5.1.1 The $\textsf{JU}$ semantics and the $\textsf{AS}$ semantics

The historically first rule update semantics in answer-set programming is the justified update semantics, or $\textsf{JU}$ semantics for short (Leite and Pereira Reference Leite and Pereira1998), with the idea to define a set of rejected rules, which depends on a stable model candidate, and then verify that the candidate is indeed a stable model of the remaining rules.

Definition 14 ( $\textsf{JU}$ Semantics Leite and Pereira 1998) Let $\boldsymbol{P} = \langle \mathcal{P}_i \rangle_{i < n}$ be a DLP and J an interpretation. We define the set $\mathsf{rej}_{[\mathsf{JU}]}({\boldsymbol{P}, J})$ of rejected rules as

\begin{equation*} \mathsf{rej}_{[\mathsf{JU}]}({\boldsymbol{P}, J}) = \{ \pi \in \mathcal{P}_i | \exists j \, \exists \sigma : i < j < n \land \sigma \in \mathcal{P}_j \land \pi \Join \sigma \land J \mathrel\mid= B(\pi)[\sigma] \} .\end{equation*}

The set $[\kern-.3ex[{\boldsymbol{P}}]\kern-.3ex]_{\mathsf{JU}}$ of $\textsf{JU}$ models of a DLP $\boldsymbol{P}$ consists of all stable models J of the program

\begin{equation*} \mathsf{all}(\boldsymbol{P}^{\mathsf{e}}) \backslash \mathsf{rej}_{[\mathsf{JU}]}({\boldsymbol{P}^{\mathsf{e}}, J}) . \end{equation*}

Under the $\textsf{JU}$ semantics, a rule $\pi$ is rejected if and only if a more recent rule $\sigma$ is in conflict with $\pi$ and the body of $\sigma$ is satisfied in the stable model candidate J. Note that the latter condition is essential – without it, rules might get rejected simply because a more recent rule $\sigma$ has a conflicting head, without a guarantee that $\sigma$ will actually be activated.

Before we illustrate the $\textsf{JU}$ semantics with an example, we first present a related semantics, which prevents rejected rules from rejecting other rules. It is dubbed the update answer-set semantics, or $\textsf{AS}$ semantics for short (Eiter et al. Reference Eiter, Fink, Sabbatini and Tompits2002):

Definition 15 ( $\textsf{AS}$ Semantics Eiter et al. 2002) Let $\boldsymbol{P} = \langle \mathcal{P}_i \rangle_{i < n}$ be a DLP and J an interpretation. We define the set of rejected rules $\mathsf{rej}_{[\mathsf{AS}]}({\boldsymbol{P}, J})$ as ^{Footnote 2}

\begin{equation*} \mathsf{rej}_{[\mathsf{AS}]}({\boldsymbol{P}, J}) = \{ \pi \in \mathcal{P}_i | \exists j \, \exists \sigma : i < j < n \land \sigma \in \mathcal{P}_j \backslash \mathsf{rej}_{[\mathsf{AS}]}({\boldsymbol{P}, J}) \land \pi \Join \sigma \land J \mathrel\mid= B(\pi)[\sigma] \}.\end{equation*}

The set $[\kern-.3ex[{\boldsymbol{P}}]\kern-.3ex]_{\mathsf{AS}}$ of $\textsf{AS}$ models of a DLP $\boldsymbol{P}$ consists of all stable models J of the program

\begin{equation*} \mathsf{all}(\boldsymbol{P}^{\mathsf{e}}) \backslash \mathsf{rej}_{[\mathsf{AS}]}({\boldsymbol{P}^{\mathsf{e}}, J}) . \end{equation*}

The definitions of the $\textsf{JU}$ and $\textsf{AS}$ semantics are fairly straightforward and reflect intuitions about rule updates better than an approach based on a belief update construction, such as the one by Alferes and Pereira (Reference Alferes and Pereira1996). As an illustration, let us look at the result of these semantics when applied to the example in the introduction:

Example 16 Consider again the program $\mathcal{P}$ from the example in the introduction which contains the rules

\begin{align*} \mathsf{goHome} \leftarrow \mathop{\sim\!} \mathsf{money}.& &\mathsf{goRestaurant} \leftarrow \mathsf{money}.& &\mathsf{money}. \end{align*}

and its update $\mathcal{U}$ with the rules

\begin{align*} \mathop{\sim\!} \mathsf{money} &\leftarrow \mathsf{robbed}. & \mathsf{robbed} &. \end{align*}

Following the discussion in the introduction, the expected stable model of the DLP $\langle \mathcal{P}, \mathcal{U} \rangle$ is $J = {\mathsf{robbed}, \mathsf{goHome}}$ . Also, $\mathsf{rej}_{[\mathsf{JU}]}({\langle \mathcal{P}, \mathcal{U} \rangle^{\mathsf{e}}, J}) = \mathsf{rej}_{[\mathsf{AS}]}({\langle \mathcal{P}, \mathcal{U} \rangle^{\mathsf{e}}, J}) = {\mathsf{money}.}$ , and J is indeed a stable model of the remaining rules in $\mathsf{all}(\langle \mathcal{P}, \mathcal{U} \rangle^{\mathsf{e}})$ . Furthermore, J is the only interpretation with these properties, so

\begin{equation*} [\kern-.3ex[{\langle \mathcal{P}, \mathcal{U} \rangle}]\kern-.3ex]_{\mathsf{JU}} = [\kern-.3ex[{\langle \mathcal{P}, \mathcal{U} \rangle}]\kern-.3ex]_{\mathsf{AS}} = {J} . \end{equation*}

Example 17 To illustrate the expansion mechanism and its interplay with the rejection mechanism, consider again the program $\mathcal{P}$ from the example in the introduction which contains the rules

\begin{align*} \mathsf{goHome} \leftarrow \mathop{\sim\!} \mathsf{money}.& &\mathsf{goRestaurant} \leftarrow \mathsf{money}.& &\mathsf{money}. \end{align*}

but now its update $\mathcal{U}$ is modified by replacing $\mathop{\sim\!} \mathsf{money}$ with $\lnot\mathsf{money}$ :

\begin{align*} \lnot \mathsf{money} &\leftarrow \mathsf{robbed}. & \mathsf{robbed} &. \end{align*}

The expanded version of both programs are:

\begin{align*}\mathcal{P}^{\mathsf{e}}& &\mathsf{goHome} &\leftarrow \mathop{\sim\!} \mathsf{money}. &\mathsf{goRestaurant} &\leftarrow \mathsf{money}.& &\mathsf{money}.\\ & &\mathop{\sim\!}\lnot\mathsf{goHome} &\leftarrow \mathop{\sim\!} \mathsf{money}. &\mathop{\sim\!}\lnot\mathsf{goRestaurant} &\leftarrow \mathsf{money}.& &\mathop{\sim\!}\lnot\mathsf{money}.\\\mathcal{U}^{\mathsf{e}}& &\lnot \mathsf{money} &\leftarrow \mathsf{robbed}. &\mathsf{robbed}.& &\\ & &\mathop{\sim\!} \mathsf{money} &\leftarrow \mathsf{robbed}. &\mathop{\sim\!}\lnot\mathsf{robbed}.& & \end{align*}

The expected stable model of the DLP $\langle \mathcal{P}, \mathcal{U} \rangle$ is $J = {\mathsf{robbed},\lnot\mathsf{money}, \mathsf{goHome}}$ . Also, $\mathsf{rej}_{[\mathsf{JU}]}({\langle \mathcal{P}, \mathcal{U} \rangle^{\mathsf{e}}, J}) = \mathsf{rej}_{[\mathsf{AS}]}({\langle \mathcal{P}, \mathcal{U} \rangle^{\mathsf{e}}, J}) = {\mathsf{money}.,\mathop{\sim\!}\lnot\mathsf{money}.}$ , and J is indeed a stable model of the remaining rules in $\mathsf{all}(\langle \mathcal{P}, \mathcal{U} \rangle^{\mathsf{e}})$ . Furthermore, J is the only interpretation with these properties, so

\begin{equation*} [\kern-.3ex[{\langle \mathcal{P}, \mathcal{U} \rangle}]\kern-.3ex]_{\mathsf{JU}} = [\kern-.3ex[{\langle \mathcal{P}, \mathcal{U} \rangle}]\kern-.3ex]_{\mathsf{AS}} = {J} . \end{equation*}

Nevertheless, problematic examples which are not handled correctly by these semantics have also been identified (Leite Reference Leite2003). Many of them involve tautological updates, the intuition being that a tautological rule (i.e., a rule whose head literal also belongs to its body) cannot indicate a change in the modelled world because it is always true. It thus follows that a tautological update should not affect the stable models of the original program. Interestingly, immunity to tautological updates is a desirable property of belief updates in classical logic, being a direct consequence of postulate $\mathsf{BU2}$ . The following example illustrates a misbehaviour of the $\textsf{AS}$ semantics

In the previous example, the $\textsf{JU}$ semantics provides an adequate solution: since it allows rejected rules to reject, the initial rule is always rejected and $[\kern-.3ex[{\boldsymbol{P}_1}]\kern-.3ex]_{\mathsf{JU}} ={{\lnot \mathsf{p}}}$ as expected. Unfortunately, there are also numerous DLPs to which the $\textsf{JU}$ semantics assigns unwanted models, as illustrated b the following example.

