Between Belief Bases and Belief Sets: Partial Meet Contraction
Yuri D. Santos and Marcio M. Ribeiro and Renata Wassermann University of Sao˜ Paulo - Brazil [email protected] [email protected] [email protected]
Abstract satisfied). Hence, in belief base contraction there is a dis- tinction between the sentences in the base and the sentences In belief revision, the result of a contraction should inferred from the base. Especially in computer science, be- be a logically closed set (closure) such that the in- lief base revision is very useful since computing the logical put is not inferred (success). Traditionally, these closure of a set may be hard if possible at all. two postulates depend on the same consequence Operations in belief bases, however, have some undesirable operator. properties due to the fact that equivalent sentences may be The present work investigates the consequences of treated differently. In the present work we explore the conse- using two different consequence operators for each quences of keeping both closure and success where the conse- of these desiderata: the classic Cn to compute suc- quence operations in these two postulates do not necessarily cess and a second weaker operator to compute the coincide. More precisely, we assume that closure is guaran- closure. We prove a representation theorem and teed only for a weaker consequence operation. This weaker some other properties for the new contraction. notion of consequence may be easy to compute and at the same time useful to avoid some undesirable consequences of 1 Introduction belief base operations. Throughout this paper, we consider the language of classi- Belief revision is the branch of knowledge representation that cal propositional logic, closed under the usual boolean con- deals with the dynamics of epistemic states. In their semi- nectives. nal paper [Alchourron´ et al., 1985], Alchourron,´ Gardenfors¨ We call consequence operator a function that takes sets of and Makinson proposed to represent the epistemic state of an formulas into sets of formulas. A consequence operator C is agent as a logically closed set of propositions called belief Tarskian if and only if it satisfies: sets. The authors focus on three main epistemic operations over belief sets: expansion, contraction and revision. Expan- (inclusion) A ⊆ C(A) sion is the simple addition of a proposition to the epistemic state, contraction is the removal of a proposition and revision (idempotence) C(A) = C(C(A)) is the consistent addition of a proposition. Each of this oper- ations is characterized by a set of rationality postulates. (monotonicity) If A ⊆ B then C(A) ⊆ C(B) Consider, for example, the following postulates for con- Cn denotes the classical consequence operator and ` de- traction: notes the associated relation: A ` α stands for α ∈ Cn(A). (closure) K − α = Cn(K − α) Lowercase Latin letters (p, q, r) stand for atoms, lower- case Greek letters (α, β) stand for formulas, uppercase Latin (success) If α∈ / Cn(∅), then α∈ / Cn(K − α) letters (A, B, K) stand for sets of formulas. The rest of the paper is structured as follows. A brief Closure guarantees the resulting epistemic state to be rep- overview of classical belief revision is given in Section 2. resented as a logically closed set of propositions, i.e., the re- Section 3 presents pseudo-contractions, the kind of operation sult of contracting a belief set is also a belief set. Success we are exploring in this paper. In Section 4, we describe our guarantees that the removed proposition α is no longer im- proposal, depicting it with examples in Section 5. Section 6 plied by the new belief set, unless it is a tautology. highlights some previous works that are related to ours. We Hansson in [Hansson, 1992a] suggests a generalization of delineate our conclusions in Section 7. Proofs for theorems the AGM framework where the epistemic state is not nec- and propositions are found in the Appendix. essarily closed under