A Logical Characterization of the Preferred Models of Logic Programs with Ordered Disjunction

Logic Programs with Ordered Disjunction (LPODs) extend classical logic programs with the capability of expressing ordered alternatives in the heads of program rules. In particular, LPODs allow the head of a program rule to be a formula $C_{1}\times\cdots\times C_{n}$, where “$\times$” is a propositional connective called “ordered disjunction” and the $C_{i}$’s are literals. The intuitive explanation of $C_{1}\times\cdots\times C_{n}$ is “I prefer $C_{1}$; however, if $C_{1}$ is impossible, I can accept $C_{2}$; $\cdots$; if all of $C_{1},\ldots,C_{n-1}$ are impossible, I can accept $C_{n}$”. Due to their simplicity and expressiveness, LPODs are a widely accepted formalism for preferential reasoning, both in logic programming and in artificial intelligence.

At present, the semantics of LPODs is defined [Brewka (2002), Brewka et al. (2004)] based on the answer set semantics, using a two-phase procedure. In the first phase, the answer sets of the LPOD are produced. This requires a modification of the standard definition of answer sets. In the second phase, the answer sets are “filtered”, and we obtain the set of “most preferred” answer sets, which are taken as the meaning of the initial program. Notice that both phases are not purely model-theoretic: the first one requires the construction of the reduct of the program and the second one is performed using the so-called “degree of satisfaction of rules”, a concept that relies on examining the rules of the program to justify the selection of the most preferred answer sets. Apart from its logical status, the current semantics of LPODs produces in certain cases counterintuitive most preferred answer sets. This discussion leads naturally to the question: “Is it possible to specify the semantics of LPODs in a purely model-theoretic way?”.

An important first step in this direction was performed by \citeNlpod-cabalar, who used Equilibrium Logic [Pearce (1996)] to logically characterize the answer sets produced in the first phase described above. However, to our knowledge, the second phase (namely the selection of the most preferred answer sets), has never been justified model-theoretically. We consider this as an important shortcoming in the theory of LPODs. Apart from its theoretical interest, this question also carries practical significance, because, as we are going to see, the present formalization of the second phase produces in certain cases counterintuitive (and in our opinion undesirable) results. The main contribution of the present paper is to provide a purely model-theoretic characterization of the semantics of LPODs. The more specific contributions are the following:

• •

  We propose a new semantics for LPODs which uses an additional truth value in order to select as most preferred models those in which a top preference fails only if it is impossible to be satisfied. We demonstrate that the proposed approach overcomes the shortcomings of the traditional semantics of LPODs. In this way, the most preferred models of an LPOD can be characterized by a preferential ordering of its models.

• •

  We demonstrate that our approach can be seamlessly extended to programs that allow both ordered and classical disjunctions in the heads of clauses. In particular, we define a natural class of such programs and demonstrate that all our results about LPODs transfer, with minimal modifications, to this new class. In this way we provide a clean semantics for a class of programs that can express not only strict levels of preference but also alternatives that are equally preferred.

Section 2 introduces LPODs and gives relevant background. Sections 3 and 4 describe the shortcomings of LPOD semantics and give an intuitive presentation of the proposed approach for overcoming these issues. The remaining sections give a technical exposition of our results. The proofs of all results have been moved to corresponding appendices.

Logic programs with ordered disjunction are an extension of the logic programs introduced by \citeNGL91, called extended logic programs, which support two types of negation: default (denoted by $\mathit{not}$) and strong (denoted by $\neg$). Strong negation is useful in applications but it is not very essential from a semantics point of view: a literal $\neg A$ is semantically treated as an atom $A^{\prime}$. For the basic notions regarding extended logic programs, we assume some familiarity with the work of \citeNGL91.

The intuitive explanation of a formula $C_{1}\times\cdots\times C_{n}$ is “I prefer $C_{1}$; however, if $C_{1}$ is impossible, I can accept $C_{2}$; $\cdots$; if all of $C_{1},\ldots,C_{n-1}$ are impossible, I can accept $C_{n}$”.

