Determining Explanations using Transmutations

1 Introduction

Information systems use information theoretic descrip­ tions to capture their view of the world. These descrip­ tions are generally incomplete and evolve over time. An intelligent information system requires a mechanism for self-modification when new information about the world is acquired.

The AGM paradigm [Alchourr6n et al. , 1985; Gardenfors, 1988; Gardenfors and Makinson, 1988] has become a standard framework for modeling information change. In particular, it provides a mechanism for the revision and contraction of information. Within the AGM paradigm, the family of revision functions and the family of con­ traction functions are constrained by rationality postu­ lates. The logical properties of a body of information are not strong enough to uniquely determine a revi­ sion or contraction function. Therefore, the principal constructions for these functions rely on some form of underlying preference relation, such as a family of se­ lection functions [Alchourr6n et al., 1985], a system of spheres [Grove, 1988], a nice preorder on models [Kat­ suno and Mendelzon, 1992; Peppas and Williams, 1995],

Maurice Pagnucco

Knowledge Systems Group Basser Department of Computer Science University of Sydney, NSW 2006 Australia

Brailey Sims

Banach Astronautics Group Department of Mathematics University of Newcastle, NSW 2308 Australia ·

or an epistemic entrenchment ordering [Gardenfors and Makinson, 1988] . In this light we consider an informa­ tion system to be composed of a set of information to­ gether with a preference relation.

We refer to the process of changing the underlying preference relation of an information system as a trans­ mutation. Based on observations made by Gardenfors [1988, p72l, Williams [1994a] generalized the conditional­ Izations of Spohn [1988] to· more general transmutations. Spohn represents an information system using ordinal conditional functions, which map possible worlds to ordi­ nals, and argues that conditionalization, which is a con­ structive method for modifying an ordinal conditional function, is a reasonable procedure. Williams explored conditionalization further and examined an alternative construction; an adjustment. Both conditionalizations and adjustments are transmutations. Intuitively, condi­ tionalization involves a relative minimal change neces­ sary to effect the revision or contraction, whilst adjust­ ment involves an absolute minimal change.

Spohn characterizes the notion of reason for in [1983] and shows how it can be captured within his framework in [1983; 1988].

The objective of this paper is, firstly, to recast Spohn's notion of reason for in a more general setting using transmutations rather than conditionalizations, and secondly, to extend this approach to determine most plausible explanations. Some preliminary work in this direction using a theory base representation can be found in [Williams, 1994b]. We also provide an intuitively appealing and transparent relationship between explanation based on abduction and Spohnian reasons based on adjustments. Finally and quite surprisingly, we find explicit conditions which capture the relationship between Spohnian reasons and the various forms of explanation identified by Boutilier and Becher [1995].

In section 2, we describe entrenchment rankings, while in section 3 we define transmutations of entrenchment rankings, and describe a particular type of transmuta­ tion, namely an adjustment. In section 4, we outline Spohn's notion of reason for, and provide conditions which allow this notion to be defined using both arbi­ trary transmutations and adjustments. In the spirit of Spohn we provide a notion of most plausible explanation. In section 5 we will illustrate some of our definitions and results with examples. In section 6 we compare our no­ tion of explanation with one based upon the process of abductive inference. We discuss related work in section

7, and summarise our position in section 8.

We begin with some technical preliminaries. Let .C denote a language which contains a complete set of Boolean connectives. We will denote sentences in .C by lower case Greek letters. We assume .C is governed by a logic that is identified with its consequence relation 1- which is assumed to satisfy the following conditions (Gardenfors, 1988]: (a) if a- is a truth­ functional tautology, then 1- a-, (b) if 1- a- --+ f3 and 1- a-, then 1- f3 (modus ponens), (c) 1- is consistent, that is, lf.l, where ..1. denotes the inconsistent theory, (d) 1- satisfies the deduction theorem, and (e) 1- is compact.

The set of all logical consequences of a set T s; .C, that is {a- : T 1- a-}, is denoted by Cn(T). A theory of .C is any subset of .C closed under Cn. We let .c~ denote the set of all consistent nontautological sentences in .C. That is, the set of contingent sentences. Finally, a well­ ranked preorder on a set r is a preorder such that every nonempty subset of r has a minimal member.

