Credibility-limited Functions for Belief Bases

In this paper we adapt the Ferm´e and Hansson model of Shielded Contraction as well as Hansson et al. Credibility- limited Revision for belief bases, to join two of the many variations of the AGM model, i.e. those in which knowledge is represented through belief bases instead of logic theories, and those in which the object of the epistemic change does not have priority over the existing information as is the case in the AGM model.


Introduction
In the logic of theory change, the AGM model has acquired the status of a standard model [1]. However, the AGM model seems to be improper in many contexts. This inspired many researchers to propose extensions and generalizations to AGM. Among these extensions, we can mention the following. 1. Nonprioritized belief revision: the AGM model always accepts the new information. This feature appears, in general, to be unrealistic, since rational agents, when confronted with information that contradicts previous beliefs, often reject it altogether or accept only parts of it. The nonprioritized models allow new functions, where the new information has no special priority (see [8] for a survey of different proposals). In the case of contraction, the input sentence is always retracted (except in the limiting case of tautologies), independently of the original set of beliefs. 2. Extensions in the representation of the belief states and of the inputs: belief sets (sets of sentences closed under logical consequence) cannot distinguish between fundamental and derived beliefs. Belief bases, sets of sentences not necessarily closed under logical consequence, have been shown to possess more expressive power than belief sets [5][6][7][12][13][14].
The objective of this paper is to present a model that combines both kinds of extensions. This model consists of the extension for belief bases of Shielded Contraction [4], in which the contraction operations are filtered by a group of sentences capable of being retracted, and Credibility-limited Revision [10] in which the new information in the revision is only added if certain acceptability conditions are fulfilled. We will enunciate the axiomatic for these models, a constructive model for each one of these operations, establishing the corresponding representation theorem.
is an arbitrary tautology and an arbitrary contradiction. A set of sentences that is closed under logical consequence is called a belief set, otherwise it is a belief base.

Belief bases functions
The use of logically closed sets to represent beliefs has received much criticism. Among them we can mention: 1. Belief sets make no distinction between basic beliefs and those which were inferred from them. 2. They are computability intractable. 3. Belief bases can distinguish between different inconsistent belief states.
Base expansion is simply a set union, i.e. · « « .
We can construct a base contraction function using the remainder sets, i.e. maximal subsets of that fail to imply «: .
Partial meet base contraction also satisfies other interesting properties as we will see in the following observation: OBSERVATION 2.4 [8] Let be a belief base and ¤ an operator on . Then: 1. If ¤ satisfies inclusion and relevance, then it satisfies vacuity (If «, then ¤« ). 2. If ¤ satisfies uniformity, then it satisfies extensionality (If «°¬, then ¤ « ¤ ¬). 3. If ¤ satisfies inclusion and relevance, then it satisfies failure (If «, then ¤ « ). In belief bases, as in belief sets, revising by « can be reduced to contraction by « and then expanded by « (Levi identity). DEFINITION 2.5 [1,6] Let be a belief base. The partial meet base revision operator on based on a selection function is the operator £ such that for all sentences «: The axiomatic characterization of partial meet base revision is: [6] Let be a belief set. An operator £ on is a partial meet revision function for if and only if £ satisfies:

Uniformity
If for all subsets ¼ ,

Shielded base contraction
In standard accounts of belief contraction, the contraction is always successful (except for the limiting case of tautologies). However, this is not intuitive in general. A rational agent can have certain nontautological beliefs which it is not disposed to resign. In the case of human agents, it is easy to find examples. In computer systems, there are integrity constraints that cannot be eliminated in the update/correction process. The basic idea of Shielded Contraction is to define a contraction function that consists of two different steps. The first one evaluates if the input sentence is suitable to be contracted. The second one contracts in the positive case, applying a partial meet base contraction function, otherwise no change is made.
The procedure to evaluate if a given sentence can be contracted is checking if it belongs to a set of sentences denominated a retractible set, whose properties are defined in the next subsection.

The set of retractible sentences
The set Ê defines which sentences can be discarded from a belief base via a contraction function. The sentences in Ê are called 'retractible' sentences and the rest 'irretractible' sentences. The construction of the retractible set is subjective and depends on how 'important' the beliefs are for an rational agent. However, we can define some rationality criteria about this set, those that will characterize the retractible sets.
The tautologies cannot be removed from the consequences of a belief base, hence all the tautologies are 'nonretractible' sentences.
We can strengthen this property in the following way: if a sentence « is irretractible, then « cannot be eliminated from the consequence of a belief base. Hence all the consequences of « cannot be retracted: Since a conjunctive sentence « ¬ can be removed from a belief base only if at least one of its conjuncts « and ¬ is removed, we should expect « ¬ not to be retractible without either « or ¬ being so.

Conjunctive completeness
The concept of 'irretractibility' means that a sentence cannot be removed from a belief base , independently of which contraction we perform, hence: where is a base contraction function. Now we can define the a base contraction function using a retractible set.

Postulates for shielded base contraction
Since the success postulate is given up, we must find weaker versions of it that capture our intuitions. When a sentence is irretractible (and then, impossible to contract), the contraction must return the original set of sentences: 1

Relative success
The 'status' of irrevocability is associated to a sentence per se. Then if a sentence ¬ cannot be removed in a contraction by itself, then it cannot be removed at all from the belief base.
In contraction, no sentences are added.
In Shielded Base Contraction, a sentence ¬ is retracted in a contraction from by « if this sentence contributes in some way to imply « from . Consequently we can use the same postulate mentioned in Theorem 2.3.
The postulate of success indicates the behaviour of a contraction function when a sentence is included in the consequence of a belief base, but does not determine the behaviour when the sentence to be contracted is not a logical consequence from the belief base. Then the minimal change to contract the sentence is to do nothing.
A natural requirement is that equivalent sentences must return the same contraction (If «°¬, then « ¬ (extensionality)). In belief bases, moreover, there are other cases in which two sentences must return the same contraction. If every subset of sentences of accepts a sentence « if and only if it accepts a sentence ¬, then both sentences are indistinguishable in the contraction operation, since the reasons to accept « are exactly the same as the ones to accept ¬.
Then it is reasonable that contracting by « must be equivalent to contracting by ¬.
If two sentences « and ¬ are both irretractible, then we should expect their conjunction to also be irretractible.

