Studies on Brutal Contraction and Severe Withdrawal

In this paper we present an axiomatic characterization for brutal contractions. Then we consider the particular case of the brutal contractions that are based on a bounded ensconcement and also the class of severe withdrawals which are based on bounded epistemic entrenchment relations that are defined by means of bounded ensconcements (using the procedure proposed by Mary-Anne Williams). We present axiomatic characterizations for each one of those classes of functions and investigate the interrelation among them.


Introduction
The central goal underlying the research area of logic of theory change (for an overview see [3]) is the study of the changes which can occur in the belief state of a rational agent when he receives new information.
The most well known model of theory change was proposed by Alchourrón, Gärdenfors, and Makinson in [1] and is, nowadays, known as the AGM model. Assuming that the belief state of an agent is modelled by a belief set (i.e. a logically closed set of sentences), this framework essentially provides a definition for contractions-i.e. functions that receive a sentence (representing the new information received by the agent), and return a belief set which is a subset of the original one that does not contain the received sentence. In the mentioned paper, the class of partial meet contractions was introduced and axiomatically characterized. Subsequently several constructive models have been presented for the class of contraction functions proposed in the AGM framework (such as the system of spheres-based contractions [8], safe/kernel contractions [2,9,14], and the epistemic entrenchment-based contractions [6,7]). Also several adaptations and variations of those constructive models have been presented and studied in the literature as it is the case, for example, of severe withdrawals (or mild contractions or Rott's contractions) [12,15] which results of simplifying the definition of epistemic entrenchment-based contractions.
Although the AGM model has quickly acquired the status of standard model of theory change, several researchers (for an overview see [3]) have pointed out its inadequacy in several contexts and proposed several extensions and generalizations to that framework. One of the most relevant of the proposed extensions of the AGM model of contraction is to use sets of sentences not (necessarily) closed under logical consequence-which are designated belief bases-rather than belief sets to represent belief states.
Hence, several of the existing models (of AGM contractions) were generalized to the case when belief states are represented by belief bases instead of belief sets. Among those we emphasize the ensconcement-based contractions and the brutal contractions (of belief bases) proposed in [17], which can be seen as adaptations to the case of belief bases of the epistemic entrenchmentbased contractions and of the severe withdrawals, respectively. In fact, the definitions of those operations are both based on the concept of ensconcement, which is an adaptation of the concept of epistemic entrenchment relation to the case of belief bases. In the mentioned paper Mary-Anne Williams has also presented a method for constructing an epistemic entrenchment from an ensconcement relation.
In the present paper we will study the interrelation among brutal contractions (of belief bases) and severe withdrawals (of belief sets). More precisely, we will present an axiomatic characterization of the class of brutal contractions. After that we devote special attention to the class of brutal contractions which are based on bounded ensconcements-the so-called bounded brutal contractions and also to the class of the so-called ensconcement-based severe withdrawals, which is formed by the severe withdrawals that are based on an epistemic entrenchment relation defined from a bounded ensconcement using Mary-Anne William's method. We shall provide axiomatic characterizations to each one of those classes of functions and study the interrelation among them. This paper is organized as follows: In Section 2 we provide the notation and background needed for the rest of the paper. In Section 3 we present axiomatic characterizations for the classes of brutal contractions and bounded brutal contractions. In Section 4 we show how to define a bounded brutal contraction from an ensconcement-based severe withdrawal and viceversa. Furthermore we present an axiomatic characterization for the class of ensconcement-based severe withdrawals. Finally, in Section 5, we briefly summarize the main contributions of the paper. In the Appendix we provide proofs for all the original results presented.

Formal Preliminaries
We will assume a language L that is closed under truth-functional operations and a consequence operator Cn for L. Cn satisfies the standard Tarskian properties, namely inclusion (A ⊆ Cn (A)), monotony (if A ⊆ B, then Cn(A) ⊆ Cn(B)), and iteration (Cn(A) = Cn (Cn(A))). It is supraclassical and compact, and satisfies deduction (if β ∈ Cn (A ∪ {α}), then (α → β) ∈ Cn(A)). A α will be used as an alternative notation for α ∈ Cn (A), α for α ∈ Cn(∅) and Cn(α) for Cn ({α}). Upper-case letters denote subsets of L. Lower-case Greek letters denote elements of L. ⊥ stands for an arbitrary contradiction.
A well-ranked preorder on a set X is a preorder such that every nonempty subset of X has a minimal element, and similarly an inversely well-ranked preorder on a set X is a preorder such that every nonempty subset of X has a maximal element. A total preorder on X is bounded if and only if it is both well-ranked and inversely well-ranked. 1

