Local Theories of InheritanceKrishnaprasad Thirunarayan �Department of Computer Science and EngineeringWright State University, Dayton, OH 45435.(513)-873-5109tkprasad@cs.wright.eduAbstractInheritance networks are not expressive enough to capture all intuitionsbehind inheritance. Hence, a number of signi�cantly di�erent semantics havebeen proposed. Here, a family of local theories of inheritance is presentedthat are conceptually simple and computationally tractable. A general model-theoretic framework for specifying direct semantics to networks and clarifyingrelationships among them is developed. It is also shown that the local con-straints specifying the semantics are satis�able when the attention is restrictedto acyclic networks. The notion of local semantics is generalized to what iscalled \ground" local semantics to further understand the relationship betweenthe local theories and the path-based theories of inheritance. A polynomial-time set-based inheritance algorithms that conform to these inheritance theoriesis also presented.�This research was supported in part by the NSF grant IRI-9009587.1

1 IntroductionIn order to exhibit intelligent behaviour, a system needs to represent and utilizeknowledge about the relevant domains. Hence, it is necessary to design repre-sentational languages that express this knowledge concisely and develop reasoningmethods to derive facts implicit in these representations. The general task of knowl-edge representation and reasoning is computationally very hard. But a signi�cantportion of common-sense reasoning can be factored out as inheritance reasoningthat can be carried out quickly. This has been the primary motivation for the useof inheritance networks for knowledge representation, and the design of e�cientspecial purpose algorithms to reason with them.Any realistic representation of the real-world knowledge must necessarily allowrepresentation of exceptions. For instance, typically, mammals are non yers. Batsare exceptional mammals because, normally, they can y. But gravely sick bats donot y. In general, properties inherited from a class, c, must dominate the propertiesinherited from any superclass of c, in case of a con ict.The representation language should also allow expression of preferential inheri-tance of a property from a class over inheritance from another class. For instance,given that Pratt is a graduate teaching assistant, that is, Pratt is both a graduatestudent and a teaching assistant, one must be able to infer that Pratt gets salaryand is required to pay taxes. Note that even though a graduate student typicallydoes not get salary, and hence is not a tax payer, Pratt \inherits" its tax payingstatus because he is a teaching assistant.There have been a number of proposals for the semantics of inheritance networks.This has been done either through a translation into a general logical formalism orby special path-based techniques. The former approach requires the speci�cation ofan algorithmic transformation of the network into a set of sentences in a logical lan-guage. The intuition about inheritance is captured indirectly through the semanticsof the logic language. The user must know the logical formalism and the transla-tion algorithm to understand the meaning of the network [5; 8; 12; 19; 20; 25; 32].Furthermore, the \declarativeness" of these approaches is arguable, as it dependson the conceptual di�culty of the algorithmic part. Since most transformations arequite complex, it is often unclear how the di�erent semantics relate to each other.2

In the second approach, the semantics is given by characterizing sets of inheritancepaths in the network [10; 13; 14; 28; 36; 39; 40].In this paper we put forth a general framework for specifying model theory of in-heritance networks directly , bypassing complex intermediate transformations. Fromthe user's perspective, this is desirable, as it makes comprehension of inheritancespeci�cations easier. Furthermore, there seem to be equally well-motivated, yet sig-ni�cantly di�erent interpretations of certain network topologies. This is, in part,due to the fact that inheritance networks are not su�ciently expressive. Our direct,model-theoretic approach enables careful study of the di�erences among some ofthese proposals.The local theories of inheritance, we propose, are conceptually simple and com-putationally attractive [18]. Informally, a semantics of inheritance networks has thelocality property if the meaning of a node depends only on the meaning assignedto its immediate neighbors and not on the meaning assigned to nodes arbitrarilyfar away. Locality accrues bene�ts such as compositionality. To understand themeaning of the whole network, one can understand the meaning of its constituentsubnetworks and then compose their meaning appropriately using only the meaningof the \interface" nodes. The locality property enables a declarative understand-ing of the network. Furthermore, the changes in the meaning of a node propagatesmoothly through the network. (We are tacitly assuming that the gluing processdoes not change the speci�city relationships among the nodes in the subnetworks.)It also supports e�cient parallel computation of the meaning of large networks.This is because, one can partition the task of computing the meaning of a networkinto smaller independent subtasks of computing the meanings of its subnetworks,and then combining these meanings to obtain the meaning of the whole network.The paper is organized as follows. Section 2.1 speci�es the syntax of preferen-tial inheritance networks which are simple generalization of inheritance networks.Section 2.2 provides an informal description of our approach. We develop a model-theoretic framework for the declarative semantics of inheritance networks in Sec-tion 2.3 and de�ne the local speci�city we espouse in Section 2.4. In Section 2.5,we study some of the salient features of the proposed semantics. Section 2.6 de�nesand analyzes a model-theoretic notion of locality in the context of inheritance net-3

works. In Section 2.7, we discuss the inheritance algorithm. Finally, we conclude inSection 3.2 Direct Semantics of Inheritance NetworksWe describe a general framework for specifying declarative semantics of preferentialnetworks directly in terms of the network [18]. In particular, our approach captures\individual ow" intuitions about inheritance [13], which capture more adequatelynotion of inheritance than the \property ow view" [21; 39]. Pragmatically, themost important payo� of this approach is that inheritance algorithms based on ourtheories are computationally tractable.2.1 Syntax of Preferential Inheritance NetworksTraditional inheritance networks are extended to preferential inheritance networksby imposing suitable orderings on the in-arcs into a node. Note that we orderthe children of each node rather than its parents because understanding of inheri-tance as the ow of individuals up the network to acquire properties captures moreadequately our intuitions about inheritance than the view of properties owingdownwards [21]. This \speci�city" ordering captures class-subclass relationship (asin the bats-example in Section 1) and supports representation of disambiguationinformation not directly inferrable from the topology (as in the GTA-example inSection 1).De�nition 1 A preferential inheritance network is an ordered directed acyclic graphconsisting of a set of individual nodes I, a set of property nodes P, a set of positivearcs E+ � (I [ P) � P, a set of negative arcs E� � (I [ P) � P, and for eachnode p 2 P, a speci�city relation �p on its in-arcs.We use p, q and r to stand for any property node in P, and i for any individualnode in I. Assume E+ \E� = ;: 4

2.2 Motivational ExamplesWe illustrate our approach using the network in Figures 1 and 2 whose arcs representthe following facts:� Clyde is an individual. Tweety is a penguin. Donald is a bird.� Penguins are Birds. Typically, birds y. Typically, penguins do not y.For our purposes, the state of the world is characterized by a description of theproperties possessed by the individuals in the world. A semantic structure for thenetwork is a triple (D, I,M), where D stands for the domain of discourse, I standsfor the individual meaning function for nodes, and M stands for the set-basedmeaning function for property nodes.The domain of discourse D is a pair of sets (Di;Dp), where Di stands for a setof individuals and Dp for a set of properties. The interpretation function I assignsto each individual node in I an individual from Di and to each property node in Pa property from Dp. The interpretation functionM assigns to every property nodep in P a pair of subsets of Di, denoted as (p+; p�), where p+ (resp. p�) containsindividuals that are believed to possess property p (resp. :p). p+ \ p� containsindividuals for which there is evidence to possess both p and :p. To accomodategeneric statements, Di can be replaced by (Di [ Dp).The following assignment to the nodes of the network given in Figure 1(a) depictsa possible interpretation of the network.� Di = fClyde;Donald; Tweetyg:Dp = fpenguins; birds; flyg:� I(Clyde) = Clyde.I(Donald) = Donald.I(Tweety) = Tweety.I(penguins) = penguins.I(birds) = birds.I( y) = fly. 5

