a contraction core for horn belief change: preliminary report

8/9/2019 A Contraction Core for Horn Belief Change: Preliminary Report

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A Contraction Core for Horn Belief Change: Preliminary Report

Richard Booth

University of LuxembourgLuxembourg

[email protected]

Thomas Meyer

Meraka Institute, CSIRSouth Africa

[email protected]

Ivan Varzinczak

Meraka Institute, CSIRSouth Africa

[email protected]

Renata Wassermann

Universidade de Sao PauloBrazil

[email protected]

Abstract

In this paper we continue recent investigations into be-lief change for Horn logic. The main contributionis a result which shows that the construction methodfor Horn contraction for belief sets based on infra-remainder sets, as recently proposed by Booth et al.,corresponds exactly to Hanssons classical kernel con-traction for belief sets, when restricted to Horn logic.This result is obtained via a detour through Horn con-traction for belief bases during which we prove thatkernel contraction for Horn belief bases produces pre-cisely the same results as the belief base version ofthe Booth et al. construction method. The use of be-lief bases to obtain the result provides evidence for theconjecture that Horn belief change is best viewed as ahybrid version of belief set change and belief basechange. One of the consequences of the link with basecontraction is the provision of a more elegant represen-tationresult for Horn contractionfor belief sets in whicha version of the Core-retainment postulate features. Thepaper focuses on Delgrandes entailment-based contrac-tion (e-contraction), but we also mention similar resultsfor inconsistency-based contraction (i-contraction) and

package contraction (p-contraction).

Introduction

In his seminal paper, (Delgrande 2008) has shed some lighton the theoretical underpinnings of belief change by weak-ening the usual assumption in the belief change commu-nity that the underlying logical formalism should be atleast as strong as (full) classical propositional logic. Del-grande investigated contraction for belief sets (sets of sen-tences closed under logical consequence) for Horn clauses.Delgrandes main contributions were threefold. Firstly, heshowed that the move to Horn logic leads to different typesof contraction, referred to as entailment-based contraction

and inconsistency-based contraction, which coincide in thefull propositional case. Secondly, Delgrande showed thatHorn contraction for belief sets does not satisfy the contro-versial Recovery postulate, but exhibits some characteristicsthat are usuallly associated with the contraction ofbases (ar-bitrary sets of sentences). And finally, Delgrande made atentative conjecture that a version of Horn contraction usu-ally referrred to as orderly maxichoice contraction is the ap-propriate method for contraction in Horn theories.

In a subsequent paper, Booth et al. (2009) showed thatwhile Delgrandes partial meet constructions are appropri-ate choices for contraction in Horn logic, they do not con-stitute all the appropriate forms of Horn contraction. Theybuild on a more fine-grained construction for belief set con-traction which we refer to in this paper as infra contraction.In addition to the two types of Horn contraction introducedby Delgrande, they also introduce a third.

However, the investigation into Horn belief contraction isnot closed yet. For one thing, Booth et al.s representationresult has the rather unsatisfactory property that it relies on apostulate which refers directly to the construction method itis intended to characterise. Moreover, as referred to earlier,although Horn contraction is defined on Horn belief sets, itseems to be related in some ways to contraction for beliefbases: an aspect which has not yet been explored properly.

In this paper we continue the investigation into contrac-tion for Horn logic, and address both the issues mentionedabove, as well as others. We bring into the picture a con-struction method for contraction first introduced by Hans-son (1994), known as kernel contraction. Although ker-

nel contraction is usually associated with base contraction,it can be applied to belief sets as well. Our main contri-bution is a result which shows that the infra contraction ofBooth et al. corresponds exactly to a version of Hanssonskernel contraction for belief sets, when restricted to Hornlogic. In order to prove this, we first take a close look at Horncontraction for belief bases, defining a base version of infracontraction and proving that this construction is equivalentto kernel contraction for Horn belief bases. Since Horn be-lief sets are not closed under classical consequence, they canbe seen as a hybrid between belief sets and belief bases.This justifies the use of belief bases to obtain results for be-lief set Horn contraction.