Epistemic Entrenchment and Severe Withdrawals
We start by recalling, in the following definition, the concept of epistemic entrenchment relation. Definition 1 ([6,7]). An ordering of epistemic entrenchment with respect to a belief set K is a binary relation ≤ on L which satisfies the following properties: We shall denote the strict part and the symmetric part of ≤ by < and = ≤ , respectively. Now we proceed to the presentation of the definition of the severe withdrawals (also known as mild contractions or Rott's contractions) which was introduced by Rott in [12]. Definition 2 ([12]). Let K be a belief set and ≤ be an epistemic entrenchment relation with respect to K. The ≤-based severe withdrawal on K is the operation ÷ ≤ defined, for any α ∈ L, by: An operation ÷ on K is a severe withdrawal if and only if there is an epistemic entrenchment relation ≤ with respect to K such that, for all sentences Severe withdrawals were axiomatically characterized independently by Rott and Pagnucco in [15] and by Fermé and Rodriguez in [5].

Observation 3 ([15]).
Let K be a belief set and ÷ be a contraction function on K. Then ÷ is a severe withdrawal if and only if it satisfies the following postulates: We note also that in [15,Proof of Lemma 1 (i)] it is shown that, in the presence of (÷1) to (÷4), the postulate (÷9) is equivalent to the following two postulates (taken together): Hence, in [15], it is also presented an alternative axiomatization of severe withdrawals consisting of the postulates (÷1) to (÷4), (÷6), (÷7a) and (÷8).
However, at this point it is worth mentioning that, as attested by the following observation, postulate (÷6) is redundant in both of the above recalled axiomatic characterizations.

Ensconcement and Brutal Contractions
We start by recalling the definition of ensconcement and some related concepts, which was originally proposed by Mary-Anne Williams [16][17][18]. Definition 5 ([17]). An ensconcement is a pair (A, ) where A is a belief base and is a transitive and connected relation on A that satisfies the following three conditions: 2 is well-ranked/inversely well-ranked, then (A, ) is a well-ranked/ inversely well-ranked ensconcement. If is both well-ranked and inversely well-ranked then (A, ) is a bounded ensconcement.
Williams has also introduced the concepts of cut and proper cut, which we recall in the following definition. • For all α ∈ Cn(A) the cut of α, denoted cut (α) is the following subset of A: • For all α ∈ L the proper cut of α, denoted cut ≺ (α) is the subset of A defined by: The following observation states that when α is an explicit belief, its proper cut consists of the set of sentences which are strictly more ensconced than α.
We notice that, when α ∈ A the proper cut cut ≺ (α) can be seen as the set formed by the sentences of A which may be considered to be strictly better than α.

(b) is well-ranked (inversely well-ranked, bounded) if and only if ≤ is well-ranked (inversely well-ranked, bounded).
Finally, we recall the definition of the brutal contraction which was introduced in [17] and is essentially based on the above presented notion of proper cut.

Definition 10 ([17]).
Let (A, ) be an ensconcement. The -based brutal contraction on A is the operation − such that: An operation − on A is a brutal contraction if and only if there is an ensconcement (A, ) such that for all sentences α:

Axiomatic Characterization of Brutal Contraction Functions
In this section we present an axiomatic characterization for the class of brutal contractions. Furthermore we also provide a representation theorem regarding the subclass of that class of functions formed by the brutal contractions that are based on a bounded ensconcement.
In the following theorem we present an axiomatic characterization of brutal contraction functions.
The first five postulates listed above are well known in the literature of belief change. Strong Inclusion, is presented in [5,15] as meaning that if α is not deducible from the set that results of contracting A by β then anything given up in removing α from A should also be given up when removing β from A. Uniform Behaviour asserts that if a sentence α that is deducible from A is such that the result of its contraction from A coincides with the result of contracting A by a sentence which is (explicitly) present in A then α should be deducible from the union of the set of all the sentences fulfilling that property with the set that results of contracting A by α. We note that this postulate is trivial when α ∈ A.
At this point it is worth to compare the above representation theorem with the axiomatic characterization for the severe withdrawals which results of combining Observations 3 and 4. We note that the postulates of relative closure, inclusion, failure and vacuity (together), success and strong inclusion can be seen as the analogues in the belief bases setting of the postulates (÷1), (÷2), (÷3), (÷4) and (÷9). Thus, the main difference among the two axiomatizations is the presence of the postulate of uniform behaviour in the characterization of brutal contractions. In this regard we recall that uniform behaviour holds trivially when α ∈ A. Therefore, this postulate can be seen as the property that captures the behaviour of brutal contractions by implicit sentences (a kind of contraction that does not occur in the belief sets setting since, in that context all beliefs are explicit).
The following observation lists some other properties that are satisfied by a brutal contraction function.
Observation 12. Let A be a belief base and − be an operator on A that satisfies success, inclusion, vacuity, failure, relative closure and strong inclusion. Then − satisfies: A brief comment concerning a couple of arguably undesirable properties of the above list is in order. Expulsiveness was first presented in [11, p. 102] as a highly implausible property of belief contraction, since according to it two unrelated sentences influence the result of the contraction by each other. Other one of the above listed postulates that also suffers from this same excessive strength is linearity, which was originally presented in [5,15]. Nevertheless, Rott and Pagnucco [13,15] argue that the concept of severe withdrawal (a contraction function that satisfies the two mentioned postulates) is still interesting and well-motivated.

Bounded Brutal Contraction Functions
In this subsection we introduce the bounded brutal contractions and obtain an axiomatic characterization for that class of functions.
Definition 13. Let A be a belief base. An operation − is a bounded brutal contraction on A if and only if it is a brutal contraction based on a bounded ensconcement.
We introduce the following postulates: Upper Bound (respectively Lower Bound) states that every non-empty set of nontautological formulae of A contains an element which is such that the result of contracting A by that sentence is a superset (respectively a subset) of any set which results of contracting A by one of the remaining sentences of that set. Clustering asserts that for any sentence β in A there exists some sentence α in A ∪ Cn(∅) such that the result of the contraction of α from A is the set consisting of the union of the result of contracting A by β with the set formed by all the sentences of A which are such that the result of contracting it from A coincides with the result of contracting A by β.
The two following observations present some interrelations among the above proposed postulates and some of the postulates included in the axiomatic characterization that was obtained for the class of brutal contraction.
Observation 14. Let A be a belief base and − be an operator on A that satisfies success, inclusion, failure, relative closure, strong inclusion and lower bound. Then − satisfies clustering.
Observation 15. Let A be a belief base and − an operator on A that satisfies failure, success, strong inclusion and clustering. Then − satisfies uniform behaviour.
We are now in a position to present an axiomatic characterization for the class of bounded brutal contractions.

Theorem 16. (Axiomatic characterization of bounded brutal contraction functions) Let A be a belief base. An operator − on A is a bounded brutal contraction on A if and only if it satisfies success, inclusion, vacuity, failure, relative closure, strong inclusion, lower bound and upper bound.
The following observation asserts that for any non-tautological sentence α which is deducible from A it holds that the result of contracting A by α coincides with the result of the contraction of A by some sentence explicitly included in A.
Observation 17. Let A be a belief base and − be an operator on A that satisfies success, inclusion, failure, relative closure, strong inclusion and lower bound. Then for all α ∈ Cn (A)

Relation Between Bounded Brutal Contraction and Ensconcement-based Severe Withdrawal
In this section we will define and axiomatically characterize a particular kind of severe withdrawals which we will show to be the contraction functions that correspond to the bounded brutal contractions in the context of belief set contractions. We start by noticing that, given a bounded ensconcement (A, ), we can combine Observation 9 and Definition 2 in order to obtain the contraction function on the belief set Cn (A) that is formally introduced in the following definition.
Definition 18. Let (A, ) be a bounded ensconcement. An operation ÷ on Cn (A) is an ensconcement-based withdrawal related to (A, ) if and only if Cn (A) where ≤ is the epistemic entrenchment with respect to Cn (A) presented in Observation 9 and ÷ ≤ is the ≤ -based severe withdrawal on Cn (A) defined by (R ≤ ).
Comparing the above definition with Definitions 10 and 13 it becomes clear that there is a strong interrelation among the ensconcement-based severe withdrawals and the (bounded) brutal contractions. That interrelation is explicitly presented in the two following theorems: Theorem 19. Let (A, ) be a bounded ensconcement, − be the -based brutal contraction, and ÷ ≤ be the ensconcement-based severe withdrawal related to (A, ), Theorem 20. Let (A, ) be a bounded ensconcement, − be the -based brutal contraction, and ÷ ≤ be the ensconcement-based severe withdrawal related to (A, ), then Cn (A) Given a bounded ensconcement (A, ), these two theorems expose how the -based brutal contraction on A can be defined from the ensconcementbased withdrawal related to (A, ) and, vice-versa, how the latter can be defined by means of the former.