Representation theorem
The following representation theorem characterizes the shielded base contraction function.
THEOREM 3.2 Let be a belief base and an operation on . Then the following conditions are equivalent: 1. satisfies relative success, persistence, inclusion, relevance, vacuity, uniformity and conjunctive constancy.
2. is the operator of shielded base contraction induced by a partial meet base contraction operator ¤ for and a set Ê Ä that satisfies nonretractability propagation, and non retractability preservation and such that:

Credibility-limited base contraction
In nonprioritized revision function, the success postulate is replaced by weaker postulates, that do not accept the new information in certain cases. In this section we will focus on credibility-limited revision for belief bases, that is a particular case of the model decision+revision mentioned by Hansson in [8]. Our work is an extension of Credibility-limited Revision, that was presented for belief sets in [10]. 2 Conceptually, in this kind of revision there are sentences that the agent is not supposed to accept even when revising by it. In these cases, this 'incredulity' is based in certain sentences that the agent is not supposed to rectify or discard.
In particular, in the credibility-limited model, the credible sentences, i.e. the sentences that the agents are dispose to accept are represented by a set . This does not mean that they are currently accepted. The concept of being 'credible' is related with the concept of retractible mentioned in Section 3 and the relation can be expressed as:

The set of credible sentences
The credibility-limited base function therefore consists of two steps: in the first one, the agent evaluates if the epistemic input is an element of . If so, the second step is to revise the belief base by the input (in our case in the AGM style), otherwise, no change is made.
The following are desirable properties for the set . A natural requirement is that the contradictions are not credible.

Element consistency
If a sentence is credible, its derived sentences must be credible too.
If two sentences are not credible, their disjunction is also not credible.

¬ ¾ , then
¬, or equivalently: The outcome of a revision must be credible.

Revision credibility
If a sentence is not credible, then its negation is always believed.
If the belief base is consistent, expansive credibility can be strengthened.
Using the set we can define a credibility-limited base revision function.

DEFINITION 4.2 Let be a belief base. The operation on is a credibility-limited base revision on if
and only if there is an partial meet base revision £ on and a set of sentences such that for all sentences «:

Postulates for credibility-limited base revision
As credibility-limited base revision is a nonprioritized revision function, then we must give up the success postulate. Since in the unsuccessful case no change is made, we need a postulate that reflects this fact.
We can not expect the outcome of the operation to be consistent in all cases, because in the unsuccessful case, the belief base remains unchanged. However, we can ensure consistency in the case of consistent belief bases.
As in AGM, we conserve the expansion function as the upper bound of the credibilitylimited base revision. This bound is reached when the input sentence is consistent with the belief base.
A second weaker version of success is used for disjunctions. The revision of a disjunction is successful if at least one of the disjuncts can be successfully revised.
A sentence ¬ can be eliminated in the revision by «, only if this sentence contributes in some way to imply « from .
Relevance If ¬ ¾ and ¬ ¾ «, then there is some ¼ such that £ « If a sentence is successfully revised, then its consequences will be accepted in the revised belief base. Therefore it is intuitive to require that these sentences can also be successfully revised.
We can strengthen this postulate to all the sentences accepted in the revised belief base.
Note that when is consistent regularity follows from strong regularity and consistency preservation. As in contraction, we strengthen the postulate of extensionality to uniformity.

Representation theorem
Let be a consistent belief base and an operation on . Then the following conditions are equivalent: 1. satisfies relative success, consistency preservation, inclusion, vacuity, disjunctive distribution, relevance, strong regularity, and uniformity. 2. is the operator of credibility-limited base revision induced by a partial meet base revision operator for and a set that satisfies element consistency, single sentence closure, disjunctive completeness, expansive credibility, revision credibility and strong revision credibility and such that Then by Ô Ö× ×Ø Ò , ´« ¬µ ¬ and ´« ¬µ «. Then ´« ¬µ « ¬, from which it follows that « ¬ ¾ Ê . Now we will define a partial meet base contraction function based on and show that it satisfies the equivalence defined in condition 2. We will first define a selection function based on , that allow the construction of a contraction function and satisfies the postulates defined for belief bases (see [6]).
We must prove.
1) is a well-defined function: in order to be a function, it must be the case that for all « ½ and « ¾ if

¾ ½µ
Let ¤ be a base contraction function that satisfies success, inclusion, relevance, and uniformity and Ê a set of sentences that satisfies conjunctive completeness, nonretractability preservation, and non retractability propagation. Let such that: We must prove that satisfies relative success, inclusion, relevance, persistence, vacuity, and uniformity.

Relative success
Due to Theorem 2.3, the function ¤ « Ì ´ µ satisfies success, consistency preservation, inclusion, relevance, and uniformity and due to Theorem 2.6, the revision defined by means of the Levi identity £ « ´ ¤ «µ « satisfies consistency, success, inclusion, relevance, and uniformity.
We will prove that

¾ ½µ
Let £ be a partial meet base revision function, a set of sentences that satisfies the properties listed in condition 2. Let be an operator such that: « £ « if « ¾ otherwise Relative success trivial by definition and £ success.

Strong regularity Let
« ¬. Then it follows by strong revision credibility that ¬ ¾ and the rest follows by £ success.