The investigation into base contraction also affords us the

opportunity to improve on the rather unsatisfactory repre-sentation result proved by Booth et al. for infra contraction.We show that a more elegant representation result can be ob-tained by replacing the postulate introduced by Booth et al.with the well-known Core-retainment postulate, which isusually associated with base contraction. The presence ofCore-retainment here further enforces the hybrid nature ofHorn belief changelying somewhere between belief setchange and base change.


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The paper focuses on Delgrandes entailment-based con-traction (e-contraction), but we also mention similar resultsfor two other relevant types of Horn contraction.

Horn logic has found extensive use in Artificial Intel-ligence, in particular in logic programming, truth mainte-nance systems, and deductive databases.1 This explains, inpart, our interest in belief change for Horn logic. Anotherreason for focusing on this topic is because of its application

to debugging and repairing ontologies in description logics(Baader et al. 2003). In particular, Horn logic can be seen asthe backbone of the EL family of description logics (Baader,Brandt, and Lutz 2005), and a proper understanding of be-lief change for Horn logic is therefore important for findingsolutions to similar problems expressible in the EL family.

The rest of the paper is structured as follows: The nextsection introduces the formal background needed. This isfollowed by discussions on base contraction and belief setcontraction, after which we briefly review the existing workon propositional Horn contraction. The following three sec-tions constitute the core of the paper. In the first one weprove that kernel contraction and infra contraction are equiv-alent on the level of bases. This enables us, in the following

section, to prove that kernel contraction and infra contrac-tion are equivalent on the Horn belief set level as well. Andfrom this we are led in the next section to provide a moreelegant characterisation for infra contraction for Horn be-lief sets than the one presented by Booth et al.. We wrapup with a section on related work which also concludes anddiscusses future directions of research.

Logical Background

We work in a finitely generated propositional language LPover a set of propositional atoms P, which includes the dis-tinguished atoms and , and with the standard model-theoretic semantics. Atoms will be denoted by p , q , . . ., pos-sibly with subscripts. We use , , . . . to denote classicalformulas. They are recursively defined in the usual way.

Classical logical consequence and logical equivalence aredenoted by |= and respectively. For X LP, the set ofsentences logically entailed by X is denoted by Cn(X). Abelief set is a logically closed set, i.e., for a belief set K,K = Cn(K). We usually denote belief sets by K, possiblydecorated by primes. P(X) denotes the power set (set ofall subsets) ofX.

A Horn clause is a sentence of the form p1 p2 . . . pn q where n 0, pi, q P for 0 i n (recall thatthe pis and q may be one of or as well). If n = 0 wewrite q instead of q. A Horn set is a set of Horn clauses.

Given a propositional language LP, the Horn language LHgenerated from LP is simply the Horn clauses occurring inLP. The Horn logic obtained from LH has the same seman-tics as the propositional logic obtained from LP, but just re-stricted to Horn clauses. A Horn belief set, usually denotedby H (possibly with primes), is a Horn set closed under log-ical consequence, but containing only Horn clauses. Hence,

1Despite our interest in Horn clauses, it is worth noting that inthis work we do not consider logic programming explicitly and wedo not use negation as failure at all.

|=, , the Cn(.) operator, and all other related notions aredefined relative to the logic we are working in (e.g. |=

PLfor

propositional logic and |=HL

for Horn logic). We shall dis-pense with such subscripts where the context makes it clearwhich logic we are dealing with.

Base Contraction

Contraction is intended to represent situations in which anagent has to give up information, say a formula , from itscurrent stock of beliefs. Other operations of interest in be-lief change are the expansion of an agents current beliefs by, where the basic idea is to add regardless of the conse-quences, and the revision of its current beliefs by , wherethe intuition is to incorporate into the current beliefs insome way while ensuring consistency at the same time.

We commence with a discussion on base contractionwhere an agents beliefs are represented as a set of sentences,also known as a base. We usually denote bases by B, possi-bly decorated with primes.

Definition 1 A base contraction for a base B is a func-tion from LP toP(LP).

Intuitively the idea is that, for a fixed base B, contraction ofa formula produces a new base B .

One of the standard approaches for constructing beliefcontraction operators is based on the notion of a remainderseta maximally consistent subset ofB not entailing (Al-chourron, Gardenfors, and Makinson 1985). Below we ap-ply this to bases, but as we shall see, it can be applied tobelief sets (closed under logical consequence) as well.