� M(penguins) = (fg; fg).M(birds) = (fg; fClydeg).M( y) = (fDonald; Tweetyg; fClydeg).However, this assignment does not seem to capture the intuitive meaning of thenetwork. For instance, it is not known from the above assignment whether Tweetyis a penguin or not, even though it can be gleaned from the given informationthat Tweety is a penguin. To rule out such an assignment as unsuitable, we makeexplicit constraints that the edges impose on the assignment of sets to nodes. Inparticular, whenever there is a positive (resp. negative) arc from an individualnode n to a property node p, we require that n 2 p+ (resp. n 2 p�). ThusTweety 2 penguins+ and Donald 2 birds+.Similarly, observe that penguins are birds, and hence, Tweety should also belongto birds+. That is, we wish to rule out interpretations of the network in which anentity is a penguin, but is not a bird. To capture the meaning of a \strict" positive(resp. negative) arc from a property node p to a property node q, we add theconstraint p+ � q+ (resp. p+ � q�). The following interpretation of the networksatis�es the afore-mentioned constraints. (Also see Figure 1(b).)� M(penguins) = (fTweetyg; fg).M(birds) = (fDonald; Tweetyg; fClydeg).M( y) = (fDonald; Tweetyg; fClyde; Tweetyg).Even though, on the basis of the given facts, we intuitively conclude that Tweetydoes not y, our formalization supports inconsistent conclusions about Tweety's ight. This is because the arcs from node bird and node penguin into node yhave been incorrectly interpreted as strict, rather than as defeasible. Given the factthat penguins are a subclass of birds, the contradiction can be resolved in favorof Tweety's inability to y. These ideas can be formalized as follows: A positive(resp. negative) defeasible arc from a property node p to a property node q istranslated as (p+ � knownexceptions) � q+ (resp. (p+ � knownexceptions) �q�), where the \known exceptions" to a rule is interpreted as stronger con ictingconclusions . For the example at hand this turns out to be: penguins+ � fly�and (birds+ � penguins+) � fly+. Note that in going from strict arcs to6

defeasible arcs the focus shifted from relating assignments to pairs of adjacent nodes,to relating assigments to a node and all of its neighboring nodes. This happensbecause the context of a node is very important in interpreting the meaning of adefeasible arc incident on it . The interpretations that satisfy these constraints arecalled models of the network. These capture states of the world consistent with thenetwork. A possible model of the network in Figure 2(a) is:� M(penguins) = (fTweetyg; fg).M(birds) = (fDonald; Tweetyg; fClydeg).M( y) = (fDonaldg; fTweety; Clydeg).From the given facts, it is also conceivable that the individual Clyde is a birdand that it ies. However, there is nothing in the input to suggest that or otherwise.To re ect this ignorance, and to avoid baseless pinning down of the characteristicsof Clyde, we introduce the notion of a minimal model. In the absence of cycles inthe network, this can be formalized by replacing � by = as shown:� (penguins+; penguins�) = (fTweetyg; fg)� (birds+; birds�) = ([fDonaldg [ penguins+]; fg)� (fly+; f ly�) = ([birds+ � penguins+]; penguins+).The unique minimal model satisfying these constraints turns out to be:� M(penguins) = (fTweetyg; fg).M(birds) = (fDonald; Tweetyg; fg).M( y) = (fDonaldg; fTweetyg).In words, Tweety is a penguin bird that cannot y, while Donald is a ying bird.Also see Figure 2(b).The afore-mentioned formalization captures the properties one can associatewith the individuals Tweety, Donald etc. To determine the set of properties thatcan be associated with a \typical" penguin, or a \typical" bird, or a \typical" yingobject, we can imagine associating with each property node a special individualnode as its child. That is, asserting p 2 p+ for each node p. Then, the properties7

� �� � � �66 ���� ......ITweetyDonaldbirds penguins yClyde(fDonald,Tweetyg, fClydeg) (fg, fg)(fg, fClydeg) (a) � �� � � �66 ���� ......ITweetyDonaldbirds penguins yClyde(fDonald,Tweetyg, fClyde,Tweetyg)(fTweetyg, fg)(fDonald, Tweetyg, fClydeg) (b)Figure 1: Interpretations of the Penguin Triangle� �� � � �66 ���� ......ITweetyDonaldbirds penguins yClyde(fDonaldg, fTweety,Clydeg)(fTweetyg, fg)(fDonald, Tweetyg, fClydeg) (a) � �� � � �66 ���� ......ITweetyDonaldbirds penguins yClyde(fDonaldg, fTweetyg) (fTweetyg, fg)(fDonald, Tweetyg, fg) (b)Figure 2: A Model and a Minimal Model8

inherited by the new individual p corresponding to the property node p gives thecollection of properties associated with a \typical" member of p. These inheritedproperties can also be interpreted as the set of generic statements supported by ournetwork.With this augmentation, the minimal model of our network becomes:� M(penguins) = (fpenguin; Tweetyg; fg).M(birds) = (fbird;Donald; penguin; Tweetyg; fg).M( y) = (ffly; bird;Donaldg; fpenguin; Tweetyg).and the corresponding conclusions supported by our network are:� Tweety is a penguin. Tweety is a bird. Tweety does not y.� Donald is a bird. Donald ies.� Penguins are birds. Penguins do not y. Birds y.Consider now the speci�city relation used to disambiguate con icting conclu-sions about Tweety's ight. We argued that the inheritance of :fly via penguinsdominates the inheritance of fly via birds. This is justi�ed because, if, on the ba-sis of the fact that x is a penguins , x is a birds , then we can regard penguins asproviding more speci�c information than birds . So when we associate a property:fly with penguins we are willing to override any evidence for fly that rests on the\implicit" property birds of the members of penguins . This notion of speci�city canbe formalized by determining whether or not penguins 2 birds+, and if so, setting8 common parents of fly : hbirds; flyi �fly hpenguins; flyi:Thus, we observe that the computation of inheritance relation and the speci�cityrelation can be integrated.We now consider another detailed example of specifying the semantics of mono-tonic inheritance [37] in our framework. (For simplicity, we do not specify reasoningwith generic statements.) Here, each arc is interpreted as strict, that is, it standsfor a statement that does not admit any exception. In the event of a con ict, thecontradictory properties may be assigned the status of inconsistent (or ambiguous)9

belief. In the Nixon diamond example the positive (resp. negative) arc from nodequaker (resp. republican) to node paci�st is interpreted as an assertion that \everyquaker is a paci�st" (resp. \every republican is not a paci�st "). Under this inter-pretation, if Nixon is both a quaker and a republican then Nixon is inferred to bea paci�st because he is a quaker, and inferred not to be a paci�st because he is arepublican. This results in an inconsistent belief about Nixon's paci�sm.The constraints that capture the semantics of monotonic inheritance are:C1 Individual i has property p if there is a positive arc from node i to node p, orindividual i has property q and there is a positive arc from node q to node p.Formally, for each property p and individual i :p+(i) if E+(i; p) _ 9q [ E+(q; p) ^ q+(i) ]:C2 Individual i does not have property p if there is a negative arc from node i tonode p, or individual i has property q and there is a negative arc from nodeq to node p, or individual i has property q and there is a negative arc fromnode p to node q, individual i has property :q and there is a positive arc fromnode p to node q.Formally, for each property p and individual i :p�(i) if E�(i; p) _ 9q [ ( E�(q; p) ^ q+(i) ) _( E�(p; q) ^ q+(i) ) _ ( E+(p; q) ^ q�(i) ) ]:Figure 3(a) depicts a network which represents the following facts:� Richard is a quaker. Nixon is a republican.� Every quaker is a paci�st. Every republican is a nonpaci�st.The following interpretation is a valid semantic structure for this network:� Di = fR:Nixon; Carterg:Dp = fquaker; republican; pacifistg:10

� �� � � �66����* ..............ypaci�st republicanquakerRichard Nixon(fR. Nixong, fR. Nixong)(fR. Nixong, fR. Nixong)(fR. Nixong, fR. Nixong) (a) � �� � � �66����* ..............ypaci�st republicanquakerPenn Bush(fPenng, fBushg) (fBushg, fPenn, Carterg)(fPenng, fBushg) Carter(b)� �� � � �66����* ..............ypaci�st republicanquakerPenn Bush(fPenng, fBushg) (fBushg,fPenng)(fPenng, fBushg) Carter(c)Figure 3: The monotonic inheritance models11

