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Belief merging within fragments of propositional logic

04/25/2014
by   Nadia Creignou, et al.
0

Recently, belief change within the framework of fragments of propositional logic has gained increasing attention. Previous works focused on belief contraction and belief revision on the Horn fragment. However, the problem of belief merging within fragments of propositional logic has been neglected so far. This paper presents a general approach to define new merging operators derived from existing ones such that the result of merging remains in the fragment under consideration. Our approach is not limited to the case of Horn fragment but applicable to any fragment of propositional logic characterized by a closure property on the sets of models of its formulae. We study the logical properties of the proposed operators in terms of satisfaction of merging postulates, considering in particular distance-based merging operators for Horn and Krom fragments.

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Introduction

Belief merging consists in achieving a synthesis between pieces of information provided by different sources. Although these sources are individually consistent, they may mutually conflict. The aim of merging is to provide a consistent set of information, making maximum use of the information provided by the sources while not favoring any of them. Belief merging is an important issue in many fields of Artificial Intelligence (AI)

[Bloch and (Eds)2001] and symbolic approaches to multi-source fusion gave rise to increasing interest within the AI community since the s [Baral, Kraus, and Minker1991, Cholvy1998, Lin1996, Revesz1993, Revesz1997]. One of today’s major approaches is the problem of merging under (integrity) constraints in order to generalize both merging (without constraints) and revision (of old information by a new piece of information). For the latter the constraints then play the role of the new piece of information. Postulates characterizing the rational behavior of such merging operators, known as IC postulates, have been proposed by Konieczny and Pino Pérez [Konieczny and Pino Pérez2002] in the same spirit as the seminal AGM [Alchourrón, Gärdenfors, and Makinson1985] postulates for revision. Concrete merging operators have been proposed according to either semantic (model-based) or syntactic (formula-based) points of view in a classical logic setting [Chacón and Pino Pérez2012]. We focus here on the model-based approach of distance-based merging operators [Konieczny, Lang, and Marquis2004, Konieczny and Pino Pérez2002, Revesz1997]. These operators are parametrized by a distance which represents the closeness between interpretations and an aggregation function which captures the merging strategy and takes the origin of beliefs into account.

Belief change operations within the framework of fragments of classical logic constitute a vivid research branch. In particular, contraction [Booth et al.2011, Delgrande and Wassermann2013, Zhuang and Pagnucco2012] and revision [Delgrande and Peppas2011, Putte2013, Zhuang, Pagnucco, and Zhang2013] have been thoroughly analyzed in the literature. The study of belief change within language fragments is motivated by two central observations:

  • In many applications, the language is restricted a priori. For instance, a rule-based formalization of expert’s knowledge is much easier to handle for standard users. In case users want to revise or merge some sets of rules, they indeed expect that the outcome is still in the easy-to-read format they are used to.

  • Many fragments of propositional logic allow for efficient reasoning methods. Suppose an agent has to make a decision according to a group of experts’ beliefs. This should be done efficiently, therefore the expert’s beliefs are stored as formulæ known to be in a tractable class. For making a decision, it is desired that the result of the change operation yields a set of formulæ in the same fragment. Hence, the agent still can use the dedicated solving method she is equipped with for this fragment.

Most of previous work has focused on the Horn fragment except [Creignou et al.2014] that studied revision in any fragment of propositional logic. However, as far as we know, the problem of belief merging within fragments of propositional logic has been neglected so far.

The main obstacle hereby is that for a language fragment , given belief bases and a constraint , there is no guarantee that the outcome of the merging, , remains in as well. Let for example, , and be two sets of formulæ and a formula expressed in the Horn fragment. Merging with typical distance-based operator proposed in [Konieczny and Pino Pérez2002] does not remain in the Horn language fragment since the result of merging is equivalent to , which is not equivalent to any Horn formula (see [Schaefer1978]).

We propose the concept of refinement to overcome these problems. Refinements have been proposed for revision in [Creignou et al.2014] and capture the intuition of adapting a given operator (defined for full classical logic) in order to become applicable within a fragment. The basic properties of a refinement aim to (i) guarantee the result of the change operation to be in the same fragment as the belief change scenario given and (ii) keep the behavior of the original operator unchanged in case it delivers a result which already fits in the fragment.

Refinements are interesting from different points of view. Several fragments can be treated in a uniform way and a general characterization of refinements is provided for any fragment. Defining and studying refinements of merging operators is not a straightforward extension of the revision case. It is more complex due to the nature of the merging operators. Even if the constraints play the role of the new piece of information in revision, model-based merging deals with multi-sets of models. Moreover applying this approach to different distance-based merging operators, each parameterized by a distance and an aggregation function, reveals that all the different parameters matter, thus showing a rich variety of behaviors for refined merging operators.

