mathit{Head}(r)$ in classical logic, for every rule $r \in P$ . The reduct of a labelled program P with respect to I, written $P^I$ , is a simple extension of the standard reduct by Gelfond and Lifschitz (Reference Gelfond and Lifschitz1988) that collects now the labelled positive rules:

\begin{eqnarray*}P^I \stackrel{\text{df}}{=} \{ \ \mathit{Lb}(r): \mathit{Head}(r) \leftarrow \mathit{Body}^+(r) \ \mid \ r \in P, \ I \models \mathit{Body}^-(r)\ \}.\end{eqnarray*}

As usual, an interpretation I is a stable model (or answer set) of a program P if I is a minimal model of $P^I$ . Note that, for the definition of stable models, the rule labels are irrelevant. We write $\mathit{SM}(P)$ to stand for the set of stable models of P.

We define the rules of a program P that support an atom p under interpretation I as $\mathit{SUP}(P,I,p) \stackrel{\text{df}}{=} \{ r \in P \mid p \in \mathit{H}(r), I \models \mathit{Body}(r)\}$ that is, rules with p in the head whose body is true w.r.t. I. The next proposition proves that, given I, the rules that support p in the reduct $P^I$ are precisely the positive parts of the rules that support p in P.

For the $\subseteq$ direction, take any $r' \in \mathit{SUP}(P^I,I,p)$ . By definition of reduct, we know that r’ is a positive rule and that there exists some $r \in P$ where $\mathit{Lb}(r)=\mathit{Lb}(r')$ , $\mathit{H}(r)=\mathit{H}(r')$ , $\mathit{B}^+(r)=\mathit{B}^+(r')$ and $I \models \mathit{Body}^-(r)$ . Consider any rule r satisfying that condition (we could have more than one): we will prove that $r \in \mathit{SUP}(P,I,p)$ . Since $r'\in \mathit{SUP}(P^I,I,p)$ , we get $I \models \mathit{Body}(r')$ but this is equivalent to $I \models \mathit{Body}^+(r)$ . As we had $I \models \mathit{Body}^-(r)$ , we conclude $I \models \mathit{Body}(r)$ and so r is supported in P given I.

Condition (i) means that there are no repeated labels in the graph, that is, $\lambda(p) \neq \lambda(q)$ for different atoms $p, q \in I$ . Condition (ii) requires that each atom p in the graph is assigned the label $\ell$ of some rule with p in the head, with a body satisfied by I and whose atoms in the positive body form all the incoming edges for p in the graph. Intuitively, labelling p with $\ell$ means that the corresponding (positive part of the) rule has been fired, “producing” p as a result. Since a label cannot be repeated in the graph, each rule can only be used to produce one atom, even though the rule head may contain more than one (when it is a disjunction). In general, program P may allow alternative ways of deriving an atom p in a model I. Thus, a same model I may have multiple support graphs under P, as we will illustrate later.

It is not difficult to see that an explanation $G=\langle I,E,\lambda \rangle$ for a model I is uniquely determined by its atom labelling $\lambda$ . This is because condition 1 about $\lambda$ in Definition (ii) uniquely specifies all the incoming edges for all the nodes in the graph. On the other hand, of course, not every arbitrary atom labelling corresponds to a well-formed explanation. We will sometimes abbreviate an explanation G for a model I by just using its labelling $\lambda$ represented as a set of pairs of the form $\lambda(p):p$ with $p \in I$ .

The name of supported model is not casual: we prove later on that, for non-disjunctive programs, the above definition coincides with the traditional one in terms of fixpoints of the immediate consequence operator (van Emden and Kowalski Reference van Emden and Kowalski1976) or as models of Clark’s completion (Clark Reference Clark, Gallaire and Minker1978). From Definition 2, it is clear that all justified models are obviously supported $\mathit{JM}(P) \subseteq \mathit{SPM}(P)$ but, in general, the opposite does not hold, as we will see later. Our main focus, however, is on justified models, since we will relate them to proofs, that are always acyclic. We can observe that not all models are justified, whereas a justified model may have more than one explanation, as we illustrate next.