Definition 2 Given a set B, X B iff (i) X B;(ii) X |= ; (iii) for all X s.t. X X B, X |= . Wecall the elements ofB the remainder sets ofB w.r.t. .

It is easy to verify that B = if and only if|= .

Since there is no unique method for choosing betweenpossibly different remainder sets, there is a presuppositionof the existence of a suitable selection function for doing so.

Definition 3 A selection function for a set B is a (par-tial) function from P(P(LP)) to P(P(LP)) such that(B) = {B} ifB = , and (B) Botherwise.

Selection functions provide a mechanism for identifying theremainder sets judged to be most appropriate, and the result-ing contraction is then obtained by taking the intersection ofthe chosen remainder sets.

Definition 4 For a selection function , the base contrac-tion generated by as follows: B =

(B)is a base partial meet contraction.

Two subclasses of base partial meet deserve special mention.

Definition 5 Given a selection function , is a basemaxichoice contraction iff (B) is always a singletonset. It is a base full meet contraction iff (B) = BwheneverB = .


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Base full meet contraction is unique, while base maxichoicecontraction usually is not.

For reasons that will become clear, it is interesting to ob-serve that the following convexity principle does not hold forbelief bases.

(Convexity) For a base B, let mc be a base maxichoicecontraction, and let fm be base full meet contraction.For every set X and s.t. (B fm ) X B mc ,there is a base partial meet contraction pm s.t. B pm = X.

The principle simply states that every set between the re-sults obtained from base full meet contraction and some basemaxichoice contraction can be obtained from some basepartial meet contraction. To see that it does not hold, letB = {p q, q r, p q r, p r q} and considercontraction by p r. It is easily verified that base maxi-choice gives either B(p r) = B = {p q, pr q}or B (p r) = B = {q r, p q r, p r q}and therefore the only other result obtained from base partialmeet contraction is that which is provided by base full meetcontraction: B (p r) = B = {p r q}. But

observe that even though it is the case that B

X B

where X = {p q r, p r q}, X is not equal to anyofB, B, or B.

Base partial meet contraction can be characterised by thefollowing postulates

(B 1) If|= , then B |= (Success)

(B 2) B B (Inclusion)

(B 3) IfB |= if and only ifB |= for all B B,then B = B (Uniformity)

(B 4) If (B \ (B )), then there is a B s.t.(B ) B B and B |= , but B {} |= (Relevance)

Theorem 1 (Hansson 1992) Every base partial meet con-traction satisfies (B 1)(B 4). Conversely, every basecontraction which satisfies (B 1)(B 4) is a base partialmeet contraction.

Kernel contraction was introduced by Hansson (1994) asa generalization ofsafe contraction (Alchourron and Makin-son 1985). Instead of looking at maximal subsets not imply-ing a given formula, kernel operations are based on minimalsubsets that imply a given formula.

Definition 6 For a base B, X B iff (i) X B;(ii) X |= ; and (iii) for every X s.t. X X, X |= .B is called the kernel set ofB w.r.t. and the elements

ofB are called the -kernels ofB.The result of a base kernel contraction is obtained by

removing at least one element from every (non-empty) -kernel ofB, using an incision function.

Definition 7 An incision function for a base B is a func-tion from the set of kernel sets of B to P(LP) such that(i) (B )

(B ); and (ii) if = X B ,

then X ((B )) = .

Definition 8 Given an incision function for a base B, thebase kernel contraction forB generated by is definedas: B = B \ (B ).

Base kernel contraction can be characterised by the samepostulates as base partial meet contraction, except that Rel-evance is replaced by the Core-retainment postulate below:

(B 5) If (B \ (B )), then there is some B B

such that B

|= but B

{} |= (Core-retainment)Theorem 2 (Hansson 1994) Every base kernel contractionsatisfies (B 1)(B 3) and (B 5). Conversely, everybase contraction which satisfies (B1)(B3) and(B5)is a base kernel contraction.

