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<Paper uid="C88-1078">
  <Title>Inheritance in Hierarchical Relational Structures</Title>
  <Section position="3" start_page="0" end_page="384" type="intro">
    <SectionTitle>
1. Introduction
</SectionTitle>
    <Paragraph position="0"> The use of structures for representing knowledge has long been acknowledged as a vital tool in AI. A consequence of the use of structures is the identification of certain kinds of hierarchically organised sub-structures. These have many useful purposes, but one of particular significance is that of avoiding repetition, by storing information at one place in the hierarchy, and then using the hierarchical structure to infer that the property is inherited by many other components of the structure. In addition to the savings in storage that this inheritance over hierarchies can offer, there is the possibility of using the hierarchy to infer new information, so that the process becomes not just a means of saving space, but also of generating new knowledge.</Paragraph>
    <Paragraph position="1"> The form of inheritance most frequently studied is what we call &amp;quot;downward inheritance&amp;quot;, because it exploits the passage from the general to the particular. There has been a considerable amount of work done about inheritance between objects via total inclusion relations, for the very good reason that this form is always valid /Findler 1979/. This form is sometimes confused with downward inheritance betwen sets, which, as we briefly discuss below, is much more problematic.</Paragraph>
    <Paragraph position="2"> We will concentrate on total hierarchical relations, between objects, classes and actions. The inheritance techniques that we consider are thus completely valid. The use of partial hierarchical relations introduces elements of plausibility which are beyond the present context. Touretzky provides a foundational analysis of partial hierarchies /Touretzky 1986/, while their use as the basic structure for semantical analogy is described in Garigliano and Long/Garigliano and Long 1988/.</Paragraph>
    <Paragraph position="3"> The concept that other relations, beyond set membership and inclusion, may have inferential properties has been investigated by Schank /Schank 1977/ and especially Wilensky /Wilensky t980/. Their inferential rules, however, tend to capture specific aspects of real world interaction: it would thus be difficult to con-A typical example of the use of downward inheritance is the argument cats are vertebrates, vertebrates have back-bones, so cats have back-bones. We have also identified other hierarchical relations for which downward inheritance is valid: embroidering is a special form of sewing (or, embroidering &amp;quot;specialises&amp;quot; sewing), sewing requires skill, so embroidering requires skill.</Paragraph>
    <Paragraph position="4"> The most common use of inheritance - so common that it is often the only inheritance rule discussed - is that which can be expressed abstractly as: every element in A has property P, B is a subset of A, so all elements of B have the property P. This is the rule exemplified by the cats and vertebrates above.</Paragraph>
    <Paragraph position="5"> A third kind of inheritance hierarchy is based on properties of sets, rather than of individuals within the sets. For example, I can count the set of first-division footballers, Liverpool United is a subset of first division footballers, so I can count the set of Liverpool United players. It is clear that this is not the same as the cats and vertebrates example, since the claim is about the set as a whole, rather than about the individual members.</Paragraph>
    <Paragraph position="6"> It is important to observe that the downward inheritance rule is not universally valid. To see this, consider the example: the Tory government was elected by the British people, tile Scots are a subset of the British, so the Tory government was elected by tile Scots. It is clear that this example leads to a false conclusion. We will not attempt to explain all the conditions under which a relation can be expected to have the downward inheritance rule for classes in this work, but we do note that one important part of such conditions can be the &amp;quot;homogeneity&amp;quot; of the class in the first relation with respect to the given property and the related entity /Garigliano and Long 1988/. By homogeneity we refer to a measure of the evenness of distribution of a property within a set.</Paragraph>