Example 1 Consider the labelled logic program P

\begin{eqnarray*}\ell_1: \ a \vee b\quad\quad\quad\ell_2: \ d \leftarrow a \wedge \neg c\quad\quad\quad\ell_3: \ d \leftarrow \neg b.\end{eqnarray*}

No model $I \models P$ with $c \in I$ is justified since c does not occur in any head, so its support is always empty $\mathit{SUP}(P,I,c)=\emptyset$ and c cannot be labelled. The models of P without c are $\{b\}$ , $\{a,d\}$ , $\{b,d\}$ and $\{a,b,d\}$ but only the first two are justified. The explanation for $I=\{b\}$ corresponds to the labelling $\{(\ell_1: b)\}$ (it forms a graph with a single node). Model $I=\{a,d\}$ has the two possible explanations:

(2) \begin{eqnarray}\ell_1:a \longrightarrow \ell_2:d\hspace{100pt}\ell_1:a \quad\quad \ell_3:d.\end{eqnarray}

Model $I=\{b,d\}$ is not justified: we have no support for d given I, $\mathit{SUP}(P,I,d)=\emptyset$ , because I satisfies neither bodies of $\ell_2$ nor $\ell_3$ . On the other hand, model $\{a,b,d\}$ is not justified either, because $\mathit{SUP}(P,I,a)=\mathit{SUP}(P,I,b)=\{\ell_1\}$ and we cannot use the same label $\ell_1$ for two different atoms a and b in a same explanation (condition (i) in Def. Definition 1). $\Box$

Definition 3 (Proof of an atom)

Let I be a model of a labelled program P, $G=\langle I,E,\lambda \rangle$ an explanation for I under P and let $p \in I$ . The proof for p induced by G, written $\pi_G(p)$ , is the derivation:

\begin{eqnarray*}\pi_G(p) & \stackrel{\text{df}}{=} & \frac{\pi_G(q_1) \ \dotsc \ \pi_G(q_n)}{p} \ \lambda(p),\end{eqnarray*}

where, if $r \in P$ is the rule satisfying $\mathit{Lb}(r)=\lambda(p)$ , then $\{q_1,\dots,q_n\}= \mathit{B}^+(r)$ . When $n=0$ , the derivation antecedent $\pi_G(q_1) \ \dotsc \ \pi_G(q_n)$ is replaced by $\top$ (corresponding to the empty conjunction). $\Box$

Example 2 Let P be the labelled logic program:

\begin{eqnarray*}\ell_1: \ p\quad\quad\quad\ell_2: \ q \leftarrow p\quad\quad\quad\ell_3: \ r \leftarrow p, q.\end{eqnarray*}

P has a unique justified model $\{p,q,r\}$ whose explanation is shown in Figure 1 (left) whereas the induced proof for atom r is shown in Figure 1 (right). $\Box$

Fig. 1.

Some results for model $\{p,q,r\}$ of program in Example 2.

The next proposition trivially follows from the definition of explanations:

It is worth mentioning that explanations do not generate any arbitrary Modus Ponens derivation of an atom, but only those that are globally “coherent” in the sense that, if any atom p is repeated in a proof, it is always justified repeating the same subproof.

In the previous examples, justified and stable models coincided: one may wonder whether this is a general property. As we see next, however, every stable model is justified but, in general, the opposite may not hold. To prove that stable models are justified, we start proving a correspondence between explanations for any model I of P and explanations under $P^I$ .

Proof Let I be a stable model of P. To prove that there is an explanation G for I under P, we can use Proposition 1 and just prove that there is some explanation G for I under $P^I$ . We will build the explanation with a non-deterministic algorithm where, in each step i, we denote the graph $G_i$ as $G_i=\langle I_i,E_i,\lambda_i \rangle$ and represent the labelling $\lambda_i$ as a set of pairs of the form $(\ell:p)$ meaning $\ell=\lambda(p)$ . The algorithm proceeds as follows:

