Perfect agreements

set preferred to K^f implies 0 V y/, A " h 0 v y / . But also 0—>y/ e K\ by K'e KA(j)C Kl(j) and yr eK~K'.^]5 7/ence K' HI/A, contradicting K'eKA[<j),y/].

If there is no prioritization, i.e., i f <l is empty and -< is C , this case is o f limited relevance. Then the epistemic entrenchment rela-tion generated by K is nearly empty i f AT is a theory, because then K'eK Ay/= ATI y^ and K"e KA<j) = Kl 0 imply K'<tK".^x(> So in this case, <p<y/ according to Definition 11 can hold only i f either y/e K and 0 £ K, or h y/ and H 0. Similarly, for theories K without prioritization, (I) is only possible i f either y/eK and <j)Ayf<£ K, or hy/, and (III) is only possible i f either (f)Vy/eK and 0 g A^, or

h 0 V y / . This corresponds to a well-known trivialization result of A l c h o u r r o n and M a k i n s o n [1, Observation 2.1] for full meet con-tractions o f theories. However, i f a theory is genuinely prioritized, the construction offered is perfectly reasonable.

6.2.2. Nebel's blown-up contractions. The second case in point is when a meet contraction is supplemented with a mechanism to enforce the recovery postulate. This is basically the suggestion of Nebel [20]:

DEFINITION 12. For any prioritized base H for K, the blown-up meet contraction — is given by

y/eK^<j> iff ( A ^ , 0) U{ 0 — / / } hy/, where A > , 0 is the meet contraction determined by H.

The set Rec- { 0 — : X ^eH) ^{I S a} recovery ticket which allows one to " u n d o " a base contraction with respect to 0 . It is easy to check that on Definition 12, K - (K— 0 ) + 0 , for every 0 in K. But since clearly 0 implies every element o f Rec, and Rec in turn implies 0—^y/((or every y / i n K), we find that y/e AT—0 according to Definition 12 iff 0 V y^G AT—0 according to Definition 2. C o m -paring (II) with (III) and realizing that / / A [ 0 , 0 V y / ] ~ / / A ( 0 V y/), we see that this is the same as y/e AT—0 according to Definitions 5

1 5 See footnote 5.

1 6 Unless K" equals K. Cf. footnote 5.

and 11. So Nebel's variation on meet contractions is captured ex-actly by large EE-contractions based on the relation < generated b y / / .

6.2.3. Revisions based on the Levi identity. Thirdly, the corre-spondence is perfect i f a contraction is only an intermediate for a revision constructed with the help o f the Levi identity. Since, by Levi and the deduction theorem, y/is in K* <p iff <p—>t//is in K^^tp, we have to check K— ~^<p only for sentences o f the form 0 — • y/. But clearly, for every relation < satisfying (EE2¹), - - 0 < (<t>-^y/) is equivalent to - ^ 0 < - > 0 V ( 0 — • (//), so Definitions 5 and 6 are equiva-lent for sentences o f the form 0 — ^ i n K — Hence, by the Inter-polation L e m m a , both large and small EE-revisions are identical with the meet revision determined by / / . '⁷

7. Conclusion

The aim o f this paper has not been to present exciting new technical results. It is rather meant to provide one more illustration for the versatility o f the concept of epistemic entrenchment in the rational reconstruction of belief change. T o the best of our knowledge, we have offered one o f the first suggestions how to apply epistemic entrenchment to belief states ("bases") which are not supposed to be logically closed (compare Dubois and Prade [4]) but which may be partitioned into various levels of priority. We also hope to have furthered the intuitive understanding of epistemic entrenchment and its relation to multiple contraction.

Our starting point has been a fixed prioritized belief base / / . It contains the reasoner's explicit beliefs o f various strengths and generates a belief set K~Cn(H). O u r concern is "syntax-based"

belief change, or belief change determined by belief bases, and we assume that it is the syntactical structure and the epistemic weight-ing of the explicit beliefs that governs the changes o f K.