5.1.2 The $\textsf{DS}$ semantics

The problem illustrated with Example 19 is addressed in the dynamic stable model semantics, or $\textsf{DS}$ semantics for short (Alferes et al. Reference Alferes, Leite, Pereira, Przymusinska and Przymusinski2000), by constraining the set of atoms that can be assumed false by default.

Definition 20 ( $\textsf{DS}$ Semantics Alferes et al. 2000) Let $\boldsymbol{P} = \langle \mathcal{P}_i \rangle_{i < n}$ be a DLP and J an interpretation. The set of rejected rules $\mathsf{rej}_{[\mathsf{DS}]}({\boldsymbol{P}, J})$ is identical to the set $\mathsf{rej}_{[\mathsf{JU}]}({\boldsymbol{P}, J})$ and we define the set of default assumptions $\mathsf{def}(\boldsymbol{P}, J)$ as

\begin{equation*} \mathsf{def}(\boldsymbol{P}, J) = \{ \mathop{\sim\!} l | l \in \mathscr{L} \land \lnot \exists \pi \in \mathsf{all}(\boldsymbol{P}) : H(\pi) = {l} \land J \mathrel\mid= B(\pi) \}.\end{equation*}

The set $[\kern-.3ex[{\boldsymbol{P}}]\kern-.3ex]_{\mathsf{DS}}$ of $\textsf{DS}$ models of a DLP $\boldsymbol{P}$ consists of all interpretations J such that

\begin{equation*} J' = \mathsf{least}( [ \mathsf{all}(\boldsymbol{P}^{\mathsf{e}}) \backslash \mathsf{rej}_{[\mathsf{DS}]}({\boldsymbol{P}^{\mathsf{e}}, J}) ] \cup \mathsf{def}(\boldsymbol{P}^{\mathsf{e}}, J) ) , \end{equation*}

where $J' = J \cup \mathop{\sim\!}\,(\mathscr{L} \backslash J)$ and $\mathsf{least}(\cdot)$ denotes the least model of the argument program with all literals treated as atoms.

Note that it follows from the definition of a (regular) stable model that J is a $\textsf{JU}$ stable model of a DLP $\boldsymbol{P}$ if and only if

\begin{equation*} J' = \mathsf{least}( [ \mathsf{all}(\boldsymbol{P}^{\mathsf{e}}) \backslash \mathsf{rej}_{[\mathsf{DS}]}({\boldsymbol{P}^{\mathsf{e}}, J}) ] \cup \mathop{\sim\!}\,(\mathscr{L} \backslash J) ) .\end{equation*}

Hence, the difference between the $\textsf{JU}$ and $\textsf{DS}$ semantics is only in the set of default assumptions that can be adopted to construct the model. In particular, if a rule that derives an objective literal l is present in $\mathsf{all}(\boldsymbol{P})$ , then $\mathop{\sim\!} l$ is not among the default assumptions in the $\textsf{DS}$ semantics although it could be used as a default assumption in the $\textsf{JU}$ semantics. In other words, according to the $\textsf{DS}$ semantics, once some objective literal l can be derived by some (older) rule, we can no longer assume $\mathop{\sim\!} l$ by default. The $\textsf{DS}$ semantics thus resolves problems with examples such as the previous one encoded by $\boldsymbol{P}_2$ , that is, it holds that $[\kern-.3ex[{\boldsymbol{P}_2}]\kern-.3ex]_{\mathsf{DS}} ={{\mathsf{p}}}$ . But even the $\textsf{DS}$ semantics exhibits problematic behaviour when tautological updates are involved, as illustrated by the following example.

5.1.3 The $\textsf{RD}$ semantics

The trouble with tautological and some other types of irrelevant updates has been discussed and finally resolved by Alferes et al. (Reference Alferes, Banti, Brogi and Leite2005) who defined the so-called refined extension principle – a principle encoding the desirable immunity to tautological, cyclic and other irrelevant updates – as well as a rule update semantics satisfying the principle. The definition of this semantics is very similar to the $\textsf{DS}$ semantics, the only difference being that in the set of rejected rules, $i \leq j$ is required instead of $i < j$ , which seems to be a technical trick with little correspondence to any intuition. The semantics is called the refined dynamic stable model semantics, or $\textsf{RD}$ semantics for short:

Definition 22 ( $\textsf{RD}$ Semantics Alferes et al. 2005) Let $\boldsymbol{P} = \langle \mathcal{P}_i \rangle_{i < n}$ be a DLP and J an interpretation. We define the set of rejected rules $\mathsf{rej}_{[\mathsf{RD}]}({\boldsymbol{P}, J})$ as

\begin{equation*} \mathsf{rej}_{[\mathsf{RD}]}({\boldsymbol{P}, J}) = \{ \pi \in \mathcal{P}_i | \exists j \, \exists \sigma : i \leq j < n \land \sigma \in \mathcal{P}_j \land \pi \Join \sigma \land J \mathrel\mid= B(\pi)[\sigma] \}.\end{equation*}

The set $[\kern-.3ex[{\boldsymbol{P}}]\kern-.3ex]_{\mathsf{RD}}$ of $\textsf{RD}$ models of a DLP $\boldsymbol{P}$ consists of all interpretations J such that

\begin{equation*} J' = \mathsf{least}( [ \mathsf{all}(\boldsymbol{P}^{\mathsf{e}}) \backslash \mathsf{rej}_{[\mathsf{RD}]}({\boldsymbol{P}^{\mathsf{e}}, J}) ] \cup \mathsf{def}(\boldsymbol{P}^{\mathsf{e}}, J) ) , \end{equation*}

where J’ and $\mathsf{least}(\cdot)$ are as before.

Due to satisfying the refined extension principle, the $\textsf{RD}$ semantics is completely immune to tautological updates. For instance, in case of the previous example encoded by $\boldsymbol{P}_3$ we obtain $[\kern-.3ex[{\boldsymbol{P}_3}]\kern-.3ex]_{\mathsf{RD}} = \emptyset$ . As we shall see, a vast majority of rule update semantics, even those that have been developed much later and are not based on causal rejection, are not immune to tautological updates.

Banti et al. (Reference Banti, Alferes, Brogi and Hitzler2005) present an alternative, equivalent, characterisation for the $\textsf{RD}$ semantics, based on level mappings, which is perhaps better in helping understand the difference w.r.t. previous semantics than the trick used in the previous definition of the set of rejected rules where $i \leq j$ is required instead of $i < j$ .

Theorem 23 ( $\textsf{RD}$ Semantics, Alternative Characterisation Banti et al. Reference Banti, Alferes, Brogi and Hitzler2005) Let $\boldsymbol{P} = \langle \mathcal{P}_i \rangle_{i < n}$ be a DLP and J an interpretation. Given a level mapping $\ell$ , let the set $\mathsf{rej}_{[\ell]}({\boldsymbol{P}, J})$ of rejected rules be defined as follows:

\begin{equation*} \mathsf{rej}_{[\ell]}({\boldsymbol{P}, J}) = \{ \pi \in \mathcal{P}_i | \exists j \, \exists \sigma : i < j < n \land \sigma \in \mathcal{P}_j \land \pi \Join \sigma \land J \mathrel\mid= B(\pi)[\sigma] \land \ell(H(\pi)[\sigma]) > \ell(B(\pi)[\sigma]) \}.\end{equation*}

Then, $J\in[\kern-.3ex[{\boldsymbol{P}}]\kern-.3ex]_{\mathsf{RD}}$ iff there exists a level mapping $\ell$ such that:

1. 1. J is a $\textsf{C}$ -model of $\mathsf{all}(\boldsymbol{P}^{\mathsf{e}}) \backslash \mathsf{rej}_{[\ell]}({\boldsymbol{P}^{\mathsf{e}}, J})$ , and

2. 2. $\forall\mathsf{p}\in J,\exists\sigma\in\mathsf{all}(\boldsymbol{P}^{\mathsf{e}}) \backslash \mathsf{rej}_{[\ell]}({\boldsymbol{P}^{\mathsf{e}}, J})$ such that $H(\pi)[\sigma]=\mathsf{p}\land, \ell(H(\pi)[\sigma]) > \ell(B(\pi)[\sigma])\land J\models B(\pi)[\sigma]$ .

This characterisation borrows from the work of Hitzler and Wendt (Reference Hitzler and Wendt2005) on uniform characterisations of different semantics for logic programs in terms of level mappings. In particular, this characterisation is based on the notion of well-supported models (Fages Reference Fages1994), an alternative view of stable models that characterises them as $\textsf{C}$ -models with the additional requirement that there exists some level mapping such all atoms in the model are supported by some rule whose head is that atom and the level of the that atom is greater than the level of the rule’s body. When extended to the case of DLPs, besides being used to decide whether some objective literal should be true, rules are also used to reject other rules. Hence, the concept of well-supportedness is also adopted to the rejection mechanism, and rules can only reject other rules with the additional constraint that the level of their heads be greater than the level of their bodies.