Observe that Core-retainment is slightly weaker than Rele-vance. And indeed, it thus follows that all base partial meetcontractions are base kernel contractions, but as the follow-ing example from Hansson (1999) shows, some base kernelcontractions are not base partial meet contractions.

Example 1 LetB = {p,pq, p q}. Then B (pq) ={{p,p q}, {p q, p q}}, from which it follows thatthere is an incision function forB s.t. (B (p q)) =

{p q, p q}, and then B (p q) = {p}. On the otherhand, B(p q) = {{p,p q}, {p q}}, from which itfollows that base partial meet contraction B (p q) yieldseither{p,p q}, or{p q}, or{p,p q} {p q} = ,none of which are equal to B (p q) = {p}.

Belief Set Contraction

In belief setcontraction the aim is to describe contraction onthe knowledge level (Gardenfors 1988) independent of howbeliefs are represented. Thus, contraction is defined only forbelief sets (i.e., sets closed under logical consequence).

Definition 9 A belief set contraction for a belief setK isa function from LP toP(LP).

By the principle of categorical matching (Gardenfors andRott 1995) the contraction of a belief set by a sentence isexpected to yield a new belief set.

Given the construction methods for base contraction al-ready discussed, two obvious first attempts to define beliefset contraction are to consider both partial meet contractionand kernel contraction restricted to belief sets. For partialmeet contraction this works well: the remainder sets (Defi-nition 2) of belief sets are all belief sets as well, the defini-tion of selection functions (Definition 3) carries over as is forbelief sets, and partial meet contraction (Definition 4) whenapplied to belief sets will always yield a belief set as a result.When applied to belief sets, we refer to the contractions inDefinition 4 as belief set partial meet contractions.

For kernel contraction matters are slightly more compli-cated. For one thing, for a belief set as input, base kernelcontraction does not necessarily produce a belief set as aresult. Of course, it is possible to ensure that a belief setis obtained by closing the result obtained from base kernelcontraction under logical consequence.

Definition 10 Given a belief setK and an incision function for K, the belief set contraction for K generated as


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follows: K = Cn(K ) , is a belief set kernelcontraction.

This is very closely related to a version of base contractionthat Hansson (1999) refers to as saturated base kernel con-traction:

Definition 11 Given a base B and an incision function forB, the base contraction forB generated as follows:

B = B Cn(B ) , is a saturated base kernelcontraction.

It is easily shown that when the set B in the definition forsaturated base kernel contraction is a belief set, the two no-tions coincide.

Observation 1 For a belief set K and an incision function forK, the saturated base kernel contraction for and thebelief set kernel contraction for are identical.

While it can be shown that some saturated base kernel con-tractions are not base partial meet contractions, this distinc-tion disappears when considering belief sets only.

Theorem 3 (Hansson 1994) LetK be a belief set. A belief

set contraction is a saturated base kernel contraction ifand only if it is a belief set partial meet contraction.

And as a result of Observation 1 and Theorem 3 we imme-diately have the following corollary:

Corollary 1 LetK be a belief set. A belief set contraction is a belief set kernel contraction if and only if it is a beliefset partial meet contraction.

Belief set contraction defined in terms of partial meet con-traction (and kernel contraction) corresponds exactly to whatis perhaps the best-known approach to belief change: the so-called AGM approach (Alchourron, Gardenfors, and Makin-son 1985). AGM requires that belief set contraction be char-acterised by the following set of postulates:

(K 1) K = Cn(K ) (Closure)

(K 2) K K (Inclusion)

(K 3) If / K, then K = K (Vacuity)

(K 4) If|= , then / K (Success)

(K 5) If , then K = K (Extensionality)

(K 6) If K, then Cn((K ) {}) = K (Recovery)

Alchourrn et al. (1985) have shown that these postulatescharacterise belief set partial meet contraction exactly.

Theorem 4 Every belief set partial meet contraction satis-fies (K 1)(K 6). Conversely, every belief set contrac-tion which satisfies

(K 1)

(K 6)is a belief set partial

meet contraction.

