A Semantic Perspective on Belief Change in a Preferential Non-Monotonic Framework

A Semantic Perspective on Belief Change in a Preferential Non-Monotonic

Framework

Giovanni Casini

CSC, Universit´e du Luxembourg Luxembourg [email protected]

Eduardo Ferm´e

Universidade da Madeira NOVA-LINCS Portugal [email protected]

Thomas Meyer

CAIR

(CSIR & Univ. of Cape Town) South Africa [email protected]

Ivan Varzinczak

CRIL Univ. Artois & CNRS

France [email protected] Abstract

Belief change and non-monotonic reasoning are usually viewed as two sides of the same coin, with results showing that one can formally be defined in terms of the other. In this paper we investigate the integration of the two formalisms by studying belief change for a (preferential) non-monotonic framework. We show that the standard AGM approach to be-lief change can be transferred to a preferential non-monotonic framework in the sense that change operations can be defined onconditionalknowledge bases. We take as a point of depar-ture the results presented by Casini and Meyer (2017), and we develop and extend such results with characterisations based on semantics and entrenchment relations, showing how some of the constructions defined for propositional logic can be lifted to our preferential non-monotonic framework.1

1 Introduction

Both belief change and non-monotonic reasoning deal with the problem of handling conflicting information. For exam-ple, suppose we know that vertebrate red-blood cells have a nucleus (v _Ñ n), that mammalian red-blood cells are ver-tebrate red-blood cells (m Ñ v), but that mammalian

red-blood cells do not have a nucleus (m_Ñ n). The existence of mammalian red-blood cells (m) then renders our

knowl-edge base inconsistent. Belief-change operators modify the existing knowledge base to preserve its consistency (in our example,v _Ñ nshould be weakened). On the other hand,

non-monotonic reasoning usually handles such conflicts by introducingdefeasibility(this amounts to stating that verte-brate red-blood cells usually have a nucleus). Both mech-anisms are deeply interconnected. Indeed, technically each of them can be considered as a re-formulation of the other (Makinson 1993).

At a first glance belief change seems to be superfluous in non-monotonic settings—a revision operator can simply be replaced with expansion, with the non-monotonic machin-ery then ensuring consistency of some kind. This view does not take into account that non-monotonic frameworks con-tain a mix of defeasible and classical (non-defeasible) infor-mation. In our example above the statement that mammalian

1_{This is a preprint version of the homonym}

paper published in the Proceedings of KR 2018 (URL:https://aaai.org/ocs/index.php/KR/KR18/paper/view/18071/17146)

red-blood cells are vertebrate red-blood cells should proba-bly not be defeasible, while the statement about vertebrate red-blood cells having a nucleus probably should. The chal-lenge thus becomes one of defining belief change for the monotonicpart of the formalism, while simultaneously en-suring that the non-monotonic part remains well-behaved.

Our goal is the characterisation of belief change in a non-monotonic setting, that is still an open problem, since the standard approaches to belief change usually assume an underlying Tarskian consequence relation which is ex-plicitly monotonic (Alchourr´on, G¨ardenfors, and Makinson 1985). This is not the first study on belief revision in a con-ditional framework, but it differs from most of the previ-ous approaches, such as those by Kern-Isberner (1999) and Wobcke (1995). These proposals give to the conditionals a subjunctive interpretation and rely on the already-known correspondence between them and the belief revision opera-tors: the conditionals are interpreted as direct expressions of the belief revision policies of the agent, and usually the main technical problem is the definition of non-trivial revision op-erators that avoid the well-known G¨ardenfors’ impossibil-ity result (G¨ardenfors 1988, Section. 7.4). Here we work on conditional knowledge bases, and we do not assume a sub-junctive interpretation of the conditionals. Rather, the condi-tional knowledge base could formalise defeasible informa-tion of a different nature, describing for example prototypi-cal behaviour or deontic constraints, and the conditionals are the objects of the belief change, not an alternative expression of the belief revision policies. As we shall see, this also im-plies that G¨ardenfors’ impossibility result does not apply in our framework. The present paper extends the results on the topic by Casini and Meyer (2017) by giving a semantic char-acterisation and defining another class of revision operators. In addition to the classical problem ofconsistency preser-vation, we also consider belief change for the preservation of a restricted version ofcoherence, as it is intended in the field of logic-based ontologies. A knowledge base is coherent if every class that has been introduced in the language can in principle be populated (Qi and Hunter 2007). In our exam-ple above the (non-defeasible) statements _tv _Ñ n,m _Ñ v,m _Ñ n_u cause the knowledge base to be incoherent w.r.t. m(there cannot be any mammalian red-blood cells),

but the knowledge base only becomes inconsistent whenm

in a propositional setting, the incoherence of a knowledge base w.r.t. an atom (or any formulaA, for that matter), cor-responds to the statement A being a consequence of our

knowledge base.

We focus on modelling belief change assuming as the un-derlying logical framework that of non-monotonic reason-ing developed by Kraus, Lehmann and Magidor (1990), in whichdefeasibleconditionals of the formC_|„Dare added

to the language of propositional logic (withCandDbeing classical propositional formulas).

The paper is structured as follows: In Section 2 we give brief summaries of both the AGM and KLM frameworks, and summarise the main results by Casini and Meyer (2017). In Section 3 we present our first contribution, namely a semantic characterisation of Casini and Meyer’s operators. Section 4 is devoted to the characterisation of a new class of change operators.

2 Preliminaries

We consider a propositional languageLgenerated by some finite set of atomsA, with lower case letters denoting atoms, and capital letters denoting elements of L. We adopt the standard semantics for propositional logic. The setVof val-uationsvare functions from the set of atoms inLtot0,1u, denoting truth and falsity. Satisfaction is denoted by_,, and entailment by(, whilev (respectively, S) indicates a for-mula uniquely characterisingv(a set of valuations_SÑ_V).

rA_sdenotes the set of valuations satisfyingA.

2.1 AGM belief change

AGM belief change (Alchourr´on, G¨ardenfors, and Makin-son 1985) assumes an underlying logic with a propositional language and a Tarskian consequence relation Cn_p¨q that is compact and satisfies disjunction in the premises (Al-chourr´on, G¨ardenfors, and Makinson 1985, p. 511–512). A knowledge baseKis assumed to be a set of formulasclosed under Cnp¨q. AGM is concerned with three types of opera-tions on knowledge bases: expansion, contraction, and revi-sion. Expansion is simply defined as adding a formula and closing under entailment:K`_{A ”def} CnpK_{Y t}A_uq. The in-tuition associated with the contraction ofK byAis that it

should result in a knowledge baseK´_A not entailingA.