We have given a reformulation o f meet contractions determined by a prioritized belief base as extended EE-contractions: Defini-tion 2 is equivalent to the combinaDefini-tion of DefiniDefini-tions 9 and 10.

Presumably because of the problems with the recovery postulate, NebeEs atten-tion switches from contracatten-tions in [20] to revisions in [21].

This representation depends on an extension o f the concept of epistemic entrenchment to sets o f sentences ("bunch ment"). W e elaborated on the basic idea of epistemic entrench-ment as comparative retractability by giving it two different read-ings. The usual "competitive" interpretation was distinguished from what we called the "minimal change interpretation" o f the phrase 'y/is harder to discard than </>'.

W e proposed a method o f extracting an epistemic entrenchment relation < from a prioritized belief base H. Discovering that Defi-nition 11 is equivalent to the combination o f DefiDefi-nitions 3 and 8, we observed a confluence of the two interpretations o f epistemic entrenchment (the "Coincidence Lemma"). It was demonstrated that for every meet contraction determined by a prioritized base H one can specify upper and lower bounds in the form o f large and small EE-contractions based on the relation < generated by H (the

"Interpolation Lemma"). In a number of interesting cases, this re-sult can be sharpened to an identity. Nebel [21, Theorem 9] proved that an epistemic entrenchment relation < can be applied as i f it were a belief base prioritization <l. W e have offered an answer to the reverse question: Given some belief base prioritized by <l, is it possible to exhibit a natural relation o f entrenchment < such that the EE-contraction with respect to < leads to the same results as the meet contraction with respect to <3?

Since the publication of Gardenfors's Knowledge in Flux, rela-tions o f epistemic entrenchment have been known to be interdefinable with belief contractions. F o r theory change by sin-gletons, the following transitions are standard in the literature:

(jxy/ iff y/eK—(QAifland <l> 2 K—(<l>Ay/) y/e K — <p iff y/e K, and 0 < </> V y /o r h 0

T o my mind, there is no denying that these bridge principles are the pivotal points of an illuminating and well-developed theory of belief change [7, 9, 25, 26, 27]. In our particular framework where everything starts from a prioritized belief base, one can prove that the principles hold, but only approximately in some cases. In order to bring this out as clear as possible, we adopt the following

notation. F o r any prioritized belief base / / , let C{H) and 2T(//)be the contraction operation and the entrenchment relation derived from H by Definitions 2 and 11 respectively. Moreover, i f < is an entrenchment relation and — is a contraction function, let C ' ( < ) and !(—) denote the contraction function and the entrenchment relation obtained by the above bridge principles (which are identi-cal with our Definitions 5 and 7). Finally, let C (<) denote the small EE-contraction with respect to <.

We are now i n a position to formulate the following summary of relationships:

OBSERVATION

10. Let H be a prioritized belief set and let — - C(H) and< - T(H). Then

(a) Z ( - ) = <,

(b) C " ( < ) C - C C W

(c) W ) ) C - C C m

(d) £ ( C "(<)) = < = £ ( C + ( < ) ) .

Proof (a) is the Coincidence Lemma, and (b) is the Interpolation Lemma; (c) follows from (a) and (b): C~(X(^)) = C (T(C(If))) = (by (a)) C{T(H)) Q (by (b)) C{H) C (by (b) C '(Tiff)) £ (by (a)) C⁺(1(C(H))) = C\T(^))\ for (d), let < ' - T(C (<)) and < " = 1(C ⁺(<)). Then, by definition, 0 < ' y/ iff⁷ <f> A y < y/ and 0 A y/ < 0 . Similarly, 0 < " yf iff 0A I / A < ( 0A I / A ) V y/ and 0AI/A i t ( 0 A ^ ) V 0 .

Both of these conditions reduce to 0<y, by Observation 5 and ( E E l ) - ( E E 3 ' ) . Q . E . D .

This shows that the bridge principles o f Gardenfors and Makinson get the transitions quite right, provided that both the contraction operation and the entrenchment relation are deter-mined by a fixed belief base. The agreement is perfect in the case of entrenchment relations and only approximate in the case o f contraction operations. Parts (c) and (d) to some extent reproduce the nice results o f Gardenfors and M a k i n s o n [9, Corollary 6].