5.1.4 Relationship between semantics based on causal rejection

The rule update semantics introduced above are strongly related to one another. The above considerations show that undesired stable models of the $\textsf{AS}$ , $\textsf{JU}$ and $\textsf{DS}$ semantics were eliminated by enlarging the set of rejected rules or by shrinking the set of default assumptions. The following theorem shows that no additional stable models were added in the process:

Theorem 24 (Leite Reference Leite2003; Alferes et al. Reference Alferes, Banti, Brogi and Leite2005) Let $\boldsymbol{P}$ be a DLP. Then,

\begin{equation*} [\kern-.3ex[{\boldsymbol{P}}]\kern-.3ex]_{\mathsf{AS}} \supseteq [\kern-.3ex[{\boldsymbol{P}}]\kern-.3ex]_{\mathsf{JU}} \supseteq [\kern-.3ex[{\boldsymbol{P}}]\kern-.3ex]_{\mathsf{DS}} \supseteq [\kern-.3ex[{\boldsymbol{P}}]\kern-.3ex]_{\mathsf{RD}} . \end{equation*}

Moreover, for each inclusion above there exists a DLP for which the inclusion is strict.

Furthermore, all of these semantics coincide when only acyclic DLPs are considered, showing that the differences in the definitions of rejected rules and default assumptions are only relevant in the presence of cyclic dependencies between literals.

Theorem 25 (Homola Reference Homola2004) Let $\boldsymbol{P}$ be an acyclic DLP. Then,

\begin{equation*} [\kern-.3ex[{\boldsymbol{P}}]\kern-.3ex]_{\mathsf{AS}} = [\kern-.3ex[{\boldsymbol{P}}]\kern-.3ex]_{\mathsf{JU}} = [\kern-.3ex[{\boldsymbol{P}}]\kern-.3ex]_{\mathsf{DS}} = [\kern-.3ex[{\boldsymbol{P}}]\kern-.3ex]_{\mathsf{RD}} . \end{equation*}

The relation between these semantics and other formalisms has also been studied. It has been shown by Eiter et al. (Reference Eiter, Fink, Sabbatini and Tompits2002) that the $\textsf{AS}$ semantics coincides with the non-disjunctive case of the semantics for inheritance programs by Buccafurri et al. (Reference Buccafurri, Faber and Leone1999).

One of the open issues with these semantics is that one cannot easily condense a DLP to a single logic program that could be used instead of the DLP to perform further updates. The first obstacle is that the stable models of a DLP may be non-minimal, as illustrated by the following example.

Since stable models of non-disjunctive programs are subset-minimal, no such program can have the set of stable models $[\kern-.3ex[{\boldsymbol{P}_4}]\kern-.3ex]_{\mathsf{RD}}$ . Condensing to a disjunctive program is also problematic because rule update semantics are constrained to non-disjunctive programs only, so after a condensation one would not be able to perform any further updates.

Nevertheless, the original definition of the four discussed $\mathsf{S}$ -semantics ( $\textsf{JU}$ , $\textsf{AS}$ , $\textsf{DS}$ and $\textsf{RD}$ semantics) was accompanied by a translation of a DLP to a single non-disjunctive program over an extended language whose stable models correspond one-to-one to the $\mathsf{S}$ -models assigned to the DLP under the respective rule update semantics. Due to the language extension, the new program cannot simply be updated directly as a substitute for the original DLP, but may serve as a way to study the computational properties of the rule update semantics and as a way to implement it using existing answer-set solvers.

In the literature, we can also find the semantics proposed by Osorio and Cuevas (Reference Osorio and Cuevas2007) which can be seen as simpler substitutes for the $\textsf{AS}$ semantics. Unlike the semantics discussed above, they are not defined declaratively, instead they are specified directly by translating the initial program and its update to a single program over the same language. The first translation essentially weakens the rules from the original program by making them defeasible, that is, a rule $l\leftarrow B(\pi)$ is transformed into the rule $l \leftarrow B(\pi), \mathop{\sim\!}\overline{l}$ . The authors have shown that the resulting semantics is equivalent to the $\textsf{AS}$ semantics if a single update is performed and the updating program contains a tautology $l \leftarrow l$ for every objective literal l. Due to its simplicity, because it only deals with a single update in a way that cannot immediately be extended to additional updates, this semantics is not sensitive to the addition and removal of tautologies, but it adopts the problematic behaviour of the $\textsf{AS}$ semantics even when the tautologies are removed from the updating program. For example, when considering the DLPs

\begin{align*} \boldsymbol{P}_5 &= \langle {\mathsf{p}., \lnot \mathsf{p}.}, \emptyset \rangle\enspace\text{and}\\ \boldsymbol{P}_5' &= \langle {\mathsf{p}., \lnot \mathsf{p}.}, {\mathsf{p} \leftarrow \mathsf{p}., \lnot \mathsf{p} \leftarrow \lnot \mathsf{p}.} \rangle\end{align*}

the $\textsf{AS}$ semantics correctly assigns no $\textsf{AS}$ model to $\boldsymbol{P}_5$ although its sensitivity to tautological updates causes $\boldsymbol{P}_5'$ to have two $\textsf{AS}$ models: ${\mathsf{p}}$ and ${\lnot \mathsf{p}}$ . The first semantics suggested by Osorio and Cuevas (Reference Osorio and Cuevas2007) assigns these two interpretations, ${\mathsf{p}}$ and ${\lnot \mathsf{p}}$ , to both $\boldsymbol{P}_5$ and $\boldsymbol{P}_5'$ , so it exhibits problematic behaviour even on DLPs that were correctly handled by the $\textsf{AS}$ semantics.

The second translation is more involved as it produces a program that may not be expressible by a non-disjunctive program. The resulting update semantics is shown to coincide with the $\textsf{AS}$ semantics in case only a single update is performed. Note that since the $\textsf{AS}$ semantics coincides with the $\textsf{JU}$ semantics on DLPs of length two, ^{Footnote 3} the above mentioned relationships between semantics by Osorio and Cuevas (Reference Osorio and Cuevas2007) and the $\textsf{AS}$ semantics also hold for the $\textsf{JU}$ semantics. Their behaviour on DLPs of length three or more has not been studied.

5.2.1 The $\textsf{PRZ}$ semantics

One rule update semantics of this type was defined by Zhang (2006) and relies on the semantics for prioritised logic programs proposed by Zhang (Reference Zhang2003). ^{Footnote 4} Generally speaking, according to Zhang (2006), the semantics is assigned to a prioritised logic program $({\mathcal{P}, \prec})$ by pruning away less preferred rules, obtaining an ordinary logic program $\mathcal{P}^\prec \subseteq \mathcal{P}$ called a reduct. Formally, $\mathcal{P}^<$ is a reduct of $({\mathcal{P}, \prec})$ if there exists a sequence of sets $\mathcal{P}_{i}$ ( $i=0,1,\ldots$ ) such that

\begin{align*}\mathcal{P}_{0} = \mathcal{P},\quad\quad\quad\quad\quad\mathcal{P}_{i+1} = \mathcal{P}_{i}-\mathcal{R}_{i},\quad\quad\quad\quad\quad\mathcal{P}^< = \bigcap_{i=0}^{\infty}\mathcal{P}_{i},\end{align*}

where

\begin{align*}\mathcal{R}_{i}=\{\pi\in\mathcal{P}_{i} \mid &\exists\pi'\in\mathcal{P}_{i}, \text{ such that }\forall\pi''\in\mathcal{R}_{i}, \pi''<\pi' \land \pi'' \triangleleft (\mathcal{P}_{i}-\mathcal{R}_{i}) \text{ and}\\[3pt]&\not\exists \mathcal{R}'_{i}\subseteq\mathcal{P}_{i} \text{ such that }\exists\pi''\in\mathcal{R}_{i},\forall\pi'\in\mathcal{R}'_{i}, \pi'<\pi'' \land \pi' \triangleleft (\mathcal{P}_{i}-\mathcal{R}'_{i}) \},\end{align*}

and $\pi\triangleleft\mathcal{Q}$ denotes that rule $\pi$ is defeated by program $\mathcal{Q}$ , which is true if there exists some objective literal $l\in I\in[\kern-.3ex[{{\mathcal{Q}}}]\kern-.3ex]_{\mathsf{SM}}$ such that $l\in B(\pi)^-$ .

A prioritised logic program may have zero or more reducts and the preferred stable models are all the stable models of all the reducts. For a detailed discussion of reducts and their properties the reader can refer to the paper by Zhang (Reference Zhang2003).

Subsequently, the update semantics defined by Zhang (2006) performs an update of a program $\mathcal{P}$ by a program $\mathcal{U}$ by executing the following steps:

1. 1. Take some stable model $J_\mathcal{P}$ of $\mathcal{P}$ .

