A reconstruction of the multipreference closure

05/05/2019 ∙ by Laura Giordano, et al. ∙ Università del Piemonte Orientale Università di Torino 0

The paper describes a preferential approach for dealing with exceptions in KLM preferential logics, based on the rational closure. It is well known that the rational closure does not allow an independent handling of the inheritance of different defeasible properties of concepts. Several solutions have been proposed to face this problem and the lexicographic closure is the most notable one. In this work, we consider an alternative closure construction, called the Multi Preference closure (MP-closure), that has been first considered for reasoning with exceptions in DLs. Here, we reconstruct the notion of MP-closure in the propositional case and we show that it is a natural variant of Lehmann's lexicographic closure. Abandoning Maximal Entropy (an alternative route already considered but not explored by Lehmann) leads to a construction which exploits a different lexicographic ordering w.r.t. the lexicographic closure, and determines a preferential consequence relation rather than a rational consequence relation. We show that, building on the MP-closure semantics, rationality can be recovered, at least from the semantic point of view, resulting in a rational consequence relation which is stronger than the rational closure, but incomparable with the lexicographic closure. We also show that the MP-closure is stronger than the Relevant Closure.

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1 Introduction

Kraus, Lehmann and Magidor in [22, 23] and Lehmann and Magidor in [23] investigate the properties that a notion of plausible inference from a conditional knowledge base should satisfy (KLM properties, for short). These properties led to the definition of the notions of preferential and rational consequence relation, as well as to the definition of the rational closure of a conditional knowledge base [23]. Although not all non-monotonic formalisms in the literature satisfy KLM properties, and although the adequacy of these properties has been and still is subject of debate (see, for instance, [3]), the rational closure construction (which is a polynomial construction) recently has been considered for defeasible reasoning in description logics [9, 11, 8, 19, 7], which are the formalisms at the basis of OWL ontologies [26].

While the rational closure provides a simple and efficient approach for reasoning with exceptions, it is well known that “it does not provide for inheritance of generic properties to exceptional subclasses” [24]. This problem was called by Pearl [27] “the blocking of property inheritance problem”, and it is an instance of the “drowning problem” in [1].

To overcome this weakness of the rational closure, Lehmann in [24] introduced the notion of lexicographic closure, as a “uniform way of constructing a rational superset of the rational closure” thus strengthening the rational closure but still defining a rational consequence relation.

In this paper, we consider another closure construction, that we call multi-preference closure (MP-closure), as it was first proposed in the context of description logics as a construction to soundly approximate the multipreferential semantics [21], a strengthening of the rational closure. Here, we consider the MP-closure in the propositional setting, and reconstruct its semantics, showing that it is a natural (weaker) variant of Lehmann’s lexicographic closure which simply uses a different lexicographic ordering. Following the pattern in [24], in the following, we will present a characterization of the closure both in terms of maxiconsistent sets and of a model-theoretic construction. In both cases, the characterization exploits a lexicographic ordering which compares tuples of sets of defaults rather than tuples of numbers (i.e., the number of defaults in the sets), as in the lexicographic ordering used by the lexicographic closure.

The MP-closure construction departs from lexicographic closure in the choice that, in case of contradictory defaults with the same rank, one tries to satisfy as many defaults as possible (where the number of defaults matters, rather than the defaults themselves). This choice was adopted by the lexicographic closure in agreement with the Maximal Entropy approach [25]. Abandoning the Maximal Entropy approach and following instead Poole’s proposal [28] (an alternative route that was also considered but not explored by Lehmann) leads to a construction which defines a preferential consequence relation rather than a rational consequence relation. Rational Monotonicity is not satisfied when reasoning under the MP-closure, but a more cautious notion of entailment is obtained with respect to reasoning under the lexicographic closure.

We believe that the MP-closure defines an interesting notion of entailment per se that may be reasonable in specific contexts, for instance, when reasoning about multiple inheritance in ontologies, where the lexicographic closure appears to be too bold. Nevertheless, we will also see that from a semantic point of view rationality can be easily regained by defining a rational consequence relation, starting from the MP-closure semantics, which is a superset of the MP-closure, while is incomparable with the lexicographic closure.