The existence of a rule $r \in P^I$ in line 5 is guaranteed because the while condition asserts $I_i \not\models P^I$ and so there must be some rule whose positive body is satisfied by $I_i$ but its head is not satisfied. We prove next that the existence of an atom $p \in I \cap \mathit{Head}(r)$ (line 5) is also guaranteed. First, note that the while loop maintains the invariant $I_i \subseteq I$ , since $I_0=\emptyset$ and $I_i$ only grows with atoms p (line 7) that belong to I (line 6). Therefore, $I_i \models \mathit{Body}(r)$ implies $I \models \mathit{Body}(r)$ , but since $I \models P^I$ , we also conclude $I \models r$ and thus $I \models \mathit{Head}(r)$ that is $I \cap \mathit{H}(r) \neq \emptyset$ , so we can always pick some atom p in that intersection. Now, note that the algorithm stops because, in each iteration, $I_i$ grows with exactly one atom from I that was not included before, since $I_i \models \neg \mathit{Head}(r)$ , and so, this process will stop provided that I is finite. The while stops satisfying $I_i \models P^I$ for some value $i=n$ . Moreover, $I_n = I$ , because otherwise, as $I_i \subseteq I$ is an invariant, we would conclude $I_n \subset I$ and so I would not be a minimal model of $P^I$ , which contradicts that I is a stable model of P. We remain to prove that the final $G_n=\langle I_n,E_n,\lambda_n \rangle$ is a correct explanation for I under $P^I$ . As we said, the atoms in I are the graph nodes $I_n=I$ . Second, we can easily see that $G_n$ is acyclic because each iteration adds a new node p and links this node to previous atoms from $\mathit{B}^+(r) \subseteq I_i$ (remember $I_i \models \mathit{Body}(r)$ ), so no loop can be formed. Third, no rule label can be repeated, because we go always picking a rule r that is new, since it was not satisfied in $I_i$ but becomes satisfied in $I_{i+1}$ (the rule head $\mathit{Head}(r)$ becomes true). Last, for every $p \in I$ , it is not hard to see that the (positive) rule $r \in P^I$ such that $\mathit{Lb}(r)=\lambda_n(p)$ satisfies $p \in \mathit{H}(r)$ and $\mathit{B}^+(r) = \{ q \mid (q,p) \in E \}$ by the way in which we picked r and inserted p in $I_i$ , whereas $I \models \mathit{Body}(r)$ because $I_i \models \mathit{Body}(r)$ , r is a positive rule and $I_i \subseteq I$ .

As a result, we get $\mathit{SM}(P) \subseteq \mathit{JM}(P) \subseteq \mathit{SPM}(P)$ , that is, justified models lay in between stable and supported.

In general, the number of explanations for a single justified model can be exponential, even when the program is Horn, and so, has a unique justified and stable model corresponding to the least classical model, as we just proved. As an example ^{Footnote 2} :

Example 3 (A chain of firing squads)

Consider the following variation of the classical Firing Squad Scenario introduced by Pearl (Reference Pearl and Kaufmann1999) for causal counterfactuals (although we do not use it for that purpose here). We have an army distributed in n squads of three soldiers each, a captain and two riflemen for each squad. We place the squads on a sequence of n consecutive hills $i=0,\dots,n-1$ . An unfortunate prisoner is at the last hill $n-1$ , and is being aimed at by the last two riflemen. At each hill i, the two riflemen $a_i$ and $b_i$ will fire if their captain $c_i$ gives a signal to fire. But then, captain $c_{i+1}$ will give a signal to fire if she hears a shot from the previous hill i in the distance. Suppose captain $c_0$ gives a signal to fire. Our logic program would have the form:

\[\begin{array}{r@{\ }l@{\quad\quad}r@{\ }l@{\quad\quad}r@{\ }l}s_0: & \mathit{signal}_0&a_i: & \mathit{fireA}_i \leftarrow \mathit{signal}_i&a'_{i+1}: & \mathit{signal}_{i+1} \leftarrow \mathit{fireA}_i \\& &b_i: & \mathit{fireB}_i \leftarrow \mathit{signal}_i&b'_{i+1}: & \mathit{signal}_{i+1} \leftarrow \mathit{fireB}_i\end{array}\]

for all $i=0,\dots,n-1$ where we assume (for simplicity) that $\mathit{signal}_{n}$ represents the death of the prisoner. This program has one stable model (the least model) making true the $3 n + 1$ atoms occurring in the program. However, this last model has $2^n$ explanations because to derive $\mathit{signal}_{i+1}$ from level i, we can choose between any of the two rules $a'_i$ or $b'_i$ (corresponding to the two riflemen) in each explanation. $\Box$

In many disjunctive programs, justified and stable models coincide. For instance, the following example is an illustration of a program with disjunction and head cycles.