An interpretation of an LPOD is a set of literals. An interpretation is called consistent if there does not exist any atom $A$ such that both $A$ and $\neg A$ belong to $I$. The notion of model of an LPOD is defined as follows.

To obtain the preferred answer sets of an LPOD, a two-phase procedure was introduced by \citeNlpod-brewka. In the first phase, the answer sets of the LPOD are produced. This requires a modification of the standard definition of answer sets for extended logic programs. In the second phase, the answer sets are “filtered”, and we obtain the set of “most preferred” ones. The first phase is formally defined as follows.

The second phase produces the “most preferred” answer sets using the notion of the degree of satisfaction of a rule. Formally:

The satisfaction degrees of rules are then used to define a preference relation on the answer sets of a program. Given a set of literals $M$, let $M^{i}(P)=\{R\in P\mid\mathit{deg}_{M}(R)=i\}$. The preference relation is defined as follows.

Notice that \citeNlpod-brewka originally defined only the preference relation of Definition 6. In the follow-up paper [Brewka et al. (2004)] two more preference relations were introduced, namely the cardinality and the Pareto, in order to treat cases for which the inclusion preference did not return the expected results. All these relations do not rely exclusively on the models of the source program, and are therefore subject to similar criticism. For this reason, in this paper we focus attention on the inclusion preference relation.

From a foundational point of view, the main issue with the semantics of LPODs is that, in its present form, it is not purely model-theoretic. Despite the simplicity and expressiveness of ordered disjunction, one can not characterize the meaning of a program by just looking at its set of models. Recall that this principle is one of the cornerstones of logic programming since its inception: the meaning of positive logic programs is captured by their minimum Herbrand model [van Emden and Kowalski (1976)] and the meaning of extended logic programs is captured by their equilibrium models [Pearce (1996)]. How can the most preferred models of LPODs be captured model-theoretically? The existing issues regarding the semantics of LPODs are illustrated by the following examples.

The main purpose of this paper is to define a model-theoretic semantics for LPODs. In other words, we would like to be able to choose the most preferred answer sets of a program using preferential reasoning on the answer sets themselves. Actually, such an approach should also be applicable directly on the models of the source program, without the need to first construct the answer sets of the program. We would expect that such an approach would also provide solutions to the shortcomings of the previous section.

But on what grounds can we compare the answer sets (or, even better, the models) of a program and decide that some of them satisfy in a better way our preferences than the others? This seems like an impossible task because the answer sets (or the models) do not contain any information related to the ordered disjunction preferences of the program.

It turns out that we can introduce preferential information inside the answer sets of a program by slightly tweaking the underlying logic. The answer sets of extended logic programs are two-valued, ie., a literal is either $T$ (true) or $F$ (false). We argue that in order to properly define the semantics of LPODs, we need a third truth value, which we denote by $F^{*}$. The intuitive reading of $F^{*}$ is “impossible to make true”.

To understand the need for $F^{*}$, consider again the intuitive meaning of $C_{1}\times C_{2}$: we prefer $C_{2}$ only if it is impossible for us to get $C_{1}$. Impossible here means that if we try to make $C_{1}$ true, then the interpretation will become inconsistent. Therefore, we seem to need two types of false, namely $F$ and $F^{*}$: $F$ means “false by default” while $F^{*}$ means “impossible to make true”. The following example demonstrates these issues.

The above discussion suggests the following semantics for “$\times$” in the proposed logic. Let $u,v\in\{F,F^{*},T\}$. Then:

The intuition of the above definition is that we return the value $v$ only if it is impossible to satisfy $u$; in all other cases, we return $u$.

We now get to the issue of how we can use the value $F^{*}$ to distinguish the most preferred answer sets: we simply identify those answer sets that are minimal with respect to their sets of atoms that have the value $F^{*}$. As we have mentioned, the value $F^{*}$ means “impossible to make true”. By minimizing with respect to the $F^{*}$ values, we only keep those answer sets in which a top preference fails only if it is impossible to be satisfied.