� I(Richard) = R:Nixon.I(Nixon) = R:Nixon.I(quaker) = quaker.I(republican) = republican.I(paci�st) = pacifist.� M(quaker) = (fR:Nixong; fR:Nixong).M(republican) = (fR:Nixong; fR:Nixong).M(paci�st) = (fR:Nixong; fR:Nixong).Observe that both individual nodes Richard and Nixon are mapped to the sameindividual R:Nixon, and that there is no explicit reference to the individual Carter.It can be veri�ed that this structure satis�es the constraints C1 and C2, and hence,it is a model of the network.In particular, R:Nixon is believed to be a quaker (resp. republican) becauseof the direct positive arc from node Richard (resp. Nixon) to node quaker (resp.republican). R:Nixon is believed to be :quaker (resp. :republican) because of thepositive (resp. negative) arc from the node quaker (resp. republican) to node paci�stand the fact that Richard Nixon is believed to be a pacifist (resp. :pacifist).Figure 3(b) depicts a network which represents the following facts:� Carter is an individual.� Penn is a quaker. Bush is a republican.� Every quaker is a paci�st. Every republican is a nonpaci�st.The following semantic structure is a Herbrand model for this network because itsatis�es the network and also the unique-name hypothesis (that is, distinct individ-ual (resp. property) nodes map to distinct individuals (resp. properties)):� Di = fPenn;Bush; Carterg:Dp = fquaker; republican; pacifistg:� I(Penn) = Penn.I(Bush) = Bush. 12

I(Carter) = Carter.I(quaker) = quaker.I(republican) = republican.I(paci�st) = pacifist.� M(quaker) = (fPenng; fBushg).M(paci�st) = (fPenng; fBushg).M(republican) = (fBushg; fPenn; Carterg)-Even though the assumption that Carter is not a republican is consistent withthe network, it seems redundant. To capture the intended meaning of the network,we formalize the notion of minimality by replacing if with i� in the constraints C1and C2. The semantic structure in Figure 3(c) is the same as that in Figure 3(b),except that the meaning of node republican is changed as follows:M(republican) = (fBushg; fPenng):This structure satis�es the minimality constraints, and hence, is a minimal modelof the network.To understand the di�ering interpretations of ambiguity, consider again theNixon's diamond shown in Figure 3 where the arcs into node paci�st are reinter-preted as defeasible (rather than as strict).� Quakers are normally paci�sts.� Republicans are normally not paci�sts.In this situation we can feign ignorance about Nixon's paci�sm. This is legitimate,since we have two equally strong con icting evidences in support of his paci�sm andthere is no way to choose one over the other to determine whether Nixon is a paci�stor not. This is the skeptical approach to ambiguity. On the other hand, it is equallyreasonable to argue that in real life Nixon is either a paci�st or a non-paci�st, andto associate a set of two minimal models with this network. In one model, Nixon isa Paci�st, while in the other, he is not. This is the credulous approach to ambiguity.Formalization of all these aspects and options is the subject matter of the followingsections. 13

2.3 Direct SemanticsWe present formal speci�cations of a family of related inheritance theories for pref-erential inheritance networks.De�nition 2 A Herbrand semantic structure for a preferential network is a pair(D, M), where D stands for the domain of discourse, and M for the set-basedmeaning function. The domain D is a pair (Di, Dp), where Di is a set of individualscorresponding to nodes in I, and Dp is a set of properties corresponding to nodes inP. The meaning function M associates with each property node p 2 P, a pair ofsubsets (p+, p�) of Di [ Dp, where p+ represents the set of individuals that possessproperty p union the set of properties by virtue of which an individual may possessp; and p� represents the set of individuals that possess :p union the set of propertiesby virtue of which an individual may possess :p.We use the letters n, m to denote individual nodes in I and n, m to denote therespective individuals in Di. The letters p, q, : : : will be used to denote propertynodes in P and p, q, : : : to denote the respective properties in Dp. We assume,for notational convenience, the equivalences: p+(n) � n 2 p+ (resp. p�; p�) and:p+(n) � n 62 p+ (resp. p�; p�).The di�erent approaches to inheritance di�er in the speci�cation of the con-straints for a semantic structure to be a model. Informally, individual n shouldinherit property p if the maximal evidence in support of the inheritance is througha positive arc to node p; and individual n should inherit property :p if the maximalevidence is through a negative arc to node p. There is maximal evidence in supportof :p(n) (resp. p(n)) if there is a negative (resp. positive) arc from n to p, or if thereis no positive (resp. negative) arc from n to p, but there is an \undefeated" negative(resp. positive) arc from t to p and n inherits t. The negative (resp. positive) arcfrom t to p is \defeated", if there is a node s that has a positive (resp. negative)arc to p such that n inherits s, and s provides more speci�c information than t.In a skeptical interpretation of ambiguity, individual n is ambiguous about in-heriting p or :p, whenever there are equally strong or incomparable evidences for pand :p. However, in a credulous interpretation of ambiguity, n inherits p in someof the models and inherits :p in the others.14

The statement that \9 maximal evidence for :p(n)" can be formally expressedas: E�(n; p) _ :E+(n; p) ^ 9t [ E�(t; p) ^ t+(n) ^:9s ( [E+(s; p) ^ s+(n) ] ^ ht; pi �p hs; pi ) ];and the statement that \9 maximal evidence for p(n)" can be formally expressedas: E+(n; p) _ :E�(n; p) ^ 9t [ E+(t; p) ^ t+(n) ^:9s ( [ E�(s; p) ^ s+(n) ] ^ ht; pi �p hs; pi ) ]:2.3.1 Skeptical SemanticsThe following constraints formalize a skeptical semantics of inheritance networksthat resembles the meaning assigned to networks by the evidence-based theory[19; 22]. In particular, they specify when an individual i can possess a propertyp. For each property p and individual n :p�(n) if \9 maximal evidence for :p(n)": (1)p+(n) if \9 maximal evidence for p(n)": (2)p�(n) � p+(n) ^ p�(n):The constraints for a property q to provide more speci�c information than prop-erty p are similar. However, observe that the preferential networks cannot representcomplementary inheritance [22].2.3.2 Minimal ModelsWe may select \preferred" models, by capturing the idea of minimality, by replacingif with i� in (1) and (2). Note that checking whether a semantic structure is amodel or a minimal model is local , i.e., for each node, it depends only on its children.15

2.3.3 An Alternative Skeptical SemanticsThe skeptical theory [13] treats \no information" and \ambiguous information" ona par. This interpretation of the ambiguity is obtained from the above semanticsby associating with each node a pair of sets (p+h ; p�h ), wherep+h = p+ � p� and p�h = p� � p�:2.3.4 Probabilistic Skeptical SemanticsA skeptical semantics that is related to inheritance theories [8; 10] can be speci�edas follows:For each property p and individual n :p�(n) i� [ \9 maximal evidence for :p(n)" (3)_ (:p+(n) ^ 9t ( E+(p; t)^ t�(n) _ E�(p; t)^ t+(n) ) ]:p+(n) i� \9maximal evidence for p(n)": (4)p�(n) � p+(n) ^ p�(n):The meaning of each property node is a pair of sets (p+g ; p�g ), wherep+g = p+ � p� and p�g = p� � p�:The speci�city relation described in Section 2.4 is similar to the one obtainedby restricting the notion of defeat for paths [10] to just links [11]. Furthermore,the conclusions obtained using the \contrapositive" component are similar to thosesanctioned by the causal semantics of inheritance networks [8].2.3.5 Credulous SemanticsThe following constraints formalize a credulous semantics of inheritance networks.For each property p and individual n :p�(n) if :p+(n) ^ \9 maximal evidence for :p(n)": (5)p+(n) if :p�(n) ^ \9 maximal evidence for p(n)": (6)p+ \ p� = ;: (7)16