Axiomatic Characterization of Ensconcement-Based Severe Withdrawals
In this subsection we will present an axiomatic characterization for the class of ensconcement-based severe withdrawals. To do that we must start by introducing the following postulate: This postulate essentially states that the result of contracting the belief set Cn (A) by any sentence α coincides with the logical closure of some subset of A. Indeed, it is not hard to see that base-reduction is equivalent to: ∀α∃A ⊆ A : Cn(A ) = Cn (A) ÷ α (which is quite similar to the postulate finitude proposed by Hansson in [10]).
The following observation relates a contraction by an implicit sentence with a contraction by an explicit sentence.
Observation 21. Let ÷ be an operator on Cn(A) that satisfies (÷1), (÷2), (÷4), (÷9), base-reduction and lower bound, then for all α ∈ Cn (A) We are now in a position to present the following axiomatic characterization for the ensconcement-based severe withdrawals.
Theorem 22. (Axiomatic characterization of ensconcement-based withdrawals) Let A be a belief base. An operator ÷ on Cn (A) satisfies (÷1) to (÷4), (÷9), base-reduction, upper bound and lower bound if and only if there exists a bounded ensconcement (A, ) such that ÷ is an ensconcement-based withdrawal related to (A, ).
Theorems 19 and 20 expose how a base contraction function can be defined from a belief set contraction function and, vice-versa. Combining those two results with the axiomatic characterizations presented in Theorems 16 and 22 we can obtain the following results which highlight the correspondence among sets of postulates for base contraction and sets of postulates for belief set contraction. The two following observations consist of a slight refinement of the right to left part of Corollary 24. These results specify more precisely which properties of the belief base contraction are needed in order to assure that the belief set contraction obtained from it as exposed in Theorem 20 satisfies certain postulates.

Conclusion
We have presented two axiomatic characterizations for brutal contractions one for brutal contractions based on a general ensconcement relation and another one for the particular case of brutal contractions that are based on bounded ensconcements. We have also introduced and axiomatically characterized the class of ensconcement-based severe withdrawals which is formed by the severe withdrawals that are based on epistemic entrenchment relations which are obtained from an ensconcement relation using the construction proposed by Mary-Anne Williams. Some results were presented concerning the interrelation among the classes of bounded brutal contractions and of ensconcement-based severe withdrawals.  Lemma 1 ([4,Lemma 11]).
Proof. If α, then from ( 2) it follows that β. Hence, the proof follows from ( 3) and Lemma 1 (a). Assume now that α and consequently that β. From left to right it follows from Lemma 1 (i). For the other direction: Let α, β ∈ A, if α ≺ β, then by Lemma 1 (g) cut ≺ (α) β and so cut ≺ (β) β which contradicts Lemma 1 (b). Due to the symmetry of the case we may conclude that β ≺ α. Since α ≺ β, β ≺ α, and is connected, we can conclude that α β and β α. Lemma 3 ([15,Lemma 2]). Let K be a belief set. If ÷ is a severe withdrawal on K, then ÷ satisfies the following postulates: Lemma 4 ([15, Observation 19(ii)]). Let K be a belief set. If ÷ is a severe withdrawal on K, then ÷ can be represented as an entrenchement-based withdrawal where the relation ≤ on which ÷ is based is obtained by (Def ≤ from ÷) α ≤ β if and only if α ∈ K ÷ β or β and ≤ satisfies (EE1) to (EE5).