logical consequences. Although in his work epistemic states are represented as arbitrary sets of sen- tences called belief bases (i.e. closure is not necessarily sat- 2 AGM Paradigm isfied), the removal of a sentence in contraction is still eval- In the AGM paradigm [Alchourron´ et al., 1985], belief states uated against the closure of the belief base (i.e. success is are typically represented by sets of sentences closed under logical consequence, the so-called belief sets. Three change (uniformity) If for all B0 ⊆ B, α ∈ Cn(B0) if and only if operations were initially defined: expansion, contraction and β ∈ Cn(B0), then B − α = B − β revision. Expansion is the simple addition of a new sentence, Furthermore, we have the following representation theo- followed by logically closing the resulting set, i.e, K + α = rem: Cn(K ∪ {α}). In the case of contraction, we have a class of operations, all delimited by rationality postulates that they Theorem 4 [Hansson, 1992b] An operation is a partial meet should satisfy. Let K be a belief set and α and β be formulas. contraction over belief bases if and only if it satisfies the pos- The original AGM basic postulates for contraction are: tulates of success, inclusion, relevance and uniformity. (closure) K − α = Cn(K − α) For logically closed sets, both characterizations are equiv- alent (this is true for classical logics, for other cases, cf. (success) If α 6∈ Cn(∅), then α 6∈ K − α [Ribeiro et al., 2013]) in the sense that all initial AGM pos- tulates are consequence of the postulates for bases [Hansson, (inclusion) K − α ⊆ K 1999].
(vacuity) If α 6∈ K, then K − α = K 3 Pseudo-Contractions Contraction operators on belief sets can be generated from (recovery) K ⊆ (K − α) + α operations on belief bases. As an example, one can define a contraction operator for belief sets as K ÷ α = Cn(B − α), (extensionality) Cn(α) = Cn(β) K −α = K −β If , then where K = Cn(B) and − is a base contraction operator. If − is a partial meet base contraction, ÷ satisfies five of the six Note that in the presence of closure, we can use α 6∈ K −α AGM postulates, but not recovery. or α 6∈ Cn(K − α) interchangeably in the success postulate. In [Hansson, 1989] a weakening of the inclusion postulate AGM revision is usually defined from contraction and ex- was proposed, called logical inclusion. pansion by means of the Levi identity: (logical inclusion) Cn(B − α) ⊆ Cn(B) K ∗ α = (K − ¬α) + α Hansson has suggested to call operations that satisfy suc- Therefore, in this paper we will focus on contraction. cess and logical inclusion pseudo-contractions. Besides defining rationality postulates for contraction, Al- Nebel has proposed a pseudo-contraction for bases that chourron´ et al.[1985] have also proposed a construction for generates a contraction that satisfies all the six AGM postu- a contraction operation. Partial meet contraction is based on lates [Nebel, 1989]. the notion of a remainder set, the set of all maximal subsets Definition 5 Let V B be the conjunction of all elements of that do not imply the element that is to be contracted. For- B. Nebel’s pseudo-contraction for the set B is the operator mally: − such that for all sentences α: Definition 1 Let B be a set and α a formula. The remainder set B⊥α is such that X ∈ B⊥α if and only if: B if α ∈ Cn(∅) B − α = • X ⊆ B T γ(B⊥α) ∪ {α → V B} otherwise • X 0 α Although the belief set operation generated from Nebel’s • For all sets Y , if X ⊂ Y ⊆ B, then Y ` α pseudo-contraction satisfies all the AGM postulates, it adds unnecessary information to the base. As shown in [Ribeiro Definition 2 A function γ is a selection function for the set and Wassermann, 2008], it suffices to add {α → V B0}, B if and only if: where B0 = B\T γ(B⊥α). As already noted in [Ribeiro and • If B⊥α 6= ∅ then ∅= 6 γ(B⊥α) ⊆ B⊥α Wassermann, 2008], there is no other intuition behind Nebel’s • Otherwise, γ(B⊥α) = {B} operation than maintaining recovery, a postulate which has been