Example 4 Let P be the program:

\begin{eqnarray*}\ell_1: p \vee q\hspace{50pt}\ell_2: q \leftarrow p\hspace{50pt}\ell_3: p \leftarrow q.\end{eqnarray*}

This program has one justified model $\{p,q\}$ that coincides with the unique stable model and has two possible explanations, $\{(\ell_1:p),(\ell_2:q)\}$ and $\{(\ell_1:q),(\ell_3:p)\}$ . $\Box$

However, in the general case, not every justified model is a stable model: we provide next a simple counterexample. Consider the program P:

\begin{eqnarray*}\ell_1: a \vee b\hspace{50pt}\ell_2: a \vee c,\end{eqnarray*}

whose classical models are the five interpretations: $\{a\}$ , $\{a,c\}$ , $\{a,b\}$ , $\{b,c\}$ and $\{a,b,c\}$ . The last one $\{a,b,c\}$ is not justified, since we would need three different labels and we only have two rules. Each model $\{a,c\}$ , $\{a,b\}$ , $\{b,c\}$ has a unique explanation corresponding to the atom labellings $\{(\ell_1:a),(\ell_2:c)\}$ , $\{(\ell_1:b),(\ell_2:a)\}$ and $\{(\ell_1:b),(\ell_2:c)\}$ , respectively. On the other hand, model $\{a\}$ has two possible explanations, corresponding to $\{(\ell_1:a)\}$ and $\{(\ell_2:a)\}$ . Notice that, in the definition of explanation, there is no need to fire every rule with a true body in I – we are only forced to explain every true atom in I. Note also that only the justified models $\{a\}$ and $\{b,c\}$ are also stable: this is due to the minimality condition imposed by stable models on positive programs, getting rid of the other two justified models $\{a,b\}$ and $\{a,c\}$ . The following theorem asserts that, for non-disjunctive programs, every justified model is also stable.

Moreover, for non-disjunctive programs, we can prove that our definition of supported model, coincides with the traditional one in terms of fixpoints of the immediate consequence operator (van Emden and Kowalski Reference van Emden and Kowalski1976). Given a non-disjunctive program P, let $T_P(I)$ be defined as $\{p \mid r \in P, I \models \mathit{Body}(r), \mathit{Head}(r)=p\}$ .

For right to left, suppose $I \models P$ and there is some support graph G of I under P. We prove both inclusion directions for $I=T_P(I)$ . For $\subseteq$ , suppose $p \in I$ . Then p is a node in G and there is a rule r such that $\lambda(p)=\mathit{Lb}(r)$ , $p=\mathit{Head}(r)$ (P is non–disjunctive) and $I \models \mathit{Body}(r)$ . But then $p \in T_P(I)$ . For $\supseteq$ , take any $p \in T_P(I)$ and suppose $p \not\in I$ . Then, we have at least some rule $r \in P$ with $I \models \mathit{Body}(r)$ and $I \not\models \mathit{Head}(r) (=p)$ , something that contradicts $I \models P$ .

To illustrate supported models in the disjunctive case, consider the program:

\begin{eqnarray*}\ell_1: a \vee b \leftarrow c\hspace{50pt}\ell_2: c \leftarrow b.\end{eqnarray*}

The only justified model of this program is $\emptyset$ which is also stable and supported. Yet, we also obtain a second supported model $\{b,c\}$ that is justified by the (cyclic) support graph with labelling $\{\ell_1:b, \ell_2:c\}$ .

3 An ASP encoding to compute explanations

In this section, we focus on the computation of explanations for a given stable model. We assume that we use an ASP solver to obtain the answer sets of some program P and that we have some way to label the rules. For instance, we may use the code line number (or another tag specified by the user), followed by the free variables in the rule and some separator. In that way, after grounding, we get a unique identifier for each ground rule.

To explain the answer sets of P, we may build the following (non-ground) ASP program x(P) that can be fed with the (reified) true atoms in I to build the ground program x(P,I). As we will prove, the answer sets of x(P,I) are in one-to-one correspondence with the explanations of I. The advantage of this technique is that, rather than collecting all possible explanations in a single shot, something that is too costly for explaining large programs, we can perform regular calls to an ASP solver for x(P,I) to compute one, several or all explanations of I on demand. Besides, this provides a more declarative approach that can be easily extended to cover new features (such as, for instance, minimisation among explanations).