The above discussion gives a description of how we can select the “most preferred (three-valued) answer sets” of an LPOD. However, it is natural to wonder whether it is possible to also characterize the answer sets model-theoretically, completely circumventing the construction of the reduct. A similar question was considered by \citeNlpod-cabalar, who demonstrated that the (two-valued) answer sets of an LPOD coincide with the equilibrium models of the program. We adapt \citeANPlpod-cabalar’s characterization to fit in our setting. More specifically, we extend our three-valued logic to a four-valued one by adding a new truth value $T^{*}$, whose intuitive meaning is “not false but its truth can not be established”. We then demonstrate that the three-valued answer sets of an LPOD $P$ are those models of $P$ that are minimal with respect to a simple ordering relation and do not contain any $T^{*}$ values. In this way we get a two-step, purely model-theoretic characterization of the most-preferred models of LPODs: in the first step we use the $T^{*}$ values as a yardstick to identify those models that correspond to answer-sets, and in the second step we select the most preferred ones, by minimizing with respect to the $F^{*}$ values.

Finally, we consider the problem of characterizing the semantics of logic programs that contain both disjunctions and ordered disjunctions in the heads of rules. This is especially useful in cases where some of our preferences are equally important. For example:

states that wine and beer are our top preferences (but we have no preference among them), and soda and juice are our secondary preferences. We consider the class of programs in which the heads of rules consist of ordered disjunctions where each ordered disjunct is an ordinary disjunction (as in the above program). We demonstrate that the theory of these programs is very similar to that of LPODs. All our results for LPODs transfer with minimal modifications to this extended class of programs. This suggests that this is a natural class of programs that possibly deserves further investigation both in theory and in practice.

In this section we provide a new definition of the answer sets of LPODs. The new definition is based on a three-valued logic which allows us to discriminate the most preferred answer sets using a purely model-theoretic approach. In Section 6 we demonstrate that by extending the logic to a four-valued one, we can identify directly the most preferred models of a program (without first producing the answer sets).

The meaning of formulas is defined over the set of truth values $\{F,F^{*},T\}$ which are ordered as $F<F^{*}<T$. Given two truth values $v_{1},v_{2}$, we write $v_{1}\leq v_{2}$ iff either $v_{1}<v_{2}$ or $v_{1}=v_{2}$.

It is straightforward to see that the meanings of “$\vee$”, “$\wedge$”, and “$\times$” are associative and therefore we can write $I(\phi_{1}\vee\cdots\vee\phi_{n})$, $I(\phi_{1}\wedge\cdots\wedge\phi_{n})$, and $I(\phi_{1}\times\cdots\times\phi_{n})$ unambiguously (without the need of extra parentheses). Moreover, given literals $C_{1},\ldots,C_{n}$ we will often write $I(C_{1},\ldots,C_{n})$ instead of $I(C_{1}\wedge\cdots\wedge C_{n})$.

The ordering $<$ (respectively, $\leq$) on truth values extends in the standard way on interpretations: given interpretations $I_{1},I_{2}$ we write $I_{1}<I_{2}$ (respectively, $I_{1}\leq I_{2}$), if for all literals $L\in\Sigma$, $I_{1}(L)<I_{2}(L)$ (respectively, $I_{1}(L)\leq I_{2}(L)$).

When we consider interpretations of an LPOD program, we assume that the underlying set $\Sigma$ is the set of literals of the program. The following definition will be needed.

We can now give the new definitions for reduct and answer sets for LPODs.

The major difference of the above definition from that of Definition 3, are the clauses of the form $C_{i}\leftarrow F^{*},A_{1},\ldots,A_{m}$. These clauses are included so as that the value $F^{*}$ can be produced for $C_{i}$ when $I(A_{1})=\cdots=I(A_{m})=T$. Notice that if these clauses did not exist, there would be no way for the value $F^{*}$ to be produced by the reduct.

