A Consistency-Based Model for Belief Change: Preliminary Report

03/11/2000 ∙ by James Delgrande, et al. ∙ Universität Potsdam 0

We present a general, consistency-based framework for belief change. Informally, in revising K by A, we begin with A and incorporate as much of K as consistently possible. Formally, a knowledge base K and sentence A are expressed, via renaming propositions in K, in separate languages. Using a maximization process, we assume the languages are the same insofar as consistently possible. Lastly, we express the resultant knowledge base in a single language. There may be more than one way in which A can be so extended by K: in choice revision, one such "extension" represents the revised state; alternately revision consists of the intersection of all such extensions. The most general formulation of our approach is flexible enough to express other approaches to revision and update, the merging of knowledge bases, and the incorporation of static and dynamic integrity constraints. Our framework differs from work based on ordinal conditional functions, notably with respect to iterated revision. We argue that the approach is well-suited for implementation: the choice revision operator gives better complexity results than general revision; the approach can be expressed in terms of a finite knowledge base; and the scope of a revision can be restricted to just those propositions mentioned in the sentence for revision A.

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1 Introduction

We describe a general framework for belief change. The approach has something of the same flavour as the consistency-based paradigm for diagnosis [Reiter1987] or the assumption-based approach to default reasoning [Poole1988], although it differs significantly in details. Informally, in revising a knowledge base by sentence , we begin with and incorporate as much of as consistently possible. There may be more than one way in which information from can be incorporated. This gives rise to two notions of revision: a choice notion, in which one such “extension” is used for the revised state, and the intersection of all such extensions. Belief contraction is defined analogously.

We mainly focus on belief revision in this paper. For revision, first a knowledge base and sentence are expressed, via renaming atomic propositions in , in separate alphabets. We next assume that as many atoms in are equivalent to the corresponding atom in , as consistently possible. A set of such equivalent atoms is used to incorporate as much of the original knowledge base as is consistently possible. In the final section we discuss the more general approach, which we show is flexible enough to express extant approaches to revision and update.

The approach is developed in a formal, abstract framework. However, we argue that it is well-suited for implementation: The notion of choice revision gives better complexity results than general revision; moreover, we argue that belief revision is an area in which choice reasoning makes sense in some cases. Second, we show how the approach can be expressed equivalently in terms of a finite knowledge base, in place of a deductively-closed belief set. Third, we show that the scope of a revision can be restricted to just those propositions common to the knowledge base and sentence for revision.

We begin by presenting a very general framework for expressing belief change. This is restricted to address revision and contraction. Following this, we show how the approach allows for a uniform treatment of integrity constraints. As well, the approach supports iterated revision, with properties distinct from approaches based on the work of Spohn [Spohn1988]. Finally we briefly explore the general framework, and suggest it is flexible enough to express extant approaches to revision and update.

2 Background

A common approach in belief revision is to provide a set of rationality postulates for revision and contraction functions. The AGM approach of Alchourron, Gärdenfors, and Makinson [Gärdenfors1988], provides the best-known set of such postulates. The goal is to describe belief change on an abstract level, independent of how beliefs are represented and manipulated. Belief states, called belief sets, are modelled by sets of sentences closed under the logical consequence operator of some logic in some language , where the logic includes classical propositional logic. For belief set , is the deductive closure of , and is called the expansion of by . is the inconsistent belief set (i.e. ). is the set of all belief sets.

A revision function is a function from to satisfying the following postulates.

()

is a belief set.

()

()

()

If then

()

iff

()

If then

()

()

If then

That is: the result of revising by is a belief set in which is believed; whenever the result is consistent, revision consists of the expansion of by ; the only time that is obtained is when is inconsistent; and revision is independent of the syntactic form of and . The last two postulates deal with the relation between revising with a conjunction and expansion.

[Katsuno & Mendelzon1992] explores the distinct notion of belief update in which an agent changes its beliefs in response to changes in its external environment. Our interests here centre on revision; however as the end of the paper, we briefly consider this approach.