Du-ally, a revision ofKbyAshould result in a consistent knowl-edge baseK˚_Afrom whichAfollows.

In the AGM approach, any appropriate contraction opera-tor is required to satisfy sixbasic postulates for contraction: Closurep´1q,Inclusionp´2q,Vacuityp´3q,Successp´4q, Extensionalityp´5q, andRecoveryp´6q. For details, we re-fer the reader to one the many publications presenting the AGM theory (Alchourr´on, G¨ardenfors, and Makinson 1985; G¨ardenfors 1988; Hansson 1999). Additionally, AGM con-traction may be required to satisfy the two supplementary postulates about conjunctions:

p´7q K´AXKB´ ÑK´pA^Bq (Conjunctive Overlap)

p´8q IfARK´_pA^Bq, thenK´_pA^BqÑK´A(Conj. Inclusion)

AGM contraction operators can be constructed using so-calledpartial-meet functions(Alchourr´on, G¨ardenfors, and

Makinson 1985, Obs. 2.5). LetKÓA be theremainder set

ofKw.r.t.A, defined as the set containing the maximal sub-setsK1ofKs.t.A RK1. That is,K1 _PK_ÓAiff (i)K1 _ÑK, (ii)A _R K1, and (iii) there is no setK2 s.t.K1 _Ä K2 _Ñ K

andA _R K2. Letpmp¨qbe a partial-meet function defined over KÓAs.t. pmpKÓAq Ñ KÓAand, if KÓA ‰ H, then

pmpK_ÓA_{q ‰ H}. A partial-meet contraction operator´is defined as:K´_A“ìpmpKÓAq.

Observation 1 (Alchourr´on, G¨ardenfors, and Makinson 1985, Obs. 2.5)An operator´onKis a partial meet con-traction iff´satisfiesp´1q–p´6q.

Similarly, AGM revision is required to satisfy six basic postulates (Alchourr´on, G¨ardenfors, and Makinson 1985; G¨ardenfors 1988): Closure p˚1q, Inclusion p˚2q, Vacuity p˚3q, Success p˚4q, Extensionality p˚5q, and Consistency p˚6q. Two supplementary postulates for revision are pro-posed:

p˚7q K˚_pA^BqÑpK˚Aq`B (Superexpansion)

p˚8q If BRK˚A, thenpKA˚q`BÑK˚_pA^Bq (Subexpansion)

Revision can be defined in terms of contraction and ex-pansion via the Levi Identity (Levi 1977):K˚A“ pK´Aq`A.

Observation 2 (Alchourr´on, G¨ardenfors, and Makinson 1985; Ferm´e 1999) Let´be an operator onK satisfying p´1q–p´5q, and let˚be defined via the Levi Identity. Then: 1. ˚satisfiesp˚1q–p˚6q.

2. If´also satisfiesp´6q,p´7qandp´8q, then˚also satis-fiesp˚7qandp˚8q.

There are multiple ways of characterising the class of the contraction operations satisfying the six basic postulates plusp´7qandp´8q. One way is throughepistemic entrench-ment relations§e, introduced by G¨ardenfors (1988) to

de-fine the properties that an order over the sentences of L should satisfy. He proposed the following set of axioms: E1 ifC§eDandD§eE, thenC§eE (Transitivity)

E2 ifC_|ùD, thenC_§eD (Dominance)

E3 EitherC§eC^DorD§eC^D(Conjunctiveness)

E4 IfK RK, thenC_RKiffC_§eDfor allD(Minimality)

E5 IfD§eCfor everyD, then|ùC (Maximality)

G¨ardenfors proposed the following connections between orders of epistemic entrenchment and operations of contrac-tion (G¨ardenfors 1988):

DPK´_CeiffDPKand eitherC†eC_Dor |ùC (1)

C_§eDiffCRKC´^Dor (C^D (2)

Observation 3 (G¨ardenfors 1988; G¨ardenfors and Makinson 1988) Given an epistemic entrenchment §e on

a consistent belief set K, then _´e, defined via _p1_q, is a contraction operator satisfyingp´1q ´ p´8q. Vice versa, if we have a contraction operator´satisfyingp´1q ´ p´8q, there is an epistemic entrenchment§es.t.´corresponds to

Rott (1991) proposed an alternative class of entrenchment-based contraction operator:

DPK´R_C iffDPKand eitherC†eDor |ùC (3)

In order to relate´eand´R, letKóCbe a set of preferred

remainder sets w.r.t. the epistemic entrenchment relation,

K_óC_”deftK1_PK_ÓC_|K´R

C ÑK1u

The operator defined in (1) corresponds to the intersection of all the elements ofK_óC.

Proposition 1 For everyKand everyC, D P L,D P K´Ce

iffD_PK1for everyK1 _PK_óC.

Proposition 1, beyond helping in further analysing the re-lation between G¨ardenfors’ and Rott’s contractions, gives an alternative definition of G¨ardenfors’ contraction in terms of full meet of the preferred remainder set. We will refer again to this characterisation later on.

The semantics of the AGM model can be characterized by a total pre-order over the set of valuations. A total pre-order

§Kon valuations, with the strict part†Kand the symmetric

part»K, is afaithful assignmentif and only if the following

conditions hold (1) Ifv _{P r}K_sandv1 _{P r}K_s, thenv _»K v1

and (2) IfvP rKsandv1R rKs, thenv†Kv1.

The notion of a faithful assignment allows us to charac-terise contraction and revision operations satisfying_p´1q ´ p´8qandp˚1q ´ p˚8q, respectively:

Observation 4 (Katsuno and Mendelzon 1991)LetKbe

a belief set, and let min

§Kpr

Asq ”deftvP rAs |v§Kufor everyuP rAsu.

An operation´on Ksatisfies p´1q ´ p´8qif and only if there is a faithful assignment §K for K such thatK´A “

rK_{s Y}min_§Kpr As. An operation ˚onK satisfiesp˚1q ´

p˚8qif and only if there is a faithful assignment§K forK

such thatK˚_A“min_§KprA_sq.