2 Entrenchment Rankings

Spohn r1988] introduced ordinal conditional functions, while Williams (1994a] introduced related structures, namely entrenchment rankings (defined below). Essen­ tially, ordinal conditional functions take possible worlds to ordinals, whereas entrenchment rankings take sen­ tences to ordinals. Both define a ranking of the respec­ tive domains which provides a response schema for all "possible consistent information" (Spohn, 1988].

The higher the ordinal assigned to a sentence by an entrenchment ranking the more firmly held it is. Throughout the remainder of this paper it will be understood that 0 is an ordinal chosen to be sufficiently large for the purpose of the discussion.

Definition:An entrenchment ranking is a function E from the sentences in .C into the class of ordinals such that the following conditions are satisfied.

(ER1) For all a-, f3 E .C, if a- 1- {3, then E(a-) ~ E(/3).

(ER2) For all a-, f3 E .C, E(a-) ~ E(a- 1\ {3) or E(/3) ~ E( a- 1\ {3).

(ER3) 1- a- iff E(a-) = 0.

(ER4) If a- is inconsistent, then E(a-) = 0.

An entrenchment ranking represents a well-ranked epistemic entrenchment ordering (Gardenfors, 1988; Gardenfors and Makinson, 19881 where the higher the ordinal assigned to a sentence tbe more entrenched it is. The ordinal assignment can be used and viewed in two ways: (i) qualitatively, that is as a specification of a relative ordering of sentences, or (ii) quantitatively, that is as an assignment of intrinsic utility for each sentence. For a discussion of various assignment scales see (Gardenfors and Makinson, 1994].

Entrenchment rankings are information systems; a set of information together with a (necessarily well­ ordered) preference relation. We denote the family of all entrenchment rankings by £. An information set represented by an entrenchment ranking is the set of sentences whose degree of acceptance is greater than zero. Formally we have the following definition (Williams, 1994a].

Definition: We define the information set represented by E E £ to be {a- E .C : E(a-) > 0}, and denote it by set(E).

We refer to information whose degree of acceptance is zero as the non-beliefs, and information with a degree of acceptance greater than zero as the beliefs. We note that set(E) is a theory, and hence the set of beliefs are closed w1th respect to logical consequence.

3 Transmutations

Peppas (1993] introduced well-behaved revision functions and provided a construction using a well-ordered system of spheres (Grove, 1988] . Williams (1994a] gives representation theorems for entrenchment rankings and well-behaved change operators. In particular, she provided conditions which characterize both well­ behaved contraction and well-behaved revision operators using an entrenchment ranking, and conversely. An extra postulate is required for the change operator to be constructed from a well-ranked preference relation.

The informational input for AGM contraction and revision functions for theories is a sentence alone. In contrast, when modifying an entrenchment ranking we require both a sentence and an ordinal. Consequently, the informational input for transmutations is a sentence a- and an ordinal i. The interpretation being rGardenfors, 19881 that a- represents the information to be accepted by the information system, and i is the degree of firmness with which this information is to be incorporated into the transmuted information system. We define a transmutation schema (Williams, 1994a] for entrenchment rankings below.

Definition: We define a transmutation schema for entrenchment rankings, *, to be a function from £ x .c~ x 0 to £, where 0 is an ordinal, such that (E, a-, i) 1-+ E* (a-, i) satisfies:

For a continl?ent sentence a- and an ordinal i, we say E*(a-, i) is an (a-, i)-transmutation of E. It was shown in {Wifliams, 1994a], that according to the definition above, E*(a-, i) is an entrenchment ranking where a­ is assigned the degree of acceptance i, and if i is greater than zero then set(E*(a-, i)) is the well-behaved revision (set(E))~ constructed in the obvious way from the epistemic entrenchment ordering derived from the relative ordering given by E. Similarly, if i is zero then set(E*(a-, i)) is the well-behaved contraction (set(E));;.