We will not elaborate in detail on these postulates, exceptfor a brief comparison with the postulates for base contrac-tion. Closure is specific to belief set contraction and has nocounterpart in base contraction. Both Inclusion and Successalso occur as postulates for base contraction. A modifiedversion of Vacuity (in which the antecedent is changed toIf B |= ) holds for base contraction. Extensionality is

a special case of the Uniformity postulate for base contrac-tion. And finally, Recovery is stronger than both Relevanceand Core-retainment, and does not hold for base contraction.

A final word on belief set contraction for full proposi-tional logic: Booth et al. (2009) have shown that the Con-vexity property holds for belief sets.

Proposition 1 (Convexity) Let K be a belief set, let mcbe a (belief set) maxichoice contraction, and let fm be

(belief set) full meet contraction. For every belief set X andsentence s.t. (Kfm ) X Kmc , there is a (be-lief set) partial meet contraction pm s.t. Kpm = X.

The result shows that every belief set between the results ob-tained from full meet contraction and some maxichoice con-traction can also be obtained from some partial meet con-traction. This means that it is possible in principle to definea version of belief set contraction based on such sets.

Definition 12 For belief sets K andK, K K if andonly if there is some K K such that (

K)

K K. We refer to the elements of K as the infraremainder sets ofK with respect to .

Note that all remainder sets are also infra remainder sets, andso is the intersection of any set of remainder sets. Indeed, theintersection of any set of infra remainder sets is also an infraremainder set. So the set of infra remainder sets contains allbelief sets between some remainder set and the intersectionof all remainder sets. This explains why infra contractionbelow is not defined as the intersection of infra remaindersets (cf. Definition 4).

Definition 13 (Infra Contraction) Let K be a belief set.An infra selection function is a (partial) function fromP(P(LP)) toP(LP) such that (K ) = K wheneverK = , and(K ) K otherwise. A belief setcontraction is a belief set infra contraction if and only ifK = (K ).

Because of Proposition 1 we can extend Corollary 1 toshow that kernel contraction, partial meet contraction, andinfra contraction all coincide for belief sets.

Corollary 2 Let K be a belief set. A belief set contraction is a belief set kernel contraction iff it is a belief set partialmeet contraction iff it is a belief set infra contraction.

Horn Contraction

While there has been some work on revision for Hornclauses (Eiter and Gottlob 1992; Liberatore 2000; Langloiset al. 2008), it is only recently that attention has beenpaid to contraction for Horn logic. In particular, Delgrande(2008) investigated two distinct classes of contraction func-

tions for Horn belief sets: e-contraction, for removing anunwanted consequence; and i-contraction, for removing for-mulas leading to inconsistency; while Booth et al. (2009)subsequently extended the work of Delgrande. Our focus inthis paper is on e-contraction, although Delgrande, as wellas Booth et al., also consider i-contraction. Delgrandes def-inition of Horn logic allows for the conjunction of Hornclauses, and we shall follow his convention in this paper.(This differs from the definition of Booth et al. although


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their version can be recast into that of Delgrande withoutany lossor gainin expressivity.) Remembering, here weare going to use H, sometimes decorated with primes, todenote a Horn belief set.

Definition 14 An e-contraction for a Horn belief setH isa function from LH toP(LH).

Delgrandes method of construction for e-contraction is

in terms of partial meet contraction. The definitions of re-mainder sets (Definition 2), selection functions (Definition3), partial meet contraction (Definition 4), as well as maxi-choice and full meet contraction (Definition 5) all carry overfor e-contraction, with the set B in each case being replacedby a Horn belief set H, and we shall refer to these as e-remainder sets (denoted by He), e-selection functions,partial meet e-contraction, maxichoice e-contraction and fullmeet e-contraction respectively (we leave out the referenceto the term Horn, since there is no room for ambiguity).As in the full propositional case, it is easy to verify thatall e-remainder sets are also Horn belief sets, and that allpartial meet e-contractions (and therefore the maxichoicee-contractions, as well as full meet e-contraction) produce

Horn belief sets.Although Delgrande defines and discusses partial meete-contraction, he argues that maxichoice e-contraction (tobe precise, a special case of maxichoice e-contraction re-ferred to as orderly maxichoice e-contraction) is the ap-propriate approach for e-contraction. Booth et al. (2009),on the other hand, argue that although all partial meet e-contractions are appropriate choices for e-contraction, theydo not make up the set of all appropriate e-contractions.The argument for appropriate e-contractions other than par-tial meet e-contraction is based on the observation that theconvexity result for full propositional logic in Proposition 1does not hold for Horn logic.