The main contributions of this paper are the following:

  • We propose to adapt known belief merging operators to make them applicable in fragments of propositional logic. We provide natural criteria, which refined operators should satisfy. We characterize refined operators in a constructive way.

  • This characterization allows us to study their properties in terms of the IC postulates [Konieczny and Pino Pérez2002]. On one hand we prove that the basic postulates (IC0–IC3) are preserved for any refinement for any fragment. On the other hand we show that the situation is more complex for the remaining postulates. We provide detailed results for the Horn and the Krom fragment in terms of two kinds of distance-based merging operators and three approaches for refinements.

Preliminaries

Propositional Logic.

We consider as the language of propositional logic over some fixed alphabet of propositional atoms. A literal is an atom or its negation. A clause is a disjunction of literals. A clause is called Horn if at most one of its literals is positive; and Krom if it consists of at most two literals. We identify the following subsets of : is the set of all formulæ in being conjunctions of Horn clauses, and is the set of all formulæ in being conjunctions of Krom clauses. In what follows we sometimes just talk about arbitrary fragments . Hereby, we tacitly assume that any such fragment contains at least the formula .

An interpretation is represented either by a set

of atoms (corresponding to the variables set to true) or by its corresponding characteristic bit-vector of length

. For instance if we consider , the interpretation and will be represented either by or by . As usual, if an interpretation satisfies a formula , we call a model of . By we denote the set of all models (over ) of . Moreover, if and ( and are equivalent) if .

A base is a finite set of propositional formulæ . We shall often identify via , the conjunction of formulæ of , i.e., . Thus, a base is said to be consistent if is consistent, is a shortcut for , stands for , etc. Given we denote by the set of bases restricted to formulæ from . For fragments , we also use .

A profile is a non-empty finite multiset of consistent bases and represents a group of agents having different beliefs. Given , we denote by the set of profiles restricted to the use of formulæ from . We denote by . The profile is said to be consistent if is consistent. By abuse of notation we write to denote the multi-set union . The multi-set consisting of the sets of models of the bases in a profile is denoted . Two profiles and are equivalent, denoted by if . Finally, for a set of interpretations and a profile we define .

Characterizable Fragments of Propositional Logic.

Let denote the set of all Boolean functions that have the following two properties111these properties are also known as anonimity and unanimity.:

  • symmetry, , for all permutations , and

  • - and -reproduction, , for all , .

Examples are the binary AND function denoted by or the ternary MAJORITY function, if at least two of the variables , and are set to 1. We extend Boolean functions to interpretations by applying coordinate-wise the original function (recall that we consider interpretations also as bit-vectors). So, if , then is defined by , where is the -th coordinate of the interpretation .

Definition 1.

Given a set of interpretations and , we define , the closure of under , as the smallest set of interpretations that contains and that is closed under , i.e., if , then also .

Let us mention some easy properties of such a closure: (i) monotonicity; (ii) if , then ; (iii) .

Definition 2.

Let . A set of propositional formulæ is a - (or characterizable fragment) if:

  1. for all ,

  2. for all with there exists a with

  3. if then .

It is well-known that is an -fragment and is a -fragment (see e.g. [Schaefer1978]).

Logical Merging Operators.

Belief merging aims at combining several pieces of information coming from different sources. Merging operators we consider are functions from the set of profiles and the set of propositional formulæ to the set of bases, i.e., . For and we will write instead of ; the formula is referred to as the integrity constraint (IC) and restricts the result of the merging.

As for belief revision some logical properties that one could expect from any reasonable merging operator have been stated. See [Konieczny and Pino Pérez2002] for a detailed discussion. Intuitively is the “closest” belief base to the profile satisfying the integrity constraint . This is what the following postulates try to capture.

If is consistent, then is consistent
If is consistent with ,
then
If and ,
then
If and , then
is consistent if and only if
is consistent
If is consistent,
then
If is consistent,
then

Similarly to belief revision, a representation theorem [Konieczny and Pino Pérez2002] shows that a merging operator corresponds to a family of total preorders over interpretations. More formally, for , and a total preorder over interpretations, a model-based operator is defined by . The model-based merging operators select interpretations that are the ”closest” to the original belief bases.

Distance-based operators where the notion of closeness stems from the definition of a distance (or a pseudo-distance222Let , a pseudo-distance is such that and if and only if .) between interpretations and from an aggregation function have been proposed in [Konieczny and Pino Pérez2002, Konieczny and Pino Pérez2011]. An aggregation function is a function mapping for any positive integer each -tuple of positive reals into a positive real such that for any , if , then , if and only if and . Let , , be a distance and be an aggregation function, we consider the family of merging operators defined by where is a total preorder over the set of interpretations defined as follows:

  • ,

  • , and

  • if .