Observe that credulous semantics di�ers from the skeptical one essentially inthe �rst conjunct of (5) and (6) (cf. (1) and (2)). Note that checking whether asemantic structure is a model is again local .The constraints for a property q to provide more speci�c information than prop-erty p are similar.2.3.6 Minimal modelsWe may �lter out \extraneous" models by capturing minimality, similarly to theskeptical case by replacing if with i� in (5) and (6). In this case, (7) would followfrom the modi�ed (5) and (6).2.3.7 An Alternative Credulous SemanticsIn addition to the conclusions sanctioned by the above approach, we may strengthennegative conclusions by taking recourse to contrapositive reasoning [20], in which ifwe know that \p's are q's" (resp. \p's are :q's"), then we also assume that \:q'sare :p's" (resp. \q's are :p's").We specify the credulous semantics that incorporates contrapositive reasoningby modifying the constraint (5) of Section 2.3.5 and augmenting it with two disjunctsas follows: p�(n) if:p+(n) ^ [ \9 maximal evidence for :p(n)"_ 9t ( E+(p; t)^ t�(n) _ E�(p; t)^ t+(n) ) ]: (8)Furthermore, the minimality constraint is obtained by changing if to i� . Note thatchecking whether a semantic structure is a model or a minimal model is still local .2.3.8 Generic StatementsThe meaning of a node can be de�ned either as a set of individuals or as a setof properties. To determine the set of properties that may be associated with a\typical" individual of a property (resp. class) by virtue of it being a member ofthat property, imagine associating with each property (resp. class) node a special17

individual node as its child. This is achieved by extending the domain of discoursefrom the set of constants Di representing individuals in I to the set of constantsDi [ Dp representing all nodes in I [ P, and adding the constraint p 2 p+, foreach node p in P. The properties inherited by the constant p corresponding tonode p gives the collection of properties associated with a \typical" member of p.These inherited properties can also be interpreted as the set of generic statementssupported by our network.2.4 The Speci�city RelationWe view the speci�city relation �p on the in-arcs into a property node p as specifyingthe relative strength of inheritance of property p through them. The speci�cityrelation itself consists of two components: the inferred speci�city constraints andthe input preferential constraints. (In production system terminology, the inferredspeci�city is analogous to meta-knowledge, while the input preference is analogousto meta-rule [26].) The former constraints correspond to class-subclass informationnormally implicit in the topology of the network, while the latter constraints areexplicit disambiguation information input by the representer. (These informationcan also be derived from the probabilistic analysis [8; 27].) These speci�city relationsare used to resolve con icting inheritance. That is, if an individual i can potentiallyinherit p via the positive arc hq; pi and :p via the negative arc hr; pi, then thespeci�city relation between hq; pi and hr; pi is used to determine which propertyinheritance prevails over the other.We brie y describe the inferred speci�city constraints for di�erent categories ofnetworks. We regard an inheritance network as well-formed , if, for each propertynode p, the speci�city relation �p associated with its in-arcs satis�es the inferredconstraints. In the sequel, we deal only with well-formed networks.1. For tree-structured hierarchies, the speci�city relation �p is \static". It mustsatisfy the following conditions:(a) An explicitly stated assertion \i is a p" must have priority over the defaultinheritance of :p by i through r . (See Figure 4(a).) That is, for arcshi; pi and hr; pi, we have hr; pi �p hi; pi:18

(b) For all individuals in q , inheritance of p (resp. :p) through a subclass qmust take precedence over inheritance of :p (resp. p) through the class r .(See Figure 4(b).) That is, for any pair of arcs hq; pi and hr; pi, such thatthere is a directed path from q to r, it is the case that hr; pi �p hq; pi:2. For \acyclic" inheritance networks, the speci�city relationship is \dynamic"[38], and can be obtained by generalizing Condition 1b. That is, it must satisfythe following conditions:(a) A fact always overrides a default conclusion. Hence,if E(i; r) then for all children q : hq; ri �r hi; ri(b) The notion of \local" speci�city we can espouse is as follows: If, on thebasis of the fact that x is a q , we can infer that x is a p, then we regardq as providing more speci�c information than p. The intuition is thatif we associate a property r with q , then we are willing to override anyevidence for :r that rests on the \implicit" property p of the membersof q . This notion of speci�city can be computed locally by the inheritancealgorithm in a natural way. In particular, q 2 p+ implies that q providesmore speci�c information than p and hence,if q 2 p+ then for all common parents r : hp; ri �r hq; ri:Note that the apparent circularity in the afore-mentioned mutually re-cursive de�nition of speci�city relation and the inheritance relation is infact well-formed for acyclic networks as demonstrated in Appendix A.3. The above de�nition of local speci�city cannot capture the skeptical semantics[13] as illustrated by the following example brought to my notice by KarlSchlecta: In Figure 5, the individual a inherits :s from p in preference to sfrom r irrespective of whether or not there is a negative arc from a to q. Thisis because r+(p) holds and hence, p is more speci�c than r. On the contrary,in the skeptical theory [13], the addition of the negative arc from a to q willrevise \a inherits :s" to \a is ambiguous about s". This is because, in the19

���� ���� ����������������3 - ����3 - ..............(a) (b)i p r q rpFigure 4: Constraints on pre-order �p

,,.......................... 6�� �������������� @@@@@@I@@@@@@I SSSSSSSSo ��������7@@@@@@I ��������SSSSSSSSo ��������7@@@@@@I �������� ssb(b)r q pab(a)r q pa Figure 5: Speci�city can be nonlocal.20

former case, the arc from a to r is regarded as a redundant evidence, while, inthe latter case, it is accorded the status of an independent evidence.We explain brie y why the skeptical semantics [13] is what we call groundlocal. (See Section 2.6 for details.) For this purpose, we derive an ordering onthe in-arcs of a property node p as a function of an individual node n fromthe reformulated de�nition of defeasible preemption [16].A word about the notation used. �(n; �1; x; �2; q) denotes a sequence of positivearcs from node n to node q via node x, and �(n; �; q)! p (resp. �(n; �; q) 6! p)denotes a path from node n to node p ending with a positive (resp. negative)arc from node q to node p. � stands for an inheritance network and � for itsextension constructed so far.A positive path �(n; �; q)! p is preempted in the context h�;�i i� there is anode x such that (i) either x = n or there is a path of the form �(n; �1; x; �2; q) 2�, and (ii) x 6! p 2 �. This translates to the fact that inheritance of :pby n through the negative arc from node x to node p is stronger than theinheritance of p through the positive arc from node q to node p. Note thatonly the existence of the paths �, �1 and �2 is required.A negative path �(n; �; q) 6! p is preempted in the context h�;�i i� there is anode x such that (i) either x = n or there is a path of the form �(n; �1; x; �2; q) 2�, and (ii) x ! p 2 �. This translates to the fact that inheritance of p by nthrough the positive arc from node x to node p is stronger than the inheritanceof :p through the negative arc from node q to node p. Here again, the detailsof the paths �, �1 and �2 are unimportant.These observations enable representation of the \skeptical" speci�city rela-tionship [13] by ordering in-arcs into node p as a function of the individualnode n (rather than being independent of n).Similar translations of the inheritance networks to defeasible logic [3] and tostrati�ed logic programs [9] have been carried out.21

2.5 Characteristics of the Local SemanticsHere we assume, without loss of generality, that the speci�city relation �p for eachproperty node p consists entirely of the inferred component. It is straight-forwardto incorporate the preferential component.2.5.1 Skeptical SemanticsIt is not necessary for the set of constraints given in Section 2.3.1 be satis�able foran arbitrary directed graph. In fact,Lemma 1 There exist directed graphs that are unsatis�able.However, even though arbitrary directed graphs can be unsatis�able the restric-tion of acyclicity ensures that inheritance networks are satis�able.Theorem 1 Every preferential inheritance network has a minimal skeptical model.We can in fact strengthen this result toTheorem 2 Every preferential inheritance network admits a unique minimum skep-tical model.2.5.2 Credulous SemanticsWe show the satis�ability of the constraints given in Section 2.3.6 and Section 2.3.7for inheritance networks. That is,Theorem 3 Every preferential inheritance network has a minimal credulous model.2.5.3 Local Speci�cityWe summarize certain important features of local speci�city given in Section 2.4arising as a consequence of de�ning the speci�city relation and the inheritancerelation mutually recursively.The constraints to be satis�ed for the inheritance of a property by an individual,and the constraints to be satis�ed for a property to provide more speci�c information22