deemed as polemic already since the 80’s [Makinson, Definition 3 Let γ be a selection function for a set of sen- 1987]. tences B. The partial meet contraction of B by a sentence α T In this work we want to further explore the possibility of is given by B − α = γ(B⊥α). working with belief bases with logical inclusion, allowing for The partial meet operation previously defined can be ap- some syntax independence without having to resort to belief plied directly over belief bases, in such a way that it satisfies sets. the following postulates: (success) If α∈ / Cn(∅), then α∈ / Cn(B − α) 4 Between Belief Sets and Belief Bases As stated earlier, the direct application of partial meet con- (inclusion) B − α ⊆ B traction over closed belief sets and over belief bases creates problems of practical (computational infeasibility) and the- (relevance) If β ∈ B \ (B − α), then there is a B0 such oretical (syntax dependence) nature, respectively. One of that B − α ⊆ B0 ⊆ B, α∈ / Cn(B0), but α ∈ Cn(B0 ∪ {β}) the aims of this study is to assess the effects of doing the traditional partial meet contraction on belief bases closed With a proof that is very similar to that of the representa- by a consequence operation that is between the classical tion theorem for partial meet contraction on bases (which can consequence operator and the identity (i.e., the base itself). be found in [Hansson, 1999]), we can prove the following Hence, we will assume that this operator (here called Cn∗) is representation theorem: Tarskian. ∗ We will study the properties of the application of the par- Theorem 8 Provided that Cn is Tarskian, subclassical and ∗ compact, an operation is a −∗ operator if and only if it satis- tial meet contraction over a set closed under Cn , i.e., the ∗ ∗ ∗ operator defined as: fies success, inclusion , relevance and uniformity . Definition 6 Let B be a set of sentences, Cn∗ a function from From this theorem and the previous corollary, it also fol- sets of sentences to sets of sentences and γ a selection func- lows: ∗ tion for Cn (B). The operator −∗ is such that, for all sen- Corollary 9 If Cn∗ is Tarskian and satisfies subclassicality, tences α: then −∗ satisfies success, logical inclusion, logical relevance \ ∗ and uniformity. B −∗ α = γ(Cn (B)⊥α) It is interesting that we have here a set of postulates that T ∗ ∗ Notice that B −∗ α = γ(Cn (B)⊥α) = Cn (B) −γ α, are independent from Cn∗. Nonetheless, these postulates do where −γ is the partial meet contraction. Since −γ satisfies not characterize the operation, and are in general weaker than the postulates of success, inclusion, relevance and unifor- the postulates with ∗. Hansson’s logical inclusion postulate mity, it follows directly (details in the Appendix) that −∗ sat- is quite reasonable for base operations, as it brings syntactic isfies success and: independence, although with inclusion∗ we already have a (inclusion∗) B − α ⊆ Cn∗(B) degree of independence, and with better preservation of the original set (since Cn∗ is subclassical), and, depending on (relevance∗) If β ∈ Cn∗(B) \ (B − α), then there is a the chosen Cn∗, we avoid the complexity problem we have B0 such that B − α ⊆ B0 ⊆ Cn∗(B), α∈ / Cn(B0), but with the closure of Cn. α ∈ Cn(B0 ∪ {β}) Another desirable property in rational contraction opera- tions is relative closure [Hansson, 1991]. (uniformity∗) If for all B0 ⊆ Cn∗(B), α ∈ Cn(B0) if and only if β ∈ Cn(B0), then B − α = B − β (relative closure) B ∩ Cn(B − α) ⊆ B − α For several applications it is important that the construction This property is a consequence of the postulate of rele- satisfies the original success postulate, and not only a starred vance, which −∗ does not satisfy. Nevertheless, relative clo- ∗ version of it: sure is satisfied, given the condition that Cn is Tarskian. ∗ ∗ ∗ (success ) If α∈ / Cn(∅), then α∈ / Cn (B − α) Proposition 10 If Cn is Tarskian, the −∗ operator satisfies relative closure. We want that the sentence to be contracted