2.2 Preferential reasoning

To introduce defeasibility we consider the languageL|„ con-sisting of conditionals of the formA_|„B (forA, B _{P L}), that can be read as ‘typically, if A then B’. The

seman-tics of L|„is based on the notion of a preferential inter-pretation: triples of the form _xW, l,_†y where W is a set

of objects (states),l is a function fromW toV (mapping states into valuations), and_†is a strict partial order onW, that also satisfy thesmoothnesscondition: for every consis-tent propositional formula A,min†prAsWq ‰ H, where

rA_sW _{”def ts P W | lpsq , Auandmin}

†prAsWq ”def

ts P rAsW _{| there is not P rAs}W _{s.t.t † su (Kraus,}

Lehmann, and Magidor 1990). s _† t is interpreted as in-dicating that the statesrepresents a more typical situation

than the statet. A defeasible conditionalA_|„B is satisfied

in a preferential interpretationP, denoted asP,A_|„B, iff min†rAsW ÑrBsW. Observe that a propositional formula Ais satisfied in all the elements ofW(i.e.,l_pw_{q ,}Afor ev-erywPW) iffP , A|„ K. This means that any classical

propositional formulaAcan be represented as the

defeasi-ble conditional A_{|„ K}. Indeed, every conditional of the formA|„ Kisnotdefeasible, and actually represents

clas-sical propositional information. In our example, the state-ment that vertebrate red-blood cells have a nucleus (v_Ñn) will be represented as pv Ñ nq |„ K. Because of this,

we sometimes abuse notation by referring to A _{|„ K} as the propositional formulaA, or using the strict conditional A _$Bto indicateA_^ B_{|„ K}. LetUindicate the set of all the preferential interpretations for our languageL|„, and P1,P2, . . .be subsets ofU. Given a preferential modelP, let|„P be the set of all the conditionals that are satisfied by P (|„P ”def tA|„B P L | P , A|„Bu), while|„P

represents the conditionals satisfied by a set of preferential modelsP(_|„P ”def tA|„B PL|for everyP PP,P , A_|„B_u).

LetBindicate a finite set of defeasible conditionals. The set ofpreferential modelsof _B, preferential interpretations satisfying B, is denoted by JBK. The obvious notion of Tarskian entailment associated with this semantics is known as preferential entailment (Lehmann and Magidor 1992), represented as(pr, whereB (pr A|„BiffJBKÑJtA|„

B_uK. The corresponding closure operator is known as pref-erential closure:PCpBq “ tA|„B | B (pr A|„Bu. We

use the notationK,K1etc. to refer to conditionalknowledge bases closed under preferential entailment. B is preferen-tially inconsistentiff_B(pr J |„ K, andBispreferentially

incoherentw.r.t. a propositional formulaAiffB(prA|„ K.

Two bases are preferentially equivalent iff they have the same preferential models.

It is widely recognised that preferential entailment is too weak to be an appropriate form of entailment for a non-monotonic framework (Kraus, Lehmann, and Magidor 1990, pp. 4, 34). This is primarily because the preferential en-tailment relation itself is monotonic (non-monotonicity oc-curs at the object-level, within defeasible conditionals). At the same time there is sufficient consensus that any accept-able form of non-monotonic entailment will be an exten-sion of preferential entailment (Kraus, Lehmann, and Magi-dor 1990; Lehmann and MagiMagi-dor 1992). Here we consider a large class of closure operations (entailment relations) ex-tending the preferential closure (preferential entailment), the supra-preferential cumulativeclosure operators (Casini and Meyer 2017, Section 3). A closure operatorCp¨q, is supra-preferential cumulativeif it can be defined as follows:A|„ B _PCpBqiff_{P ,}A_|„B, for every preferential model_{P P} spJBKq, wheresis a choice function s.t. (i)spJBKq Ñ JBK; (ii) ifJBK‰ H, thens_pJBKq ‰ H; (iii)s_pJBKq “s_ps_pJBKqq; and (iv) ifs_{pJBKq Ñ JB}1_{K Ñ JBK}, thens_pJB1_{Kq “} s_pJBKq. The closure operators definable using such choice functions are the ones that extend preferential closure, respect consis-tency preservation (if J |„ K R PCpBq, then J |„ K R CpBq), and satisfy cumulativity, that is, for everyB,_B1, if B Ñ B1 Ñ CpBq, thenCpB1_{q “ CpBq. We refer to the}

clo-sure operators satisfying these properties as spc-operators (as supra-preferential cumulative). Most of the prominent non-monotonic closure operators proposed in the preferen-tial framework, e.g. (Kraus, Lehmann, and Magidor 1990; Lehmann and Magidor 1992; Lehmann 1995; Casini and

Straccia 2013), arespc-operators. Our goal is to analyse be-lief change for the class ofspc-operators.

2.3 Previous results

As mentioned in Section 1, belief change in a non-monotonic framework seems superfluous at a first glance, since the non-monotonic machinery usually takes care of ‘readjusting’ the inferences in order to preserve consistency and coherence facing new unexpected evidence. Neverthe-less, that is not always the case, as shown in Example 1. Example 1 Let _A “ ta,m,n,vu, with the propositions

standing for, respectively, “(being) an avian red-blood cell”, “(being) a mammalian red-blood cell”, “(being) a verte-brate red-blood cell”, and “(having) a nucleous”. Assume a knowledge baseB “ tv $n,a $v,m $ v,m $ nu.

Consider the following situations:

1. m _{P CpBq}. But mammalian red-blood cells exist, and we want to enforce such information. In propositional be-lief change we would remove some piece of information, presumably eitherm $ v,v $ n, orm $ n. In the

framework we propose we can resolve the situation by in-troducing defeasibility. In contracting m, we would like

to end up, for example, withB1 “ tv_|„n,a _$ v,m _$ v,m $ nuin which, instead of eliminatingv $ n, we

have just made it defeasible.