For each rule in P of the form (1), x(P) contains the set of rules:

(3) \begin{eqnarray}sup(\ell) & \leftarrow & as(q_1) \wedge \dots \wedge as(q_n) \wedge as(p_i) \wedge \neg as(s_1) \wedge \dots \wedge \neg as(s_j),\end{eqnarray}

(4) \begin{eqnarray}& & \ \wedge \ \neg\neg as(t_1) \wedge \dots \wedge \neg\neg as(t_k),\end{eqnarray}

(5) \begin{eqnarray}\{ f(\ell,p_i) \} & \leftarrow & f(q_1) \wedge \dots \wedge f(q_n) \wedge as(p_i) \wedge sup(\ell),\end{eqnarray}

(6) \begin{eqnarray}\bot & \leftarrow & f(\ell,p_i) \wedge f(\ell,p_h),\end{eqnarray}

for all $i,h=1\dots m$ and $i \neq h$ , and, additionally x(P) contains the rules:

(7) \begin{eqnarray}f(A) & \leftarrow & f(L,A) \wedge as(A),\end{eqnarray}

(8) \begin{eqnarray}\bot & \leftarrow & \hbox{not } f(A) \wedge as(A),\end{eqnarray}

(9) \begin{eqnarray}\bot & \leftarrow & f(L,A) \wedge f(L',A) \wedge L \neq L' \wedge as(A).\end{eqnarray}

As we can see, x(P) reifies atoms in P using three predicates: as(A) which means that atom A is in the answer set I, so it is an initial assumption; f(L,A) means that rule with label L has been “fired” for atom A, that is, $\lambda(A)=L$ ; and, finally, f(A) that just means that there exists some fired rule for A or, in other words, we were able to derive A. Predicate $sup(\ell)$ tells us that the body of the rule r with label $\ell$ is “supported” by I, that is, $I \models \mathit{Body}(r)$ . Given any answer set I of P, we define the program $x(P,I) \stackrel{\text{df}}{=} x(P) \cup \{as(A) \mid A \in I\}$ . It is easy to see that x(P,I) becomes equivalent to the ground program containing the following rules:

(10)

(11)

(12) \begin{eqnarray}f(a) & \leftarrow f(\ell,a) & \quad \quad \text{for each $a \in I,$}\end{eqnarray}

(13) \begin{eqnarray}\bot & \leftarrow \hbox{not } f(a) & \quad \quad \text{for each $a \in I,$}\end{eqnarray}

(14) \begin{eqnarray}\bot & \leftarrow f(\ell,a) \wedge f(\ell',a) & \quad \quad \text{for each } a \in I, \ \ell \neq \ell'.\end{eqnarray}

Constraint (14) guarantees that atoms $f(\ell,a)$ have a functional nature, that is, we never get two different labels for a same atom a. This allows defining the labelling function $\lambda(a)=\ell$ iff $f(\ell,a) \in J$ . We remain to prove that conditions (i)-(iii) in Definition 1 hold. Condition (i) requires that $\lambda$ is injective, something guaranteed by (11). Condition (ii) requires that, informally speaking, the labelling for each atom a corresponds to an activated, supported rule for a. That is, if $\lambda(a)=\ell$ , or equivalently $f(\ell,a)$ , we should be able to build am edge (q,a) for each atom in the positive body of $\ell$ so that atoms q are among the graph nodes. This is guaranteed by that fact that rule (10) is the only one with predicate $f(\ell,a)$ in the head. So, if that ground atom is in J, it is because $f(q_i)$ are also in J that is, $q_i \in I$ , for all atoms in the positive body of rule labelled with $\ell$ . Note also that (10) is such that $I \models \mathit{Body}(r)$ , so the rule supports atom p under I, that is, $r \in \mathit{SUP}(P,I,p)$ . Let E be the set of edges formed in this way. Condition (iii) requires that the set E of edges forms an acyclic graph. To prove this last condition, consider the reduct program $x(P,I)^J$ . The only difference of this program with respect to x(P,I) is that rules (10) have now the form:

(15) \begin{eqnarray}f(\ell,p) \leftarrow f(q_1) \wedge \dots \wedge f(q_n),\end{eqnarray}