Adjustments (defined below) were introduced in (Williams, 1994a]. They are transmutations which adopt an absolute measure under the the principle of minimal change. In particular, an entrenchment ranking is changed or disturbed, in an absolute sense, as little as necessary to give a- the degree of acceptance i. That is, each sentence is reassigned an ordinal as close as possible to its previous ordinal assignment such that the resultant structure is an entrenchment ranking. An adjustment is based on intuition similar to that used by Boutilier (19931 to specify his natural revision; namely the principle ot minimal change. According to Boutilier "when certain beliefs must be given up, it seems natural to try to keep not only important beliefs, but as much of the ordering as possible".

4 Explanation

Spohn [1983] gave the following interpretation of a being a reason for ,8:

(1?.1) a is a reason for .B iff raising the epistemic rank of a would raise the epistemic rank of ,8.

Gii.rdenfors [1990] translated Spohnian reasons into the AGM paradigm. However, in contrast to the translation we give later based on transmutations his translation based on revision and contraction alo~e is somewhat artificial since the epistemic rank after a revision or contraction is undetermined. Essentially his interpretation is that a is a reason for .B if and only if .B is not retained in the contraction of a . In the same paper Gii.rdenfors identifies two problems with this translation. Firstly he demonstrates that difficulties with. the circularity of reasons arise, and secondly that multiple reasons for .B cannot be modeled satisfactorily. These problems are discussed further by Olsson [1994] where an alternative solution to ours can be found, based on a comparative measure of epistemic entrenchment orderings for different theories. We note that the solution we present based on transmutations does not suffer from either of these problems.

Before rec~ting (1?.1) using transmutations, some remarks are m order. Firstly, the determination of this condition is dependent on the type of transmutation employed since it will determine changes in epistemic rank for all sentences. We contend that an adjustment is appropriate for determining reasons on the basis that it is a transmutation which performs an absolute minimal cha!'ge. In particu_lar, eve!y sentence is reassigned a new ordmal as close to 1ts previOus assignment as is consistent with the desired change. When the epistemic rank of a is increased it seems reasonable to require that this change should disturb the background information system as little as possible.

Secondly, we wish to distingish two types of reasons: (i) ordinary reasons, for which there exists an ordinal to which a can be raised, leading to an increase in the degree of .B, and (ii) strong reasons, for which raising the degree of a by any amount leads to an increase in the degree of ,8.

~ollowing qardenfors' lead, we can rewrite (1?.1) usmg the notiOn of transmutation, and the degree of acceptance to capture both types of reasons:

• (1?.2) Given an entrenchment ranking E E £, a is a reason for .B iffE*(a,i)(.B) > E(.B) for some i > E(a) .
• (1?.3) Given an entrenchment ranking E E £, a is a strong reason for .B iff E*(a, i)(.B) > E(.B) for all i > E(a) .

Theorems 1 and 3, below, provide conditions that describe both reasons and strong reasons when the transmutation employed is an adjustment .

Theorem 1: Let E E £ . Then a is a reason for .B determmed by an adjustment iff (i) E(.B) < E(-.a V ,8), and (u) E(-.a) < E(-.a V ,8).

Corollary 2:Let E E £ . Then a is a reason for .B determmed by an adjustment iff (i) -.a E (set(E))~ 13 , and (ii) .B E (set(E))~ where · is the revision function unzquely determined by Giirdenfors and Makinson's

{1988] construction based on the relative ordering of sentences giVen by E.

Theorem 3:Let E E £. If a is a reason for .B determined by an adjustment, then E(a) ~ E(.B).

Theorem 4:Let E E £. Then a is a strong reason for .B determmed by an adjustment iff (i) E(.B) < E( a) < E(-.a V .B), and (ii) E(-.a) < E(-.a v .B): -

Corollary 5:Let E E £ . Let* be an adjustment. Then a IS a strong reason for .B iff a is a reason for .B and E(a) = E(,B) .