We conclude the paper establishing some relationships with the multi-preference semantics in [21] and with the Relevant Closure, a notion of closure proposed by Casini et al. [6] as a weaker alternative to the lexicographic closure. We show that the Relevant Closure is also weaker than the MP-closure.

The paper is organized as follows. In Section 2 we recall the definition of the rational closure and its semantics and, in Section 3, the definition of the lexicographic closure and discuss some examples which motivate the interest in investigating an alternative notion of closure, abandoning the Maximal Entropy approach. In Section 4 we reformulate the MP-closure construction in the propositional setting in terms of maxiconsistent sets and, then, we study its model-theoretic semantics, its properties and its relations with the lexicographic closure. In Section 5 a rational consequence relation is defined, which is a superset of the MP-closure but neither stronger nor weaker than the lexicographic closure. The relationships with the Relevant Closure and the multipreference semantics are also investigated. Section 6 concludes the paper.

2 The rational closure

In this section we recall the definition of the rational closure by Lehmann and Magidor [23] and its semantics, that we will exploit to define the semantics of the MP-closure.

Let the language be defined from a set of propositional variables , the boolean connectives and the conditional operator . Following the presentation in [19], as in [12, 4] and with a minor deviation from the original presentation in [23], here we consider the conditionals as formulas belonging to the object language.

The formulas of are defined as follows: if is a propositional formula, ; if and are propositional formulas, ; if is a boolean combination of formulas of , then . A knowledge base is a set of conditional assertions . In the following, we will restrict our attention to finite knowledge bases over a finite language.

The semantics of conditional KBs is defined by considering a set of worlds equipped with a preference relation . Intuitively the meaning of is that is more typical/more normal/less exceptional than . We say that a conditional is true in a model if holds in all most normal worlds where is true, i.e. in all -minimal worlds satisfying . In [23] Lehmann and Magidor introduce ranked models as a family of preferential models [22].

Definition 1 (Preferential models and ranked models)

A model is a triple where:

  • is a non-empty set of worlds;

  • is an irreflexive, transitive relation on satisfying the Smoothness condition defined below;

  • is a function , which assigns to every world the set of atoms holding in that world. If is a boolean combination of formulas, its truth conditions (, ) are defined as for propositional logic. Let be a propositional formula; we define and , implies . Moreover:

    iff for all , if then .

At this point we can define the Smoothness condition: if , then either or there is such that .

A model is a preferential model for which the relation is modular: for all , if then either or .

Validity and satisfiability of a formula are defined as usual. We say that a formula is satisfiable in the preferential (rational) semantics if there is a preferential (ranked) model and a world such that . We say that a formula is valid in a preferential (ranked) model , and we write , if, for all , it holds that . We say that a formula is valid in the preferential (rational) semantics if it is valid in all preferential (ranked) models, i.e. if, for all preferential (ranked) models , it holds that .

Given a set of formulas of and a model , we say that is a model of , written , if for every and every , we have that . preferentially entails a formula , written if is valid in all preferential models of . rationally entails a formula , written if is valid in all rational models of .

As a consequence of Theorems 6.8 and 6.9 in [17], if a set of formulas is satisfiable in a ranked model, then it is satisfiable in a finite ranked model. In the following, we will restrict our consideration to ranked models with a finite set of worlds.

Given a (finite) ranked model , we can define the rank of a world in .

Definition 2 (Rank of a world in )

Given a (finite) ranked model , the rank of a world , written , is the length of the longest chain from to a minimal (i.e. there is no such that ).

Hence, the preference relation of a ranked model defines a ranking function (this just is a special case of the general result in [23] where there is no restriction to finite models). Observe that, according to [23], here we might have forgotten the smoothness condition, which is satisfied in any well-founded model and, in particular, in any finite model. Notice also that Definition 2 makes sense even if the relation is not modular and that, for a modular relation on a finite set, all maximal chains 111A chain is maximal if there is no element such that for some it holds . from an element to a minimal have the same length.

The previous definition defines from a rank function . The opposite is also possible and can be defined from a ranking function by letting if and only if (this is similarly stated in [23], where a ranking function over a possibly infinite set is considered, since there is no restriction to finite models).

The rank of a formula in a model depends on the rank of the worlds satisfying the formula.

Definition 3 (Rank of a formula in a model)

The rank of a formula in a model is . If there is no such that , then has no rank in .