ceases to be logically (classically) implied by the resulting set after the Kernel contraction [Hansson, 1994] is an alternative con- contraction. In this case, the role of the logic Cn∗ is just to struction for contraction, which instead of considering maxi- give a degree of syntactic independence to the operation. mal sets not implying a given formula as in partial meet con- As our purpose here is to make the contraction on a set traction, considers minimal sets implying it. It works by find- closed by a Cn∗ that does not generate as many consequences ing the minimal sets (called α-kernels) that imply the element as the classic Cn, the following property is desirable: α being contracted and then selecting at least one element of each α-kernel to remove from the belief base. The operation (subclassicality) Cn∗(A) ⊆ Cn(A) is characterized by the same postulates as partial meet con- ∗ Clearly, if Cn (A) = A (identity), we have that −∗ is the traction on bases, except for relevance, which is weakened to usual operation of partial meet contraction on bases. Simi- core-retainment: ∗ larly, for all Tarskian Cn that also satisfies subclassicality, 0 ∗ (core-retainment) If β ∈ B \ (B − α), then there is a B applying −∗ to belief sets (i.e., K = Cn (K) = Cn(K)) 0 0 0 yields the usual AGM partial meet contraction on belief sets. such that B ⊆ B, α∈ / Cn(B ), but α ∈ Cn(B ∪ {β}) Following the same idea as logical inclusion, in [Ribeiro Kernel contraction may have erratic behaviour due to its and Wassermann, 2008] we have a weakening of relevance: non-satisfaction of relevance [Hansson, 1999]. For instance, (logical relevance) If β ∈ B \ (B − α), then there is a consider the logically independent sentences p and q, and let B0 such that B − α ⊆ B0 ⊆ Cn(B), α∈ / Cn(B0), but A = {p, p ∨ q, p ↔ q}. The kernel contraction A − (p ∧ q) = α ∈ Cn(B0 ∪ {β}) {p} is possible, whereas partial meet contraction cannot have this outcome. As observed by Hansson, it is not sensible to The next corollary follows: give up p ∨ q, since p was kept. Corollary 7 If Cn∗ is Tarskian and satisfies subclassical- Would the weakening of relevance to relevance∗ or logi- ity, then an operation that satisfies success, inclusion∗, cal relevance be enough so as to make these behaviours show ∗ ∗ relevance and uniformity also satisfies logical inclusion, up in the −∗ operator? Kernel contraction does not satisfy logical relevance and uniformity. any of these last two postulates. Furthermore, Hansson had already noticed that the lack of relative closure also con- Any operator satisfying inclusion will satisfy this postulate tributes to these unnecessary removals in contraction. The trivially. If we break {α → V B} into the set of sentences operator −∗, as shown above, satisfies relative closure. {α → β | β ∈ B}, Nebel’s pseudo-contraction will not sat- ∗ A property of our pseudo-contraction is enforced closure . isfy core-addition. Clearly the −∗ operator does not satisfy 0 ∗ ∗ it also (neither does −∗), and it does not satisfy vacuity as (enforced closure ) B − α = Cn (B − α) well. In the effort to fix these two problems, the operations of general partial-meet pseudo-contraction and ∆-partial-meet ∗ Proposition 11 If Cn is Tarskian and satisfies subclassical- pseudo-contraction, proposed in [Ribeiro and Wassermann, ∗ ∗ ity, an operator that satisfies inclusion and relevance also 2008], seem to be viable solutions. satisfies enforced closure∗. Whenever A ⊂ Cn∗(A), this postulate will imply that 5 Examples vacuity is not satisfied, which is an essential postulate in the In this section we are going to show some concrete examples point of view of rational contractions (that respect the princi- of Cn∗ functions that can be useful in the solution of practi- ple of minimal change). It has as effect that the belief base ∗ cal problems. The first example we mention is the Cleopatra will always end up closed by Cn after the contraction, even example, adapted from [Ribeiro and Wassermann, 2008]. though the original base was not closed. Example 1 Consider a language with three propositional (vacuity) If α∈ / Cn(B), then B − α = B letters, p, q and r and a belief base B = {p ∧ q}, where p stands for Cleopatra had a son and q, Cleopatra had a daugh- Nevertheless, our construction does satisfy a weaker form ter. If we want to contract by p, applying a partial meet con- of vacuity: traction produces B − p = ∅. This is not always the expected (vacuity∗) If α∈ / Cn(B), then B − α = Cn∗(B) result, because the loss of faith in the belief that Cleopatra had a son also made us lose faith in the belief that she had a Proposition 12 If Cn∗ is Tarskian and satisfies subclassical- daughter. ∗ ∗ ity, an operator that satisfies inclusion and relevance also With the classic partial meet construction for bases we ∗ satisfies vacuity . would have B⊥p = {∅}, so the selection function needs to Corollary 13 If Cn∗ is Tarskian and satisfies subclassicality, choose {∅}, causing the overall contraction process to pro- ∗ ∗ duce ∅ as final result. then the operator −∗ satisfies enforced closure and vacuity . On the other hand, if we take B to represent the belief set A simple way to restore vacuity is to redefine the −∗ op- K = Cn(B), then K contains both p and q and the belief that erator in the following manner: Cleopatra had a daughter (q) may survive the contraction, i.e., Definition 14 Let B be a set of sentences, Cn∗ a function we may have q ∈ K − p. But then we would also have q ∨ r, from sets of sentences to sets of sentences and γ a selection r → q and many other irrelevant formulas in the resulting set, ∗ 0 since it is closed under Cn. function for Cn (B). The operator −∗ is such that, for all sentence α: Let us consider an intermediate consequence operator: ∗ Cn1(A) = {α | α ∈ A or for any formulas β, δ, B if α∈ / Cn(B) α ∧ β ∈ A or β ∧ α ∈ A or β ∧ α ∧ δ ∈ A} B −0 α = ∗ T γ(Cn∗(B)⊥α) otherwise ∗ We can use Cn1 with the −∗ operator to solve the problem Cn∗(B) = 0 of the preceding example. In this case, we have 1 Observation 15 The −∗ operator satisfies success, {p ∧ q, p, q} Cn∗(B)⊥p = {{q}} ∗ ∗ ∗ and 1 , hence the selection inclusion , uniformity and vacuity. If Cn is Tarskian function would choose the whole remainder set, {{q}}, and, and subclassical, −0 also satisfies logical inclusion, ∗ accordingly, B −∗ p = {q}. uniformity and relative closure. Although the usefulness of this consequence operation is The proof of the observation above is not given, but can be dubious, its use already brings better results than the typical trivially obtained from theorem 8, corollary 7 and proposition base contraction in some cases, as in the former example. ∗ 10. Note that relevance and logical relevance were lost. Example 2 Suppose I believe that the town of Juazeiro do Although we have attained vacuity, we have only partly Norte is located in the state of Pernambuco (j → p) and that ∗ gotten rid of enforced closure (just when α∈ / Cn(B)). the state of Pernambuco is located in Brazil (p → b). Speak- If on one hand logical inclusion seems to make more sense ing with a colleague, I found that this town is not located in than inclusion for base contractions, by allowing some syn- his state (Pernambuco), that is, I contract j → p from my tactic independence, effects such as enforced closure∗ illus- base. The outcome is B − (j → p) = {p → b}. So, I no trate the need to refrain from careless additions of sentences longer know whether Juazeiro do Norte is located in Brazil. in the contraction. Here we should mention the postulate of core-addition [Ribeiro and Wassermann, 2008]: In this example, as well as in the previous one, one can blame the poor codification of the belief base for the prob- (core-addition) If β ∈ (B − α) \ B, then there is a β0 ∈ lems. The knowledge that Juazeiro do Norte is located in B \ (B − α) and a B0 ⊆ B − α such that α → β0 ∈/ Cn(B0) Brazil, if obvious, perhaps would be individually