than another property, are practically identical. So, if an inheritance network � isaugmented with a new individual node i and a new positive arc E+(i; q), then q ismore speci�c than p i� i inherits p. Similarly, q is potentially disjoint from p i� iinherits :p. This provides an intuitive justi�cation for our notion of speci�city.An individual c inherits a property p if there is a direct arc from c to p, or if cinherits p through a child of p. An individual c can be thought to inherit a propertyp via a node q, if c inherits q, and among all the children of p that c inherits, qis the most speci�c. Additionally, if individual nodes c and d inherit from nodes qand r, and q is more speci�c than r, then both c and d inherit from q in preferenceto r. The property inherited by c can di�er from that inherited by d only if there isanother \con icting" node t which is not dominated by q for c or d. In particular,q being more speci�c than r does not depend on c or d.All these observations follow directly from the semantics speci�cation in Sec-tion 2.3 and the local speci�city in Section 2.4.2.6 LocalityThe locality principle is at the heart of language de�nition techniques [1] and is alsofundamental to e�cient knowledge representation [30]. For instance, a propositionallogic language can be speci�ed using a context-free grammar, and its semanticsde�ned by giving truth-tables for the boolean operators and a truth assignmentto all proposition symbols in the language. This semantics speci�cation exhibitslocality property by de�nition. The meaning of an arbitrary formula depends onlyon the meanings of its \immediate" subformulae in the derivation tree, and does notdepend on how that meaning is arrived at. The locality property of the semanticsallows a \context-free" application of an inference rule. Given a Prolog rule andthe fact that all the body literals in the rule are proven, the head of the rule can beasserted without any further deliberations about what else is present or provablefrom the program. Similarly, given the credibility of each MYCIN rule, the certaintyfactors associated with the body literals, and the combination function, the certaintyof the head can be determined [30].In probabilistic reasoning [30], conditional probabilities are not amenable tosuch \local" treatment. This is because a conditional probability P (p j q) does not23

impose a de�nitive constraint on the probability of p, given the probability of q, inthe presence of some additional evidence e. Determining the applicability of each arcof an inheritance network independently falls prey to the same problem because thecontext in which the arc appears has not been considered explicitly [8; 27]. However,belief propagation in causal networks requires only local computation. The valueof a belief parameter associated with a node depends only on the correspondingparameter values associated with its children/parent nodes and the local conditionalprobability matrix. Similarly, it is possible to specify the contexual \support" and\defeat" information in an inheritance network by relating the meanings of a nodeand its neighbors using local constraints as shown in Section 2.3. Furthermore,the belief propagation algorithm requires a number of passes through the beliefnetwork for convergence to a stable assignment. In contrast, the local semanticsof inheritance network can be computed in a monotonic fashion in just two passes,because arcs of the form :p to q (resp. :q) are absent, as described in Appendix B.In tree-structures, the meaning of a tree can be de�ned in terms of the meaningsof its \immediate" subtrees, which are always disjoint from each other. In the caseof DAG structures such a recursive decomposition is not always possible. Howeverthe concept of locality can still be de�ned using the notion of neighborhood asdescribed below. (These ideas were �rst reported in [23]. They are reproduced herefor the sake of completeness.)2.6.1 Locality in Inheritance NetworksWe can formalize the notion of local semantics in the context of inheritance networksby making precise the statement | the meaning of a node is constrained only bythe meaning assigned to the nodes in its immediate neighborhood.Let � be a preferential inheritance network . A node q is a neighbor of a node pif there is a direct arc between p and q. A node-environment �p corresponding tothe node p of an inheritance network � is the subnetwork of � consisting of the nodep and all its neighbors. A speci�city relation wrt a node p is a binary asymmetricrelation among the arcs of the node-environment of p. We say that there exists anisomorphism between the node-environments �1p and �2q if there exists a bijectionbetween the nodes of �1p and �2q , such that the adjacency relation among the nodes24

and the speci�city relation among the arcs of the node-environments are preserved .For our purposes, a semantics of inheritance networks is an assignment of sets tothe nodes of the networks that satisfy a set of constraints C. (See Section 2.3 foran example.) When the set of constraints C is implicit in the discussion, we justsay that the semantics of a network is an assignment that satis�es the network. Asatisfying assignment to the network � is denoted by M�, and the meaning thatM� assigns to a node p of � is denoted byM�(p). The notationM�[M�(p) S]stands for the assignment of sets to the nodes of � obtained fromM� by replacingM�(p) with S.De�nition 3 A semantics of inheritance networks exhibits the property of localityif for any pair of networks �1 and �2 containing isomorphic node-environments�1p and �2q, whenever the meaning assigned to all the neighbors of node p in �1p isidentical to the meaning assigned to the \corresponding" neighbors of node q in �2q ,it is the case that the assignmentM2�[M2�(q) M1�(p)] to the network �2 satis�esthe network �2.The theories of inheritance given in Section 2.3 have the locality property byde�nition. We further illustrate the concept of locality by showing that the creduloussemantics [39] is nonlocal . Figure 6(1) and Figure 6(2) depict two networks, andtheir \upward" meanings [39]. (Note that this example has been chosen to illustratethe importance of a \suitable view" of the semantics to ascertain its locality.) Eachnode p is assigned a pair of sets of nodes (p+, p�), where q 2 p+ stands for qpossesses p and q 2 p� for q possesses :p. Figure 6(3) depicts the application ofthe locality condition to p. The assignment to the network in Figure 6(3) doesnot satisfy the network according to Touretzky's semantics because n and b do notinherit p similarly. This shows that the semantics [39] may be nonlocal.But one may argue that the semantics may in fact be local if only one \looked atit in the right perspective". For instance, one may consider the assignment of sets(p+, p�) to node p to mean the following: q 2 p+ stands for p possesses q (instead ofq possesses p as above) and q 2 p� for p possesses :q (instead of q possesses :p asabove). In that case, one can verify that the satisfying \downward" assignments tothe networks in Figure 6 do not violate locality. But it can be shown that this view25

of the semantics is still nonlocal by considering the assignments to networks givenin Figure 7. Figure 7(1) displays a model of the network, in which it is not knownwhether t is a w. The assignment in Figure 7(2) is a model of the modi�ed network,in which t is a w by virtue of being a b. However, the assignment in Figure 7(3)obtained by the application of the locality condition is not a model according to thesemantics [39].Another consequence of the lack of locality property is the counter-intuitiveinterpretation of certain network topologies [33; 40]. Consider the following example[40]: C's and D's are E's. B's are C's and D's. A's are B's and not D's. Accordingto [39], there is an ambiguity about whether A's are E's ; while the semantics inSection 2.3 accords support to the conclusion | A's are E's , which is intuitivelysatisfactory.It must also be mentioned that a translation is said to be local if incrementalchanges to the network results in incremental changes to its translation [32]. Forinstance, addition of an arc to a network results in the addition of a fact to thetranslation [32]. This is because complex meta-level axioms capture the computationof speci�city relations from the assertions describing the network. On the otherhand, our notion of locality is semantic in that it is based on the relationship amongthe meanings of adjacent nodes. We regard a semantics of inheritance networks aslocal if the meaning of a node in the network depends only on the meanings of itsimmediate neighbours (where the meaning of a node is tacitly assumed to be a pairof sets of nodes and not a pair of sets of paths).It has also been assumed that the speci�city relation used to disambiguate inher-itance con icts wrt a common property by an individual via a set of nodes, dependsonly on the set of nodes, and not on the individual inheritance paths . But one canargue that this is counter-intuitive, as illustrated below. Consider Figure 5(a) [38].A p is a q and a q is an r, and hence, p is more speci�c than r. One can thenview the arcs from a and b to r as being redundant. So, both a and b, inherit :sfrom p. On the other hand, in Figure 5(b) [38], a is not a q. So the arc from a tor can be interpreted as contributing an independent evidence for a to be an r. Inthis situation, there is an ambiguity about whether a is an s, because we cannotdisambiguate the con icting evidence for inheriting s through r and for inheriting26

j ii j@@I��� ���*HHHY���* HHHY��k rq pn(1)a b(fa, bg,;)(fa, b, ng, ;)(fa, b, ng, ;)(fa, b, n, qg, frg) j ii���* HHHY��k rq pa b (fa, b, ng, ;)(fa, b, ng, ;)(fa, n, qg, fb, rg)jn(2)����>ZZZZ} (fa, bg, ;)����DDDO PPi ��1 j ii j@@I��� ���*HHHY���* HHHY��k rq pn(3)a b(fa, bg,;)(fa, b, ng, ;)(fa, b, ng, ;)(fa, n, qg, fb, rg)Figure 6: NonLocality of Touretzky's Semantics Iiii 6��1PPi 6wfb t(1) (fwg,;)(fw, fg,;) (fbg,ffg) iii 6-��1PPi 6wfb t(2) (fwg,;)(fw, fg,;) (fw, bg,ffg)���� iii 6-��1PPi 6wfb t(3) (fwg,;)(fw, fg,;) (fw, bg,ffg)Figure 7: NonLocality of Touretzky's Semantics II27