Definition 3.

A counting distance between interpretations is a function defined for every pair of interpretations by where is a nondecreasing function such that if and only if . If for every , we call a drastic distance and denote it via . If for all , we call the Hamming distance and denote it via . If for every interpretations and we have , then we say that the distance satisfies the triangular inequality.

Observe that a counting distance is indeed a pseudo-distance, and both, the Hamming distance and drastic distance satisfy the triangular inequality.

As aggregation functions, we consider here , the sum aggregation function, and the aggregation function defined as follows. Let and , be two interpretations. Let , where , be the vector of distances between and the belief bases in . Let be the vector obtained from by ranking it in decreasing order. The aggregation function is defined by , with if , where denotes the lexicographical ordering.

In this paper we focus on the and operators where is an arbitrary counting distance. These operators are known to satisfy the postulates , as shown in [Konieczny, Lang, and Marquis2004] generalizing more specific results from [Konieczny and Pino Pérez2002, Lin and Mendelzon1998]. Finally, we define certain concepts for merging operators and fragments.

Definition 4.

A basic (merging) operator for is any function satisfying for each . We say that satisfies an postulate () in if the respective postulate holds when restricted to formulæ from .

Refined Operators

Let us consider a simple example to illustrate the problem of standard operators when applied within a fragment of propositional logic.

Example 1.

Let , and such that , , and . Consider the distance-based merging operators, and . The following table gives the distances between the interpretations of and the belief bases, and the result of the aggregation functions and .

Hence, we have . Thus, for instance, we can give as a result of the merging for both operators. However, there is no with (each satisfies the following closure property in terms of its set of models: for every , also )). Thus, the result of the operator has to be “refined”, such that it fits into the Horn fragment. On the other hand, it holds that , and also the result is in Krom. This shows that different fragments behave differently on certain instances. Nonetheless, we aim for a uniform approach for refining merging operators.

We are interested in the following: Given a known merging operator and a fragment of propositional logic, how can we adapt to a new merging operator such that, for each and , ? Let us define a few natural desiderata for inspired by the work on belief revision. See [Creignou et al.2014] for a discussion.

Definition 5.

Let be a fragment of classical logic and a merging operator. We call an operator a -refinement for if it satisfies the following properties, for each and .

  1. consistency: is consistent if and only if is consistent

  2. equivalence: if and then

  3. containment:

  4. invariance: If , then , where denotes the set of formulæ in for which there exists an equivalent formula in .

Next we introduce examples of refinements that fit Definition 5.

Definition 6.

Let be a merging operator and . We define the -based refined operator as:

where .

We define the -based refined operator as:

where is a function that selects the minimum from a set of interpretations with respect to a given and fixed order.

We define the -based refined operator as:

The intuition behind the last refinement is to ensure a certain form of fairness, i.e. if no model is selected from the profile, this carries over to the refinement.

Proposition 1.

For any merging operator , and a -, the operators , and are -refinements for .

Proof.

Let , and . We show that each operator yields a base from and moreover satisfies consistency, equivalence, containment and invariance, cf. Definition 5.

: since by assumption is a - and thus closed under . Consistency holds since and iff . Equivalence holds since implies . Containment: let , i.e.  and . By monotonicity of , then . Since then therefore . Invariance: let , i.e.  and . By hypothesis , therefore .

: if (i.e. ) then satisfies all the required properties as shown above; otherwise consistency, equivalence and containment hold since . Moreover, by definition each fragment contains a formula with where is an arbitrary interpretation. thus also holds in this case.

: satisfies the required properties since and satisfy them. ∎

Example 2.

Consider the profile , the integrity constraint given in Example 1, the distance-based merging operator , and let be the binary AND function. Let us have the following order over the set of interpretations on : . The result of merging is . The -based -refined operator, denoted by , is such that . The -based -refined operator, denoted by , is such that . The same result is achieved by the -based -refined operator since .

In what follows we show how to capture not only a particular refined operator but characterize the class of all refined operators.

Definition 7.

Given , we define a -, , as an application which to every set of models and every multi-set of sets of models associates a set of models such that:

  1. ( is closed under )

  2. if , then

  3. If , then .

The concept of mappings allows us to define a family of refined operators for fragments of classical logic that captures the examples given before.

Definition 8.

Let be a merging operator and be a -fragment of classical logic with . For a -mapping we denote with the operator for defined as . The class contains all operators where is a -mapping and such that is a -fragment.

The next proposition is central in reflecting that the above class captures all refined operators we had in mind, cf. Definition 5.

Proposition 2.