2. IfCp¨qis a well-behaved non-monotonic closure operator, we should havea_|„n _{P CpB}1_q, since, with the informa-tion we have, we can treat avian cells as typical verte-brate cells. Now assume we are informed thata_|„ n: B2 “ tv|„n,m$ v,m $ n,a $v,a|„ nu. In this

case, sincea_|„nis a presumptive conclusion drawn by the non-monotonic machinery, the entailment relation it-self takes care of doing away with such a conclusion once faced with conflicting evidence. In this case, the introduc-tion ofa_|„ nshould correspond to a simpleexpansion. 3. We are then informed thata_|„nactually holds, which

di-rectly conflicts witha|„ nin our base. This kind of

con-flict cannot be handled by the non-monotonic machinery, sincea_|„ nis a (trivial) necessary conclusion fromB2. So, we are forced to conclude that avian red-blood cells do not exist ( a). We have two choices: either we are in-terested just in general consistency (not derivingJ |„ K), and in such a case the addition ofa_|„nis just an expan-sion and we conclude that birds do not have red-blood cells; or we are also interested in coherence and perform a revision in order to ‘make room’ fora|„nwithout being

forced to conclude a.

What is shown in the example is that belief change in a non-monotonic framework needs to distinguish between thecertainportion of our belief state (corresponding to the monotonicpart of our reasoning) and theuncertain, defea-sibleportion (thenon-monotonic part): if new information creates a conflict, the dynamics should be different, accord-ing to whether the conflict is with the certain portion of our state (hence we need to operate a non-trivial change) or with its uncertain portion (just expand the information, and the non-monotonic machinery will take care of resolving

the conflict). Hence, a main difference between modelling belief change in a Tarskian or in a non-monotonic frame-work is the one shown in Figure 1: while in the former case we move from a closed theory K, representing our belief

state, to a new closed theoryK1, in the latter, to describe our belief state, we need to distinguish between its monotonic part, a theory K closed under a Tarskian operator, and its non-monotonic part, the information obtained closingK un-der a non-monotonic operatorCp¨q. Also, such a distinction should be preserved at the end of the operation.

Figure 1: Classical vs. Non-monotonic Belief Change Hence, it is essential to our approach that, consider-ing some non-monotonic entailment relation, we identify its “monotonic part”, that is, the biggest monotonic entail-ment contained in it. We call a closure operator Cl_p¨q the monotonic coreof a non-monotonic operatorCp¨qif, for ev-ery B,_B1 we have (i)B Ñ B1 implies Cl_{pBq Ñ} Cl_pB1_q; (ii)ClpBq Ñ CpBq; (iii) for every closure operator Cl1p¨q

satisfying (i) and (ii), Cl1_{pBq Ñ} Cl_pBq. It turns out that the preferential closurePC is the monotonic core of every supra-preferential closure operator (Casini and Meyer 2017, Proposition 1).

Casini and Meyer (2017) provide a characterisation of non-monotonic revision for spc-operators. Preferential ex-pansionis defined asKA`|„B”defPCpKY tA|„Buq.

Basic preferential contraction ´is characterised by the following postulates (where”P refers to preferential equiv-alence): p´1q K´_A|„B“PCpK_A´_|„Bq p´2q K´_A|„BÑK p´3q IfK({prA|„B, thenK´_A|„B“K p´4q If({prA|„B, thenA|„BRK_A´_|„B p´5q IfA|„B ”P A1|„B1, thenK´_A|„B“K´_A1_|„B1 p´6q KÑpK_A´_|„Bq`_A|„B

The class of such operators corresponds to the class of pm-contractions defined over the set of the preferential theories (Casini and Meyer 2017, Theorem 3).

Two kinds of revision operations are introduced for the monotonic corePCp._q, one to preserve consistency, the other aimed at the preservation of coherence. Basic preferential revisions‚for the preservation ofconsistencybehave in such a way that, adding a new conditionalA|„B, we obtain a

new preferential theoryK‚A|„B that containsA|„B and is

logically consistent (_{J |„ K RK}‚

A|„B). This class of

opera-tors is defined by the following postulates: p‚1q K‚_A|„B “PCpK‚_A|„Bq p‚2q K‚_A|„B ÑK`<sub>A|„B</sub> p‚3q IfKY tA|„Bu ({ J |„ Kpr , thenK`_A|„B ÑK‚A|„B p‚4q A|„BPK‚A|„B p‚5q IfA|„B”P A1|„B1, then_K‚_A|„B“_K‚_A1_|„B1 p‚6q IfA|„B({ J |„ Kpr , thenK‚_A|„B ({ J |„ Kpr p‚`q K‚A|„B“ pK‚J|„AÑBq`A|„B

This class of revision operators is characterised using the class of Basic Preferential Contraction operators via the following re-formulation of the Levi Identity (Casini and Meyer 2017, Theorem 4):

K‚A|„B “ pK´_J|„A^ Bq`A|„B (4)

On the other hand,basic preferential revision˝aimed at the preservation of coherence models a change such that, adding a new conditional A _|„ B, we obtain a new pref-erential theory K˝A|„B containing A |„ B and that is

co-herent w.r.t. A (A |„ K R K˝A|„B). This class of

opera-tors is characterised by a set of postulates_p˝1q–p˝6q, where p˝1q,p˝2q,p˝4q,p˝5qare the reformulation for˝of the cor-responding‚-postulates, whereasp˝3qandp˝6qare: p˝3q IfK({prA|„ B, thenK`_A|„BÑK_A˝_|„B

p˝6q IfA|„B({prA|„ K, thenK˝_A|„B({prA|„ K

This class of revision operators is characterised using a super-class of the basic preferential contraction opera-tors, thepreferential withdrawals(satisfying the Postulates p´1q–p´5q) via the following re-formulation of the Levi Identity (Casini and Meyer 2017, Proposition 3):

K˝A|„B :“ pKA´|„ Bq`A|„B (5)

The results above serve as a springboard for the definition of belief change for thespc-operators. Let Cp¨qbe an spc-operator and letK now refer to knowledge bases that are closed underCp¨q. We define revision for the non-monotonic closure operatorCp¨qin terms of its monotonic core. Hence, we distinguish between_Kand its monotonic core (see Fig-ure 1) that we indicate asKPC_;KPC _{is a preferential} the-ory s.t. _K “ CpKPC_{q. The agent must be aware of such} a distinction from the beginning, and our operators allow to preserve it at every step; if we are dealing with theories initially generated by a conditional base B, the distinction would be evident, since we would have K “ CpBq and KPC _{“ PCpBq. The postulates for the class of basic} revi-sions w.r.t.Cp¨qfor the preservation of consistency (Casini and Meyer 2017, Section 6), are:

p˚1q K˚_A|„B “CpK_A˚_|„Bq p˚2q There is a K1 s.t. CpK1q “ CpK˚_A|„Bq and K1 Ñ pKPC_q` A|„B p˚3q If pKPCq`_A|„B ({ J |„ Kpr , then CppKPCq`_A|„Bq Ñ K˚A|„B p˚4q A|„B PK˚_A|„B p˚5q IfA|„B”P A1|„B1, thenK˚_A|„B “K˚_A1_|„B1 p˚6q IfA|„B({ J |„ Kpr , thenJ |„ K RK˚_A|„B