:s through p. Thus, we see that even though both a and b possess properties p andr, p is more speci�c than r wrt a, while p and r are unrelated wrt to b. This prob-lem stems from the popular view that an explicit arc which is implicitly supportedby the network is regarded as redundant, and not as contributing an additionalindependent support.This example also shows that the semantics of heterogeneous networks [14] andhomogeneous networks [13] are intrinsically nonlocal according to our de�nition.Furthermore, we observe that the speci�city relation wrt node s for individuals aand b are di�erent | node p is more speci�c than node r wrt node a, while node pand node r are unrelated wrt to node b. This motivates us to generalize the notionof locality to ground locality where the speci�city relation is further parameterizedby the \inheriting" node. This extension is similar in spirit to the generalization ofthe notion of strati�cation to local strati�cation in logic programming [31].The ground speci�city relation for node b wrt node p is a binary asymmetricrelation among the arcs of the node-environment of p for b. We say that there existsa ground isomorphism between the node-environments �1p and �2q if there exists abijection between the nodes of �1p and �2q , such that the adjacency relation amongthe nodes, and for all nodes b, the ground speci�city relation among the arcs of thenode-environments are preserved .De�nition 4 A semantics of inheritance networks exhibits the property of groundlocality if for any pair of networks �1 and �2 containing ground isomorphic node-environments �1p and �2q, whenever the meaning assigned to all the neighbors ofnode p in �1p is identical to the meaning assigned to the \corresponding" neighborsof node q in �2q, it is the case that the assignment M2�[M2�(q) M1�(p)] to thenetwork �2 satis�es the network �2.In conjunction with the speci�city notion described in Section 2.4, the con-straints in Section 2.3.3 capture the skeptical semantics [13]; and the constraintsgiven in Section 2.3.6 capture the credulous semantics [15]. Thus, these theoriesare ground local according to our de�nition. Similarly, the family of inheritancetheories [35; 36] are ground local. However, the semantics [39] is still ground non-local as it is impossible to de�ne a satisfactory ordering on the in-arcs of node p28

in Figure 6(2), and on the out-arcs of node t in Figure 7(1). In Figure 6(2), theorderings �ap and �bp for the individuals a and b respectively cannot be independentbecause of coupling; while in Figure 7(1), a suitable ordering �at cannot exist.2.7 Inheritance AlgorithmsIn Appendix B, we give an inheritance algorithm to compute a minimal model of thenetwork according to the speci�cation in Section 2.3.7. The inheritance algorithmscorresponding to other local theories given in Section 2.3 can be obtained by simplemodi�cations to portions of this algorithm.We have also designed special-purpose algorithms to compute� whether or not an individual n inherits a property p.� whether or not an individual n inherits a property :p.� all properties possessed by an individual n.� all individuals that possess a property p.The general strategy for designing these special purpose algorithms is as follows:We take the whole inheritance network, and mark relevant portions of the networkby looking at the form of the query. To compute all the properties of an individuali, we mark all the ancestors of i; to compute all individuals that possess a propertyp, we mark all descendants of p. To determine whether an individual i possesses p,we temporarily mark all the ancestors of i �rst, and then assign permanent marksto those nodes that are descendants of p too. The marking procedure gets a littlecomplex, when the inheritance algorithm supports contrapositive reasoning. Thegeneral inheritance algorithm is invoked only on this smaller marked network toanswer the query.Here, we brie y describe the computational complexity of these inheritance al-gorithms.After a careful analysis, it is clear that a realistic characterization of their com-plexity should be given in terms of parameters like the total number of nodes n,the total number of edges e, the depth of the network h, the maximum number of29

children c, the maximum number of descendants d etc. In general, the maximumsize of the sets computed as the meaning of a node is O(n). But, in the absence ofcontrapositive reasoning, the maximum size of the sets that arise as the meaningof a node is only O(d). The worst-case time complexity for computing a minimalmodel can be obtained by observing the following facts: The cost of the initializa-tion step is O(n); that of degree calculation is O(e); that of determining if q+(r)holds of two children q and r of the same node p is O(d); that of computing thespeci�city relation �p at each node p is O(c2d); that of computing the meaning ofa property node is O(c2d2). Thus, the time complexity of the inheritance algorithmis O(n c2d2). Similarly, one can argue that the parallel time complexity of thealgorithm is O(h c2d2).In contrast, computing credulous expansion according to [10; 39] is NP-hard.However, the skeptical semantics [13] and the inheritance theories [36] are polynomial-time computable. To be precise, the time complexity of the parallel marker passingalgorithm [13] for determining whether p inherits q is proportional to the depthof the network and the number of con icted nodes. (A direct comparison of thisresult with our inheritance algorithm is di�cult because of the di�erences in theform of the query and the underlying machine architecture, which, for instance,permits concurrent writes and bit-vector representation of sets etc.) Furthermore,given a polynomial-time algorithm to determine speci�city, a polynomial-time algo-rithm to compute ideally skeptical semantics (i.e., the intersection of all credulousextensions) can be obtained [17; 35].3 Conclusions and Future WorkA satisfactory solution to the inheritance problem is necessary to build a largesystem that can represent and utilize human experience and expertise for problem-solving in diverse areas like medical diagnosis, intelligent query answering, planning,decision-making, and common-sense reasoning about the world in general. In thispaper, we proposed a family of local inheritance theories that are conceptually sim-ple and computationally e�cient. The design space for inheritance reasoners [40]has been mapped out, and a number of choices available to the designers of such30

systems considered. It is emphasized that tractability and e�ciency aspects areimportant for the design of viable inheritance reasoners [34]. Because the languageof inheritance network is not su�ciently expressive to incorporate the various in-tuitions about inheritance, we propose to investigate alternative annotations to thenodes and to the arcs of the network to accomodate these various interpretations ofinheritance networks in a uni�ed framework. Such a language, we believe, will ben-e�t a knowledge engineer because it will permit him to decide what knowledge canbe encoded in the system, and give him understandable formal guarantees aboutthe quality of the conclusions that will be generated [2].References[1] A. V. Aho and J. D. Ullman, Principles of Compiler Design, Addison-WesleyPublishing Co., Inc., 1977.[2] F. Bacchus, A modest, but semantically well-founded, inheritance reasoner,In Proceedings of the Eleventh International Joint Conference on Arti�cialIntelligence, 1104-1109, 1989.[3] D. Billington, K. De Coster, and D. Nute, A modular translation from defeasi-ble nets to defeasible logics, Journal of Experimental and Theoretical Arti�cialIntelligence, 2, 151-177 (1990).[4] J. P. Delgrande, An approach to default reasoning based on a �rst-orderconditional logic: Revised report, Arti�cial Intelligence, 36, 63-90 (1988).[5] D. Etherington and R. Reiter, On inheritance hierarchies with exceptions,In Proceedings of the Second National Conference on Arti�cial Intelligence,104-108, 1983.[6] S. E. Fahlman, NETL: A System for Representing and Using Real-WorldKnowledge, The MIT Press, Cambridge, MA, 1979.[7] H. Ge�ner and J. Pearl, A framework for reasoning with defaults, In Proceed-ings of Society for Exact Philosophy Conference, 1988.31

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[28] L. Padgham, A model and representation for type information and its use inreasoning with defaults, In Proceedings of the Seventh National Conferenceon Arti�cial Intelligence, 409-414, 1988.[29] L. Padgham, Negative reasoning using inheritance, In Proceedings of theEleventh International Joint Conference on Arti�cial Intelligence, 1086-1092,1989.[30] J. Pearl, Probabilistic Reasoning in Intelligent Systems: Networks of PlausibleInference, Morgan Kaufmann Publishers Inc., San Mateo, 1988.[31] T. C. Przymusinski, On the declarative semantics of strati�ed deductivedatabases and logic Programs, In Foundations of Deductive Databases andLogic Programming , J. Minker (ed.), Morgan Kaufmann, 193-216, 1988.[32] H. Przymusinska and M. Gelfond, Formalization of inheritance reasoning inautoepistemic logic, Fundamenta Informaticae, 13(4), 403-444 (1990).[33] E. Sandewall, Nonmonotonic inference rules for multiple inheritance with ex-ceptions, Proceedings of the IEEE , 74(10), 1345-1353, 1986.[34] B. Selman and H.J. Levesque, The tractability of path-based inheritance,In Proceedings of the Eleventh International Joint Conference on Arti�cialIntelligence, 1140-1145, 1989.[35] L. A. Stein, Skeptical inheritance: Computing the intersection of credulousextensions, In Proceedings of the Eleventh International Joint Conference onArti�cial Intelligence, 1153-1158, 1989.[36] L. A. Stein, Resolving ambiguity in nonmonotonic inheritance hierarchies,Arti�cial Intelligence, 55, 259-310 (1992).[37] R. Thomason, J. Horty, and D. Touretzky, A calculus for inheritance in mono-tonic semantic nets, in: Z. Ras and M. Zemankova (eds.), Proceedings of theSecond International Symposium on Methodologies for Intelligent Systems ,280{287, 1987. 34