Recently there has been interest in iterated belief revision, a topic that the AGM approach by-and-large leaves open. Representative work includes [Boutilier1994, Williams1994, Lehmann1995, Darwiche & Pearl1997]. We discuss Darwiche and Pearl’s approach here. They employ the notion of an epistemic state that encodes how the revision function changes following a revision. denotes an epistemic state; denotes the belief set corresponding to . So now the result of revising an epistemic state is another epistemic state (from which the revised belief set may be determined using . Darwiche and Pearl propose the following postulates that “any rational system of belief change should comply with” (p. 2). Following their practice, we use to stand for when it appears as an argument of .

:

If then .

:

If then .

:

If then .

:

If then .

[Nayak et al.1996] propose a variant of along with the following postulate:

:

If then

where indicates that the change in following revision by depends in part on . This postulate is shown to be strong enough to derive , , and in the presence of the other postulates.

There has also been work on specific approaches to revision based on the distance between models of a knowledge base and a sentence to be incorporated in the knowledge base. This work includes [Dalal1988, Forbus1989, Satoh1988, Winslett1988]. In these approaches, models of the new knowledge base consist of models of the sentence to be added that are closest (based on “distance” between atomic sentences) to models of the original knowledge base.

Our approach differs from previous work first, in that we provide a specific, albeit general, framework in which approaches may be expressed. As well, the general framework allows the incorporation of different forms of integrity constraints. Also, given that it falls into the “consistency-based” paradigm, the approach has a certain syntactic flavour. However, notably, our approach is independent of the syntactic form of the knowledge base and sentence for revision.

Our technique of maximizing sets of equivalences of propositional letters bears a superficial resemblance to the use of such equivalences in [Liberatore & Schaerf1997] (based in turn on the technique developed in [de Kleer & Konolige1989]). However the approaches are distinct; in particular and in contradistinction to these references, we employ disjoint alphabets for a knowledge base and revising sentence. As well, the approach bears a resemblance to that of [del Val1993]. However, unlike del Val, we provide a single approach which may be restricted to yield extant approaches; also, we place no a priori restrictions on the form of a knowledge base.

3 Formal Preliminaries

We deal with propositional languages and use the logical symbols , , , , , , and to construct formulas in the standard way. We write to denote a language over an alphabet of propositional letters or atomic propositions. Formulas are denoted by the Greek letters , , , …. Knowledge bases or, equivalently, belief sets are initially identified with deductively-closed sets of formulas and are denoted , , …. So we have , where is the deductive closure of the formula or set of formulas given as argument. Later we relax this restriction.

Given an alphabet , we define a disjoint alphabet as . Then, for , we define as the result of replacing in each proposition from by the corresponding proposition in (so implicitly there is an isomorphism between and ). This is defined analogously for sets of formulas.

We define a belief change scenario in language as a triple , where are sets of formulas in . Informally, is a knowledge base that will be changed such that the set will be true in the result, and the set will be consistent with the result. For a base approach to revision we take and for a base approach to contraction we take .

In the definition below, “maximal” is with respect to set containment (rather than set cardinality). The following is our central definition.

Definition 1

Let be a belief change scenario in . Define as a maximal set of equivalences such that

Then

is a consistent definitional extension of .

Hence, a consistent definitional extension of is a modification of in which is true, and in which is consistent. We say that underlies the consistent definitional extension of . We let stand for .

Clearly, for a given belief change scenario there may be more than one consistent definitional extension. We will make use of the notion of a selection function that for any set has as value some element of . In Definition 2 and 3, these primitive functions can be regarded as inducing selection functions on belief change scenarios, such that has as value some consistent definitional extension of . This is a slight generalisation of selection functions as found in the AGM approach.

4 Revision and Contraction

Definition 1 provides a very general framework for specifying belief change. In the next two definitions we give specific definitions for revision and contraction. We develop these specific approaches and then, at the end of the paper, we return to the more general framework of Definition 1 and discuss how it can be used to express other approaches.

Definition 2 (Revision)

Let be a knowledge base and a formula, and let be the family of all consistent definitional extensions of . Then

  1.  is a choice revision of by with respect to some selection function with .

  2.  is the (skeptical) revision of by .

Table 1 gives examples of (skeptical) revision. The first column gives the original knowledge base, but with atoms already renamed. The second column gives the revision formula, while the third gives the set(s) and the last column gives the results of the revision. For the first and last column, we give a formula whose deductive closure gives the corresponding belief set.

Table 1: (Skeptical) revision examples.