We say an information system given by E is in a state of coherence if whenever a is a reason for .B then E(a) ~ E(,B). That is, the degree of acceptance of {3 is at. le<~;St ~ high as_ the degree of acceptance of a. The JUstificatiOn for t~Js can be made along similar lines to that of (ER1) [Gardenfors 1988, p89] . · Intuitively the explanandum should be at least as plausible as the explanation. Consequently, if .B is an observation not yet accepted then the only reasons for it can be non­ beliefs when the background information is represented by a coherent entrenchment ranking. In practical terms this su~gests that in the case where E( a) > E(.B) ~nd a IS supposed to be a reason for .B then not only IS the entrenc~ment .. ranking incoherent . but the agent ~u~t reorga_nzse [Gardenfors 1988; Williams, 1994a] 1ts mformat10n by transmuting it to a coherent state, and th~n use the resultant entrenchment ranking to determme reasons. Theorem 3 shows that adjustments respect this notion of coherence.

An appealing property of adjustments as the transmu­ tation to specify reasons, apart from its adherence to the p~incil?le of minimal change, is that only the relative orde_nng g1ven by the ~ackground information system it­ self IS needed. In particular, only the relative ranking of three sentences need ever be compared!

We say an explanation is a nontrivial reason. That is a is an explanation for .B iff a is a reason for .B and not logically equivalent to it .

In t~e spirit of Spohn we define a most plausible ex­ planatiOn to be an explanation which is capable of in­ creasing the degree of acceptance of the explanandum as much as any other explanation during a transmutation.

Definition:Let E E £ . Let A be a set of explanations for a sentence ,8. Define a E A to be a most plausible (strong) explanation for .B ifE*(-y,i)(.B) ~ E*(a, i)(.B) for all -y E A, and for all ordinals i > E( a) .

Theorem 6: Let * be an adjustment and used for the determination of reasons. Let A be a set of (strong) explanations for a sentence ,8. Then a is a most plausib{e (strong) explanation in A for .B iff E(-y- .B) < E(a-+ .B) fpr all-y E A . -

In section 7 we contrast this notion of plausibility with that of Boutilier and Becher (1995].

5 Examples

• a : wet grass

6: it is raining

• ,8: water main is broken ry : grass is under cover
• -y: sprinkler is on

,P: E = m2

The proposition a represents an observation while the other propositions constitute possible explanations for this observation.

Example 1: Let E E £ be such that 0 = E(,B) , E(7J), E(-.a), E(-y), E(-.6) < E(-.6 V ry) , E(-.ry), E(-.-y), E(a), E(6) , E(ry- a), E(,P- a), E(6 /\ fJ- a), E(-.,8) < E(-y- a) < E(.B- a) < E(6 /\ ...,Ti- a) < E(,P)

The grass is uncovered, the sprinkler IS off, and it IS raining.

TJ and ¢J are not reasons for a .

a, {3, 'Y, 6 /1. -.TJ are reasons for a. a and 6 /1. -.TJ are both strong reasons for a . {3 is more plausible than 'Y as an explanation for a .

6/1. ...,TJ is the most plausible (strong) explanation for a. Example 2: Let E E E be such that 0 = E({3) , E('Y), E(-.a), E(-.{3), E(-.'Y), E(-.6), E(6) , E(-.TJ), E(TJ) E(-.6VTJ) < E(a), E(TJ _.a), E(6!1.ry _.a)< E('Y _.a) < E(f3 _.a)< E(6 /1. -.TJ-+ a)< E(I/J)

It is not known whether the grass is covered, whether the sprinkler is on, or whether it is raining.

TJ and ¢J are not reasons for a.

a , {3, 'Y, 6 /1. -.TJ are reasons for a .

a is the only strong reason for a .

6 /1. -.TJ is a more plausible explanation for a than {3 or 'Y·

6 Abductive Reasoning

One method currently gaining popularity for defining the notion of explanation is that of abductive reasoning [Paul , 1993; Stickel , 1991] . Abduction is a form of logical mference that aims to derive plausible explanations for information and is, in fact, often described as inference to the best explanation.

An abductive inference proceeds by proposing or generating hypotheses which would account for , or explain a group of facts . A common way of defining explanation as an abductive inference is the following (see [Paul, 1993], for example- however, note that we do not adopt his restriction to abducibles ).