2. 2. Determine the set $\mathit{Update}(J_\mathcal{P},\mathcal{U})$ of interpretations resulting from updating $J_\mathcal{P}$ by $\mathcal{U}$ , in a way similar to to the approach of Marek and Truszczynski (Reference Marek and Truszczynski1994, Reference Marek and Truszczynski1998), given by:

   \begin{equation*} \mathit{Update}(J_\mathcal{P},\mathcal{U}) = { I \in [\kern-.3ex[ {\mathcal{U}} ]\kern-.3ex]_{\mathsf{C}} | \lnot \exists J \in [\kern-.3ex[ {\mathcal{U}} ]\kern-.3ex]_{\mathsf{C}} : (J \div J_\mathcal{P}) \subsetneq (I \div J_\mathcal{P}) },\end{equation*}
   where $\div$ denotes set-theoretic symmetric difference. Choose any interpretation from $\mathit{Update}(J_\mathcal{P},\mathcal{U})$ and denote it by $J_{\langle \mathcal{P}, \mathcal{U} \rangle}$ .

3. 3. Extract a maximal subset $\mathcal{P}'$ of $\mathcal{P}$ that is coherent with $J_{\langle \mathcal{P}, \mathcal{U} \rangle}$ , that is, such that there exists a stable model of $\mathcal{P}' \cup {l. | l \in J_{\langle \mathcal{P}, \mathcal{U} \rangle}}$ .

4. 4. The set $({\mathcal{P}' \cup \mathcal{U}, \mathcal{P}' \times \mathcal{U}})^<$ of reducts of the prioritised logic program $({\mathcal{P}' \cup \mathcal{U}, \mathcal{P}' \times \mathcal{U}})$ is the result of updating $\mathcal{P}$ by $\mathcal{U}$ .

As explained byZhang (2006), the intuition behind the first two steps is that simply taking a maximal subset of $\mathcal{P}$ coherent with $\mathcal{U}$ is too crude an operation because it does not take into account the source of a conflict.

Example 27 (Intuition For Steps 1. and 2. Zhang Reference Zhang2006) Consider the programs

\begin{align*} \begin{aligned} \mathcal{P}: && \mathsf{p} &. \\ && \mathsf{q} &\leftarrow \mathsf{r}. \end{aligned} && \text{and} && \begin{aligned} \mathcal{U}: && \mathsf{r} &\leftarrow \mathsf{p}. \\ && \lnot \mathsf{q} &\leftarrow \mathsf{r}. \end{aligned} \end{align*}

Since $\mathcal{P} \cup \mathcal{U}$ is incoherent, some part of $\mathcal{P}$ needs to be eliminated to regain coherence. There are two maximal subsets of $\mathcal{P}$ that are coherent with $\mathcal{U}$ : ${\mathsf{p}.}$ and ${\mathsf{q} \leftarrow \mathsf{r}.}$ . However, intuition suggests that the former set is preferable since the direct conflict between rules $(\mathsf{q} \leftarrow \mathsf{r}.)$ and $(\lnot \mathsf{q} \leftarrow \mathsf{r}.)$ provides a justification for eliminating the rule $(\mathsf{q} \leftarrow \mathsf{r}.)$ and thus keeping the fact $(\mathsf{p}.)$ .

The approach taken, then, is to first consider a stable model of $\mathcal{P}$ and update it by $\mathcal{U}$ , obtaining a new interpretation $J_{\langle \mathcal{P}, \mathcal{U} \rangle}$ that reflects the new information in $\mathcal{U}$ . Afterwards, a maximal set of rules from $\mathcal{P}$ coherent with $J_{\langle \mathcal{P}, \mathcal{U} \rangle}$ is used to form a prioritised logic program that prefers rules from $\mathcal{U}$ over rules from $\mathcal{P}$ . The reducts of this program form the result of the update.

Due to the possibility of having multiple reducts as possible results of the update, it is not completely clear how updates can be iterated. Do we choose one reduct and commit to it? Which one do we choose, then? Or do we simply consider all of the reducts and all possible evolutions? Due to these unresolved issues, we formally define this semantics only for DLPs of length two. We call it preference-based Zhang’s semantics, or $\textsf{PRZ}$ semantics for short.

One distinguishing feature of the $\textsf{PRZ}$ semantics is that by relying on a stable model of $\mathcal{P}$ for conflict resolution, it is unable to detect “latent” conflicts between rules that have not been “triggered” in the initial stable model or its update. This is illustrated in the following example: ^{Footnote 5}

Example 29 (Undetected Latent Conflicts in the $\textsf{PRZ}$ semantics) Consider the programs

\begin{align*} \begin{aligned} \mathcal{P}: && \mathsf{p} &\leftarrow \mathsf{r}. \\ && \mathsf{q} &\leftarrow \mathsf{r}. \end{aligned} && \text{and} && \begin{aligned} \mathcal{U}: && \mathsf{r}&. \\ && \lnot \mathsf{p} &\leftarrow \mathsf{q}. \end{aligned} \end{align*}

and let $\boldsymbol{P} = \langle \mathcal{P}, \mathcal{U} \rangle$ . The single stable model of $\mathcal{P}$ is $J_\mathcal{P} = \emptyset$ and its update by $\mathcal{U}$ results in the interpretation $J_{\langle \mathcal{P}, \mathcal{U} \rangle} = {\mathsf{r}}$ which is coherent with $\mathcal{P}$ . The resulting prioritised logic program $({\mathcal{P} \cup \mathcal{U}, \mathcal{P} \times \mathcal{U}})$ has only one reduct, $\mathcal{P} \cup \mathcal{U}$ , that has no stable model. In other words, $[\kern-.3ex[{\boldsymbol{P}}]\kern-.3ex]_{\mathsf{PRZ}} = \emptyset$ and the conflict between $\mathcal{P}$ and $\mathcal{U}$ remained unresolved. Note also that $[\kern-.3ex[{\boldsymbol{P}}]\kern-.3ex]_{\mathsf{AS}} = [\kern-.3ex[{\boldsymbol{P}}]\kern-.3ex]_{\mathsf{JU}} = [\kern-.3ex[{\boldsymbol{P}}]\kern-.3ex]_{\mathsf{DS}} = [\kern-.3ex[{\boldsymbol{P}}]\kern-.3ex]_{\mathsf{RD}} = {{\lnot \mathsf{p}, \mathsf{q}, \mathsf{r}}}$ .

The $\textsf{PRZ}$ semantics is also sensitive to tautological updates:

Example 30 (Tautological Updates in the $\textsf{PRZ}$ semantics) Consider the programs

\begin{align*} \begin{aligned} \mathcal{P}: && \mathsf{p} &\leftarrow \mathop{\sim\!} \lnot \mathsf{p}. \\ && \lnot \mathsf{p} &\leftarrow \mathop{\sim\!} \mathsf{p}. \end{aligned} && \text{and} && \begin{aligned} \mathcal{U}: && \mathsf{p} &\leftarrow \mathsf{p}. \end{aligned} \end{align*}

Both stable models ${\mathsf{p}}$ and ${\lnot \mathsf{p}}$ of $\mathcal{P}$ remain unchanged after an update by $\mathcal{U}$ and thus both rules of $\mathcal{P}$ are retained in the resulting prioritised logic program $({\mathcal{P} \cup \mathcal{U}, \mathcal{P} \times \mathcal{U}})$ . Its only reduct, however, is the program ${\mathsf{p} \leftarrow \mathop{\sim\!} \lnot \mathsf{p}., \mathsf{p} \leftarrow \mathsf{p}.}$ , which has a single stable model ${\mathsf{p}}$ . The tautological update has thus discarded one of the stable models of $\mathcal{P}$ .

5.3.1 The $\textsf{RVS}$ semantics

Sakama and Inoue (Reference Sakama and Inoue2003) have proposed a rule update semantics that is clearly based on ideas from belief revision, similarly as formula-based belief update operators. In particular, they define that a program $\mathcal{P}' \cup \mathcal{U}$ achieves the update of $\mathcal{P}$ by $\mathcal{U}$ if $\mathcal{P}'$ is a maximal subset of $\mathcal{P}$ such that $\mathcal{P}' \cup \mathcal{U}$ is coherent, that is, it has a stable model.

As with the $\textsf{PRZ}$ semantics, we define the semantics of Sakama and Inoue (Reference Sakama and Inoue2003) only for DLPs of length two because it is not clear how one should deal with multiple results of an update. We call the resulting semantics the $\textsf{RVS}$ semantics:

Similarly to the $\textsf{PRZ}$ semantics, as discussed in Example 27, the approach adopted by the $\textsf{RVS}$ semantics pays no attention to the source of conflicts – any solution of a conflict is as good as any other as long as only a minimal set of rules is eliminated. Another consequence is that conflicts are removed at any cost, even if there is no plausible way to explain why the update should restore coherence. This has been criticised by Leite (Reference Leite2003), who argued that every conflict has several causes and each type of conflict should be dealt with accordingly. One consequence of this is that an empty or tautological update may restore coherence (and consistency) of an initial program. If we compare this to belief change principles and operators, such a behaviour is typical of revision but is not desirable for updates. Garcia et al. (Reference Garcia, Lefèvre, Stéphan, Papini and Würbel2019) introduced a family of revision operators for logic programs similar in spirit to the $\textsf{RVS}$ semantics, except that they allow both for the addition and/or removal of rules from a logic program to achieve coherence. As with the $\textsf{RVS}$ semantics, these operators are sensitive to empty or tautological updates.