justified, but α → β0 ∈ Cn(B0 ∪ {β}). and so it would deserve to be explicitly in the base. At this point the syntactic independence dilemma reappears. In some 6 Related Work cases we want to have it, but without having to generate in- There have been several attempts in the literature to study finitely many derivative sentences with little utility. AGM-like contraction operations based on different conse- However, when working with ontologies, for instance, it is quence operations. possible that the user does not want to make explicit every Concerning belief bases, Hansson and Wassermann 2002 possible relationship, trusting the transitivity of some proper- have shown that the original representation results for char- ties (i.e., he would be more concerned with his ontology on acterizing partial meet contraction only depended on the un- the knowledge level than on the syntactic level). One may derlying logic being compact and monotonic. The main dif- also want a knowledge base with little redundancy. ference from what we are proposing in this paper is that the Again, neither the belief base nor the belief set approach underlying consequence operator C is used for computing the would give us the desired result. ∗ remainder sets and everywhere in the postulates. So, for ex- Returning to the foregoing example, we could use a Cn2 ample, if we take as underlying logic one for approximate that adds to the base the transitive closure of →: reasoning, as suggested in [Chopra et al., 2001], the remain- Cn∗(A) = A ∪ {α → α | for some β, ders will be different than the ones we use for our −∗ opera- 2 1 2 [ α → β, β → α ∈ A}. tion. If the approximation is done from below (cf. Schaerf 1 2 and Cadoli, 1995]), then each element of the approximate re- ∗ In that case, we would have Cn2(B) = B ∪ {j → b}, mainder will contain an element of the classical remainder. ∗ what results in Cn2(B)⊥(j → p) = {{p → b, j → b}}. Success will also be computed in terms of the chosen under- As in the last example, the selection function must choose the lying consequence operation, so the outcome of the contrac- only member of the remainder set, therefore B −∗ (j → p) = tion by α may still classically imply α. {p → b, j → b}. It is interesting to note that we could also More recently, there has been a series of papers consider- ∗ [ use here Cn20 (B) to be the set of all Horn consequences of ing Horn Contraction Delgrande, 2008; Booth et al., 2011; B. Delgrande and Wassermann, 2013], which again, replace all The following example was adapted from [Hansson, 1993]. classical reasoning by Horn reasoning. Contraction of belief sets has also been studied for non-classical logics [Ribeiro et Example 3 Suppose I believe, for good and independent rea- al., 2013; Ribeiro, 2013] along the same lines of the work on sons, that Andy is son of the mayor (a) and Bob is son of the belief bases. mayor (b). Then I hear the mayor say: “I certainly have The approach of [Meyer, 2001] is similar to ours in the fact nothing against our youth studying abroad. My only son did that it “weakens” the formulas that must be removed from the it for three years”. I then have to retract a ∧ b from my base base, thus it is also a pseudo-contraction. However, their pur- B = {a, b}. But it is reasonable to retain a belief that ei- pose is different since they do not take computational costs ther Andy or Bob is the son of the mayor, i.e., the result of the into account (they define base contraction from theory con- contraction should be {a ∨ b}. traction, using logical closure), whereas it is one of our main The remainder set for the operation above is B⊥(a ∧ b) = concerns in this paper. {{a}, {b}}. So, the resulting partial meet contraction is either The closest proposal to the one described in this paper {a}, {b} or ∅, the first two being odd since we do not seem to is the idea of disjunctively closed belief bases, proposed by have reasons to prefer a over b or vice-versa. Hansson [1993], where a single closure is studied, the one we In the same paper where he presented the example above, used in Example 3. Hansson has done an extensive study of partial meet contrac- To the best of our knowledge, [Ribeiro and Wassermann, ∗ 2008] was the first proposal where general alternative conse- tion on disjunctively closed bases. If we define Cn3(A) as the disjunctive closure of A, as defined by Hansson, that is, quence operations are used only to extend the set of formulas ∗ being considered on contraction. The present work follows Cn3(A) is the set of sentences that are either elements of A A the line of [Ribeiro and Wassermann, 2008] (even borrowing or disjunctions of elements of , we can manage to get the ∗ desired result. the Cn notation), but with a different focus. While the idea there was to study forms of recovery, here we are interested ∗ W Cn3(A) = A ∪ { αi|αi ∈ A} in characterizing partial meet operations where the initial be- ∗ lief state is closed under some subclassical consequence op- We have Cn3(B) = {a, b, a ∨ b}. Then, the remainder ∗ erator. set is Cn3(B)⊥(a ∧ b) = {{a, a ∨ b}, {b, a ∨ b}}, and so the selection function may choose both sets, producing the 7 Conclusion expected result in the lack of evidence for a or b: B −∗ (a ∧ b) = {a ∨ b}. In this paper, we have proposed an operation of pseudo- Consider now the case where the language has three propo- contraction, denoted by −∗. We have provided a character- sitional letters (a, b, and c). If we take the belief set K = ization of this operation in terms of postulates which were Cn(B), we have that K contains a ∨ c and b ∨ c. It is adapted from the classical ones to use subclassical operators not hard to see that there are two remainder sets containing of consequence, which we denote by Cn∗. {a, a ∨ b, a ∨ c, b ∨ c}, {b, a ∨ b, a ∨ c, b ∨ c} and hence, these One of the drawbacks of the construction of −∗ is that it two formulas may survive contraction, even if the original set satisfies enforced closure∗, i.e., the result of contraction is ∗ did not mention c. always closed under Cn , causing −∗ to violate vacuity. We have found a way to partially circumvent the problem Part 2: For γ to be a selection function it remains to be 0 ∗ ∗ by defining the −∗ operator. We have also noted that it would proven that if Cn (A)⊥α is not empty, then γ(Cn (A)⊥α) be worthwhile that the operator could satisfy the postulate is not empty as well. Then, assuming Cn∗(A)⊥α 6= ∅, we of core-addition, that avoid needless inclusions. Finally, we know that there is at least one X ∈ Cn∗(A)⊥α, and we must conclude that these two flaws would be possibly amended by show that at least one of these X contains A −∗ α. Since the use of the pseudo-contractions proposed in [Ribeiro and Cn∗(A)⊥α is not empty, α∈ / Cn(∅), and by success, α∈ / ∗ ∗ Wassermann, 2008]. Cn(A −∗ α). By inclusion , A −∗ α ⊆ Cn (A), then, by Despite these problems, we have shown some practical ex- the upper bound property [Alchourron´ and Makinson, 1981], 0 0 0 ∗ amples of use of this operator in which it does better than the there is an A such that A −∗ α ⊆ A and A ∈ Cn (A)⊥α. direct application of partial meet contraction for bases. The By the construction of γ, γ(Cn∗(A)⊥α) is non-empty. practical usefulness of this operator is limited on isolation, Part 3: Case 1, α ∈ Cn(∅). Then, by logical relevance, but the importance of the study of its properties can be better since there is no A0 such that α∈ / Cn(A0), no element is in ∗ ∗ understood in the context of the pseudo-contraction operators A \ A −∗ α, then, using inclusion , A ⊆ A −∗ α ⊆ Cn (A). suggested in [Ribeiro and Wassermann, 2008]. We know that Cn∗(A)⊥α = ∅, then T γ(Cn∗(A)⊥α) = ∗ ∗ ∗ Future work involves exploring the use of different Cn to Cn (A). We need to show that Cn (A) ⊆ A −∗ α. By ∗ ∗ model real problems encountered in reasoning with ontolo- relevance , we know that Cn (A) \ A −∗ α = ∅, then ∗ gies and with Horn