    <Paragraph position="7"> The downward inheritance rule for actions is also dependent on certain conditions being satisfied by the first relation, as can be seen by the example: rolling pins are used in cooking, frying specialises cooking, so rolling pins are used in frying. This is plainly false, so we may infer that there is some condition which &amp;quot;specialises&amp;quot; satisfies, but is violated by &amp;quot;are used in&amp;quot;. The solution to this particular example is actually found in the discussion below. There are other examples for which the solution is not quite so readily identified, but it is possible that tile solution is analogous to that for classes, in adopting some measurement of horn ogeneity.</Paragraph>
    <Paragraph position="8"> A more dramatic example in which downward inheritance fails is the following: cows eat plants, cacti are a subset of plants, so cows eat cacti. The inference is certainly false, but this example is of considerable interest since it appears to follow the same pattern as the cats and vertebrates example. If we follow the direction of Schank and Wilensky, adopting a set of primitives from natural language as the basis of our knowledge representation, we cannot, it appears, identify the inheritance properties of those primitives by a simple syntactic check. This follows from the observation that the syntactic pattern of the cats and vertebrates  example was essentially identical to that of the cows ~md plants, yet the inheritance rule is valid only in the first case.</Paragraph>
    <Paragraph position="9"> 3. Upward inheritance We cm~ intuitively understand upward inherit~mce as a form of inheritanc(: that goes from the specific to the general. An exampie of such inheritance is the argument: A is smaller than B, B is a subset of C, so A is smaller than C.</Paragraph>
    <Paragraph position="10"> Further examples are: a camera creates pictures, pictures are representalions, hence a canrera creates representations. A pan is for cooking, cooking is a specialization of processing food, hence a pan is for processing food.</Paragraph>
    <Paragraph position="11"> Of coorse, the higher we go up the hierarchy, tile less usefitl the information derived may appear: for example, if we substitute Doing Something for Processing Food, the above inference is still valid, but not very nsefid. It is impo~tant to note, however, that when ihe relation is an upward one, then the deriving inference is valid, not simply plausible. There is no possibility of this inferen,-e being some kind of induction: the explauation for it is to be fontal elsewhere.</Paragraph>
    <Paragraph position="12"> As we mentioned before, .just as not all relations allow downward inheritance, so too not all relations allow upward inheritance. Here are some examples when the inference fails: John is allergic to cats, cats ~u'e vertebrates, hence John is allergic to vertebrates.</Paragraph>
    <Paragraph position="13"> 1 can coun\[ the size of a football team, a football temu is a sub. set of the world population, hence I can couut the size of the world population.</Paragraph>
    <Paragraph position="14"> Cats avoid swimming, swimming specifies moving, hence cats avoid moving.</Paragraph>
    <Paragraph position="15"> All these e:~amples emphasize the hnpossibility of nsiug a syntactic check to decide which relations offer a hierarchical inheritance, or, if they offer such an inheritance, which direction it is valid for.</Paragraph>
    <Paragraph position="16"> The issue turns around the particular relalions used; this clemly calls for au analysis of the underlying structure of these relations. null 4. When inheritance is valid We must now attentpt to identify what property of a relation it is that enables it to be used for upward or downward inheritance.</Paragraph>
    <Paragraph position="17"> First let us explore upward inheritance. There is a strong clue available to us in our search in the following example: if x is a member of A and A is a subset of B then x is a member of 11.</Paragraph>
    <Paragraph position="18"> Ilere we scc a set theoretic property which actually obeys the upward inhedtauce rule. This is a very important example - it is not hard to see why it works. The reason that the property is inherited is i:hat although the relation &amp;quot;is a member&amp;quot; relates an object to a class, at the stone time it singles out a very specific part of that class. This part of the class must be carried through to any new class containing the original one. Thus, the significant feature of the relation, &amp;quot;is a member&amp;quot;, is that it carries an implicit act of restricting the part of the class to which it refers. Consider a further example: John owns a cat, cats are animals, so John owns an animal. This is an instance of ~l upward inheritance rule that works, m~d the relation