Definition:An abductive explanation of a sentence {3 with respect to a set of sentences T is a sentence a such that {i) T U {a} 1- {3, and (ii) T U {a} 'rf .L

Part (i) of the definition corresponds to having the sentence {3 accepted whenever the theory is expanded with a, whilst (ii) restricts a to being consistent with T . In other words, a is an abductive explanation for {3 if {3 E T+ , and -.a ¢ Tt.

Within the framework of transmutations, our domain theory is obtained from a given entrenchment ranking using proper cuts, defined below.

Definition: Given an E E £ . We define a proper cut of E with respect to a sentence a , denoted by cut:<,(E, a), to be the set of sentences {,B E .C : E( a) < E(f3)} .

It is not hard to see that all proper cuts are theories, and that set(E) = cut< (E, a /1. -.a) that is, the largest proper cut is precisely the accepted information.

The idea of a proper cut has proven to be an expedient concept which has been used for several purposes elsewhere. For example, Rott (1991] uses them to describe theory contraction and revision , Williams [1994b] uses them to define change operators for theory bases, and Gii.rdenfors and Makinson [1994] use them to specify their nonmonotonic inference relation based on expectation orderings. More particularly, if we allow relative orderings of sentences given by E to determine the epistemic entrenchment ordering then , using the construction of Gii.rdenfors and Makinson (1988], we have (set(E))~ = (cut<(E, -.a))t and (set(E));; = (cutdE, a))~a n set(E) .

The following theorem shows that abductive explana­ tion is closely related to Spohnian reasons.

Theorem 7:Let E be an entrenchment ranking and a , {3 E .co··. Let * be an adjustment and used for the determination of reasons. Then a is a reason for {3 iff a is an abductive explanation of {3 with respect to the theory cut< (E, {3) n cut< (E, -.a) .

Theorem 7 says, a is a reason for {3 if and only if a is an abductive explanation of {3 with respect to the smaller of the two proper cuts, cut< (E, {3) or cut< (E, -.a). The former being the set of sentences strictly greater than {3 itself, and the latter being the largest proper cut which is consistent with a .

Corollary 8:Let E be an entrenchment ranking and a, {3 E .C""'. Let * be an adjustment and used for the determination of reasons. Then a is a reason for {3 iff {3 E (cut< (E, {3))~.

Moreover, depending on the relative status of the sentences determined by E the reason may be a strong one. In particular, if cut<(E, a) = cut dE, {3) then the abductive explanation described in Theorem 7 is a strong reason since this condition will hold iff E({J) = E( a) .

The definition of explanation, through the process of abductive inference, will in general classify many sentences as explanations. Often we are interested in determining a single best explanation or , at the very least , reducing the number of possible explanations. This is usually done by adopting some form of preference criteria (e.g., cost-based measures, probabilities, etc. - see [Paul, 1993] for an overview) .

In the framework discussed here we are equipped with a preference ordering over the sentences; namely, an entrenchment ranking which assigns a degree of acceptance to each sentence. Given this interpretation of entrenchment rankings it would seem natural to adopt this ordering to characterize best abductive explanations. Formally we have the following theorem.

Theorem 9:Let E be an entrenchment ranking and a, {3 E .C""'. Let * be an adjustment and used for the determination of reasons. Let A be a set of {strong) explanations for a sentence {3. Then a is a most plausible (strong) explanation in A for {3 iff a is an abductive explanation of {3 with respect to cut< (E, {3) n cut<(E, -.a), and there is no abductive explanation 'Y in A of {3 with respect to cutdE, {3) n cut<(E, -.'Y) for any 'YEA such that E(a-+ {3) < E('Y-+ {3).

Another important consideration when determining ab­ ductive explanations is the specificity of the explana­ tion . The notion of most (and least) specific abduction was discussed by Stickel (1991] . It is described within a resolution-based system for performing abduction. A most specific abduction only allows pure literals (literals that resolve with no other clause in the information set) to be used in an abduction. On the other hand, a least specific abduction for a sentence is just the sentence it­ self. Stickel suggests that most specific abductions are suited to diagnostic tasks while least specific abductions are more appropriate for natural language interpreta­ tion. The specificity of an abductive explanation is given below:

Definition: An abductive explanation a of {3 with respect to a theory T is more specific than an abductive

explanation 'Y of {3 with respect to T iff T U {a} 1- 'Y·

Equating the theory T of the definition above with cut< (E, {3) n cut< (E, -.a) motivates the following defi­ nition of specificity for reasons based on adjustments.