5.3.2 The $\textsf{RVD}$ semantics

Similar ideas form the basis of the rule revision semantics proposed by Delgrande (Reference Delgrande2010). Note that since the distinction between program update and revision, as these terms are used in the literature, is somewhat blurry, in the following we also refer to this semantics as an update semantics. Informally, the stable model of a sequence of programs is constructed by first keeping all rules from the last program and committing to a minimal set of default literals used to derive one of its stable models. Subsequently, a maximal coherent subset of the previous program is added and further commitments are made. This process is iterated until the first program of the sequence is processed, as illustrated by the following example.

Example 34 Consider the DLP

\begin{equation*} \boldsymbol{P} = \langle {\mathsf{p}.}, {\mathsf{q}.}, {\mathsf{r} \leftarrow \mathop{\sim\!} \mathsf{p}., \mathsf{r} \leftarrow \mathop{\sim\!} \mathsf{q}.} \rangle . \end{equation*}

We start with the last program of the sequence which has the stable model ${\mathsf{r}}$ . This stable model can be derived either using the literal $\mathop{\sim\!} \mathsf{p}$ or $\mathop{\sim\!} \mathsf{q}$ and we need to choose one of these and commit to it. If we pick the former, the overall set of literals we commit to at this stage is ${\mathop{\sim\!} \mathsf{p}, \mathsf{r}}$ . We then proceed to the second program and realise that it is coherent with our commitments as well as the rules from the last program. We thus add $\mathsf{q}$ to our set of commitments and the fact $(\mathsf{q}.)$ to the set of rules that we are going to keep. Proceeding to the first program of the sequence, the rule within it is inconsistent with our commitment to $\mathop{\sim\!} \mathsf{p}$ , so the rule needs to be discarded. The set of objective literals we committed to until now, namely ${\mathsf{q}, \mathsf{r}}$ , forms one stable model of $\boldsymbol{P}$ . Note that if we initially commit to $\mathop{\sim\!} \mathsf{q}$ , we obtain the stable model ${\mathsf{p}, \mathsf{r}}$ .

To formalise this construction, Delgrande (Reference Delgrande2010). uses three-valued interpretations, defined as pairs of disjoint interpretations $J = (J^+,J^-)$ , which we dub three-valued d-interpretations to distinguish from the three-valued interpretations used elsewhere in this paper, based on which a modified notion of reduct of a program $\mathcal{P}$ without default negation in the head is defined as

\begin{equation*}\mathcal{P}^J = \{\langle S^+[H(\pi)] \leftarrow S^+[B(\pi)]\cup\mathop{\sim\!}S^-[B(\pi)]\backslash J^- \rangle | \pi \in \mathcal{P} \land J^+ \cap S^-[B(\pi)] = \emptyset\}.\end{equation*}

This reduct is used to define a special notion of three-valued answer-sets of a program $\mathcal{P}$ which are those three-valued d-interpretations $J = (J^+,J^-)$ such that $\mathsf{least}(\mathcal{P}^{J^+})=\mathsf{least}(\mathcal{P}^{J})=J^+$ and for any $I = (J^+,I^-)$ such that $I^- \subset J^-$ we have that $\mathsf{least}(P^{I}) \neq J^+$ , where $\mathsf{least}(\cdot)$ is as before. Additionally, Delgrande (Reference Delgrande2010). defines a concept of canonical program corresponding to a three-valued d-interpretation $J = (J^+,J^-)$ as

\begin{equation*}Pgm(J)=\{ \mathsf{p}. \mid \mathsf{p} \in J^+ \}\;\cup\;\{ \leftarrow \mathsf{p}. \mid \mathsf{p} \in J^- \}.\end{equation*}

Then, given a DLP without default negation in heads of rules $\boldsymbol{P} = \langle \mathcal{P}_i \rangle_{i < n+1}$ , an interpretation J is an r-answer-set of $\boldsymbol{P}$ iff there is a sequence $\langle (\mathcal{P}^r_i,J_i) \rangle_{i < n+1}$ such that

1. 1. $\mathcal{P}^r_n = \mathcal{P}_n$ and $J_n$ is a three-valued answer-set of $\mathcal{P}_n$ ;

2. 2. for $i < n$ , $\mathcal{P}^r_i$ is a maximal set of rules of $\mathcal{P}_i$ consistent with $\mathcal{P}^r_{i+1} \cup Pgm(J_{i+1})$ (or $\mathscr{L}$ if $\mathcal{P}^r_{i+1} \cup Pgm(J_{i+1})$ is inconsistent) and $J_i$ is a three-valued answer-set of $\mathcal{P}^r_i$ ;

3. 3. $J=J^+_1$ .

We refer to this semantics as the $\textsf{RVD}$ semantics:

Similarly as the $\textsf{RVS}$ semantics, the $\textsf{RVD}$ semantics resolves conflicts at any cost, a consequence of which is that empty and tautological updates restore coherence and consistency. Furthermore, it exhibits the same behaviour as the $\textsf{PRX}_i$ semantics in Example 32, that is, it prefers to satisfy default assumptions in further programs to satisfying earlier facts. It actually goes even further than the $\textsf{PRX}_i$ semantics, as illustrated in the following example:

Example 36 (Default Assumptions vs. Facts in the $\textsf{RVD}$ semantics) Consider the programs

\begin{align*} \begin{aligned} \mathcal{P}: && \mathsf{p} &. \end{aligned} && \text{and} && \begin{aligned} \mathcal{U}: && \mathsf{q} &\leftarrow \mathop{\sim\!} \mathsf{p}. \end{aligned} \end{align*}

and let $\boldsymbol{P} = \langle \mathcal{P}, \mathcal{U} \rangle$ . We obtain $[\kern-.3ex[{\boldsymbol{P}}]\kern-.3ex]_{\mathsf{RVD}} = {{\mathsf{q}}}$ , as opposed to $[\kern-.3ex[{\boldsymbol{P}}]\kern-.3ex]_{\mathsf{AS}} = [\kern-.3ex[{\boldsymbol{P}}]\kern-.3ex]_{\mathsf{JU}} = [\kern-.3ex[{\boldsymbol{P}}]\kern-.3ex]_{\mathsf{DS}} = [\kern-.3ex[{\boldsymbol{P}}]\kern-.3ex]_{\mathsf{RD}} = [\kern-.3ex[{\boldsymbol{P}}]\kern-.3ex]_{\mathsf{PRZ}} = [\kern-.3ex[{\boldsymbol{P}}]\kern-.3ex]_{\mathsf{PRX_i}} = [\kern-.3ex[{\boldsymbol{P}}]\kern-.3ex]_{\mathsf{RVS}} = {{\mathsf{p}}}$ . This indicates that the default assumptions in the updating program are given preference over facts from the initial program even more aggressively than in case of the $\textsf{PRX}_i$ semantics. If we interpret $\mathsf{p}$ as $\mathsf{dog}(\mathit{bo})$ and $\mathsf{q}$ as $\lnot \mathsf{canBark}(\mathit{bo})$ , then this example shows that if initially $\mathsf{dog}(\mathit{bo})$ is known to be true and later we learn that

\begin{equation*} \lnot \mathsf{canBark}(\mathbf{x}) \leftarrow \mathop{\sim\!} \mathsf{dog}(\mathbf{x}). , \end{equation*}

meaning that, by default, individuals that are not dogs cannot bark, then this immediately modifies our knowledge about $\mathit{bo}$ : we no longer know whether $\mathsf{dog}(\mathit{bo})$ is true or not and, in addition, we conclude that $\mathsf{canBark}(\mathit{bo})$ is (explicitly) false. Poor $\mathit{bo}$ …

A methodology based on maximal subsets of the initial program coherent with its update was also used by Osorio and Zepeda (Reference Osorio and Zepeda2007) for updating programs under the pstable model semantics. The idea is used indirectly by augmenting the bodies of original rules with additional literals and using an abductive framework to minimise the set of rules “disabled” by falsifying the added literal. We do not further consider this semantics because it diverges from the standard notion of a stable model and uses pstable models instead.

Finally, there also exist approaches based on a semantic framework that directly encodes literal dependencies induced by rules, and performs changes on the dependencies instead of on the rules themselves. The advantage over dealing with rules is that the dependency relation is monotonic, so AGM postulates and operators can be applied to it directly (KrÜmpelmann and Kern-Isberner Reference Krümpelmann, Kern-Isberner, Meyer and Ternovska2010; KrÜmpelmann Reference Krümpelmann2012). In the work of SefrÁnek (2006, 2011), the dependency framework is used for specifying irrelevant updates, an instance of which are tautological updates, and designing update semantics immune to such irrelevant updates.