From Postulates to Brutal Contraction:
Let − be an operator on A that satisfies success, inclusion, vacuity, failure, relative closure, strong inclusion and uniform behaviour. In order to prove that − is a brutal contraction we must prove that there exists a transitive, connected relation that satisfies ( 1) -( 3) and such that where cut is defined in terms of .
Let be defined for α, β ∈ A as follows: Let α ∈ A and γ ≺ α, then, according to our construction, there are two possible cases to consider (the other two cases are excluded by failure, success and inclusion): Since α ∈ A and A−γ ⊆ A−α then, by Observation 12 (a), it follows that α ∈ A − γ.
Hence H ⊆ A − γ. Therefore, by success, it follows that H γ.
( 2) Let α, β ∈ A such that α and β. We need to prove that α β and β α. Both follow from the definition of .
( 3) Follows trivially from the definition of . ( is connected) By the definition of , it follows that α β if and only if (A − β ⊆ A − α or α) and β. We will prove by cases: Case 1) α and β. It follows from ( 2) that β α.

(f ) Follows by strong inclusion.
(g) We will prove by cases: Case 1) α, it follows from failure that A−α = A and so (by inclusion) Case 2) β, due to the symmetry of the case, it follows that (h) Follows by success and (g).
(i) If α ∧ β it follows trivially from failure. Assume now that α ∧ β. It follows from α ↔ β that α, β, α → β and β → α. Then, due to success, A − β α and A − α β. Hence, by strong inclusion, If β, then it follows, from failure, that A − β = A and that H = A. Then it is enough to consider α = β. Assume now that β. We will prove by cases: We will consider two cases: Thus we have two cases to consider: from Observation 12 (f) and relative closure, Proof of Theorem 16. From bounded brutal contraction to postulates: Let − be a bounded brutal contraction operator on A. By Theorem 11 − satisfies success, inclusion, vacuity, failure, relative closure and strong inclusion. It remains to show that − satisfies upper bound and lower bound.
Upper Bound Let X ⊆ A be a non empty set of non-tautological formulae. Since is well ranked there exists β ∈ X such that β α for all α ∈ X. Hence, by Lemma 1 (f), there exists β ∈ X for all α ∈ X such that cut ≺ (α) ⊆ cut ≺ (β). Therefore, by definition of − there exists β ∈ X for all Lower Bound Analogous to upper bound.

From postulates to bounded brutal contraction:
Let − be an operator on A that satisfies success, inclusion, vacuity, failure, relative closure, lower bound, upper bound and strong inclusion. From Observation 14 and Observation 15 it follows that − satisfies uniform behaviour. Let be defined by: According to the Postulates to Construction part of the proof of Theorem 11 satisfies ( 1) - ( 3) and is such that It remains to prove that is bounded. To do so we must prove that is well-ranked and inversely well-ranked. ( is well-ranked) Let X = ∅ and X ⊆ A. We will prove by cases: Case 1) All formulae in X are tautologies. Let β be one of those formulas. Hence by ( 3) β α for all α ∈ X. Case 2) All formulae in X are non-tautological. By upper bound there exists β ∈ X such that A − α ⊆ A − β for all α ∈ X. Hence, by definition of , there exists β ∈ X such that β α for all α ∈ X. Case 3) There are some formulae in X, that are tautological and others that are not. Consider X = X \ Cn(∅). Hence, by the previous case, there exists β ∈ X such that β α for all α ∈ X . Therefore, it follows from ( 3) that β α for all α ∈ X. ( is inversely well-ranked) Let X = ∅ and X ⊆ A. We will prove by cases: Case 1) There are some β ∈ X ∩ Cn(∅). Then, by definition of , α β for all α ∈ X. Case 2) All formulae in X are non-tautological. By lower bound there exists β ∈ X such that A − β ⊆ A − α for all α ∈ X. Hence, by definition of , there exists β ∈ X such that α β for all α ∈ X.
We have proved that is a bounded ensconcement. Let ≤ be the bounded epistemic entrenchment related to Cn (A) defined from as exposed in Observation 9. It remains to show that Cn (A) where ÷ ≤ is defined (as in (R ≤ )) by: (A) and α Cn (A) otherwise According to Lemma 4 and since ÷ is a severe withdrawal function, the epistemic entrenchment ≤ on which ÷ is based on is such that: α ≤ β if and only if α ∈ Cn(A)÷β or β. Thus to prove that Cn (A) (A), or ii) α, β ∈ Cn (A) and cut (β) ⊆ cut (α). We will prove by cases: Case 1) α ∈ Cn (A). Then, by (÷2), α ∈ Cn(A) ÷ β. Case 2) α, β ∈ Cn (A) and cut (β) ⊆ cut (α). It follows trivially if β. Assume now that β. The proof for ÷ lower bound is analogous to the one presented for ÷ upper bound.