[38] R. Thomason and J. Horty, Logics for inheritance theory, In Non-MonotonicReasoning , M. Reinfrank, J. de Kleer, M. Ginsberg, and E. Sandewall (eds.),Springer-Verlag, 1989.[39] D. Touretzky, The Mathematics of Inheritance Systems , Morgan Kaufmann,Los Altos, 1986.[40] D. Touretzky, J. Horty, and R. Thomason, A Clash of intuitions: The currentstate of nonmonotonic multiple inheritance systems, In Proceedings of theTenth International Joint Conference on Arti�cial Intelligence, 476-482, 1987.A Proofs of TheoremsDe�nition 5 The degree of a node in a preferential inheritance network is thelength of a longest directed path from a leaf node to the node. That is, the degreeof a leaf node is 0. The degree of a nonleaf node is one more than the maximum ofthe degrees of its children.De�nition 6 The depth of a node in a preferential inheritance network is thelength of a longest directed path from the node to a terminal node. (A terminalnode is a property node with no out-arcs.) That is, the depth of a terminal node is0. The depth of a nonterminal node is one more than the maximum of the depthsof its parents.Lemma 1 There exist directed graphs that are unsatis�able.Proof: Consider the cyclic network in Figure 8 from [15]. There is an arc from pto q so q+(p) must hold. In the absence of any prospect for a con ict at nodes rand t, p can propagate from node q to nodes r and t. Hence, r+(p) and t+(p) musthold. There is an arc from t to p, hence, p+(t) also holds. Thus, according to ourde�nition of speci�city, p is more speci�c than t and t is more speci�c than p.Now consider the individual node n. p+(n) must hold because there is an arcfrom n to p. As both positive and negative arcs are incident on q there is potentialfor a con ict to occur. In particular, we want to determine whether or not n is a q.35

We analyze the two possible scenerios | the case when t+(n) holds and the casewhen t+(n) does not hold. If t+(n) holds then both q+(n) and q�(n) cannot hold.This is because, both hp; qi �q ht; qi and ht; qi �q hp; qi hold, and hence, there isneither a maximal support for q+(n) nor a maximal support for q�(n). If q+(n)does not hold, then there is no support for r+(n), and hence, there is no supportfor t+(n). Thus t+(n) cannot hold which is a contradiction. But, if we assumethe other possibility that t+(n) does not hold, then there is clearly no chance of acon ict at q wrt n. Hence, q+(n), r+(n) and t+(n) must all hold which is contraryto our assumption. So the network in Figure 8 is unsatis�able according to oursemantics. �Theorem 1 Every preferential inheritance network has a minimal skeptical model.Proof: We prove that there exists an interpretation that satis�es the constraintsgiven in Section 2.3.2. (One can modify this interpretation trivially to construct amodel that satis�es constraints in Section 2.3.3.)Let C denote the set of constraints, Pi denote the set of property nodes withdegree i or less, and Ci the set of constraints corresponding to nodes in Pi. Recallthat for every property node p in the network, there are two i� -constraints |corresponding to p+ and p� respectively. We can impose a natural ordering onC, based on the DAG structure of the network. In particular, C can be strati�edbased on the degree of the property node associated with the constraint. We show,by induction on degree, that a model M for C can be constructed from the chainof partial models M0, M1, : : :that satisfy C0, C1, : : : respectively. In particular,the meaning assigned to nodes in Pi by Mi and M are identical. In what followswe assume that the local speci�city relation required at any node is completelydetermined by the inheritance relation computed for its children thus far. Thecomplete proof involves a straight-forward simultaneous induction.Basis: C0 is satis�ed by the interpretationM0 that maps every property node p ofdegree 0 to (;,;). This is because the right hand side of the constraints correspondingto such nodes is identically false. In addition, for the same reason, M must alsoassign the same meaning to nodes in P0.Induction Hypothesis: Assume that Ci is satis�ed by the interpretationMi andthe meaning assigned to nodes in Pi byMi andM are identical.36

Induction Step: We show thatMi can be extended toMi+1 that satis�es Ci+1.First of all observe that the constraints in the set Di+1 = Ci+1 � Ci, correspond toproperty nodes of degree i+1. The right hand side of the constraints in Di containreferences only to nodes in Pi because inheritance networks are acyclic. This impliesthat meaning of each node p of degree i+1 can be completely determined using theconstraints for p in Di+1, the partial modelMi, and the local speci�city informationw.r.t. children of p (which is assumed to be known from the simultaneous inductionhypothesis about the the meaning of nodes in Pi). Thus the partial model Mi+1extendsMi by interpreting nodes in Pi+1�Pi such that they satisfy Di+1. Observethat these new assignments are not relevant to the satis�ability of the constraintsin Ci. Thus we see that a model M satisfying C can be constructed iteratively bydetermining [iMi as indicated. �Theorem 2 Every preferential inheritance network admits a unique minimum skep-tical model.Proof: We show that the constraints in Section 2.3.2 can be satis�ed in only oneway, by contradiction.Assume that there exist two distinct minimal models |M1 andM2 | satis-fying the constraints in Section 2.3.2 for an inheritance network. M1 andM2 mustagree on the meaning of nodes in P0 because the right hand side of the constraintscorresponding to such nodes is identically false. So, if M1 and M2 di�er, thenthere must exist a smallest i such that they di�er on a property node p in Pi wherei > 0, and they agree on all nodes in [j<iPj . But this is a contradiciton becausethe meaning of the nodes in Pi is completely determined by the meaning of thenodes in [j<iPj and the local constraints in Di. Hence every inheritance networkhas a unique minimum skeptical model. �Theorem 3 Every preferential inheritance network has a minimal credulous model.Proof: We prove that there exists an interpretation that satis�es the constraintsof Section 2.3.7. (This proof can be modi�ed trivially to demonstrate the same forthe constraints in Section 2.3.6.) 37

Analogously to the skeptical case, we try to impose a natural ordering on the setof constraints C based on the DAG structure of the network. However this is morecomplicated in the credulous case because, the body of a constraint can refer to themeaning of the node and its parents in addition to that of the children. So, for thepurpose of stratifying the constraints, we associate with every property node p threepredicates/sets | p+f , p�f , and p�b . Intuitively, p+f (resp. p�f ) is that component ofp+ (resp. p�) which is determined by the children of p, while p�b is that componentof p� which is determined by the parents of p. (The latter component is due tocontrapositive reasoning.) The set Sd denotes the set of predicates of the form p+f ,p�f corresponding to property nodes with degree d , and the set Smaxdegree+h denotesthe set of predicates of the form p�b corresponding to property nodes of depth h.The constraints associated with a property node p of degree d+ 1 and depth h+ 1can be expressed abstractly as follows where cond is some condition that dependson the bracketed predicates:p+f (n) i� :p�f (n) ^ cond[[j�dSj ](n):p�f (n) i� :p+f (n) ^ cond[[j�dSj ](n)::[p+f (n) ^ p�f (n)]p�b (n) i� cond[[j�dSj [ [j�hS(maxdegree+j)](n)::[p+f (n) ^ p�b (n)]Let Ci correspond to the constraints associated with the components in [j�iSj . Ccan be strati�ed using the degree and the depth of the node corresponding to theleft hand side of a constraint as follows, where Di+1 = Ci+1 � Ci:D0 � D1 � : : :� Dmaxdegree+maxdepthWe show, by induction on the strati�cation level of a constraint, that a modelM forC can be constructed from a chain of partial modelsM0,M1, : : : ,Mmaxdegree + maxdepththat satisfy C0, C1, : : : , Cmaxdepth+maxdepth respectively. In particular, we see thatthe components of the meaning assigned to a node byM can be obtained directlyfrom the corresponding components assigned byMi.38