As the following observation shows, revision forCp¨qcan be defined in terms of revision of the monotonic core. Observation 5 (Casini and Meyer 2017, Th. 6)˚is a revi-sion operator satisfyingp˚1q–p˚6qiff there is a preferential revision‚satisfyingp‚1q–p‚6qs.t.K_A˚_|„B“CppKPCq‚_A|„Bq.

The postulates for the revision operations_bfor preserv-ing coherence underspc-operators arepb1q,pb2q,pb4qand pb5q, which are direct reformulations of, respectively,p˚1q, p˚2q,p˚4q, andp˚5q, plus the following two:

pb3q IfKPC({prA|„ B, thenCppKPCq`_A|„BqÑKb_A|„B

pb6q IfA|„B ({prA|„ K, thenKb_A|„B*C A|„ K

And then we obtain the analogue of Observation 5:

Observation 6 (Casini and Meyer 2017, Theorem 7)An operatorbis a revision operator satisfyingpb1q–pb6qiff there is a preferential revision ˝ satisfying p˝1q–p˝6q s.t. KbA|„B “CppKPCq˝A|„Bq.

The latter Observation is valid only for the spc-operators that satisfy an extra property, that is satisfied by most of the prominent entailment relations in the preferential framework (Casini and Meyer 2017, Sect. 6): for everyAand every_K, A|„ K PKiffA|„ K PKPC_.

3 A Semantics for the Basic Operations

In this section we provide a semantic characterisation of the classes of operators presented in Section 2.3. Once we have characterised preferential contraction and expansion, the re-sults for the revision operators will follow immediately from the Observations in the previous section.

As it is usually the case, the semantic characterisation of the expansion operator is straightforward:

K`c

A|„B”def tC|„D|P ,C|„D

for everyP PJKKXJA|„BKu.

whereJKKandJA_|„B_K are the sets of all preferential models ofKandA|„B, respectively.

Proposition 2 For every conditional theory K and every conditionalA_|„B,_KA`<sub>|„</sub>c<sub>B“K</sub>`_A|„B.

The proof is immediate from the fact thatK`A|„cB contains

allthe preferential models satisfyingKandA|„B.

Mov-ing to the class of preferential contractions, its semantic characterisation is quite straightforward; characterisations of basic contraction on the same line have been done also in the propositional case (Grove 1988; Hansson 1999). Let sA|„Brdenote the set of thecounter-modelsofA|„B:

Letc :PpL|„q ˆ pL|„q›ÑPpUqbe a choice function. For everyKand every pair of conditionalsA_|„B,C_|„D,

cpK, A|„Bqis such that: pc1_q If_K({prA|„B, thencpK, A|„Bq “ H; pc2_q IfK(prA|„Band({prA|„B, thenH ‰cpK, A|„ BqÑsA|„Br; pc3_q IfA_|„B_”P C_|„D, then_|„cpK,A|„Bq“|„cpK,C|„Dq; pc4_q If(prA|„B,cpK, A|„Bq “ H.

Notice that, under Proposition 4 below, in Property (c3) above we refer to the conditionals holding in the selected interpretations (|„cpK,A|„Bq), instead of referring to the

se-lected interpretations themselves (c_pK, A_|„B_q), since dis-tinct sets of models can characterise the same set of condi-tionals. Using the functionc, we can define a semantic con-traction operator´cas follows:

Definition 1 Given a knowledge base_K and a conditional

A|„B, for every conditionalC|„D:

C|„DP_KA´_|„c_B iff _P ,C|„Dfor every

P PJKKYc_pK, A_|„B_q

The following representation theorem connects´to´c. Theorem 1 For every basic preferential contraction oper-ator´there is a semantic contraction operator´cs.t. for

everyKand every conditionalA_|„B: K´A|„B“K´

c A|„B

Conversely, for every semantic contraction operator ´c

there is a basic preferential contraction operator´s.t., for everyKand every conditionalA|„B:

K´A|„cB“K´A|„B

It is also possible to give a characterisation of the class ofpreferential withdrawals, i.e., the contraction operations satisfying p´1q–p´5q, but not p´6q(recovery). It is suffi-cient to define a class of choice functionscw_{p¨qthat differ}

fromcp¨qin thatpc4qis dropped andpc2qis reformulated. pcw_{1q if} K({prA|„B, thencpK, A|„Bq “ H; pcw_{2q if} K (pr A|„B and({prA |„B, thencpK, A|„ BqXsA|„Br‰ H. pcw_{3q ifA|„B”P C|„D, then|„} cpK,A|„Bq“|„cpK,C|„Dq.

Proposition 3 For every preferential withdrawal opera-tor ´w _{there is a semantic contraction operator ´c}w _s.t.

for everyKand everyA_|„B: K´A|„wB“K´

cw

A|„B

Conversely, for every semantic contraction operator ´cw

there is a basic preferential contraction operator´w_{s.t. for}

everyKand everyA|„B:

K´A|„cwB“K´

w

A|„B

Once we have representation results for expansion and basic contraction, the analogous results for both classes of preferential revision presented by Casini and Meyer (2017) are straightforward. We just need to consider a semantic difference between the preferential interpretations and the propositional ones: the former ones do not necessarily sat-isfy completetheories, so there are pairs of interpretations P1,P2 s.t.|„P1 Ä|„P2. It is an immediate consequence of the monotonicity ofPC and the representation result by Kraus et al. (1990, Theorem 3), proving that for every pref-erential theory there is a prefpref-erential model characterising it. As a consequence, we have that a preferential theory is not characterised by a unique set of interpretations.

Proposition 4 There are sets of preferential modelsR1,R2 such thatR1‰R2butR1andR2satisfy exactly the same set of conditionals, i.e.,|„R1 “|„R2.