In detail, for the last example, we wish to determine

(1)

We find maximal sets such that

We get two such sets of equivalences, namely and . Accordingly, we obtain

In addition to , we get , jointly implying .

In this example we get two choice extensions, and . This raises the question of the usefulness of choice revision compared to general revision. An apparent limitation of a choice reasoner is that it might draw overly strong conclusions. However, in belief revision this may be less of a problem than, say, in nonmonotonic reasoning: the goal in revision is to determine the true state of the world; if a (choice) revision results in an inaccurate knowledge base, then this inaccuracy will presumably be detected and rectified in a later revision. So choice revision may do no worse than a “skeptical” operator with respect to “converging” to the true state of the world. In addition, as we later show, it may do so significantly more efficiently and with better worst-case behaviour. Hence for a land vehicle exploring a benign environment, choice revision might be an effective part of a control mechanism; for something like flight control, or controlling a nuclear reactor, one would prefer skeptical revision.

Contraction is defined similarly to revision.

Definition 3 (Contraction)

Let be a knowledge base and a formula, and let be the family of all consistent definitional extensions of . Then

  1.  is a choice contraction of by with respect to some selection function with .

  2.  is the (skeptical) contraction of by .

Table 2 gives examples of (skeptical) contraction, using the same format and conventions as Table 1.

Table 2: (Skeptical) contraction examples.

In detail, for the first example we wish to determine

(2)

We compute the consistent definitional extensions of . We rename the propositions in and look for maximal subsets of such that

We obtain yielding

4.1 Properties of Revision and Contraction

With respect to the AGM postulates, we obtain the following.

Theorem 1

Let and be defined as in Definition 2. Then and satisfy the basic AGM postulates to , as well as .

For () we have instead the weaker postulate:

()

iff:

We obtain analogous results for and with respect to the AGM contraction postulates:

Theorem 2

Let and be defined as in Definition 3. Then satisfies the basic AGM postulates to , , and . In addition, satisfies the basic AGM postulates to , .

We also obtain the following interdefinability results:

Theorem 3 (Levi Identity)

Theorem 4 (Partial Harper Identity)

The following example shows that equality fails in the Harper Identity: if and , then while Similar results are obtained for choice revision and choice contraction by appeal to appropriate selection functions.

Iterated belief change:

The approach obviously supports iterated revision. Since we use a “global” metric, and since we can assume that every revision result, given and , can be determined, we continue to use here rather than Darwiche and Pearl’s for an epistemic state. That is, for us, we don’t need to refer to epistemic states, since we have completely specified how should behave on all arguments. Nonetheless, neither operator in Definition 2 satisfies any of the Darwiche-Pearl postulates for iterated revision. Nor in our opinion should they. For example, for , if we have

(3)

then in our approach we obtain that

(4)

[Darwiche & Pearl1997, Nayak et al.1996] assert that these results should be equal. However, it is possible (contra ) that there are cases where revising by yields and a subsequent revision by then gives , but revising by would yield . Which is to say, a significant difficulty in the area of belief revision is that different people have conflicting intuitions. However, Darwiche and Pearl argue that all rational revision functions should obey . Consequently they would need to argue that in all cases, having (4) result from (3) is irrational.

More seriously, an instance of (letting be and be , whence ) is the following:

:

.

Thus if you revise by () and then revise by the negation of some of this information (), then the other original information () is lost. So, in a variant of an example from [Darwiche & Pearl1997], consider where I see a new bird in the distance and come to believe that it is red and flies. If on closer examination I see that it is yellow, then according to and so , I also no longer believe that it flies. This seems too strong a condition to want to adopt. We conjecture (but have no proof) that approaches based on [Spohn1988], such as [Darwiche & Pearl1997], are subject in some form to such a “blanketing” result.

On the other hand, there are nontrivial results concerning iterated revision that hold for the present approach. For example, we have:

Theorem 5

Let be defined as in Definition 2. Then:

Semantics:

The operator provides a (near) syntactic counterpart to the minimal-distance-between-models approach of [Satoh1988]. For two sets and , let be the symmetric difference, . For formulas , define

where is the set of all models of , each of which is identified with a set of propositions. Then, we have:

Theorem 6

Let be a belief change scenario in where , and let be the family of all sets of equivalences, as defined in Definition 1.