The previous definition defines from a rank function . The opposite is also possible and in general in ranked models the rank function and can be defined from each other by letting if and only if (this is similarly stated by [23] where a rank function over a possibly infinite set is used, since there is no restriction to finite models).

Lehmann and Magidor proved that, for a knowledge base which is a set of positive conditional assertions of the form the rational (ranked) entailment is equivalent to the preferential entailment. Also, the rational entailment does not define a rational consequence relation (i.e., a consequence relation which also satisfies the property of Rational Monotonicity). A possible formulation of Rational Monotonicity is the following:

i.e., if belongs to the consequence relation and does not, then must belong as well to the consequence relation.

In order to strengthen rational entailment, Lehmann and Magidor in [23] introduce the notion of rational closure, which provides a solution to both the problems above and can be seen as the “minimal” (in some sense) rational consequence completing a set of conditionals. In the following we recall the definition of the rational closure.

Definition 4 (Exceptionality of formulas)

Let be a knowledge base (i.e. a finite set of positive conditional assertions) and a propositional formula. is said to be exceptional for if and only if . A conditional formula is exceptional for if its antecedent is exceptional for . The set of conditional formulas of which are exceptional for will be denoted as .

It is possible to define a non increasing sequence of subsets of , by letting and, for , the set of conditionals of exceptional for , i.e. . Observe that, being finite, there is an such that or for all . The sets are used to define the rank of a formula, as in the next definition. Notice that if there is an such that , then for all , it will hold that (indeed )).

Definition 5 (Rank of a formula)

A propositional formula has rank (for ), written , if and only if is the least natural number for which is not exceptional for . If is exceptional for all then has no rank, and we let .

A conditional has rank equal to , and is the set of conditionals (defaults) in having rank .

Example 1

Let be the knowledge base containing the conditionals:

1.
2.
3.

stating that normally students do not pay taxes and are young, while employed students normally are students and pay taxes. It is possible to see that, from the definition of exceptionality above:


.

In particular, , as is non-exceptional for , while , as is exceptional w.r.t. the property that students typically are not taxpayers. Thus, the third conditional describing the properties of employed students has rank and is more specific than the conditionals describing the properties of students, which have rank .

Rational closure builds on the notion of exceptionality. Roughly speaking a conditional is in the rational closure of if is less exceptional than . We recall the construction of the rational closure for admissible knowledge bases in [23], remembering that we are considering finite knowledge base, and any finite knowledge base is admissible.

Definition 6 (Rational closure)

Let be a (finite) knowledge base. The rational closure of is defined as:

where and are propositions in the language of .

Referring to Example 1, is in the rational closure of , as . Similarly, is in .

Lehmann and Magidor in [23] develop a model theoretic semantics for the rational closure, by a canonical model construction. In [19] it was shown that a semantic characterization of the rational closure can also be given in terms of minimal canonical ranked models. In such models the rank of worlds is minimized to make each world as normal as possible. This is expressed by the following definitions corresponding to the fixed interpretations minimal semantics, , in [19], where only models with the same set of worlds and valuation function are comparable.

Definition 7 (Minimal ranked models)

Let and be two ranked models. is preferred to with respect to the fixed interpretations minimal semantics (and we write ) if:  , and

for all , and
there exists such that .

Given a knowledge base , we say that is a minimal model of with respect to if is a model of and there is no such that is a model of and .

In [19] it was also shown that, a notion of canonical model is needed when reasoning about the (relative) rank of the propositions in a model of : it is important to have them true in some world of the model, whenever they are consistent with the knowledge base.

Given a knowledge base and a query , let be the set of all the propositional variables of occurring in or in the query , and let be the restriction of the language to the propositional variables in .

A truth assignment is with , if there is no propositional formula such that and (where is extended as usual to arbitrary propositional formulas over the language ).

Definition 8 (Canonical models)

A model satisfying a knowledge base is said to be canonical if it contains (at least) a world associated with each truth assignment compatible with , that is to say: if is compatible with , then there exists a world in such that, for all propositional formulas , if and only if .

Definition 9 (Minimal canonical ranked models)

is a minimal canonical ranked model of , if it is a canonical ranked model of and it is minimal with respect (see Definition 7) among the canonical ranked models of .

We define a notion of minimal entailment w.r.t. minimal canonical ranked models of . minimally entails a formula , and we write , if is true in all the minimal canonical ranked models of .