Given this result, we introduce additional notation: for a given a set of interpretationsP, withJP_Kwe denote the set of all models of the conditionals that are satisfied by all in-terpretations inP(JPK”def J|„PK).

We start by addressing the class of operators‚, which add a conditional to a knowledge base while preserving logical consistency. We define the corresponding semantic revision operator as follows: K‚Ac|„B”defpK´_J|„cA^ Bq` c A|„B (6) That is, K‚c A|„B “JpJKKYcpK,J |„A^ BqKXJA|„BK (7)

Proposition 5 For every revision operator_‚ there is a se-mantic revision operator‚cs.t. for everyKandA_|„B:

K‚A|„B“KA‚c|„B

Conversely, for every semantic revision operator‚cthere is a revision operator‚s.t. for everyKand everyA|„B:

K‚c

A|„B“KA‚|„B

Notice that, unlike in the propositional setting, under Proposition 4 we need to apply the closure operation J¨K

on the interpretations selected by the contraction (that is, using JpJKK Y c_pK,_{J |„} A _^ B_qK instead of simply

JKKYcpK,J |„A^ Bq) before intersecting withJA|„BK. Otherwise we could end up with an empty set of inter-pretations, violating Postulatep‚6q. The following example should clarify the issue.

Example 2 Let P2 be built by a single propositional val-uation w, whileP1 is composed by a pair of distinct val-uations w, v, withw _† v. It follows that|„ P1 Ä|„ P2. Let P1 “ tP1u and P2 “ tP1,P2u. P1 ‰ P2, but |„ P1 “|„ P2. Let K “|„ P1 “|„ P2, and assume we want to expandKwithv|„ K. BothP1andP2characterise the same preferential theory_K, andv_{|„ K}is preferentially

consistent withK, since we haveP2satisfying bothK and

v |„ K. However, from the semantical point of view, if we

modelK`A|„BasP1XJA|„BKwe obtain a consistent result

We proceed in a analogous way if we are interested in modelling a revision operation that, while adding A_|„B, preserves not only consistency but also coherence w.r.t. the formulaA. As seen in Section 2.3, the class of operations_˝

was characterised only w.r.t. the class of preferential with-drawals (Casini and Meyer 2017, Proposition 3), not w.r.t. the preferential basic contractions. We reformulate the class of the basic revision operators preserving coherence using semantic contractions: that is, as

K˝Ac|„wB ”def JpJBKYc w

pK, A_|„ B_qKXJA_|„BK (8) Proposition 6 For every revision operator_˝ there is a se-mantic revision operator˝cw_{s.t., for every}

KandA_|„B,

K˝_A|„B“K˝_Ac_|„w_B.

Conversely, for every semantic revision operator˝cw _there

is a revision operator˝s.t., for everyKand everyA_|„B,

K˝Ac|„wB“K˝A|„B.

Also, using this semantic characterisation, it becomes very easy to strengthen the previous representation result (Casini and Meyer 2017, Proposition 3), using the class of preferential basic contractions _´ instead of simply using preferential withdrawals. First, we can prove that we can use the class of choice functionsc instead of the choice

func-tionscw_{, still referring to the same semantic construction:}

KA˝c|„B ”defJpJBKYcpK, A|„ BqKXJA|„BK (9)

Theorem 2 For every revision operator˝there is a seman-tic revision operator˝cs.t., for everyKandA|„B,

K˝_A|„B“K˝c A|„B.

Conversely, for every semantic revision operator˝cthere is a revision operator˝s.t., for everyKand everyA|„B,

K˝c

A|„B“K˝A|„B.

An easy consequence of Theorems 1 and 2 is a strenghten-ing of Proposition 3 by Casini and Meyer (2017), usstrenghten-ing ba-sic preferential contractions (satisfyingp´1q–p´6q) instead of preferential withdrawals (p´1q–p´5q).

Theorem 3 For every revision operator ˝ there is a pref-erential basic contraction operator´s.t., for everyKand everyA|„B,

K˝A|„B “ pK´A|„ Bq`A|„B

Vice versa, for every preferential basic contraction opera-tor ´ there is a revision operator ˝ s.t., for every K and everyA|„B,

K˝_{A|„B “ p}K´_{A|„ Bq}`_A|„B

Regarding the class ofspc-closures, we have seen in Sec-tion 2.2 that each closure Cp¨q corresponds on the seman-tic side to a choice function sC : PpUq ›Ñ PpUq s.t.

A _|„ B _{P CpKq} iff A _|„ B is satisfied by every prefer-ential interpretation in sCpJKKq. Given a theory K, closed

under Cp¨q, and its monotonic core KPC_{, the class of} re-vision operators ˚ for consistency preservation is charac-terised using the class of preferential revision operators ‚ as _K˚

A|„B “ CppKPCq‚A|„Bq(see Observation 5). Givenp6q

and Proposition 5, it is immediate to conclude that the class of non-monotonic revision operators_˚for aspc-closure_Cp¨q can be characterised via the class of preferential contractions ´cas K˚Ac|„B”defCpppKPCq´ c J|„A^ Bq` c A|„Bq (10) That is,_K˚c A|„Bis equal to: sCpJpJKPCKYcpK,J |„A^ BqKXJA|„BKq (11) We proceed analogously for the class of non-monotonic revisions b to preserve coherence, characterised via ˝ as KbA|„B “CppKPCq˝A|„Bq(see Observation 6).

Given p9q and Theorem 2, the class of revision opera-tors bfor an spc-closureCp¨qcan be characterised via the class of preferential contractions´cas

KbA|„cB”defCpppKPCq´ c A|„ Bq` c A|„Bq (12) That is, K˚c A|„B“sCpJpJKPCKYcpK, A|„ BqqKXJA|„BKq (13)

4 Beyond the Basic Postulates

We move to explore possible classes of interesting operators beyond the basic forms of contraction and revision. As seen in Section 2, in the AGM approach there are two extra pos-tulates for contraction,_p´7qandp´8q, and for revision,p˚7q andp˚8q, that can be associated with an ordering between the formulas (entrenchment) and an order between the interpre-tations.p´7qandp´8qdefine a way of preferring some traction operations to others, in particular favouring the con-traction operators that respect the relevance of certain pieces of information, corresponding to more entrenched formulas and/or more preferred interpretations. We aim at an analo-gous characterisation in the conditional framework of oper-ations respecting the relevance of the information.