Example 2 (Booth, Meyer, and Varzinczak 2009) LetH = CnHL({p q, q r}). It is easy to ver-ify that, for the e-contraction of p r , maxichoice yields either H1mc = CnHL({p q}) or H

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mc =CnHL({q r, p r q}) , that full meet yields Hfm =CnHL({p r q}), and that these are the only three par-tial meet e-contractions. Now consider the Horn beliefset H = CnHL({p q r, p r q}). It is clear thatHfm H H

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mc , but there is no partial meet e-contraction yielding H.

In order to rectify this situation, Booth et al. propose that ev-ery Horn belief set between full meet and some maxichoicee-contraction ought to be seen as an appropriate candidatefor e-contraction.

Definition 15 (Infra e-Remainder Sets) For Horn beliefsets H andH, H He iff there is some H Hes.t. (

He) H H. We refer to the elements of

He as the infra e-remainder sets ofH w.r.t. .

As with the case for full propositional logic, e-remaindersets are also infra e-remainder sets, and so is the intersectionof any set ofe-remainder sets. Similarly, the intersection ofany set of infra e-remainder sets is also an infra e-remainder

set, and the set of infra e-remainder sets contains all Hornbelief sets between some e-remainder set and the intersec-tion of all e-remainder sets. As in the full propositional case,this explains why e-contraction is not defined as the intersec-tion of infra e-remainder sets (cf. Definition 4).

Definition 16 (Horn e-Contraction) Let H be a Horn be-lief set. An infra e-selection function is a (partial) func-tion fromP(P(LH)) toP(LH) such that(He ) = H

wheneverHe = , and(He ) He otherwise.An e-contraction is an infra e-contraction if and only ifH = (He ).

Booth et al. show that infra e-contraction is captured pre-cisely by the six AGM postulates for belief set contraction,except that Recovery is replaced by the following (weaker)postulate (H e 6), and the Failure postulate (below) isadded.

(He 6) If H \ (H ), then there exists an X s.t.(He) X H and X |= , but X {} |=

(He 7) If|= , then He = H (Failure)

More formally (Booth, Meyer, and Varzinczak 2009):

Theorem 5 Every infra e-contraction satisfies postulates(K1)(K5), (He6) and(He7). Conversely, everye-contraction which satisfies (K 1)(K 5), (H e 6)and(He 7) is an infra e-contraction.

Observe firstly that (H e 6) bears some resemblance tothe Relevance postulate for base contraction. Observe alsothat it is a somewhat unusual postulate in that it refers di-rectly to e-remainder sets. It is possible to provide a more el-egant characterisation of infra e-contraction as we shall see.Before we do so, however, we first take a detour throughbase contraction.

Base Infra Contraction

In the section on base contraction we have considered re-mainder sets for bases and kernel sets for bases, but not in-fra remainder sets for bases. We commence this section withthe definition ofbase infra remainders sets.

Definition 17 (Base Infra Remainder Sets) For bases Band B, B B iff there is some B B s.t.(

B) B B. We refer to the elements of B asthe base infra remainder sets ofB with respect to .

Observe that the definition of base infra remainder sets isthe same as for infra e-remainder sets, differing only in that(i) it deals with belief bases and not belief sets; and ( ii) itis defined in terms of remainder sets for bases, and not for(Horn) belief sets.

Base infra remainder sets can clearly be used to define aform of base contraction in a way that is similar to that inDefinitions 13 and 16.

Definition 18 (Base Infra Contraction) A base infra se-lection function is a (partial) function from P(P(LP))to P(LP) s.t. (B ) = B whenever B = , and(B ) B otherwise. A base contraction gen-erated by as follows: B = (B ) is a base infracontraction.


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A natural question to ask is how base infra contractioncompares with base partial meet contraction and base kernelcontraction. The following fundamental result, which playsa central role in this paper, shows that base infra contractioncorresponds exactly to base kernel contraction.

Theorem 6 A base contraction for a base B is a base kernelcontraction forB iff it is a base infra contraction forB.