Definition:Let E E £ . Let * be an adjustment and used for the determination of reasons. Let A be a set of (strong) explanations for a sentence {3 . Then a E A is a more specific (strong) explanation for {3 than '"'( E A iff a is a {strong) reason for 'Y·

Theorems 6. and 9 allow us t? say that if a and 1 are both explanat10ns for (3 and a 1s also an explanation for y then the more specific explanation a is more plausible 1f E(1 ..... (3) ~ E(a ..... (3) . This can be generalised for longer chains of explanation to yield a method for determining the preferred level of specificity.

7 Related Work

Recently Gardenfors [1988] as well as Boutilier and B~_cher f1995] have given related accounts of explanation. Gardenlors uses changes in probability, whilst we use changes in plausibility. The relationship between Boutilier and Becher's work and that presented in this paper is somewhat unexpected because both approaches draw their intuitions from seemingly different directions indeed opposite directions. In particular, our analysi~ focuses on changes at the preference relation level whilst Boutilier and Becher's analysis considers the behaviour at the informational content level. However their underlying similarity appears to stem from the commi~t.ment to the principle of minimal change.

Boutiher and Becher use an explicit assumption that explanations for beliefs are beliefs, and explanations for non-beliefs are non-beliefs. By way of justification they say that observations in their framework are not !"ccepted into an information set until some explanation IS found and accepted. Our idea of coherence is related to this assumption. For instance, according to it the only explanations for non-beliefs are non-beliefs. However, it does allow beliefs to have non-beliefs as explanations, a trait which is more in keeping with the usual interpretation of abductive explanations. Using Corollary 2 and the connections below one can show that Boutilier and Becher's explanations respect our idea of coherence, and therefore their assumption is stronger th~n that of coherence. due to the restriction that only behefs can be explanat10ns for beliefs.

Boutilier and Becher's definition of a explains (3, although restricted to the propositional case and based on the assumption above, is closely related to a is a reason fo_r (3 based on adjustments. In particular, if their expla.<Jat10ns are based on a well-ranked CO revision model structure [Boutilier, 1993] then we have:

a is a predictive explanation for f3 iff

• (i) a is a.~eason for (3 , (ii) a E set(E) iff f3 E set(E), . and (m) -,a E set(E) iff -,13 E set(E). a IS a hypothetical explanation for f3 iff

(i) a is a reason for (3, (ii) a, f3 ¢ set(E), and

• . (iii) ..,a E set(E) iff -,(3 E set(E) . a IS a factual explanation for f3 iff

(i) a is a reason for (3, (ii) a , f3 E set(E) , and

. (iii) ..,a E set(E) iff -,(3 E set(E).

a IS a counterfactual explanation for f3 iff

(i) a is a reason for (3 , and

(ii) if (3 E set(E) then a E set(E) .

We note, hypothetical explanations are strong reasons since E(a) = E(/3) = 0.

In contradistinction to reasons based on adjustments Boutilier and Becher's explanations preclude non-belief~ being ~xplanations. ~or accepted beliefs. In particular, accordmg to Bout1her and Becher if a ¢ set(E) and f3 E set(E) then a cannot explain (3 , but on our account a could be a reason for (3 . Consequently, the notion of reason for based on adjustment is more general.

Clearly, the connections described above may not hold if transmutations other than adjustments are used to determine reasons. In this sense, Spohnian reasons speci~ed by ('R2) are far more general than the explanatiOns of Boutilier and Becher because their notion of explanation essentially corr~ponds to reasons based on a particular transmutation· namely an adjustment . ' '

Boutilier and Becher argue that preferred explanations are those that are most plausible; they require the least change _in the informati?n ~et in order to be accepted. In particular, they mamtam as much information as possible, including as much default information as possible.