theories. Cn (A) ⊆ A −∗ α. Case 2, α∈ / Cn(∅). Cn∗(A)⊥α is non-empty and Appendix by part 2, γ(Cn∗(A)⊥α) is non-empty as well. Since ∗ A −∗ α is a subset of all elements of γ(Cn (A)⊥α), Proof of Theorem 8: T ∗ A −∗ α ⊆ γ(Cn (A)⊥α). We need to show that Construction-to-postulates: We know that A −∗ α = T ∗ T ∗ ∗ γ(Cn (A)⊥α) ⊆ A −∗ α. γ(Cn (A)⊥α) = Cn (A) −γ α, where −γ is the par- ∗ Take ε∈ / A −∗ α. If ε∈ / Cn (A), obviously ε∈ / tial meet contraction. We also know that −γ satisfies success, T ∗ ∗ γ(Cn (A)⊥α). If ε ∈ Cn (A) \ A −∗ α, then by inclusion, relevance and uniformity. So, we have: ∗ 0 0 ∗ relevance there is an A such that A −∗ α ⊆ A ⊆ Cn (A), ∗ 0 0 • If α∈ / Cn(∅), then α∈ / Cn(Cn (A) −γ α) α∈ / Cn(A ) but α ∈ Cn(A ∪ {ε}). It follows from the 00 00 • Cn∗(A) − α ⊆ Cn∗(A) upper bound property that there is an A such that A ⊆ A γ and A00 ∈ Cn∗(A)⊥α. From A ⊆ A00, α ∈ Cn(A0 ∪ ε) ∗ ∗ 0 • If β ∈ Cn (A)\(Cn (A)−γ α), then there is a B such and ε ∈ A00 we conclude that α ∈ Cn(A00), so we must have ∗ 0 ∗ 0 that Cn (A) −γ α ⊆ B ⊆ Cn (A), α∈ / Cn(B ), but ε∈ / A00. By our definition of γ, A00 ∈ γ(Cn∗(A)⊥α), and 0 00 T ∗ α ∈ Cn(B ∪ {β}). since ε∈ / A , we conclude that ε∈ / γ(Cn (A)⊥α). • B0 ⊆ Cn∗(A) α ∈ Cn(B0) If for all , if and only if Proof of Proposition 10: We know that A −∗ α = β ∈ Cn(B0) Cn∗(A) − α = Cn∗(A) − β ∗ , then γ γ . Cn (A) −γ α, where −γ is the partial meet contraction. ∗ [ ] Since Cn (A) −γ α = A −∗ α, we are done. Since partial meet satisfies relative closure Hansson, 1999 , Cn∗(A)∩Cn(Cn∗(A)− α) ⊆ Cn∗(A)− α is valid. From Postulates-to-construction: This part is is almost trivially γ γ this we have Cn∗(A)∩Cn(A− α) ⊆ A− α. By the inclu- obtained from the proof of the representation theorem for par- ∗ ∗ sion property of Cn∗ (which is Tarskian) and set theory we tial meet contraction for bases, which can be found in [Hans- get A ∩ Cn(A − α) ⊆ Cn∗(A) ∩ Cn(A − α). son, 1999]. ∗ ∗ ∗ Let −∗ be an operation for A that satisfies success, Proof of Proposition 11: Since Cn is Tarskian, by in- inclusion∗, relevance∗ and uniformity∗. From the last proof clusion, A − α ⊆ Cn∗(A − α). We want to show that ∗ and corollary 7 we conclude that −∗ also satisfies logical rel- Cn (A − α) ⊆ A − α. Suppose by contradiction that evance and uniformity. Let γ be a function such that: β ∈ Cn∗(A − α) \ (A − α). From inclusion∗, monotonicity ∗ ∗ ∗ ∗ ∗ and idempotence of Cn we obtain β ∈ Cn (A) \ A − α. • If Cn (A)⊥α = ∅, then γ(Cn (A)⊥α) = {Cn (A)}. Relevance∗ guarantees that there is an A0 such that A − α ⊆ ∗ ∗ 0 ∗ 0 0 • Otherwise γ(Cn (A)⊥α) = {X ∈ Cn (A)⊥α | A −∗ A ⊆ Cn (A), α∈ / Cn(A ) but α ∈ Cn(A ∪ {β}). α ⊆ X} By subclassicality of Cn∗ and β ∈ Cn∗(A − α) we have β ∈ Cn(A−α). By A−α ⊆ A0 and by the inclusion property We need to show that (1) γ is a well-defined function, (2) 0 0 0 T ∗ of Cn, β ∈ Cn(A ). So, we have Cn(A ) = Cn(A ∪ {β}), γ is a selection function and (3) γ(Cn (A)⊥α) = A −∗ α for all α. which is a contradiction. Part 1: For γ to be a well-defined function, for all Proof of Proposition 12: Assume α∈ / Cn(B). α and β, if Cn∗(A)⊥α = Cn∗(A)⊥β, we must have By inclusion∗ we already have B−α ⊆ Cn∗(B). To finish T γ(Cn∗(A)⊥α) = T γ(Cn∗(A)⊥β). Suppose that the proof it is sufficient to prove that Cn∗(B) \ (B − α) = ∅. Cn∗(A)⊥α = Cn∗(A)⊥β. It follows from observation By relevance∗, if β ∈ Cn∗(B)\(B −α), then there is a B0 1.39 in [Hansson, 1999] that any subset of Cn∗(A) implies such that B − α ⊆ B0 ⊆ Cn∗(B) and α ∈ Cn(B0 ∪ {β}). ∗ ∗ 0 α if and only if it implies β. By uniformity, Cn (A) −∗ From this and subclassicality of Cn we get B ∪ {β} ⊆ ∗ ∗ α = Cn (A) −∗ β. By the definition of γ we have Cn (B) ⊆ Cn(B) and then by monotony and idempotence γ(Cn∗(Cn∗(A))⊥α) = γ(Cn∗(Cn∗(A))⊥β). Since Cn∗ of Cn we have Cn(B0 ∪ {β}) ⊆ Cn(B). Since α∈ / Cn(B) is Tarskian, by idempotence, the result follows. by assumption, we cannot have such α. References [Nebel, 1989] B. Nebel. 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