It has been shown that, for any satisfiable knowledge base, a finite minimal canonical ranked model exists (see [19], Theorem 1), and that minimal canonical ranked models are an adequate semantic counterpart of rational closure. The correspondence between minimal canonical ranked models and rational closure is established by the following theorem.

Theorem 2.1 ([19])

Let be a knowledge base and be a minimal canonical ranked model of . For all conditionals :

if and only if ,

where is the rational closure of .

Furthermore, when is finite, the rank of a proposition in any minimal canonical ranked model of is equal to the rank assigned by the rational closure construction. Otherwise, and proposition is not satisfiable in any ranked model of (in any ranked model of , has no rank).

Observe that, by Theorem 2.1, the set of conditionals minimally entailed from coincide with the set of conditionals true in any (arbitrarily chosen) minimal canonical ranked model of . In the following, we will restrict our consideration to the finite minimal canonical models of the knowledge base (which, as said above, always exist when is consistent), and we denote their set by .

Example 2

Considering again the knowledge base in Example 1, we can see that conditional assertions  and   are satisfied in all the minimal canonical models of . For the first conditional, in all the minimal canonical models of , has rank , while has rank . Thus, in all the minimal canonical models of each typical Italian student must be an instance of . Similarly for the second conditional assertion.

Instead, the conditional is not minimally entailed from and, hence, it does not belong to the rational closure of . Indeed, the proposition is exceptional for , as it violates the property of students that normally they do not pay taxes and, then, in all models , . Furthermore, both and , hence nothing can be concluded about the typical employed students being young or not. Employed students do not “inherit“ any of the more general defeasible properties of students, not even the property that students are normally young. In general, the rational closure “does not provide for inheritance of generic properties to exceptional subclasses” [24].

In particular, the rational closure does not satisfy (among others) the desirable condition called by Lehmann the presumption of typicality. By Rational Monotony, if the rational closure of a KB contains then it must contain either or . But which one? Lehmann suggests that “in the absence of convincing reason to accept the latter, we should accept the former”. This and other desirable conditions led to the definition of the lexicographic closure as a “uniform way of constructing a rational superset of the rational closure” [24], thus strengthening the rational closure but still providing a rational consequence relation.

3 From the Lexicographic closure to the MP-closure

To overcome the weakness of rational closure, Lehmann introduced the notion of lexicographic closure [24], which strengthens the rational closure by allowing, roughly speaking, a class to inherit as many as possible of the defeasible properties of more general classes, giving preference to the more specific properties. In the example above, the property of students being young should be inherited by employed students, as it is consistent with all other default properties of employed students (i.e., with default 4) and, by “presumption of independence” [24], even if typicality is lost with respect to one consequent (the property that typically students are not taxpayers), we may still presume typicality of employed students with respect to other typical properties of students, such as the property of being young.

Let us recap the definition of the lexicographic closure in [24]. In order to compare alternative sets of defaults, in [24] a seriousness ordering among sets of defaults is defined by associating with each set of defaults a tuple of numbers , where is the oreder of , i.e. the least finite such that (i.e. there is no defaults with finite rank or rank higher than , but there is at least one default with rank ). The tuple is constructed considering the ranks of defaults in the rational closure. is the number of defaults in with rank and, for , is the number of defaults in with rank .

For instance, in the example Example 1 above, the set of defaults (that we will denote, synthetically as ) is associated with the tuple meaning that contains: no default with rank , one default with rank (default ) and one default with rank (default ).

A modular order among sets of defaults is obtained from the natural lexicographic order over the tuples . This order gives preference to those sets of defaults containing more specific defaults. Notice that the numbers in the tuple are in decreasing order w.r.t. the rank of the defaults, and the highest is the rank, the more specific is the default.

Lehmann defines a notion of basis for a formula in a knowledge base . A basis for is a set of defaults in such that is consistent with , the material counterpart of , and is maximal w.r.t. the seriousness ordering for this property222The material counterpart of , , is the set containing a material implication , for each conditional in ..

In the example above, the set of defaults forms a basis for , as its materialization is consistent (in the propositional calculus) with , and is maximal w.r.t. the seriousness ordering among the sets having this property. is actually the unique basis for .