Let be a basic merging operator and a characterizable fragment of classical logic. Then, is the set of all -refinements for .

Proof.

Let be a -fragment for some . Let . We show that is a -refinement for . Let and . Since there exists a -mapping , such that . By Property 1 in Definition 7 is indeed in . Consistency: If then by Property 4 in Definition 7. Otherwise, by Property 2 in Definition 7, we get . Equivalence for is clear by definition and since is defined on sets of models. Containment: let , i.e., and . We have by monotonicity of . By Property 2 of Definition 7, . Since we have . Thus, , i.e., . Invariance: In case , we have since is a -fragment. By Property 3 in Definition 7, we have . Thus as required.

Let be a -refinement for . We show that . Let be defined as follows for any set of interpretations and a multi-set of sets of interpretations: . For , if then , otherwise if there exists a pair such that and , then we define . If there is no such then we arbitrarily define as the set consisting of a single model, say the minimal model of in the lexicographic order. Note that since is a -refinement for , it satisfies the property of equivalence, thus the actual choice of the pair is not relevant, and hence is well-defined. Thus the refined operator behaves like the operator .

We show that such a mapping is a -mapping. We show that the four properties in Definition 7 hold for . Property 1 is ensured since for every pair , is closed under . Indeed, either if is closed under , or and since its set of models is closed under , or consists of a single interpretation, and thus is also closed under . Let us show Property 2, i.e., for any pair . It is obvious when (then ), as well as when is a singleton and when is closed and thus . Otherwise and since satisfies containment . Therefore in any case we have . Property 3 follows trivially from the definition of when is closed under . Property 4 is ensured by consistency of . ∎

Note that the -mapping which is used in the characterization of refined merging operators differs from the one used in the context of revision (see [Creignou et al.2014]). Indeed, our mapping has two arguments (and not only one as in the case of revision). The additional multi-set of sets of models representing the profile is required to capture approaches like the -based refined operator, which are profile dependent.

IC Postulates

+ - + - +
, - - - + -
, - - - - -
Table 1: Overview of results for for refinements in the Horn and Krom fragment (, ).

The aim of this section is to study whether refinements of merging operators preserve the IC postulates. We first show that in case the initial operator satisfies the most basic postulates (), then so does any of its refinements. It turns out that this result can not be extended to the remaining postulates. For we characterize a subclass of refinements for which this postulate is preserved. For the four remaining postulates we study two representative kinds of distance-based merging operators. We show that postulates and are violated for all of our proposed examples of refined operators with the exception of the -based refinement. For and the situation is even worse in the sense that no refinement of our proposed examples of merging operators can satisfy them neither for nor for . Table 1 gives an overview of the results of this section. However, note that some of the forthcoming results are more general and hold for arbitrary fragments and/or operators.

Proposition 3.

Let be a merging operator satisfying postulates , and a characterizable fragment. Then each -refinement for satisfies in as well.

Proof.

Since is characterizable there exists a , such that is a -fragment. Let be a -refinement for . According to Proposition 2 we can assume that is an operator of form where is a suitable -mapping. In what follows, note that we can restrict ourselves to and to since we have to show that satisfies in .

: Since satisfies , . Thus, by monotonicity of the closure. Hence, , since and is a -fragment. According to Property 2 in Definition 7 we have , and therefore by definition of , , which proves that .

: Suppose satisfiable. Since satisfies , is satisfiable. Since is a -refinement (Proposition 2), is also satisfiable by the property of consistency (see Definition 5).

: Suppose is consistent with . Since satisfies , . We have . Since (observe that it is here necessary that the profiles are in the fragment) by Property 3 of Definition 7 we have .

: Let and with and . Since satisfies , . By the property of equivalence in Definition 5 we have . ∎

A natural question is whether refined operators for characterizable fragments in their full generality preserve other postulates, and if not whether one can nevertheless find some refined operators that satisfy some of the remaining postulates.

First we show that one can not expect to extend Proposition 3 to . Indeed, in the two following propositions we exhibit merging operators which satisfy all postulates, whereas some of their refinements violate in some fragments.

Proposition 4.

Let be a merging operator with , where is an arbitrary counting distance. Then the -based refined operator violates postulate in and . In case is a drastic distance, violates postulate in every characterizable fragment .

Proof.

First consider is a drastic distance. We show that violates postulate in every characterizable fragment . Since is a characterizable fragment there exists such that is a -fragment. Consider a set of models that is not closed under and that is cardinality-minimum with this property. Such a set exists since is a proper subset of . Observe that necessarily . Let , consider the knowledge bases and such that and . By the choice of both and are in , whereas is not. Let . Since the merging operator uses a drastic distance it is easy to see that