is &amp;quot;owns a&amp;quot;. If we examine this relation we find that it has Ihe imalogons property for objects that &amp;quot;is a mmnber&amp;quot; has for classes -it implicitly restricts that object to which it refers out of all tile objects. John does not own all cats, but only a single cat, in tile same way that not every element in A is x, but only one. Again, a frying-p~m is fi~r frying, frying specialises cooking, so a fi'ying-pan is for cooking is an example of an upward inheritance over action.,;, using &amp;quot;is for&amp;quot;. Here, too, we find that what is being said implicitly is that there is a conceivable instance of frying for which one conld use a frying-pan. It does not mean that fi'ying must always be done with a frying-pan - we could use a deep-fat fryer or a wok tor example. Thus, when we extend frying to cooking we are actuo ally referring to the same instance of frying ill which a frying-pan could be used, and ushlg the fact that this is also an instance of cooking because frying specialises cooking.</Paragraph>
    <Paragraph position="19"> So, we have three examples of relations which have the upward inheritance property mid seem 1o have an analogous property -for objects this property is that the relation specilies a particular object of all the possible objects in a class. For classes the pro-petty is that the relation highlights a subset of the class (in tile &amp;quot;is a member&amp;quot; example this subset is'{x} ) and for actions the property is that the relation s~ccities a p,'u'ticular instance (though possibly hypothetical) of an action in which a certain condition hohls.</Paragraph>
    <Paragraph position="20"> After a more formal analysis of these examples (considered in detail in /Long and Garigliano 1988/) we lind tha! for a tlansitire and rellexive hierarchical relation, such ~, &amp;quot;subset&amp;quot; or &amp;quot;speci-alises&amp;quot;, a second relation has the upward inheritance property it&amp;quot; it contains an implicit existential quantifier - lhat is. if the relation implies a property of a limited parl of the class of objects to which it refers, in sonte sense. This sense has been made more formal in /Long and Garigli~mo 1988/. Furthermore, this condilion is shown to be both necessary and sntlicienl.</Paragraph>
    <Paragraph position="21"> Alter seeing tile pattern for npward inheritance, it is not difficult to lind tile pattern fk~r downward inheritance. In this case, the inheritance works for reflexive and transitive hierarchies if and only if the inherited property contai~s an implicit universal quantifier, in a sense which has been made formal. For example, when we say &amp;quot;vertebrates have backbones&amp;quot;, we a~:t~ally mean all vertebrates have a backbone.</Paragraph>
    <Paragraph position="22"> The analysis indicates that there is a very simple role lha.I allows \]us to convert any relation into one for which tile upward int~crii tance rule operates and, conversely, for checLin,~,, it&amp;quot; ~n inhcri lance rule will work for a given relation. F For exeanple, suppose we take tile relation used above, John is allmgic to tats. Now, if we modify &amp;quot;is allergic to&amp;quot; according t~ the analysis, we may build the new relation, &amp;quot;is allergic to some&amp;quot;, Thus, John is allergic to some cats, cats are vertebrates~ so John is allergic to some vertebrates. It is quite clear that this new relation has the property of upward inheritance, unlike the original relation.</Paragraph>
    <Paragraph position="23"> As an example of tile process of checking whether upward inheritance will apply to a given relation, consider the following. We have the relation &amp;quot;is smaller than&amp;quot;, at)plied between sets, as used in ~m example above. In order to confirm that upward inheri.truce call be applied with this relation we must lind the construction that yields &amp;quot;is smaller than&amp;quot;, given some starting relation. Consider the relation &amp;quot;one up on&amp;quot;, defined by: A is one up on B ifflAl+ 1 = IBt Now, &amp;quot;is smaller thmL&amp;quot; can be defined by: A is smaller than B fit&amp;quot; there is ,,;ome subset of B which is one up on A.</Paragraph>
    <Paragraph position="24"> Thus, &amp;quot;is smaller .than&amp;quot; has upward inheritance with respect to the subset hierarchy.</Paragraph>
    <Paragraph position="25"> ltl this case, the relation is a mathematical one, so proving that the inheritance is valid is not difficult, even without tile insighl we have gained, However, for more general relations, borrowed directly from natural language (in a Schank or Wilensky style for primitives) the insight provides the only tool we have to for-really prove an inheritance property.</Paragraph>