From the section on base contraction, and from Exam-ple 1, specifically, we know that base kernel contraction ismore general than base partial meet contractionevery basepartial meet contraction is also a base kernel contraction, butthe converse does not hold. From Theorem 6 it thereforefollows that base infra contraction is more general than basepartial meet contraction as well. This is not surprising, giventhat a similar result holds for partial meet e-contraction andinfra e-contraction as we have seen in the section on Horncontraction (cf. Example 2).

Theorem 6 has a number of other interesting conse-quences as well. On a philosophical note, it provides cor-roborative evidence for the contention that the kernel con-traction approach is more appropriate than the partial meet

approach. The fact that kernel contraction is at least as gen-eral as partial meet contraction for both base and belief setcontraction is already an argument favouring it over partialmeet contraction. Theorem 6 adds to this by showing that aseemingly different approach to contraction (infra contrac-tion), which is also at least as general as partial meet con-traction for both base and belief set contraction, turns outto be identical to kernel contraction. As we shall see in thenext section, Theorem 7 is also instrumental in lifting thisresult to the level of Horn belief sets.

Kernel e-contraction = Infra e-contraction

Through the work of Booth et al. (2009) we have alreadyencountered partial meet contraction and infra contraction

for Horn belief sets (partial meet e-contraction and infra e-contraction), but we have not yet defined a suitable versionof kernel contraction for this case.

Definition 19 Given a Horn belief set H and an incisionfunction for H , the Horn kernel e-contraction for H , ab-breviated as the kernel e-contraction for H is defined asH e = CnHL(H ), where is the base kernelcontraction for obtained from .

Given the results on how kernel contraction, partial meetcontraction and infra contraction compare for the base case(kernel contraction and infra contraction are identical, whileboth are more general than partial meet contraction), onewould expect similar results to hold for Horn belief sets.

And this is indeed the case. Firstly, infra e-contraction andkernel e-contraction coincide.

Theorem 7 Given a Horn belief setH, an e-contraction forH is an infra e-contraction forH if and only if it is a kernele-contraction forH.

Proof sketch: Consider a base B and a formula . FromTheorem 6 it follows that the set of base infra remainder setsofB w.r.t. (i.e., the set B ) is equal to the set of results

obtained from the base kernel contraction of B by , call itKCB . Now let B be such that is a set of Horn clauses closedunder Horn consequence (a Horn belief set) and a Hornclause. The elements of KCB are not necessarily closedunder Horn consequence, but if we do close them, we obtainexactly the set of results obtained from kernel e-contractionwhen contracting B by (by the definition of kernel e-contraction). Let us refer to this latter set as CnHL(KC

B ).

I.e., CnHL(KCB ) = {CnHL(X) | X KC

B }. Also, theelements of B are not closed under Horn consequence,

but if we do close them the resulting set (refer to this set asCnHL(B )) contains exactly the infra e-remainder sets ofB w.r.t. . I.e., CnHL(B ) = B e (why this is the case,will be explained below). But since B = KCB , it is also

the case that CnHL(B ) = CnHL(KCB ), and therefore

CnHL(KCB ) = B e. [QE D]

To see why CnHL(B ) = B e , observe that sinceB is closed under Horn consequence, the (base) remaindersets ofB w.r.t. (i.e., the elements ofB) are also closedunder Horn consequence. So the elements of B are allthe sets (not necessarily closed under Horn consequence) be-

tween

(B) and some element ofB. Therefore theelements of CnHL(B ) are all those elements of B that are closed under Horn consequence. That is, exactly theinfra e-remainder sets ofB with respect to .

From Theorem 7 and Example 2 it follows that par-tial meet e-contraction is more restrictive than kernel e-contraction. When it comes to Horn belief sets, we thereforehave exactly the same pattern as we have for belief baseskernel contraction and infra contraction coincide, while bothare strictly more permissive than partial meet contraction.Contrast this with the case for belief sets for full proposi-tional logic where infra contraction, partial meet contractionand kernel contraction all coincide.