Then,

This correspondence is interesting, but is of limited use beyond supplying a semantics for one instance of the approach. The choice approach, and (below) considerations on implementation and integrity constraints, are not readily expressed in Satoh’s model-based semantics. As well, a contraction function is straightforwardly obtained in Satoh’s approach only by using the Harper Identity (which doesn’t fully obtain here). Further, in the last section, we show how other approaches can be expressed in our general framework.

4.2 Integrity Constraints

Definitions 2 and 3 are similar in form, differing only in how the formula is mapped onto the sets and in Definition 1. Clearly one can combine these definitions, allowing simultaneous revision by one formula and contraction by another. This in-and-of-itself isn’t overly interesting, but it does lead to a natural and general treatment of integrity constraints in our approach.

There are two standard definitions of a knowledge base satisfying a static integrity constraint . In the consistency-based approach of [Kowalski1978], satisfies iff is satisfiable. In the entailment-based approach of [Reiter1984], satisfies iff . [Katsuno & Mendelzon1991] show how entailment-based constraints can be maintained across revisions: given an integrity constraint and revision function , a revision function which preserves is defined by: In our approach, we can define revision taking into account both approaches to integrity constraints.

Corresponding to Definition 2 (and ignoring the choice approach) we obtain:

Definition 4

Let be a knowledge base, a formula, and , sets of formulas. Let be the family of all consistent definitional extensions of . Then  is the revision of by incorporating integrity constraints (consistency-based) and (entailment-based).

Theorem 7

Let be defined as in Definition 4. Then .  If then is satisfiable.

Finally, and in contrast with previous approaches, it is straightforward to add dynamic integrity constraints, which express constraints that hold between states of the knowledge base before and after revision. The simplest way of so doing is to add such constraints to the set in Definition 1. To state that if is true in a knowledge base before revision then must be true afterwards, we would add to . Note however that the addition of dynamic constraints may lead to an operator that violates some of the properties of . For example with dynamic constraint leads to an inconsistent revision.

5 Implementability Considerations

We claimed at the outset that the approach is well-suited for implementation. To this end, we first consider the use of choice belief revision. Second we consider the problem of representing the results of revision in a finite, manageable representation. Lastly, we address limiting the range of .

Complexity:

From [Eiter & Gottlob1992] and Theorem 6 it follows that deciding, for given , , , whether is However, the analogous problem for choice revision is lower in the polynomial hierarchy.

Theorem 8

Given a selection function , formulas , and a set of equivalnces . Then, we have:

  1. Deciding whether determines a choice revision of and is in .

  2. Deciding is in .

We have not yet addressed restrictions on the syntactic form of or ; but see [Eiter & Gottlob1992].

Finite representations:

Definitions 12, and 3 provide a characterisation of revision and contraction, yielding in either case a deductively-closed belief set. Here we consider how the same (with respect to logical equivalence) operators can be defined, but where a knowledge base is given as an arbitrary, finite set of formulas. It follows from the discussion below that, for knowledge base and formula , we can defined choice revision so that for any selection function .

Informally the procedure is straightforward, although the technical details are less so. A knowledge base is now represented by a formula (or set of formulas). Via Definitions 1 and 2 we consider maximal sets where is consistent. For each such set , we replace each in by where and we replace each in by where . The result of these substitutions into is a sentence of size and whose deductive closure is equivalent to (some) choice revision. The disjunction of all such sentences (and so considering all possible sets ) is equivalent to .

As opposed to the computation of the sets , the result of revising or contracting a formula can be captured without an explicit change of alphabet. We start by observing that any set of equivalences induces a binary partition of its underlying alphabet , namely with and . Given a belief change scenario along with a set of equivalences (according to Definition 1), we define for , that is the result of replacing in each proposition by its negation .

For generality, let be a set of formulas:

Definition 5

Let be a belief change scenario in and let be the family of all sets of equivalences, as defined in Definition 1.

Define as and as for selection function corresponding to .

For revision, we define as the finite representation of , and analogously as the finite representation of .

Theorem 9

Let and be formulas. Then, we have

Consider example (1): So is the belief change scenario. We obtain:

which is equivalent to . For the other examples in Table 1, if is the formula corresponding to in the first column, then revising by the given via Theorem 9 is the formula given in the last line (up to permutation of symbols and elimination of definitional equivalents).