for each rule $r \in P$ like (1), $I\models \mathit{Body}(r)$ , $p \in \mathit{H}(r) \cap I$ as before, but additionally $f(\ell,p) \in J$ so the rule is kept in the reduct. Yet, the last condition is irrelevant since $f(\ell,p) \in J$ implies $f(p) \in J$ so $p \in \mathit{At}(J) = I$ . Thus, we have exactly one rule (15) in $x(P,I)^J$ per each choice (10) in x(P,I). Now, since J is an answer set of x(P,I), by monotonicity of constraints, it satisfies (11), (13), and (14) and is an answer set of the rest of the program P’ formed by rules (15) and (12). This means that J is a minimal model of P’. Suppose we have a cycle in E, formed by the (labelled) nodes and edges $(\ell_1:p_1) \longrightarrow \dots \longrightarrow (\ell_n:p_n) \longrightarrow (\ell_1:p_1)$ . Take the interpretation $J'=J\setminus \{f(\ell_1,p_1), \dots, f(\ell_n,p_n), f(p_1), \dots, f(p_n)\}$ . Since J is a minimal for P’, there must be some rule (15) or (11) not satisfied by J’. Suppose J’ does not satisfy some rule (11) so that $f(a) \not\in J'$ but $f(\ell,a) \in J' \subseteq J$ . This means we had $f(a) \in J$ since the rule was satisfied by J so a is one of the removed atoms $p_i$ belonging to the cycle. But then $f(\ell,a)$ should have been removed $f(\ell,a)\not\in J'$ and we reach a contradiction. Suppose instead that J’ does not satisfy some rule (15), that is, $f(\ell,p) \not\in J'$ and $\{f(q_1),\dots,f(g_n)\} \subseteq J' \subseteq J$ . Again, since the body holds in J, we get $f(\ell,p) \in J$ and so, $f(\ell,p)$ is one of the atoms in the cycle we removed from J’. Yet, since $(\ell:p)$ is in the cycle, there is some incoming edge from some atom in the cycle and, due to the way in which atom labelling is done, this means that this edge must come from some atom $q_i$ with $1\leq i\leq n$ in the positive body of the rule whose label is $\ell$ . But, since this atom is in the cycle, this also means that $f(q_i) \not\in J'$ and we reach a contradiction.

Proof Let I be an answer set of P and $G = \langle I,E,\lambda \rangle$ be some explanation for I under P and let us define the interpretation:

\begin{eqnarray*}J := \{f(a) \mid a \in I\} \cup \{f(\ell,a) \mid \lambda(a)=\ell \}.\end{eqnarray*}

We will prove that J is an answer set of x(P,I) or, in other words, that J is a minimal model of $x(P,I)^J$ . First, we will note that J satisfies $x(P,I)^J$ rule by rule. For the constraints, J obviously satisfy (7) because it contains an atom f(a) for each $a \in I$ . We can also see that J satisfies (11) because graph G does not contain repeated labels, so we cannot have two different atoms with the same label. The third constraint (14) is also satisfied by J because atoms $f(\ell,a), f(\ell',a)$ are obtained from $\lambda(a)$ that is a function that cannot assign two different labels to a same atom a. Satisfaction of (11) is guaranteed since the head of this rule f(a) is always some atom $a \in I$ and therefore $f(a) \in J$ . For the remaining rule, (10), we have two cases. If $f(\ell,p) \not\in J$ , then the rule is not included in the reduct and so there is no need to be satisfied. Otherwise, if $f(\ell,p) \in J$ then the rule in the reduct corresponds to (15) and is trivially satisfied by J because its only head atom holds in that interpretation. Finally, to prove that J is a minimal model of $x(P,I)^J$ , take the derivation tree $\pi_G(a)$ for each atom $a \in I$ . Now, construct a new tree $\pi$ where we replace each atom p in $\pi_G(a)$ by an additional derivation from $f(\ell,p)$ to f(p) through rule (12). It is easy to see that $\pi$ constitutes a Modus Ponens proof for f(a) under the Horn program $x(P,I)^J$ and the same reasoning can be applied to atom $f(\ell,a) \in J$ that is derived in the tree $\pi$ for f(a). Therefore, all atoms in J must be included in any model of $x(P,I)^J$ .

4 Related work

The current approach constitutes the formal basis of the new version of the explanation tool xclingo (Cabalar and Muñiz Reference Cabalar, Muñiz, Arias, Batsakis, Faber, Gupta, Pacenza, Papadakis, Robaldo, Rückschloß, Salazar, Saribatur, Tachmazidis, Weitkämper and Wyner2023) which also uses the ASP encoding from Section 3 to compute the explanations. Theorems 4 and 5 prove, in this way, that the tool is sound and complete with respect to the definition of explanation provided in the current paper.