Since Postulates p´7qandp´8q, and p˚7qandp˚8quse conjunction, their translation into the conditional framework is not immediate. G¨ardenfors’ entrenchment contraction (1) uses disjunction, but Proposition 1 gives us an alternative characterisation in terms of partial-meet contraction. Kat-suno and Mendelson’s (1991) semantic characterisation (see Section 2 here) looks more immediately implementable in the conditional framework as a ranking of preferential mod-els, but we have to take under consideration Proposition 4 in the definition of the ranking.

Given the difficulties in reformulating the postulatesp´7q andp´8qin the conditional framework, we leave them aside for the moment. Still, our goal here is to define classes of contraction and revision operators that constrain the basic operators in a form that is analogous to what we obtain in the propositional case when adding_p´7qandp´8q. Here we propose to bypass the postulates and investigate the corre-spondence between an entrenchment relation over the con-ditionals and a preference relation over the preferential in-terpretations, reformulating in the conditional environment

G¨ardenfors’ entrenchment and Katsuno and Mendelson’s ranking of interpretations. In this way, we can give a char-acterisation of a class of contraction operators constraining the class of basic contraction operators in a similar way as postulatesp´7qandp´8qdo in the propositional framework.

Conditional Entrenchment. We define in the conditional framework an entrenchment relation•e:L|„ˆL|„satisfying

the following constraints:

PE1 ifC|„D §e E |„F andE|„F §e G|„H, then

C_|„D_§eG|„H (Transitivity)

PE2 iftC1|„D1, . . . , Cn|„Dnu (pr E|„F , then there

is ani,1_§i_§n, s.t.Ci|„Di§eE|„F (Dominance)

PE3 EitherC |„ D §e E |„ F or E |„ F §e C |„ D

(Connectivity)

PE4 IfKis consistent, thenC|„DRKiffC|„D§eE|„

Ffor everyE_|„F (Minimality)

PE5 IfE|„F §eC|„Dfor everyE|„F, then(prC|„

D (Maximality)

LetA |„ B°e ”def t_C |„ _D | _A |„ _B †

e C |„ Du.

By Compactness of PCp¨q and Dominance,A _|„ B°e “

PCpA|„B°eq, that is,_A|„_B°eis a preferential theory.

For lack of expressivity, we substitute Conjunctiveness withConnectivity. In G¨ardenfors’ entrenchment the latter is implied by (E1)-(E3) (G¨ardenfors 1988, Lemma 4.20). Con-nectivity, (E1) and the property

IfC§eDandC§eE, thenC§eD^E

together implyConjunctiveness(G¨ardenfors 1988, Lemma 4.21). A restricted forms of the above property and of the Conjunctivenessproperty are derivable fromPE1-PE3. Proposition 7 The following two properties are a con-sequence of PE1-PE2. For every tuple of propositions

A, B, C, D, E:

(a) IfA_|„B _§eC|„DandA|„B §eC|„E, thenA|„

B§eC|„D^E;

(b) EitherA_|„B_§eA|„B^CorA|„C§eA|„B^C.

LetK°e

C|„D ”def tE |„ F | C |„ D †e E |„ Fuand

KóC|„Dbe the set of the preferred remainder sets w.r.t.

the entailment relation, that is:

KóA|„B”deftK1PKÓA|„B|K°e

A|„BÑK1u

We define the contraction´eusing preferential

entrench-ment as follows:

K´e A|„B”def

£

K_óA|„B (14)

This entrenchment-based conditional contraction´eis a reformulation in the conditional framework of G¨ardenfors’ entrenchment-based propositional contraction _´e: K°e

A|„B

corresponds to Rott’s contractionp3qin the conditional en-vironment, and consequently_p14_qis a conditional reformu-lation of the propositional entrenched-based contraction´e, following the version used in Proposition1.

Semantic Characterisation. On the semantic side, we in-troduce a ranking of preferential interpretations in the spirit

of Katsuno and Mendelzon (1991). With Ubeing the set of all the preferential interpretation for language L|„, let

r : U _{ބ N} be a function associating with every model a natural number, and let Li indicate the i-th layer of the

ranking (Li ”def tP P U | rpPq “ iu). Let Li be the

set of conditionals satisfied by all the interpretations inLi.

Given a set of conditionals B and an interpretation P, let BP _{”deftC|„DPK|P ,C|„Du. We want the ranking} to satisfy the following constraints:

(r1) @i_PNandi_°0, ifLi‰ H, thenLi´1‰ H; (r2) @P PU, if_{P ,}L0, thenP PL0; (r3) @P,_P1,_P˚ _PUandi_°0, if_{P P}LiandP1 PLi´1, then ifL0P ˚ ÖL0PXL0P 1

, andP˚RLjfor anyj†i,

thenP˚PLi.

Let us considerL0as our knowledge baseK, and let, for everyP,P1 P U, P §r P1 iffrpPq § rpP1q. It is very

easy to check that_§ris a faithful assignment (as described

in Sect. 2.1). That is, (1) if _{P ,} L0 andP1 , L0, then P »rP1; and (2) ifP ,L0andP1,{L0, thenP †rP1.

(r3)is necessary to deal with the peculiarity of preferen-tial models, as indicated by Proposition 4, and it is equiva-lent to the combination of the following two properties: (r3’) @P,_P˚ _PUandi_°0, ifP PLi,L0P ÑL0P

˚

, and P˚RLjfor anyj†i, thenP˚PLi.

(r3”) @P,_P1 _P Uand i _° 0, if P P Li andR1 P Lj,

j_§i, then there is an interpretationP˚PLis.t.L0P

˚

“ L0PXL0P

1

.

Proposition 8 The conditionpr3qis equivalent to the com-bination of the conditions_pr31_qand_pr32_q.

Given a set of interpretationsP _ÑU, a rankingr, and a

preferentially closed set of conditionalsK, letminrpPq “

tP PPXLi|PXLi‰ Hand@j†i,PXLj“ Huand

maxKpPq “ tP PP| EP1PPs.t.KP

1

ÅKPu.