Basis: C0 is satis�ed by the interpretation M0 that maps each predicate/setin S0 to ;. This is because the right hand side of the corresponding constraint isidentically false.Induction Hypothesis: Assume that Ci is satis�ed by the interpretationMi.Induction Step: We show thatMi can be extended toMi+1 that satis�es Ci+1.First of all, let i + 1 � maxdegree. Observe that the right hand side of theconstraints in Di+1 contains an explicit reference to p�f and the cond[] refers to thepredicates in [j�iSj . By induction hypothesis, the meaning of cond[] is �xed. Ifcond[](n) corresponding to p+f (n) (resp. p�f (n)) is true then p+f (n) (resp. p�f (n))can hold. But the disjointness condition dictates that if both p+f (n) and p�f (n)can hold, then only one of p+f (n) or p�f (n) may hold | the choice between thembeing arbitrary. This implies that meaning of each node p of degree i + 1 can becompletely determined using the constraints for p in Di+1, the partial model Miand the local speci�city information w.r.t. children of p (which is assumed from thesimultaneous induction hypothesis about the the meaning of the children.) Thusthe partial model Mi+1 extends Mi by interpreting predicates in Si as described.Note also that these new assignments are not relevant to the satis�ability of theconstraints in Ci, and hence,Mi+1 is a model for Ci+1.Now consider the remaining cases when i+ 1 � maxdegree. Observe that theright hand side of the constraints in Di+1 contains the cond[] that refers only topredicates in [j�iSj . By induction hypothesis, the meaning of cond[] is �xed anddetermines the predicates in Si+1. Also note that whenever p+f (n) holds cond[](n)does not. So, the disjointness condition is automatically satis�ed. Thus the inter-pretationMi+1 extendsMi to a model of Ci+1.So a model M of C can be constructed iteratively by determining [iMi asindicated. �A study of these proofs shows that Theorem 3 implies Theorem 1. In particular,the skeptical model Mmaxdegree is closely related to the credulous partial modelMmaxdegree. 39

B Design of an Inheritance AlgorithmIn Section 2.3, we gave a precise logical speci�cation for when an individual ninherits a property p. We will now develop systematically, an algorithm to computea minimal model of the network according to the speci�cation in Section 2.3.7. Theproof of correctness of this algorithm follows from the results in Appendix A.To specify precisely the order in which the meaning of the nodes is computed,we make the following de�nition. An arc from an individual node to a propertynode is considered strict , while an arc from a property node to a property nodeis considered defeasible. A path is de�ned as a sequence of defeasible arcs in thenetwork. The degree of a node is the length of the longest path into that node.We describe the precise ow of individuals through the network acquiring prop-erties now. There is both a forward (or upward) ow of individuals in the directionof the arrows, and a backward (or downward) ow of individuals in the reversedirection because of contraposition. The forward ow contains two components.One subcomponent is due to individuals inheriting properties positively throughthe positive arcs and the other subcomponent is due to individuals inheriting prop-erties negatively through the negative arcs. The backward ow also contains twocomponents, each contributing to the inheritance of properties negatively. One sub-component is due to individuals owing backwards through positive arcs by virtueof possessing a property negatively, and the other subcomponent is due to indi-viduals owing backward through negative arcs by virtue of possessing a propertypositively.To be precise, the positive component of the meaning of a node p, i.e., theset p+, is computed from the positive components of the meaning of its \positive"children minus the negative subcomponent of the meaning of p, i.e., a subset of p�computed due to \more speci�c" classes. The negative component of the meaningof a node p, i.e., the set p�, is the union of two subcomponents. The forward owsubcomponent of p� is computed from the positive component of the meaning ofall its \negative" children minus the positive subcomponent of p, i.e., the subsetof p+ computed due to \more speci�c" classes. The backward ow subcomponentof p� is computed from the positive component of the meaning of all its parentsthrough the negative arcs, and from the negative component in the meaning of all40

its parents through the positive arcs.In what follows, we present a set-based algorithm for computing a minimalmodel of an inheritance network developed above. With each node n we associate aconstant n, and with each property node p we associate a pair of sets (p+E ; p�E ), wherep+E (resp. p�E ) represents the set of individuals that inherit p (resp. :p) in a model E .Intuitively, the constant i associated with a node i2 I represents an individual in theworld, while the constant p associated with a node p2 P represents a \hypothetical"typical individual of the collection that possesses property p. The computation iscarried out for all individuals simultaneously. In case of an ambiguous network,one of several possible minimal models is chosen non-deterministically. (We usethe keyword pardo explicitly in some of the for-loops, to indicate that the bodyof the loop can be executed for each value of the \index" variable, in parallel,simultaneously.)We formalize the bottom-up processing of nodes in the ascending order of theirdegrees. The constant max-degree stands for the maximum degree of a node.beginbegin /* initialize-facts. */foreach property node p pardo (p+; p�) := (fpg; ;);foreach positive arc hn; pi do p+ := p+ [ fng;foreach negative arc hn; pi do p� := p� [ fng;end;begin /* compute-forward- ow-component. */foreach degree d := 1 to max-degree doforeach node p of degree d pardocompute-forward- ow-component-of-meaning-of-p;end;begin /* compute-backward- ow-component. */foreach degree d := max-degree { 1 downto 0 doforeach node p of degree d pardocompute-backward- ow-component-of-meaning-of-p;end;end; 41

We need some notation to re�ne this further. Recall that the arcs into each nodeare ordered. Let graph Hp stand for the diagram corresponding to the relation �p.The nodes of Hp are children of p. There is an edge from a node r to a node q inHp if either q+(r) or hq; pi �p hr; pi. If q+(r) is true then r is more speci�c thanq. Also, we refer to a node in relation to another node w.r.t. Hp, instead of thewhole network, by pre�xing a term with an \Hp-". For instance, an edge in Hp willbe referred to as an Hp-edge, the degree of q in Hp as the Hp-degree etc. Let p+q(resp. p+q ) denote the contribution of node q towards p+ (resp. p�) and p+ (resp.p�) denote the set of individuals that can possibly inherit p (resp. :p). The set ofindividuals that inherit p ambiguously is p+ \ p�. To choose a speci�c minimalmodel, we partition the set p+ \ p� into two sets p+a and p�a . The set p+a (resp.p�a ) stands for the set of those individuals that inherit p ambiguously, but in thechosen model E they inherit p (resp. :p).begin /* compute-forward- ow-component-of-meaning-of-p. */foreach Hp-degree d := 0 to Hp-max-degree dobeginforeach q of degree d and positive arc hq; pi pardobegin /* compute-contribution-of-q-to-p+. */ p+q := q+;foreach negative arc hr; pi such that hr; qi is an Hp-edge dop+q := p+q � r+E ;end;foreach r of degree d and negative arc hr; pi pardobegin /* compute-contribution-of-r-to-p�. */ p�r := r+;foreach positive arc hq; pi such that hq; ri is an Hp-edge dop�r := p�r � q+E ;end;/* compute-individuals-that-can-inherit-p" */ p+ := p+ [ Sq:hq;pi2E+ p+q ;/*compute-individuals-that-can-inherit-:p; */ p� := p� [ Sr:hr;pi2E� p�r ;/* choose a minimal model in case of ambiguity. */ partition(p+ \ p�; p+a ; p�a );/* compute-individuals-that-may-inherit-p;/* p+e := p+ � p�a ;/*compute-individuals-that-may-inherit-:p;*/ p�e := p� � p+a ;end; 42

end;begin /* \compute-backward- ow-component-of-meaning-of-p" */foreach positive arc hp; qi do p�e := p�e [ (q�E � p+e );foreach negative arc hp; qi do p�e := p�e [ (q+E � p+e );p+E := p+e ;p�E := p�e ;end;Inheritance algorithms to compute minimal models of other theories in Sec-tion 2.3 can be obtained by modifying this algorithm. For example, the theory inSection 2.3.6 can be obtained by deleting \compute-backward- ow-component-of-meaning-of-p". Similarly, one can compute the skeptical semantics by modifying theprocedures \compute-individuals-that-may-inherit-p" and \compute-individuals-that-may-inherit-:p" in the obvious way.

43

� ��� ��� ��� �� bb � �������� AAAAK����*����*������: t rqpn Figure 8: Unsatis�able Cyclic Network44