Inheritance in Hierarchical Relational Structures 
Derek P. LONG 
Dept. of Computer Science, 
University College London, 
Gower Street, 
London, 
ENGLAND. 
Roberto GARIGLIANO 
School of Engineering and 
Applied Science, 
University of Durham, 
Durham, 
ENGLAND. 
Abstract 
A brief survey is conducted of the inheritance principle - the 
conveyance of properties between components within a hierarch- 
ical relational structure. The standard form of inheritance is con- 
sidered, using the subset (is-a) relation and highlighted as an 
example of downward inheritance. Downward inheritance is 
extended to specialisation of actions, and cases are presented in 
which the rule fails. 
sider them on tile same plane of generality and abstractness as 
the standard hierarchical inheritance rules. Upward inheritance, 
on the other hand, is formally on the same plane as the more 
common downward form. Our analysis shows that the explana- 
tion for this powerful tool resides in one semantical aspect of a 
large class of very common relations. 
2. Downward inheritance 
An alternative and less well-known form of inheritance is intro- 
duced - upward inheritance. Several examples in which upward 
inheritance is valid and others in which it is not valid are given, 
in a treatment highlighting the analogy with downward inheri- 
tance. The validity of the rule, in those case in which it 
operates, is underlined, to distinguish it from induction. 
A brief acconnt is given of tile search for the underlying reasons 
for the validity of inheritance rules and these are then given. 
The solution turns out to be due to a hidden or implicit 
quantifier within the relations that are used. The semantical 
nature of the problem and of its solution are stressed, emphasis- 
ing the impossibility of a purely syntactic analysis and solution to 
the problem. Various points of interest arising from the analysis 
,are listed and discussed. 
1. Introduction 
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 infor- 
mation, so that the process becomes not just a means of saving 
space, but also of generating new knowledge. 
The form of inheritance most frequently studied is what we call 
"downward inheritance", 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 inclu- 
sion 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. 
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/. 
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 "specialises" sew- 
ing), sewing requires skill, so embroidering requires skill. 
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. 
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. 
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 rela- 
tion 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 "homogeneity" 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. 
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 
"specialises" satisfies, but is violated by "are used in". The solu- 
tion 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. 
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 pat- 
tern as the cats and vertebrates example. If we follow the direc- 
tion 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 
384 
example was essentially identical to that of the cows ~md plants, 
yet the inheritance rule is valid only in the first case. 
3. Upward inheritance 
We cm~ intuitively understand upward inherit~mce as a form of 
inheritanc(: that goes from the specific to the general. An exam- 
pie of such inheritance is the argument: A is smaller than B, B 
is a subset of C, so A is smaller than C. 
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. 
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 infer- 
ence 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. 
As we mentioned before, .just as not all relations allow down- 
ward inheritance, so too not all relations allow upward inheri- 
tance. Here are some examples when the inference fails: 
John is allergic to cats, cats ~u'e vertebrates, hence John is aller- 
gic to vertebrates. 
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. 
Cats avoid swimming, swimming specifies moving, hence cats 
avoid moving. 
All these e:~amples emphasize the hnpossibility of nsiug a syntac- 
tic check to decide which relations offer a hierarchical inheri- 
tance, or, if they offer such an inheritance, which direction it is 
valid for. 
The issue turns around the particular relalions used; this clemly 
calls for au analysis of the underlying structure of these rela- 
tions. 
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. 
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. 
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 "is a member" 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, "is a member", is that it carries an impli- 
cit 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 inheri- 
tance rule that works, m~d the relation is "owns a". If we exam- 
ine this relation we find that it has Ihe imalogons property for 
objects that "is a mmnber" 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 "is 
for". 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. 
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 
"is a member" 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. 
After a more formal analysis of these examples (considered in 
detail in /Long and Garigliano 1988/) we lind tha! for a tlansi- 
tire and rellexive hierarchical relation, such ~, "subset" or "speci-- 
alises", a second relation has the upward inheritance property it" 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 condi- 
lion is shown to be both necessary and sntlicienl. 
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 "vertebrates have backbones", we a~:t~ally mean all 
vertebrates have a backbone. 
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~cri- 
i tance rule operates and, conversely, for checLin,~,, it" ~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 "is allergic to" according t~ the 
analysis, we may build the new relation, "is allergic to some", 
Thus, John is allergic to some cats, cats are vertebrates~ so John 
is allergic to some vertebrates. It is quite clear that this new rela- 
tion has the property of upward inheritance, unlike the original 
relation. 
As an example of tile process of checking whether upward inher- 
itance will apply to a given relation, consider the following. We 
have the relation "is smaller than", 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 construc- 
tion that yields "is smaller than", given some starting relation. 
Consider the relation "one up on", defined by: 
A is one up on B ifflAl+ 1 = IBt 
Now, "is smaller thmL" can be defined by: 
A is smaller than B fit" there is ,,;ome subset of B which is one 
up on A. 
Thus, "is smaller .than" has upward inheritance with respect to 
the subset hierarchy. 
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. 
3~15 
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 "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. 
Now~ the use of "the" indicates a specification of an exact group 
of fruit, F say, and of apples, say A, so "the apples are among 
the frnit" now means: 
every member of the set of apples, A, is in the set of fruit, F. 
What makes the inference correct is the peculiarity of the use of 
"the" 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. 
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~" Scots. 
This is false, so the "elected by forty per-cent of" relation does 
not have downward inheritance over subsets. 
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. 
Again, this is false. 
There are other hierarchies, such as "has part", which holds 
between an object and each of its parts; for which very few rela- 
tions 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. 
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 struc- 
ture 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. 
like "is for" or "causes", to the mathematical relations such as "is 
a member of" or "smaller than". 
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. 
The final point we wish to stress is that this analysis strengthens 
the argument in favour of the semantical approach to inferences. 
As we have pointed out before, the distinction between the rela° 
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. 
5. Conclusions 
In this paper we have described and analysed a technique, the 
nse of inheritance, for transferring properties between com- 
ponents within total hierarchical ,~tructnres. It is a powerful tech- 
nique, because it allows valid inferences, as well as being very 
natural, since it models sentences that could occur in everyday 
use of language. At the same time, it is widely applicable, using 
relations which range from the common and simple such as 
"drink" or "wear", through the fundamental, primitive, relations 
3B6 

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