6.1.1 Revision based on $\textsf{HT}$ models

For a rule revision operator $\mathbin{\otimes}$ and programs $\mathcal{P}$ , $\mathcal{Q}$ , $\mathcal{U}$ , $\mathcal{V}$ , acording to Delgrande et al. (Reference Delgrande, Schaub, Tompits and Woltran2008, Reference Delgrande, Schaub, Tompits and Woltran2013), the KM postulates for belief revision are adapted to answer-set program revision using $\textsf{HT}$ models as follows:

1. ( $\textsf{PR1}_{\textsf{HT}}$ ) $\mathcal{P} \mathbin{\otimes} \mathcal{U} \mathrel\mid=_{\mathsf{HT}} \mathcal{U}$ .

2. ( $\textsf{PR2}_{\textsf{HT}}$ ) If $[\kern-.3ex[{\mathcal{P}\mathbin{\dot{\wedge}}\mathcal{Q}}]\kern-.3ex]_{\mathsf{HT}}\neq \emptyset$ , then $\mathcal{P} \mathbin{\otimes} \mathcal{U} \equiv_{\mathsf{HT}} \mathcal{P}\mathbin{\dot{\wedge}}\mathcal{Q}$ .

3. ( $\textsf{PR3}_{\textsf{HT}}$ ) If $[\kern-.3ex[{\mathcal{U}}]\kern-.3ex]_{\mathsf{HT}} \neq \emptyset$ , then $[\kern-.3ex[{\mathcal{P} \mathbin{\otimes} \mathcal{U}}]\kern-.3ex]_{\mathsf{HT}} \neq \emptyset$ . [ $\textsf{PR4}_{\textsf{HT}}$ ] If $\mathcal{P} \equiv_{\mathsf{HT}} \mathcal{Q}$ and $\mathcal{U} \equiv_{\mathsf{HT}} \mathcal{V}$ , then $\mathcal{P} \mathbin{\otimes} \mathcal{U} \equiv_{\mathsf{HT}} \mathcal{Q} \mathbin{\otimes} \mathcal{V}$ .

4. ( $\textsf{PR5}_{\textsf{HT}}$ ) $(\mathcal{P} \mathbin{\otimes} \mathcal{U}) \mathbin{\dot{\wedge}} \mathcal{V} \mathrel\mid=_{\mathsf{HT}} \mathcal{P} \mathbin{\otimes} (\mathcal{U} \mathbin{\dot{\wedge}} \mathcal{V})$ .

5. ( $\textsf{PR6}_{\textsf{HT}}$ ) If $[\kern-.3ex[{(\mathcal{P} \mathbin{\otimes} \mathcal{U})\mathbin{\dot{\wedge}}\mathcal{V}}]\kern-.3ex]_{\mathsf{HT}}\neq \emptyset$ then $\mathcal{P} \mathbin{\otimes} (\mathcal{U}\mathbin{\dot{\wedge}}\mathcal{V}) \mathrel\mid=_{\mathsf{HT}} (\mathcal{P} \mathbin{\otimes} \mathcal{U})\mathbin{\dot{\wedge}}\mathcal{V}$ .

As discussed by Delgrande et al. (Reference Delgrande, Schaub, Tompits and Woltran2008, Reference Delgrande, Schaub, Tompits and Woltran2013) no $\mathbin{\otimes}$ can be immune to tautologies or empty updates – whenever $\mathcal{P}$ is unsatisfiable and $\mathcal{U}$ is tautological or empty, immunity to tautologies and immunity to empty updates conflicts with $\textsf{PR3}_{\textsf{HT}}$ . As also discussed by Delgrande et al. (Reference Delgrande, Schaub, Tompits and Woltran2008, Reference Delgrande, Schaub, Tompits and Woltran2013) immunity to tautologies and empty updates could have been adopted instead of $\textsf{PR3}_{\textsf{HT}}$ .

Analogically to belief revision, a constructive characterisation of rule revision operators satisfying conditions $\textsf{PR1}_{\textsf{HT}}$ – $\textsf{PR6}_{\textsf{HT}}$ is based on an order assignment. Since the set of $\textsf{HT}$ models of any program $\mathcal{P}$ must be well-defined, that is, $({I, J})\in[\kern-.3ex[{\mathcal{P}}]\kern-.3ex]_{\mathsf{HT}}$ implies that $({J, J})\in[\kern-.3ex[{\mathcal{P}}]\kern-.3ex]_{\mathsf{HT}}$ , not every order assignment characterises a rule revision operator. Slota and Leite (Reference Slota and Leite2010, Reference Slota and Leite2014) additionally define well-defined order assignments as those that do.

Definition 53 (Rule Revision Operator Characterised by an Order Assignment) Let $\mathbin{\otimes}$ be a rule revision operator and $\omega$ a preorder assignment over the set of all logic programs. We say that $\mathbin{\otimes}$ is characterised by $\omega$ if for all programs $\mathcal{P}$ , $\mathcal{U}$ ,

\begin{equation*} [\kern-.3ex[{\mathcal{P} \mathbin{\otimes} \mathcal{U}}]\kern-.3ex]_{\mathsf{HT}} = \min \left( [\kern-.3ex[{\mathcal{U}}]\kern-.3ex]_{\mathsf{HT}}, \leq^{\mathcal{P}}_{\omega} \right) . \end{equation*}

A preorder assignment over over the set of all programs is well-defined if some rule update operator is characterised by it.

Similarly as with belief revision, the following definition captures a set of conditions on the assigned orders.

• $\text{If }I,J\in[\kern-.3ex[{\mathcal{P}}]\kern-.3ex]_{\mathsf{HT}}\text{, then }I\not<^{\mathcal{P}}_{\omega}J$ ;

• $\text{If }I\in[\kern-.3ex[{\mathcal{P}}]\kern-.3ex]_{\mathsf{HT}}\text{ and }J\not\in[\kern-.3ex[{\mathcal{P}}]\kern-.3ex]_{\mathsf{HT}}\text{, then }I<^{\mathcal{P}}_{\omega}J$ ;

• $\text{If }[\kern-.3ex[{\mathcal{P}}]\kern-.3ex]_{\mathsf{HT}}=[\kern-.3ex[{\mathcal{Q}}]\kern-.3ex]_{\mathsf{HT}}\text{, then }<^{\mathcal{P}}_{\omega}=<^{\mathcal{Q}}_{\omega}$ .

The representation theorem of Delgrande et al. (Reference Delgrande, Schaub, Tompits and Woltran2013) states that operators characterised by faithful order assignments over the set of all programs are exactly those that satisfy $\textsf{PR1}_{\textsf{HT}}$ – $\textsf{PR6}_{\textsf{HT}}$ .

• The operator $\mathbin{\otimes}$ satisfies conditions $\textsf{PR1}_{\textsf{HT}}$ – $\textsf{PR6}_{\textsf{HT}}$ .

• The operator $\mathbin{\otimes}$ is characterised by a faithful total preorder assignment over the set of all programs.

Schwind and Inoue (Reference Schwind and Inoue2016) provided an alternative constructive characterisation of logic programs revision operators in terms of preorders over interpretations. Delgrande et al. (Reference Delgrande, Schaub, Tompits and Woltran2008, Reference Delgrande, Schaub, Tompits and Woltran2013) present two different revision operators, one analogous to the so-called set-containment based revision proposed by Satoh (Reference Satoh1988), and the other analogous to the so-called cardinality-based revision proposed by Dalal (Reference Dalal1988). Here, we recap the latter one, because, unlike the former, it obeys all six postulates $\textsf{PR1}_{\textsf{HT}}$ – $\textsf{PR6}_{\textsf{HT}}$ .