One conclusion to be drawn from this is that the restric-tion to the Horn case produces a curious hybrid between be-lief sets and belief bases for full propositional logic. On theone hand, Horn contraction deals with sets that are logicallyclosed. But on the other hand, the results for Horn logic ob-tained in terms of construction methods are close to thoseobtained for belief base contraction.

Either way, the new results on base contraction haveproved to be quite useful in the investigation of contractionfor Horn belief sets. In the next section we provide anotherresult for Horn contraction inspired by the new results onbase contraction in this paper.

An Elegant Characterisation for e-Contraction

In the section on Horn contraction we remarked thatthe characterisation of infra e-contraction, specifically the(He 6) postulate, is somewhat unusual in that it refers di-rectly to an aspect of the construction method (e-remaindersets) that it is meant to characterise. In this section we showthat it is possible to provide a more elegant characterisationof infra e-contractionone that replaces (He 6) with theCore-retainment postulate which we encountered in the sec-


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tion on base contraction, and that is used in the characterisa-tion of base kernel contraction.

Theorem 8 Every infra e-contraction satisfies (K 1)-(K 5) Core-retainment and (H e 7). Conversely, ev-ery e-contraction which satisfies (K 1)-(K 5) , Core-retainment and(He 7) is an infra e-contraction.

This result was inspired by Theorem 6 which shows that

base kernel and base infra contraction coincide. Given thatCore-retainment is used in characterising base kernel con-traction, Theorem 6 shows that there is a link between Core-retainment and base infra contraction, and raises the ques-tion of whether there is a link between Core-retainment andinfra e-contraction. The answer, as we have seen in Theorem8, is yes. This result provides more evidence for the hybridnature of contraction for Horn belief setsin this case theconnection with base contraction is strengthened.

Related Work and Concluding Remarks

We have mentioned the work on revision for Horn clausesby Eiter and Gottlob (1992), Liberatore (2000) and Lan-glois (2008). Apart from the work of Delgrande (2008) and

Booth et al. (2009) which were discussed extensively in ear-lier sections, there is some recent work on obtaining a se-mantic characterisation of Horn contraction by Fotinopou-los (2009). Billington (1999) considered revision and con-traction for defeasible logic which is quite different fromHorn logic in many respects, but nevertheless has a rule-likeflavour to it which has some similarity to Horn logic.

In bringing Hanssons kernel contraction into the picture,we have made useful and meaningful contributions to theinvestigation into contraction for Horn logic. More specif-ically, the main contributions of this paper are (i) a resultwhich shows that infra contraction and kernel contractionfor the base case coincide; (ii) lifting the previous resultsto Horn belief sets to show that infra contraction and ker-nel contraction for Horn belief sets coincide; and (iii) usingthese results as a guide to the provision of a more elegantcharacterisation of the representation result by Booth et al.for infra contraction as applied to Horn belief sets.

In addition to e-contraction, Delgrande investigated a ver-sion of Horn contraction he refers to as inconsistency-basedcontraction (or i-contraction) where the purpose is to mod-ify an agents Horn belief set in such a way as to avoid incon-sistency when a sentence is provided as input. That is, ani-contraction i should be such that (Hi ) {} |=HL. In addition Booth et al. considered a version of pack-age contraction (or p-contraction) by a set of sentences ,for which none of the sentences in should be in the re-

sult obtained from p-contraction. Although it seems that thenew results presented in this paper can be applied to both i-contraction and p-contraction, this still has to be verified indetail.

Finally, we have seen that kernel e-contraction and in-fra e-contraction are more general than partial meet e-contraction. But there is evidence that even these forms ofHorn contraction may not be sufficient to obtain all mean-ingful answers, as can be seen from the following example.

Example 3 Consider again our Horn belief set exampleCnHL({p q, q r}) encountered in Example 2. If weview basic Horn clauses (clauses with exactly one atomin the head and the body) as representative of arcs in agraph, in the style of the old inheritance networks, then onepossible desirable outcome of a contraction by p r isCnHL(q r). However, as we have seen in Example 2,this is not an outcome supported by infra e-contraction (and

therefore not by kernel e-contraction either).Ideally, a truly comprehensive e-contraction approach for

Horn logic would be able to account for such cases as well.

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a contraction core for horn belief change: preliminary report

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