However, it is hard to make intuitive sense of the occurrences of

For any two contraction functions —, and —, over K< —, C —i s of course short for "K—{<p C K—,<j> for every <jT.

fcA' and ^fcV' in the above bridge principles. This is why I suggest a more transparent way to think o f the interdefinability between epistemic entrenchment and belief change.

<t><yf iff ^ € /L —(0, y/) and (0, y^) ( D e f i n i t i o n s ) M ^ r base specialization: i f ~ is the meet contraction de-termined by a prioritized base H, then, by Observations 4 and 6, 0 < y/ is definable by HA y/ • « / / A 0

Singleton reformulation: in so far as K~{<p,yf) is identical with K^-(<pAy/), 0 < y if and only i f y/eK— ( 0 A y / ) and 0 £ K^(<j)A yf)

y/eK—<l> iff and [ 0 ] ^ [ 0 , y / ] or h 0 (Definition 10) Meet base specialization: i f is the bunch entrenchment generated by a prioritized base H, then, by Observation 3, meet contraction determined by H coincides with E E E -contraction, and yfeK—cpis definable by HA[(p,yf]^<

/ / A [ 0 ]

Singleton interpolation: in so far as <p<yf implies [0]<^

[0, yf], and this in turn implies 0 < 0V I / A, large and small EE-contractions can serve as upper and lower bounds of EEE-contractions (Observation 9)

Our deviation from the standard account is clear. We invoke sets with two elements as arguments for contraction operations and entrenchment relations. M o r e specifically, we replace, in the direction from belief change to epistemic entrenchment, the con-tractions with respect to conjunctions by pick concon-tractions, and in the direction from epistemic entrenchment to belief change, the en-trenchments o f disjunctions by bunch enen-trenchments.

W h a t is the reward for this exercise? First and foremost, we get a better understanding of the relevant interrelations. They some-times happen to reduce to the standard definitions. But what is really meant by the latter is, I submit, precisely what is made ex-plicit by the new definitions. In one direction, I should think there is virtually no difference: A T — ( 0 A y / ) appears to be intuitively

indistinguishable from K—((j),y/}. In the other direction, however, it is only the restricted context o f theory change by singletons that makes our new definition reduce to the old one: [01^10, y] may — and must!—then be identified with 0 < 0 V y.

Secondly, we manage without reference to any particular con-nective of the object language. Thus the theory of epistemic en-trenchment becomes applicable to systems using a severely re-stricted language. F o r instance, we can speak of the entrenchment of the nodes in inheritance nets or reason maintenance systems (also called "truth maintenance systems"). There ought to be a corresponding connective-free formulation o f the so-called Gardenfors postulates for contraction operations [7, Section 3.4].

The obvious suggestion is to replace occurrences of "K — (0Ay/)' by 'K—^yf. The elimination o f connectives, however, works only for belief contractions. Belief revisions constructed according to the Levi identity make use of negations, and there does not seem to be a straightforward way to avoid this.

A t last, we should like to give two warnings. The connective-free formulation o f the theory o f epistemic entrenchment relations and theory contractions is only a by-product o f this paper, slightly improving on the presentation in [27]. It is not necessary for the analysis o f syntax-based belief change which turns essentially on the syntactical structure o f the items in a belief base. There is no immediate transfer o f insights from belief base update to updates in inheritance networks or reason maintenance systems ( R M S s ) with their unstructured "nodes". It may be expedient for some purposes to identify R M S "justifications" with H o r n clauses. But this certainly does not suffice for nonmonotonic systems. O u r Cn is supposed to be monotonic.

Multiple contraction and extended epistemic entrenchment have been found to be an appropriate means for analyzing base traction. However—and this is the second warning—, the con-cepts of multiple contraction and extended epistemic entrench-ment themselves, cut loose from the special context of contraction functions determined by prioritized belief bases, are still very much in need o f a thoroughgoing analysis. This is evidently beyond the scope of the present paper.

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Received on 18 October 1992

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