Notice that the least model of $P_{\times}^{M}$ in the above definition, can be constructed using the following immediate consequence operator $T_{P_{\times}^{M}}:(\Sigma\rightarrow\{F,F^{*},T\})\rightarrow(\Sigma\rightarrow\{F,F^{*},T\})$:

Notice that since the set $\{F,F^{*},T\}$ is a complete lattice under the ordering $\leq$, it is easy to see that the set of interpretations is also a complete lattice under the ordering $\leq$. Moreover, the operator $T_{P_{\times}^{M}}$ is monotonic over the complete lattice of interpretations; this follows from the fact that the meanings of conjunction (namely, $\min$) and that of disjunction (namely, $\max$), are monotonic. Then, by Tarski’s fixed-point theorem, $T_{P_{\times}^{M}}$ has a least fixed-point, which can be easily shown to be the $\leq$-least model of $P_{\times}^{M}$.

The following lemma guarantees that our definition is a generalization of the well-known one for extended logic programs [Gelfond and Lifschitz (1991)].

The following lemmas, which hold for extended logic programs, also extend to LPODs.

The answer sets of extended logic programs are minimal with respect to the classical truth ordering $F<T$. As it turns out, the answer sets of LPODs are minimal but with respect to an extended ordering, which is defined below.

As in the case of the original semantics of LPODs, we now need to define a preference relation over the answer sets of a program. Intuitively, we prefer those answer sets that maximize the prospect of satisfying our top choices in ordered disjunctions. This can be achieved by minimizing with respect to $F^{*}$ values. More formally, we define the following ordering:

The intuition behind the definition of $\sqsubset$ is that we prefer those answer sets that minimize the need for $F^{*}$ values. In other words, an answer set will be most preferred if all the literals that get the value $F^{*}$, do this because there is no other option: these literals must be false in order for the program to have a model. We now examine the examples of Section 3 under the new semantics introduced in this section.

We close this section by stating a result that establishes a relationship between the answer sets produced by our approach (Definition 4) and those ones produced by the original formulation [Brewka (2002), Brewka et al. (2004)].

In other words, there is a bijection between the answer sets produced by our approach and the original ones. Moreover, each three-valued answer set only differs from the corresponding two-valued one in that some literals of the former may have a $F^{*}$ value instead of an $F$ value. However, these $F^{*}$ values play an important role because they allow us to distinguish the most preferred answer sets.

In this section we demonstrate that the answer sets of LPODs can be characterized in a purely logical way, namely without even the use of the reduct. In particular, we demonstrate that the answer sets of a given program $P$ coincide with a well-defined subclass of the minimal models of $P$ in a four-valued logic. This logic is an extension of the three-valued one introduced in Section 5 and minimality is defined with respect to a four-valued relation $\preceq$ that extends the three-valued one of Definition 12. The new logic is based on four truth values, ordered as follows:

The value $T^{*}$ can be read as “not false but its truth can not be established”. The connections of this logic with Equilibrium Logic [Pearce (1996)] are discussed in Section 8.

An interpretation is now a function from $\Sigma$ to the set $\{F,F^{*},T^{*},T\}$. The semantics of formulas with respect to an interpretation $I$ is defined identically as in Definition 8. The notions of interpretation, consistent interpretation, and model are defined as in Definition 9. Moreover, we extend the three-valued $\preceq$ relation of Definition 12, as follows:

The following special kind of interpretations plays an important role in our logical characterization.

We can now state the logical characterization of the answer sets of an LPOD.

In conclusion, given an LPOD we can purely logically characterize its most preferred models by first taking its consistent $\preceq$-minimal models that are solid and then keeping the $\sqsubset$-minimal ones (see Definitions 13 and 14).