The operator is based on a notion of closeness that is given in terms of cardinality:

Definition 56 Let $\mathcal{M}$ and $\mathcal{N}$ be sets of either interpretations or three-valued interpretations. Then,

\begin{equation*}\sigma_{\vert\vert}(\mathcal{M},\mathcal{N})=\{X\in\mathcal{M}\vert\exists Y\in\mathcal{N}\text{ such that }\forall X'\in\mathcal{M},\forall Y'\in\mathcal{N},\vert X'\div Y'\vert\not<\vert X\div Y\vert\},\end{equation*}

where $\div$ denotes the symmetric difference, extended for three-valued interpretations $\langle I, J \rangle$ and $\langle K, L \rangle$ as follows:

\begin{equation*}\langle I, J \rangle\div \langle K, L \rangle = \langle I\div K, J\div L \rangle,\end{equation*}

and

\begin{align*}\vert\langle I, J \rangle\vert\leq\vert\langle K, L \rangle\vert\text{ iff }\vert J\vert\leq\vert L\vert\text{ and if }\vert J\vert = \vert L\vert\text{ then }\vert I\vert\leq\vert K\vert;\\[4pt]\vert\langle I, J \rangle\vert < \vert\langle K, L \rangle\vert\text{ iff } \vert\langle I, J \rangle\vert\leq\vert\langle K, L \rangle\vert\text{ and } \vert\langle K,L \rangle\vert\not\leq\vert\langle {I, J} \rangle\vert.\end{align*}

The operator is then defined as follows:

Definition 57 (Cardinality-based Program Revision Delgrande et al. 2008;2013) Let $\mathcal{P}$ and $\mathcal{U}$ be two logic programs. The cardinality-based revision operator $\mathbin{\otimes}_c$ is defined, up to equivalence of its input and output, as a logic program such that:

\begin{align*}[\kern-.3ex[{\mathcal{P}\mathbin{\otimes}_c\mathcal{Q}}]\kern-.3ex]_{\mathsf{HT}}=&[\kern-.3ex[{\mathcal{Q}}]\kern-.3ex]_{\mathsf{HT}},&\text{if }[\kern-.3ex[{\mathcal{P}}]\kern-.3ex]_{\mathsf{HT}}=\emptyset;\\[4pt][\kern-.3ex[{\mathcal{P}\mathbin{\otimes}_c\mathcal{Q}}]\kern-.3ex]_{\mathsf{HT}}=&\{\langle I, J\rangle\vert J\in\sigma_{\vert\vert}([\kern-.3ex[ {\mathcal{P}} ]\kern-.3ex]_{\mathsf{C}},[\kern-.3ex[ {\mathcal{Q}} ]\kern-.3ex]_{\mathsf{C}}),I\subseteq J,\\[4pt]&\enspace\text{ and if }I\subset J\text{ then } \langle {I, J} \rangle\in\sigma_{\vert\vert}([\kern-.3ex[{\mathcal{P}}]\kern-.3ex]_{\mathsf{HT}},[\kern-.3ex[{\mathcal{Q}}]\kern-.3ex]_{\mathsf{HT}})\},&\text{otherwise}.\end{align*}

6.1.2 Updates based on $\textsf{HT}$ models

To adapt the KM postulates for updates to answer-set program updates, Slota and Leite (Reference Slota and Leite2010, Reference Slota and Leite2014) substitute the notion of a complete formula used in $\mathsf{BU7}$ with the notion of a basic program, that is, a program that either it has a unique $\textsf{HT}$ model (J, J), or a pair of $\textsf{HT}$ models (I, J) and (J, J). In the former case, the program exactly determines the truth values of all atoms – the atoms in J are true and the remaining atoms are false. In the latter case, the program makes atoms in I true, the atoms in $J \backslash I$ may either be undefined or true, as long as they all have the same truth value, and the remaining atoms are false. The latter case needs to be allowed in order to make the new postulate applicable to three-valued interpretations (I, J) with $I \subsetneq J$ because no program has the single $\textsf{HT}$ model (I, J).

Using $\textsf{HT}$ models, the adaptation of the KM postulates for updates to answer-set program updates, which, for a rule update operator $\mathbin{\oplus}$ and programs $\mathcal{P}$ , $\mathcal{Q}$ , $\mathcal{U}$ , $\mathcal{V}$ become:

1. ( $\textsf{PU1}_{\textsf{HT}}$ ) $\mathcal{P} \mathbin{\oplus} \mathcal{U} \mathrel\mid=_{\mathsf{HT}} \mathcal{U}$ .

2. ( $\textsf{PU2}_{\textsf{HT}}$ ) If $\mathcal{P} \mathrel\mid=_{\mathsf{HT}} \mathcal{U}$ , then $\mathcal{P} \mathbin{\oplus} \mathcal{U} \equiv_{\mathsf{HT}} \mathcal{P}$ .

3. ( $\textsf{PU3}_{\textsf{HT}}$ ) If $[\kern-.3ex[{\mathcal{P}}]\kern-.3ex]_{\mathsf{HT}} \neq \emptyset$ and $[\kern-.3ex[{\mathcal{U}}]\kern-.3ex]_{\mathsf{HT}} \neq \emptyset$ , then $[\kern-.3ex[{\mathcal{P} \mathbin{\oplus} \mathcal{U}}]\kern-.3ex]_{\mathsf{HT}} \neq \emptyset$ . ( $\textsf{PU4}_{\textsf{HT}}$ ) If $\mathcal{P} \equiv_{\mathsf{HT}} \mathcal{Q}$ and $\mathcal{U} \equiv_{\mathsf{HT}} \mathcal{V}$ , then $\mathcal{P} \mathbin{\oplus} \mathcal{U} \equiv_{\mathsf{HT}} \mathcal{Q} \mathbin{\oplus} \mathcal{V}$ .

4. ( $\textsf{PU5}_{\textsf{HT}}$ ) $(\mathcal{P} \mathbin{\oplus} \mathcal{U}) \mathbin{\dot{\wedge}} \mathcal{V} \mathrel\mid=_{\mathsf{HT}} \mathcal{P} \mathbin{\oplus} (\mathcal{U} \mathbin{\dot{\wedge}} \mathcal{V})$ .

5. ( $\textsf{PU6}_{\textsf{HT}}$ ) If $\mathcal{P} \mathbin{\oplus} \mathcal{U} \mathrel\mid=_{\mathsf{HT}} \mathcal{V}$ and $\mathcal{P} \mathbin{\oplus} \mathcal{V} \mathrel\mid=_{\mathsf{HT}} \mathcal{U}$ , then $\mathcal{P} \mathbin{\oplus} \mathcal{U} \equiv_{\mathsf{HT}} \mathcal{P} \mathbin{\oplus} \mathcal{V}$ .

6. ( $\textsf{PU7}_{\textsf{HT}}$ ) If $\mathcal{P}$ is basic, then $(\mathcal{P} \mathbin{\oplus} \mathcal{U}) \mathbin{\dot{\wedge}} (\mathcal{P} \mathbin{\oplus} \mathcal{V}) \mathrel\mid=_{\mathsf{HT}} \mathcal{P} \mathbin{\oplus} (\mathcal{U} \mathbin{\dot{\lor}} \mathcal{V})$ .

7. ( $\textsf{PU8}_{\textsf{HT}}$ ) $(\mathcal{P} \mathbin{\dot{\lor}} \mathcal{Q}) \mathbin{\oplus} \mathcal{U} \equiv_{\mathsf{HT}} (\mathcal{P} \mathbin{\oplus} \mathcal{U}) \mathbin{\dot{\lor}} (\mathcal{Q} \mathbin{\oplus} \mathcal{U})$ .

Analogically to belief updates, a constructive characterisation of rule update operators satisfying conditions $\textsf{PU1}_{\textsf{HT}}$ – $\textsf{PU8}_{\textsf{HT}}$ is based on an order assignment, but this time over the set of all three-valued interpretations $\mathscr{X}$ . Since the set of $\textsf{HT}$ models of any program $\mathcal{P}$ must be well-defined that is, $({I, J})\in[\kern-.3ex[{\mathcal{P}}]\kern-.3ex]_{\mathsf{HT}}$ implies that $({J, J})\in[\kern-.3ex[{\mathcal{P}}]\kern-.3ex]_{\mathsf{HT}}$ , not every order assignment characterises a rule update operator. Slota and Leite (Reference Slota and Leite2010, Reference Slota and Leite2014) additionally define well-defined order assignments as those that do.

Definition 59 (Rule Update Operator Characterised by an Order Assignment) Let $\mathbin{\oplus}$ be a rule update operator and $\omega$ a preorder assignment over $\mathscr{X}$ . We say that $\mathbin{\oplus}$ is characterised by $\omega$ if for all programs $\mathcal{P}$ , $\mathcal{U}$ ,

\begin{equation*} [\kern-.3ex[{\mathcal{P} \mathbin{\oplus} \mathcal{U}}]\kern-.3ex]_{\mathsf{HT}} = \bigcup_{X \in [\kern-.25ex[{\mathcal{P}}]\kern-.25ex]_{\mathsf{HT}}} \min \left( [\kern-.3ex[{\mathcal{U}}]\kern-.3ex]_{\mathsf{HT}}, \leq^{X}_{\omega} \right) . \end{equation*}

A preorder assignment over $\mathscr{X}$ is well-defined if some rule update operator is characterised by it.

Similarly as with belief updates, order assignments are required to be faithful, that is to consider each three-valued interpretation the closest to itself.

Definition 60 (Faithful Order Assignment Over Three-valued Interpretations) A preorder assignment $\omega$ over $\mathscr{X}$ is faithful if for every three-valued interpretation X the following condition is satisfied:

\begin{equation*} \text{For every } Y \in \mathscr{X} \text{ with } Y \neq X \text{ it holds that } X <^{X}_{\omega} Y . \end{equation*}

Interestingly, faithful assignments characterise the same class of operators as the larger class of semi-faithful assignments, defined as follows:

Definition 61 (Semi-Faithful Order Assignment) A preorder assignment $\omega$ over $\mathscr{X}$ is semi-faithful if for every three-valued interpretation $X = ({I, J})$ the following conditions are satisfied (where $X^* = ({J, J})$ ):

\begin{align*} &\text{For every }Y \in \mathscr{X} \text{ with }Y \neq X\text{ and }Y \neq X^*\text{ , either }X <^{X}_{\omega} Y\text{ or }X^* <^{X}_{\omega} Y.\\[4pt] &\text{If }X^* \leq^{X}_{\omega} X \text{, then }X \leq^{X}_{\omega} X^*. \end{align*}

Preorder assignments need to satisfy one further condition, related to the well-definedness of sets of SE models of every program. It can naturally be seen as the semantic counterpart of $\textsf{PU7}_{\textsf{HT}}$ .