Their notion of plausibility is very different to the no~ion of plausibility advocated in this paper, where we cla1m that the most plausible explanation is one that is capable of increasing the degree of acceptance of the explanandum the most. For them, if a and 1 are both beliefs and explanations for another belief f3 then they are equally preferred. In other words within Boutilier and Becher's framework factual expla~ations can not be ran~ed on the _b~is <?f plausibility. Consequently, our notion .o~ plaus1b1hty IS s_ubs~antially more discerning.

Boutiher and Becher h1ghhght the desirable defeasible character of. their explanations. Spohnian reasons based on adJustments also exhibit this feature and moreover they give rise to consistency preserving rational consequ~nce relations. In particular, if we define af--..(3 to be a IS a reason for f3 based on an adjustment then f- is con.sistenc~ preserving . and rational. Conseq~ently, ther~ eXIsts a mce preference structure [Gardenfors and Makmson, 19941 and an expectation ordering which induces reasons based on adjustments.

Given the transparent connection between entrench­ ment rankings . and expectation orderings [Williams, 1995b] we can g1ve the following intuitive interpretation of _reason for in terms of expectations. We can say that a 1s a _reason for f3 if -,a V f3 is strictly more expected th~n. e1ther. -:-a or (3 , and that a is a strong reason for f3 1f m add1tion a and f3 are equally expected. Similar readings can be given for plausibility and specificity.

Williams [1995a] gives a computational model for adjustments which can be used to implement Spohnian reasons, abductive, predictive, factual, hypothetical and counterfactual explanations. · ·

Spoh_n_ [1988J.rev~sed h~s notion of reason for, and gave a co~d.Jtion wh1ch. 1~ equivalent [Williams, 1994a] to the condition below (1t IS related to conditionalization) .

a is a reason for f3 iff

(n) -E(a) + E(a Y -,(3) > -E(-,a) + E(...,a Y -,(3) .

(~) -E(-,a) + E(-,a Y /3) > -E(a) + E(a V (3), or

Intuitively, this condition says that a is a reason for f3 iff (3 _h~ a h_igher degree of acceptance, or degree of plaus1b1hty g1ve_n a ra:t~er than given -,a (Spohn, 1988] . However, th1s cond1t10n leads to counterintuitive results; sentences turn out to be reasons simply by virtue of the fact they are strictly more plausible than the explanandum. Therefore we can say that reasons based on the condition above do not respect the notion of coherence introduced in section 4.

8 Discussion

Transmutations of an information system describe not only how a set of information is revised but how the ~n~erlyi~g preference structure is revised. 'consequently, 1t 1s poss1b~e to capture Spohn's notion of reason using transmutat10ns. Indeed we would obtain a different characterization of a Spohnian reason for every type of transmutation.

An adjustment is a transmutation that involves an ab­ solute minimal change. That is, the underlying prefer­ ence relation is changed as little as necessary to effect the required change in epistemic rank. For this reason we have argued that an adjustment is an appropriate transmutation for determining explanations because in the determination of an explanation we would normally require that such a change disturb the background in­ formation system as little as necessary. The suitability of using an adjustment for Spohnian reasons is further supported by its perspicuous connection with the com­ prehensive work of Boutilier and Becher.

We showed how an adjustment can be used to determine a most plausible explanation, by selecting an explanation that is capable of raising the epistemic rank of the explanandum the most . This notion is more discerning than that given by Boutilier and Becher whose method suffers from the problem of not being able to rank the plausibility of factual explanations.

We contrasted Spohnian reasons based on adjustment with a common definition of explanation in terms of ab­ ductive reasoning. We noted that the determination of reasons based on adjustments are carried out qualita­ tively using the relative ordering of the background in­ formation system itself, and the associated nonmon<r tonic consequence relation is consistency preserving and rational.

Future work will explore the relevance of competing ex­ planations (Giirdenfors, 1993] and the strengths of com­ peting explanations using the quantitative information encapsulated in an entrenchment ranking. Preliminary results suggest these are promising directions to pursue.

Acknowledgements

The authors have benefited from the helpful suggestions of Peter Giirdenfors, Abhaya Nayak, Eric Olsson, Judea Pearl, Pavlos Peppas and Hans Rott .