A conditional is in , the lexicographic closure of , if , for any basis for . In the example, belongs to the lexicographic closure of , as , for the unique basis for . Thsi is what is expected, as the property of typical students of being young is inherited by employed students by presumption of independence.

In the following we will consider two variants of the knowledge base in Example 1 to illustrate the lexicographic closure and, later, to describe its common points and differences with the MP-closure.

Example 3

Let be the knowledge base containing the conditionals:

1.
2.
3.
4.

Here, Students and Employee have a conflicting property: students normally do not pay taxes, while employees normally do pay taxes. Furthermore, students are normally bright and employed students are normally busy.

According to the rational closure, the formulas and have both rank , while the formula has rank . Therefore, conditionals have rank , while conditional has rank . It is easy to see that the conditionals and do not belong to the the rational closure of . The same can be said about the conditional , which also is not in the rational closure of , although we would like to conclude it, as the property of typical student of being bright is not conflicting with other properties of typical employees and of typical employed students.

In this example, there are two bases for : and . They represent two alternative scenarios, the first one in which typical employed students inherit from typical students the property of not paying taxes, and the second one in which typical employed students inherit from typical employees the property of paying taxes. It is easy to see that and are not comparable with each other, i.e. none of them is more serious than the other (that is, and ), as the tuples and , associated with and (respectively), are not comparable in the lexicographic order.

Both the bases contain the default that normally students are bright and, as intended, this property extends to employed students. It is easy to see that is in the lexicographic closure of . Instead, the lexicographic closure neither contains the conditional nor the conditional , as each of them is false in one of the two bases (they are conflicting).

The following variant of Example 3 has a single basis and may suggest that the lexicographic closure is sometimes too bold.

Example 4

Let the knowledge base contain the following conditionals:

1.
2.
3.
4.

Again, defaults 1, 2 and 3 have rank in the rational closure, while default 4 has rank . As a difference with the previous example, the lexicographic closure has a single basis, . Indeed, of the two sets of defaults and , whose materializations are both consistent with , have the associated tuples and and, therefore, is less serious than (). As a consequence, there is a single basis for , and we can conclude that typical employed students are not only busy, but (like typical students) they are also young and do not pay taxes. The conditional

(1)

is in the lexicographic closure of , as , and is the only basis for .

The result above is in line with the choice of the lexicographic closure that, in the case of contradictory defaults with the same rank, as many as possible defaults should be satisfied, a choice taken in agreement with the Maximal Entropy approach [25]. However, the reason to accept that typical employed students are not young and pay taxes (rather than the converse) may be questioned and, in this last example, the lexicographic closure appears to be too bold. Indeed, the conclusion that normally employed students are young and do not pay taxes, i.e. conditional 1, here follows from the accidental fact that the properties of Employees are expressed by a single default, while the properties of Students are expressed by two defaults. Notice that, if we replace default with the two defaults and , there would be two bases in the lexicographic closure, and one would not be allowed to conclude any more that typical employed students are young and do not pay taxes. As observed by Lehmann, the lexicographic closure construction is “extremely sensitive to the way defaults are presented” and “the way defaults are presented is important” [24].

In the following section, we will consider a different notion of closure, the MP-closure, that departs from Maximal Entropy assumption of the lexicographic closure and, for instance, in Example 4, it considers both the sets of defaults and to be maximally serious, and it does not conclude conditional (1). Although also the MP-closure is somewhat syntax dependent, in this case, differently from the lexicographic closure, it treats in the same way the two different formulations of the knowledge base above. We will show that the MP-closure is stronger than the rational closure but weaker than the lexicographic closure. Abandoning the Maximal Entropy assumption leads to a construction which defines a preferential consequence relation, rather than a rational consequence relation, and which defines a more cautious notion of entailment (with respect to the lexicographic closure), that does not satisfy the property of Rational Monotonicity. We will see later that, however, rationality can be recovered, at least from the semantic point of view, by considering a rational extension of the MP-closure, which provides another (different) solution to the technical problem, risen by Lehmann, of defining a rational consequence relation which is a rational superset of the rational closure.

Following the pattern in [24], in Section 4, we present both a characterization of the MP-closure in terms of maxiconsistent sets and a model-theoretic construction. In Section 5.1 we will then exploit the semantic construction to define a rational consequence relation which is a superset of the MP-closure and is neither stronger nor weaker than the lexicographical closure.