Definition 62 (Organised Preorder Assignment) A preorder assignment $\omega$ is organised if for all three-valued interpretations X, Y and all well-defined sets of three-valued interpretations $\mathcal{M}$ , $\mathcal{N}$ the following condition is satisfied:

\begin{align*} & \text{If } Y \in \min ( \mathcal{M}, \leq^{X}_{\omega} ) \cup \min ( \mathcal{M}, \leq^{X^*}_{\omega} ) \text{ and } Y \in \min ( \mathcal{N}, \leq^{X}_{\omega} ) \cup \min ( \mathcal{N}, \leq^{X^*}_{\omega} ), \\[3pt] & \text{then } Y \in \min ( \mathcal{M} \cup \mathcal{N}, \leq^{X}_{\omega} ) \cup \min (\mathcal{M} \cup \mathcal{N}, \leq^{X^*}_{\omega} ). \end{align*}

Just as with updates in classical logic, the following representation theorem provides a constructive characterisation of rule update operators satisfying the postulates, making it possible to define and evaluate any operator satisfying the postulates using an intuitive construction.

1. 1. The operator $\mathbin{\oplus}$ satisfies conditions $\textsf{PU1}_{\textsf{HT}}$ – $\textsf{PU8}_{\textsf{HT}}$ .

2. 2. The operator $\mathbin{\oplus}$ is characterised by a semi-faithful and organised preorder assignment.

3. 3. The operator $\mathbin{\oplus}$ is characterised by a faithful and organised partial order assignment.

One of the benefits of dealing with rule updates on the semantic level is that semantic properties that are rather difficult to show for syntax-based update operators are much easier to analyse and prove. As we have seen, one of the most widespread and counterintuitive side-effects of syntax-based rule update semantics is that they are sensitive to tautological updates. In case of the semantic update operators characterised in the previous theorem, such a behaviour is impossible given that the operators satisfy $\textsf{PU2}_{\textsf{HT}}$ and $\textsf{PU4}_{\textsf{HT}}$ .

Slota and Leite (Reference Slota and Leite2014) additionally defined a concrete update operator that can be seen as a counterpart to the belief update operator by Winslett (Reference Winslett1988), which we do not present here.

6.1.3 Problems with revision and update operators based on $\textsf{HT}$ models

The most important contribution by Slota and Leite (Reference Slota and Leite2010, Reference Slota and Leite2014) is not the adaptation of the postulates to answer-set program updates, the representation theorem, nor even the concrete update operator they defined, but rather the uncovering of a serious drawback that extends to all rule update and revision operators based on KM postulates and on $\textsf{HT}$ models. In particular, it turns out that these operators are incompatible with the properties of support and fact update which are at the core of logic programming and their updates.

The following theorem shows that every rule update operator satisfying $\textsf{PU4}_{\textsf{HT}}$ violates either support or fact update, while its proof illustrates why.

Proof. Let $\mathbin{\oplus}$ be a rule update operator that satisfies $\textsf{PU4}_{\textsf{HT}}$ and $\mathcal{P}$ , $\mathcal{Q}$ and $\mathcal{U}$ the following programs:

\begin{align*} \mathcal{P}:\quad \mathsf{p}&. & \mathcal{Q}:\quad \mathsf{p} &\leftarrow \mathsf{q}. & \mathcal{U}:\quad \mathop{\sim\!} \mathsf{q}. \\ \mathsf{q}&. & \mathsf{q}&. \end{align*}

Since $\mathcal{P}$ is strongly equivalent to $\mathcal{Q}$ , by $\textsf{PU4}_{\textsf{HT}}$ we obtain that $\mathcal{P} \mathbin{\oplus} \mathcal{U}$ is strongly equivalent to $\mathcal{Q} \mathbin{\oplus} \mathcal{U}$ . Consequently, $\mathcal{P} \mathbin{\oplus} \mathcal{U}$ has the same answer-sets as $\mathcal{Q} \mathbin{\oplus} \mathcal{U}$ . It only remains to observe that if $\mathbin{\oplus}$ respects fact update, then $\mathcal{P} \mathbin{\oplus} \mathcal{U}$ has the unique answer-set ${\mathsf{p}}$ . But then ${\mathsf{p}}$ is an answer-set of $\mathcal{Q} \mathbin{\oplus} \mathcal{U}$ in which $\mathsf{p}$ is unsupported by $\mathcal{Q} \cup \mathcal{U}$ . Hence $\mathbin{\oplus}$ does not respect support.

So, any answer-set program update operator based on $\textsf{HT}$ models and the KM approach to belief update, as materialised in the fundamental principle $\textsf{PU4}_{\textsf{HT}}$ , cannot respect two basic and desirable properties: support and fact update. This is a major drawback of such operators, severely diminishing their applicability.

Moreover, the principle $\textsf{PU4}_{\textsf{HT}}$ is also adopted as $\textsf{PR4}_{\textsf{HT}}$ for revision of answer-set programs based on $\textsf{HT}$ models by Delgrande et al. (Reference Delgrande, Schaub, Tompits and Woltran2013). This means that Theorem 64 extends to semantic program revision operators, such as those defined by Delgrande et al. (Reference Delgrande, Schaub, Tompits and Woltran2013): whenever support and fact update are expected to be satisfied by a rule revision operator, it cannot be defined by purely manipulating the sets of $\textsf{HT}$ models of the underlying programs.

One question that suggests itself is whether a weaker version of the principle $\textsf{PU4}_{\textsf{HT}}$ can be combined with properties such as support and fact update. Its two immediate weakenings, analogous to the weakenings of $\mathsf{BU4}$ by Herzig and Rifi (Reference Herzig and Rifi1999), are as follows:

1. ( $\textsf{PU4.1}_{\textsf{HT}}$ ) If $\mathcal{P} \equiv_{\mathsf{HT}} \mathcal{Q}$ , then $\mathcal{P} \mathbin{\oplus} \mathcal{U} \equiv_{\mathsf{HT}} \mathcal{Q} \mathbin{\oplus} \mathcal{U}$ .

2. ( $\textsf{PU4.2}_{\textsf{HT}}$ ) If $\mathcal{U} \equiv_{\mathsf{HT}} \mathcal{V}$ , then $\mathcal{P} \mathbin{\oplus} \mathcal{U} \equiv_{\mathsf{HT}} \mathcal{P} \mathbin{\oplus} \mathcal{V}$ .

In case of $\textsf{PU4.1}_{\textsf{HT}}$ , it is easy to see that the proof of Theorem 64 applies in the same way as with $\textsf{PU4}_{\textsf{HT}}$ , so $\textsf{PU4.1}_{\textsf{HT}}$ is likewise incompatible with support and fact update.

On the other hand, principle $\textsf{PU4.2}_{\textsf{HT}}$ , also referred to as weak independence of syntax (WIS) (Osorio and Cuevas Reference Osorio and Cuevas2007), does not suffer from such severe limitations. It is, nevertheless, violated by syntax-based rule update semantics that assign a special meaning to occurrences of default literals in heads of rules, as illustrated in the following example:

Example 65 Let the programs $\mathcal{P}$ , $\mathcal{U}$ and $\mathcal{V}$ be as follows:

\begin{align*} \mathcal{P}:\quad \mathsf{p}&. & \mathcal{U}:\quad \mathop{\sim\!} \mathsf{p} &\leftarrow \mathsf{q}. & \mathcal{V}:\quad \mathop{\sim\!} \mathsf{q} &\leftarrow \mathsf{p}. \\ \mathsf{q}&. \end{align*}

Since $\mathcal{U}$ is strongly equivalent to $\mathcal{V}$ , $\textsf{PU4.2}_{\textsf{HT}}$ requires that $\mathcal{P} \mathbin{\oplus} \mathcal{U}$ be strongly equivalent to $\mathcal{P} \mathbin{\oplus} \mathcal{V}$ . This is in contrast with the $\textsf{JU}$ , $\textsf{DS}$ , and $\textsf{RD}$ semantics where a default literal $\mathop{\sim\!} \mathsf{p}$ in the head of a rule indicates that whenever the body of the rule is satisfied, there is a reason for $\mathsf{p}$ to cease being true. A consequence of this is that an update of $\mathcal{P}$ by $\mathcal{U}$ results in the single answer-set ${\mathsf{q}}$ while an update by $\mathcal{V}$ leads to the single answer-set ${\mathsf{p}}$ . Thus, when considering the principle $\textsf{PU4.2}_{\textsf{HT}}$ , benefits of the declarativeness that it brings with it need to be weighed against the loss of control over the results of updates by rules with default literals in their heads.