An extended study of the properties of the proposed four-valued logic is outside the scope of the present paper. Figure 1 lists some useful equivalences; the first column is for classical connectives, while the second column contains equivalences involving the “$\times$” operator. We note the interaction of $\times$ with $\vee$ because this will be the central theme of the next section. It is easy to see that:

We note that this equivalence does not hold, a fact which will be referenced in the next section.

We now extend the ideas of the previous sections to programs that also allow standard disjunctions in the heads of rules. The case of disjunctive LPODs (DLPODs), was initially considered by \citeNtowards-dlpods and reexamined by \citeNlpod-cabalar.

The main idea of using disjunctions in the heads of LPOD rules is described [Kärger et al. (2008)] as follows: “we use ordered disjunction to express preferences and disjunction to express indifferences”.

towards-dlpods consider rules whose heads are arbitrary combinations of atoms and the operators $\times$ and $\vee$. A set of transformations is then used in order to bring the heads of rules into “Ordered Disjunctive Normal Form (ODNF)”. More specifically, each head is transformed into a formula of the form ${\cal C}_{1}\vee\cdots\vee{\cal C}_{n}$ where each ${\cal C}_{i}$ is an ordered disjunction of literals. The resulting normalized rules are then used to obtain the preferred answer sets of the original program.

However, as observed by \citeNlpod-cabalar, one of the transformations used by \citeNtowards-dlpods to obtain the ODNF, can not be logically justified: the formula $(\phi_{1}\vee\phi_{2})\times\phi_{3}$ is not logically equivalent to the formula $(\phi_{1}\times\phi_{3})\vee(\phi_{2}\times\phi_{3})$ in terms of the logic of Here-and-There. As discussed at the end of Section 6, these two formulas are also not equivalent under our four-valued logic.

As an alternative approach to the semantics of DLPODs, \citeNlpod-cabalar proposes to use the logical characterization on rules with heads that are arbitrary combinations of disjunctions and ordered disjunctions. We could extend this approach to get a logical characterization of the most preferred models of arbitrary DLPODs: given such a program $P$, we could at first consider all the $\preceq$-minimal models of $P$ that are solid, and then select the $\sqsubset$-minimal among them. Such an approach is certainly general. However, we believe that not every such program carries computational intuition. A good example of this is given in [Cabalar (2011), Example 2], where a program with both disjunction and ordered disjunction is given, and whose computational meaning is far from clear.

In the following, we define a class of programs which, as we claim, have a clear computational interpretation and at the same time retain all properties that we have identified for LPODs. Intuitively, we allow the head of a program rule to be a formula ${\cal C}_{1}\times\cdots\times{\cal C}_{n}$ where each ${\cal C}_{i}$ is an ordinary disjunction of literals. Notice that the program in Example 8 belongs to this class, while the program in Example 9, does not. We believe that the programs of this class have a clear preferential interpretation. Intuitively, the rule heads of the programs we consider, denote a hierarchy of preferences imposed by the $\times$ operator; in each level of this hierarchy, we may have literals that have equal preference (this is expressed by standard disjunction).

It is important to stress that if we allowed arbitrary combinations of disjunctions and ordered disjunctions, the preferential intuition would be lost. To see this, consider for example the formula (a $\times$ b) $\vee$ (c $\times$ d). This gives us the information that (a $\times$ b) is at the same level of preference as (c $\times$ d), and that a is more preferred than b and c is more preferred than d; however, for example, it gives us no information of whether a is more preferred than c. On the other hand, a formula of the fragment we consider, such as (a $\vee$ b) $\times$ (c $\vee$ d) gives us a total order of a, b, c, and d.

As it turns out, the answer sets of such programs can be defined in an almost identical way as those of LPODs.

As it turns out, all the results we have obtained in the previous section, hold for DLPODs. The proofs of these results are (surprisingly) almost identical (modulo some minor notational differences) to the proofs of the corresponding results for LPODs. For reasons of completeness, the corresponding proofs are given in the appendix. The extended results are stated below.