4 The MP-closure revisited

The multipreference closure (MP-closure, for short), was preliminarily introduced in the technical report [16] as a construction which soundly approximates the multipreference semantics proposed by Gliozzi [20, 21] for the description logic with typicality, thus defining a refinement of the rational closure of . This semantics was originally proposed for separately reason about the inheritance of different properties and, hence, to provide a solution to the drowning problem related to the rational closure.

We believe that the interest of the MP-closure construction goes beyond description logics and that its definition and semantics can be reconstructed and significantly simplified in the context of propositional logic. The MP-closure can be regarded as the natural variant of the lexicographic closure, if we are ready to abandon the Maximal Entropy approach (as illustrated by Example 4 in the previous section), an alternative route already considered but not explored by Lehmann in [24]. In this section we reformulate the MP-closure construction from [16] in the propositional setting and, then, we focus on its semantics, its properties and its relations with the lexicographic closure. Further relationships, and in particular the relationships with the Relevant closure, will be investigated in Section 5.

4.1 The MP-closure construction

Given a finite knowledge base , and a formula whose rank in the rational closure of is finite, we let be the maximum finite rank for a conditional assertion (default) in the rational closure of . In the following, we exploit the MP-closure construction to define the plausible consequences of a formula . Observe that, for any formula with infinite rank, i.e. such that , the conditional is in the rational closure of , for any .

Given a subset of the conditional assertions in (a set of defaults), we let be the set of defaults in with finite rank , and be the set of defaults in with rank . The tuple , associated with , defines a partition of , according to the ranks of the defaults in the rational closure of .

We define a preference relation among sets of defaults, by comparing the tuples associated to these sets according to the natural lexicographic order on such tuples, defined inductively as follows. Given two tuples and of sets of defaults in , we let:

As the (strict) subset inclusion relation among sets is a strict partial order, the lexicographic order on the tuples of sets of defaults is a strict partial order as well. This lexicographic order provides a new seriousness ordering among sets of defaults.

Definition 10 (MP-seriousness ordering)

( is less serious than w.r.t. the MP-seriousness ordering) iff

.

Notice that the relation defines a seriousness ordering among sets of defaults, which is different from the seriousness ordering used by the lexicographic closure, where the corresponding tuple associated with would be (see Section 3), i.e. the cardinality of the sets matters, rather than the defaults in . As the lexicographic order on the tuples of sets of defaults is a strict partial order, is a strict partial order as well, although it is not necessarily modular.

The difference of the seriousness ordering between lexicographic closure and the MP-closure has an impact on the kind of conclusions one draws in the two cases, as we will see below. Let us first give a characterization of the MP-closure in terms of bases.

Definition 11 (MP-basis)

Given a finite knowledge base , and a formula with finite rank, a set of defaults is a basis for if is consistent with (the material counterpart of ) and is maximal w.r.t. the MP-seriousness ordering for this property.

Notice that the definition of a basis is exactly the same as in the lexicographic closure [24], but for the fact that it uses a different lexicographic ordering.

Definition 12 (MP-closure)

A default is in , the MP-closure of a knowledge base , if for all the MP-bases for :

where is logical consequence in the propositional calculus and is the materialization of .

Consider again Example 4, the two sets of defaults and are now incomparable using the preference relation, as the tuples associated to the sets and are respectively: and and neither nor (instead, as we have seen above, in the lexicographic closure, is more serious than ). Thus, there are two MP-bases for , namely and . Therefore, neither nor are in the MP-closure of . In this example, the MP-closure is less bold than the lexicographic closure, which, as we have seen, includes the default , as is the only basis for in the lexicographic closure.

Concerning Examples 1 and 3 above, it is easy to see that in both of them the MP-closure has the same bases as the lexicographic closure, as well as the same consequences.

It can be proved that the MP-closure is stronger than the rational closure, but weaker the lexicographic closure. We prove the second result, while postponing the proof of the first one after the introduction of the semantics of the MP-closure in Section 4.2. We prove that is coarser than .

Proposition 1

is coarser than , that is, for all the sets of defaults and , if then .

Proof

Given a knowledge base and two sets of conditionals , let us assume that . As is less serious than in the MP-ordering, it must be that:

.

consider the highest , with , such that . For such a , it must be that , while , for all such that .