There exist many other approaches for explanation and debugging in ASP (see the survey by Fandinno and Schulz (Reference Fandinno and Schulz2019)). The closest approach to the current work is clearly the one based on causal graphs (Cabalar et al. Reference Cabalar, Fandinno and Fink2014). Although we conjecture that a formal relation can be established (we plan this for future work), the main difference is that, in causal graphs, we get a unique, complete, and individual explanation for each atom, whereas the current work obtains the multiple explanations of a given model (a set of atoms). For instance, in the firing squads example, the causal-graph explanation for the derivations of atoms $\mathit{signal}_4$ and $\mathit{signal}_8$ would contain algebraic expressions with all the possible derivations for each one of those atoms. In the current approach, however, we would get an individual derivation in each support graph, but additionally, the proof we get for $\mathit{signal}_4$ has to be the same one we use for that atom inside the derivation of $\mathit{signal}_8$ .

Justifications based on the positive part of the program were also used before by Erdem and Oztok (Reference Erdem and Oztok2013). There, the authors implemented an ad hoc approach to the problem of solving biomedical queries, rather than a general ASP explanation tool.

Other examples of general approaches are the formal theory of justifications (Denecker et al. Reference Denecker, Brewka, Strass, Calimeri, Ianni and Truszczynski2015), off-line justifications (Pontelli and Son Reference Pontelli, Son, Etalle and Truszczyński2006), LABAS (Schulz and Toni Reference Schulz and Toni2016) (based on argumentation theory (Bondarenko et al. Reference Bondarenko, Dung, Kowalski and Toni1997; Dung et al. Reference Dung, Kowalski and Toni2009)) or s(CASP) (Arias et al. Reference Arias, Carro, Chen and Gupta2020). All of them provide graph or tree-based explanations for an atom to be (or not) in a given answer set. The formal theory of justifications was also extended to deal with nested graph-based justifications (Marynissen Reference Marynissen2022) and is actually a more general framework that allows covering other logic programming semantics. System xASP (Trieu et al. Reference Trieu, Son, Balduccini, Gottlob, Inclezan and Maratea2022) generates explanation graphs previously proposed by Pontelli and Son (Reference Pontelli, Son, Etalle and Truszczyński2006) and also uses an ASP meta-programming encoding. In the case of s(CASP), it proceeds in a top-down manner, building the explanation as an ordered list of literals extracted from the goal-driven satisfaction of the query. An important difference with respect to this last group of approaches is that their explanations consider dependences through default negation. To illustrate the effect, take the program:

\begin{eqnarray*}\ell_1: switch & \\\ell_2: light & \leftarrow & switch, \hbox{not } ab \\\ell_3: ab & \leftarrow & blown\_fuse \\\ell_4: ab & \leftarrow & broken\_bulb \\\ell_5: ab & \leftarrow & blackout, \hbox{not} generator.\end{eqnarray*}

The only stable model is $\{switch,light\}$ and its unique explanation is the support graph

\begin{eqnarray*}\ell_1: switch \longrightarrow \ell_2:light,\end{eqnarray*}

that is, the light is on because we toggled the switch. Adding negative information would lead us to explain $\hbox{not} ab$ and obtain two explanations: one in which we also add that there is no blown fuse, no broken bulb and no blackout; the second one is similar, but instead of no blackout, we have a doubly negative dependence on generator: that is, nothing prevents having a generator, even though we do not have it. Note how these explanations may easily get complicated: we could have to negate multiple alternative ways of breaking the bulb, even when none of them have happened ^{Footnote 3} . Our approach consists, instead, in explaining the information that currently holds, assuming that other states of affairs will arise in terms of other alternative answer sets. In other words, we refrain from using facts for which we have no evidence or reason to believe in our current model.

Another distinctive feature of our approach is that it provides explanations for disjunctive programs and, moreover, it has also allowed us to define supported and justified models for that case. In fact, we plan to study potential connections between justified models and other approaches for disjunction not based in minimal models such as in the works by Aguado et al. (Reference Aguado, Cabalar, Fandinno, Pearce, Pérez and Vidal2019) or Shen and Eiter (Reference Shen and Eiter2019).

Other ASP explanation approaches have to do with comparing stable models or explaining their non-existence. For instance, Gebser et al. (Reference Gebser, Pührer, Schaub and Tompits2008) use a meta-programming technique to explain why a given model is not an answer set of a given program. More recently, Eiter et al. (Reference Eiter, Saribatur and Schüller2019) considered the explanation of ASP programs that have no answer sets in terms of the concept of abstraction (Saribatur et al. Reference Saribatur, Eiter and Schüller2021). This allows spotting which parts of a given domain are actually relevant for rising the unsatisfiability of the problem. We plan to explore formal relations to these approaches or to study potential combinations with some of them.