Contraction is handled somewhat differently. This is not altogether surprising, given that revision and contraction are not fully interdefinable (Theorem 4). Whereas for revision we replaced each atomic proposition in by its negation in , for contraction replacements in are done over all truth values of atomic propositions in . Formally, given a belief change scenario , a corresponding set of equivalences (according to Definition 1) along with its induced partition of , and a function , we define for , as the result of replacing in each proposition by . Note that each set of equivalences induces a whole set of such mappings , viz. , amounting to all possible truth assignments to .

Definition 6

Let and be defined as in Definition 5.

Define as and as for some selection function with .

We define as the finite representation of , and analogously as the finite representation of .

Theorem 10

Let and be formulas. Then, we have

Consider example (2): We obtain

For the examples in Table 2, if is the formula corresponding to , then in contracting by the given , the result of the contraction via Theorem 10 is the formula given in the last line (up to permutation of symbols and elimination of definitional equivalents).

Theorems 9 and 10 are interesting in that they show that revision and contraction can be defined with respect to syntactic objects (viz. sentences representing the knowledge base) yet are essentially independent of syntactic form. Hence in a certain sense the approach combines the advantages of base revision [Nebel1992] and syntax-independent approaches.

Limiting the range of :

Intuitively, if an atomic sentence appears in a knowledge base but not in a sentence for revision , or vice versa, then that atomic sentence plays no part in the revision. This is indeed the case here, as the next result demonstrates. Let be the atomic sentences in . We obtain:

Theorem 11

Let be a set of formulas and a formula. Let be a consistent definitional extension of belief change scenario .

Then

So for belief change, we need consider just the atomic sentences common to and , and can ignore (with regards ) other atomic sentences. As detailed in the full paper, this result allows one to limit the primed atomic propositions in to those occurring in .

6 The General Approach

Definition 1 is quite general; in Definitions 2 and 3 we narrow the scope to specific approaches to belief change. We note however, briefly, that other approaches are expressible in this framework. Belief update is a distinct form of belief change, suited to a changing world. Update and its dual operator erasure are studied in [Katsuno & Mendelzon1992] where sets of postulates characterising the operators are given.

Definition 7 (Prime Implicate)

A consistent set of literals is a prime implicate111Note that this is the dual of prime implicant. of iff: and for we have .

Definition 8 (Update)

Let be a knowledge base and a formula and let be the set of prime implicates of . For each , , let be the consistent definitional extensions of . Then  is the update of by .

We do not define choice update here, given space limitations.

Theorem 12

satisfies the update postulates of [Katsuno & Mendelzon1992].

We show in the full paper that the operator provides a syntactic counterpart for Winslett’s update operator [Winslett1988]. We can also take a different notion of maximal in Definition 1, and base the definition on set cardinality, rather than set containment. We show that based on this measure we can capture the revision approaches of [Dalal1988] and [Forbus1989]. Lastly a minor modification to Definition 1 allows one to use the framework to capture the merging of knowledge bases.

7 Conclusion

We have presented a general consistency-based framework for belief change, having the same flavour as the consistency-based paradigms for diagnosis or default reasoning. We focus on a specific approach, in which a knowledge base and sentence are expressed, via renaming propositions in , in separate alphabets. Given this, we assume that as many corresponding atoms in each language are equivalent insofar as is consistently possible. Lastly, we express the resultant knowledge base in a single language. For the revision of by , for example, we begin with and incorporate as much of as consistently possible. This gives rise to two notions of revision: a choice notion, in which one such “extension” is used for the revised state, and the intersection of all such extensions.

The approach is well-suited for implementation: The notion of a choice extension gives better complexity results than general revision; also, belief revision is an area in which choice reasoning may be useful. Second, we show how the approach can be expressed in terms of a finite knowledge base, and that the scope of a revision can be restricted to those propositions common to the knowledge base and sentence for revision.

The approach allows for a uniform treatment of integrity constraints, in that belief change may take into account both consistency-based and entailment-based static constraints, as well as dynamic constraints. As well, it supports iterated revision. Finally, the framework is applicable to other approaches to belief change.

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