Let us now consider the two tuples of numbers

and

associated with and , respectively, in the lexicographic closure construction. Notice that, , for all , and , for all . Furthermore, is the set of all the conditionals with rank in the rational closure and, hence, . For all such that , as , it must be that . Also, from , we get . Thus, using the lexicographic ordering on numbers:

and, therefore, .

As a consequence of this result, it is easy to prove the following corollaries.

Corollary 1

Let be a knowledge base, a formula and a set of defaults. If is a basis for in the lexicographic closure, then is a basis for in the MP-closure.

Proof

Let be a basis for in the lexicographic closure, i.e. is consistent with and is maximal w.r.t. -seriousness ordering for this property.

We show that is also a basis in the MP-closure. If not, there is a set of defaults such that is consistent with and . But, then, by Proposition 1, , and is not a maximal w.r.t. among the sets of default whose materialization is consistent with , which contradicts the hypothesis that is a basis for in the lexicographic closure.

Corollary 2

Let be a knowledge base and a formula. If is in the MP-closure of , then is in the lexicographic closure of .

Proof

If is in the MP-closure of , then, in all the bases for in the MP-closure, . Let be any basis for in the lexicographic closure of . As, by Corollary 1, all the bases for in lexicographic closure of are also MP-bases for , . Hence, for all the bases for in the lexicographic closure of , , and is in the lexicographic closure of .

To conclude this section, we show that the MP-closure does not define a rational consequence relation. Let be the set of conditionals in the MP-closure of . The following counterexample shows that does not satisfy the property of Rational Monotonicity, and is a reformulation of Lehmann’s musician example[24].

Example 5

The following knowledge base

1.
2.
3.
4.

The conditionals 1, 2 and 3 have rank 0 in the rational closure, while conditional 4 has rank 1. There are two bases for in the MP-closure of , and , and the conditional is in the MP-closure of (in ). Instead, the conditional is not in (as does not hold in the basis ), that is, . By the property of Rational Monotonicity, the conditional

should be in . Instead, the last conditional is not in the MP-closure of . In fact, there are two bases for , namely and , and the formula does not hold in the first basis, as .

Notice that the example above is not a counterexample to Rational Monotonicity for the lexicographic closure, which in known to define a rational consequence relation. In fact, is the only basis for in the lexicographic closure of and, hence, the conditional is in the lexicographic closure of .

4.2 A semantic characterization for the MP-closure

A semantics for the MP-closure is defined in [14] building on the preferential semantics for rational closure of , introducing a notion of refined, bi-preference interpretation, which contains two preference relations, let us call them and : the first one plays the role of the preference relation in a model of the RC in , while the second one is built from exploiting a specificity criterium, and represents a refinement of .

In this section we define a simpler semantic characterization of the MP-closure of a propositional knowledge base, starting from the propositional models of the rational closure. This simplified setting, that corresponds to the one considered by Lehmann in his semantic characterization of the lexicographic closure [24], also allows an easy comparison among the two semantics.

Given a finite satisfiable knowledge base , in the following we define the semantics of the MP-closure by means of some preferential models of (that we call MP-models) and, then, we prove a characterization result. To this purpose, we introduce a functor associating a preferential interpretation to each finite minimal canonical ranked model characterizing the rational closure of according to Theorem 2.1. As we will see, is a model of the MP-closure.

Definition 13 (Functor )

Given a minimal canonical ranked interpretation in , we let such that: and

(2)

where, for , is the set of defaults in which are violated by (i.e., the set of conditionals such that ).

As , introduced in Definition 10, is a strict partial order, it is easy to see that in the definition above is a strict partial order as well. Indeed, is irreflexive: if then and, by irreflexivity of , . Hence, does not hold. Also, is transitive: if and , then, by definition of , and . From the transitivity of , . Therefore, .

Hence, , in Definition 13, is a preferential interpretation. Propositions 2 and 4 below will show that , i.e. the preference relation is finer than the modular preference relation , and that is a model of . Thus, is a preferential model of which is the refinement of the model of the rational closure of (in the sense that the preference relation in is finer than the preference relation in ).

Proposition 2

For all and such that , it holds that , i.e., the preference relation is finer than .

Proof

We show that, for all