    <Paragraph position="26">  There is a way o~ modifying a relation with an hnplicit existential quantifier so that it becomes a relation with an implicit universal quantifier, in addition to simply adding the universal quantifier * on top of the relation - we can restrict the domain of the relation by inserting &amp;quot;the, in front of each group of objects in the domain. For example: Johri eats fruit, apples are an~ong fruit so John eats apples is not valid. Once we modify the domain, however, we obtain: John eats the fruit, the apples are among the fruit so John eats the apples.</Paragraph>
    <Paragraph position="27"> Now~ the use of &amp;quot;the&amp;quot; indicates a specification of an exact group of fruit, F say, and of apples, say A, so &amp;quot;the apples are among the frnit&amp;quot; now means: every member of the set of apples, A, is in the set of fruit, F.</Paragraph>
    <Paragraph position="28"> What makes the inference correct is the peculiarity of the use of &amp;quot;the&amp;quot; in conjunction with the relation. It means that the relation is true of every element in the group - so John eats every piece of fluit in F.</Paragraph>
    <Paragraph position="29"> For several hierarchies we have studied there are perfectly raeaningful and uncontrived relations which have neither the upward or downward inheritance rule. For example, consider the following: the Tortes were elected, for by forty per-cent of British, Scots are a snbset of British, so the Tortes were elected by forty per-cent o~&amp;quot; Scots.</Paragraph>
    <Paragraph position="30"> This is false, so the &amp;quot;elected by forty per-cent of&amp;quot; relation does not have downward inheritance over subsets.</Paragraph>
    <Paragraph position="31"> The Tortes were elected by forty per-cent of British, British are a subset of Europeans, so the Tortes were elected by forty per-cent of Europeans.</Paragraph>
    <Paragraph position="32"> Again, this is false.</Paragraph>
    <Paragraph position="33"> There are other hierarchies, such as &amp;quot;has part&amp;quot;, which holds between an object and each of its parts; for which very few relations have any inheritance properties. In this case it is because there are very few things that can be said of an object which mnst be true of all its p~ats, or things which can be said of a part of an object which must be true of the object as well. Positional relations seem to be the only useful relations - all parts of an object mast be where the ohject is, though an object does not have to be where one of its parts is. For example, a tyre must be there where there is a car, but a car does not have to be there where there is a tyre.</Paragraph>
    <Paragraph position="34"> Finally, we must highlight a consequence of these findings, which is that those relations for which upward inheritance holds occur in sentential forms with precisely the same type and structure as those for which the rule fails. The only distinguishing feature that we have discovered and that accounts for every example we have considered is the implicit quantifier within the semantic definition of the relation. This leads us to conclude that the power of this rule of inheritance can only be available following a semantical analysis of the relations involved and will not yieldto auy kind of syntactical or grammatical analysis.</Paragraph>
    <Paragraph position="35"> like &amp;quot;is for&amp;quot; or &amp;quot;causes&amp;quot;, to the mathematical relations such as &amp;quot;is a member of&amp;quot; or &amp;quot;smaller than&amp;quot;.</Paragraph>
    <Paragraph position="36"> We have, in our language, some relations which carry an iraplicit existential quantifier and others which carry an implicit universal quantifier, but we can identify no particular reason that any of these relations should have developed in that way. The impor~ rant thing is that we are now able to build relations for which inheritance will work, and also recognise those relations, amongst those we already have, for which the rule applies.</Paragraph>
    <Paragraph position="37"> The final point we wish to stress is that this analysis strengthens the argument in favour of the semantical approach to inferences.</Paragraph>
    <Paragraph position="38"> As we have pointed out before, the distinction between the reladeg tions which allow upward inheritance, and those which do not, cannot be identified by pt, rely syntactical means, even if that dis u tinction appears evident to the human ear, and easily recognis~ able by a semantical analysis.</Paragraph>
  </Section>
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