The similarity of the definitions and of the theoretical results of DLPODs to those of standard LPODs, makes us believe that this is indeed an interesting class of programs that deserves further attention.

The work on LPODs is closely related to “Qualitative Choice Logic” (QCL) [Brewka et al. (2004)]. QCL is an extension of propositional logic with the preferential connective “$\times$”, which has the same intuitive meaning as in LPODs: $A\times B$ is read “if possible $A$, but if $A$ is impossible then at least $B$”. Essentially, QCL is the propositional logic underlying LPODs. It is worth noting that the semantics of QCL is based on the “degree of satisfaction” of formulas, which is connected to the idea of the degree of satisfaction of the rules of LPODs (Definition 5). Moreover, as remarked by one of the reviewers of the present paper, the DLPODs introduced in Section 7 are closely connected to the “basic choice formulas” of QCL [Brewka et al. (2004), Section 3.1, Definition 8]. It would be interesting to investigate whether our four-valued logic can be used to provide an alternative semantics for QCL.

The work reported in this paper is closely connected to the work of \citeNlpod-cabalar, who first considered the problem of expressing logically the semantics of LPODs. The key difference between the two works is that ours provides a characterization of both phases of the production of the most preferred models of an LPOD, while Cabalar’s work concentrates on the first one.

It is important to stress here that both our work as well as the work of \citeNlpod-cabalar, are influenced by the work of \citeNPearce96 who first gave a logical characterization of the answer sets of extended logic programs, using Equilibrium Logic. This is a non-monotonic logic which is defined on top of the monotonic logic of Here-and-There [Heyting (1930)], using a model preference approach. The technique we have proposed in this paper, when applied to a consistent extended logic program $P$, produces the standard answer sets of $P$; this is a direct consequence of Theorem 1 and Lemma 1. Therefore, for extended logic programs, the Equilibrium Logic gives the same outcome as our approach which is based on a four-valued logic and $\preceq$-minimal models that are solid. We believe that a further investigation of the connections of our approach with that of Equilibrium Logic is a worthwhile topic.

Our work is the first to provide a purely model-theoretic characterization of the semantics of LPODs. To our knowledge, the four-valued logic we have utilized does not appear to be a well-known variant/extension of Here-and-There. However, some seemingly related logics have been used in the literature of answer set extensions. The original definition of Equilibrium Logic included a second constructive negation, which corresponds to Nelson’s strong negation [Nelson (1949)]. This gave rise to a five-valued extension of Here-and-There, called ${\cal N}_{5}$. Also, a logic called ${\cal X}_{5}$, that is closely connected to ${\cal N}_{5}$, was recently proposed by \citeNACF0PV19 in order to capture the semantics of arbitrary combinations of explicit negation with nested expressions. Both ${\cal N}_{5}$ and ${\cal X}_{5}$ appear to be connected to our four-valued logic due to the different notions of false and true that they employ in order to capture aspects that arise in answer set semantics. However, the ordering of the truth values and the semantics of the logical connectives are different, and the exact correspondence (if any) between these logics and the present one, is not straightforward to establish. This is certainly an interesting topic for further investigation.

Another promising topic for future work is the characterization of the notion of strong equivalence [Lifschitz et al. (2001)] for LPODs and DLPODs. When two logic programs are strongly equivalent, we can replace one for the other inside a bigger program without worrying that the semantics of the bigger program will be affected. Characterizations of strong equivalence for LPODs have already been obtained by \citeNFaberTW08. It would be interesting to investigate if the logical characterization of the semantics of LPODs and DLPODs developed in the present paper, can offer advantages compared with their work.

We would like to thank the three anonymous reviewers of our paper for their careful and insightful comments.

This research is co-financed by Greece and the European Union (European Social Fund- ESF) through the Operational Programme “Human Resources Development, Education and Lifelong Learning 2014- 2020” in the context of the project “Techniques for implementing qualitative preferences in deductive querying systems” (5048151).