We define the choice functioncrover this ranking as:

crpL0, C|„Dq ”defmax

L0 pmin

r psC|„Drqq

That is, given a knowledge base _K and a conditional

C|„Dto be contracted, we first pick the preferred models

that do not satisfyC_|„D(minrpsC|„Dr)). Between such

models, we choose only the ones that satisfy more condition-als fromL0(max_L0pminrpsC|„Drqq); the use ofmax_L0p.q

is due to the property of preferential interpretations indicated in Proposition 4. Having defined the selection functioncr,

we can define contraction in the usual way.

C_|„D_PK´cr

A|„B iffP ,C|„Dfor every

P P rKs YcrpK, A|„Bq.

Now we can prove the correspondence between the two classes of contraction operations we have defined using the entrenchment and semantic rankings, respectively.

Theorem 4 For every entrenchment contraction operator ´ethere is a semantic contraction operator´crs.t., for

ev-ery KBKand every conditionalA|„B,

K´e

A|„B“K´ cr

A|„B.

Conversely, for every semantic contraction operator _´cr

there is an entrenchment contraction operator ´e s.t., for every KBKand every conditionalA|„B,

K´cr

A|„B“K´ e A|„B.

Our aim was for a full characterisation of AGM-like op-erations in the framework of preferential conditional logic. Ideally, we would also have provided postulates correspond-ing top´7q, p´8q,p˚7q, and p˚8q, but since they refer to the conjunction of formulae, their translation in a condi-tional language is not immediate, and we have omitted that. We have considered two alternative and equivalent charac-terizations of the class of contraction operations satisfying p´1q ´ p´8q, using Gardenfors’ entrenchment and Katsuno and Mendelson’s rankings, and we have reformulated them accordingly for the conditional framework, showing that the correspondence is preserved, as Theorem 4 proves.

We can easily give a semantic characterisation of the revi-sion operators, using the choice functionscrdefined earlier

in this section, instead of the choice functionsc(which were defined in Section 3). First the ones for preferential revision. The only constraint is that, starting from a preferential the-oryK, the rankingrmust be defined withL0“JKK.

JK‚cr A|„BK“JpJKKYcrpK,J |„A^ BqKXJA|„BK That is, JK‚cr A|„BK“JpJKKYmax_K pmin_r psJ |„A^ BrqqqKXJA|„BK which implies JK‚cr A|„BK“Jmax_K pmin_r psJ |„A^ BrqqKXJA|„BK

Proceeding analogously for˝, we obtain

JK˝cr

A|„BK“JpJKKYmax

K pminr psA|„ BrqqqKXJA|„BK

Given a theoryKclosed under aspc-closureC, the choice functionsC associated atC, and the monotonic coreKPC, using the two last equations we can define the two corre-sponding classes of non-monotonic revision operators. The two classes of operators˚andbcan then be defined as fol-lows.K˚cr

A|„Bcorresponds to the conditionals satisfied by the

set of interpretations

sCpJmax

KPCpminr psJ |„A^ BrqqKXJA|„BKq

Kbcr

A|„Bcorresponds to the conditionals satisfies by the set

s_CpJpJKKYmax

KPCpminr psA|„ BrqqqKXJA|„BKq

Since the contraction is to be defined onKPC_{, the only} constraint is for the rankingrto be built withL0“JKPCK.

5 Conclusions

This paper, in combination with previous work by Casini and Meyer (2017), provide the basis for the definition of belief change in a (preferential) non-monotonic framework. Build-ing on the characterisation in terms of entrenchment and the ranked semantics of the AGM approach, we define and char-acterise preferential expansion, contraction and revision (for both consistency and coherence preservation) on the mono-tonic core of the class of thespc-operators.

Most of the previous works dealing with the revision of conditionals, such as those by Kern-Isberner (1999) and Wobcke (1995), is connected to the revision of subjunctive conditionals, which are used to represent the revision poli-cies themselves. Here, instead, we do not consider the con-ditionals as linked to the revision policies, and leave their interpretation quite open (see Section 1). Other recent con-tributions to revision in a non-monotonic framework have come from Hunter (2016), dealing with highly implausible conditionals, and Delgrande and others (2013), that analysed belief change in the non-monotonic framework of ASP.

Regarding future developments, there are some obvious steps, such as modelling iterated revision and some forms of base revisions. An initial step in this direction has been made by Casini and Meyer (2016). In recent years there has been particular interest in preferential non-monotonic rea-soning in the framework of Description Logics (Britz, Hei-dema, and Meyer 2008; Britz, Meyer, and Varzinczak 2011; Casini and Straccia 2010; 2013; Casini et al. 2014; Giordano et al. 2013; 2015; Lukasiewicz 2008), and in such a frame-work the notion of coherence, as we use it here, is very rel-evant. Hence a major goal is to reformulate the present ap-proach for Description logics.

An important aspect of belief change in non-monotonic frameworks is contraction. The present paper and the pous work by Casini and Meyer (2017) are focused on revi-sion for non-monotonic theories, and we have reduced it to revision of its monotonic core, that in turn can be analysed in terms of contraction and expansion of the monotonic core. Hence, the only form of contraction that we have analysed up to this point is the one related to the monotonic core, while we have still to develop an analysis of the contraction of a conditional from a non-monotonic closure. The inter-esting aspect is that if we want to eliminate some piece of defeasible information, we have two possibilities: either we contract information from the monotonic core (as our con-traction operations do), or we can actuallyaddinformation to it. Consider Example 1, Step 2, wherea|„n P CpB1q.

Assume we are just interested in eliminatinga_|„nfrom our

conclusions. In a non-monotonic framework we can proceed in two ways. Either we eliminate some pieces of information fromB1, preventing the derivation ofa_|„n(e.g., eliminating

v|„nora$v), or we can add some information toBthat conflicts witha_|„n(e.g., addinga_|„ n, as it happens in the example). To the best of our knowledge, the only attempt to at modelling this kind of contraction (in the KLM frame-work) is by Booth and Paris (1998), in which they introduce negated conditionalsA |⇢ B into the language. However,

a proper analysis of this kind of contraction operation still needs to be done.

Acknowledgments

G.C. and T.M. have received funding from the EU Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agr. No. 690974 (MIREL). The work of T.M. has been supported in part by the National Research Foundation of South Africa (grant No. UID 98019).

E.F. was supported by FCT through project UID/CEC/04516/2013 (NOVA LINCS) and project PTDC/CCI-COM/30990/2017.

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