Negative Reasoning Using Inheritance
L i n Padgham
Department of Computer and Information Science
Linkoping University
S-581 83 Linkoping, Sweden
[email protected]
Abstract
This paper presents methods of default
reasoning which allow us to draw negative
conclusions that are not available in some of
the models for inheritance reasoning. Some of
these negative conclusions are shown to be
logically required, while others result from an
extension of the model to include the notion of
a default negative assumption. The negative
default
assumption
operator
is
exactly
symmetrical to positive default assumption,
and supports the drawing of extra negative
conclusions. It is argued that in some domains
negative conclusions are extremely important.
An example is given from the medical domain
to illustrate the usefulness of techniques for
deducing negative facts. A formal definition of
the inheritance model used, which in an earlier
paper by the same author [Padgham 88] was
shown to resolve a number of the classical
problems in the literature on inheritance
reasoning, is also given.
negative information, i.e. contra-indications, or
incompatibilities. It is often just this negative
information which is not quite so immediate and may
need to be deduced, using in part an inheritance
reasoning mechanism.
We present a method for doing default reasoning with
an inheritance schema which allows both default and
strict reasoning about negative conclusions in the
same way as it does about positive conclusions. In
order to present the negative reasoning we need first
to present the basic model, and the reasoning
methods
for
positive
and
explicit
negative
information. Deduction of extra negative information
then follows from these. We will illustrate with some
examples which are a little more complex than the
usual examples found in the literature because one
needs the added complexity in order to benefit from
the added reasoning. We will also attempt to relate
the methods to some standard examples, in order to
at least explain the results within a familiar context.
However the real power of these mechanisms will be
seen in more complex real-world reasoning systems,
for which the methods are of course intended.
1. Introduction
Most of the literature on inheritance reasoning
[Etherington 87, Fahlman 79, Touretzky 86,
Sandewall 86, etc.] focusses on methods for collecting
inherited information regarding an entity, in the
presence of conflicting information. Conflicting
information is usually of the type X is a Y, and X is
not a Y.
This negative information regarding what X is not is
often extremely important. For example in medical
diagnosis, it is often equally important to rule out
certain diseases as it is to obtain a positive diagnosis.
Similarly if one is reasoning about prescriptions for
medical drugs, one is extremely interested in the
This work was sponsored by the Swedish Board for Technical
Development.
1086
Commonsense Reasoning
2.
Overview of The Inheritance Model
T h e basic model is t h a t b o t h objects and types can be
described by sets of characteristics, w h i c h can be
p a r t i a l l y ordered according to the n o t i o n of more
i n f o r m a t i o n . T h u s for an object to be a member of a
p a r t i c u l a r class/type it m u s t contain as much
i n f o r m a t i o n as is required for the t y p e , or more; t h a t
is it m u s t contain at least a l l the characteristics
required by the t y p e . S i m i l a r l y a t y p e A is a subtype
of type B if A contains all the characteristics of B,
plus some e x t r a characteristics.
F o r m a l l y types are points in a lattice of descriptors
w i t h i n w h i c h a p a r t i a l order
and lattice operations
and
are defined. In the simplest case we assume
C to be the set of all characteristics, choose
d e s c r i p t o r s as subsets of C, define A
a n d define
respectively.
and
as
and
B as
operations,
We i n t r o d u c e t h e n o t i o n t h a t a t y p e is defined by
t w o d e s c r i p t o r s , t h e type core a n d the type default
T h e t y p e d e f a u l t is t h e set of characteristics we w o u l d
expect t o f i n d i n a t y p i c a l object o f t h a t t y p e . T h e
t y p e core is a subset of t h e t y p e d e f a u l t a n d consists
of those c h a r a c t e r i s t i c s s t r i c t l y necessary for an object
t o b e o f t h a t t y p e . W e i n t r o d u c e the n o t a t i o n X d and
X c to refer to t h e d e f a u l t a n d core of X respectively.
I t i s u n i m p o r t a n t w h e t h e r w e can i n fact enumerate
all t h e c h a r a c t e r i s t i c s i n these t w o sets, t h e i m p o r t a n t
p o i n t is t h a t t h e o r e t i c a l l y there are t w o such sets for
every t y p e . W e can t h e n have and reason about the
p a r t i a l i n f o r m a t i o n w h i c h i s available concerning the
descriptors a n d t h e r e l a t i o n s h i p s between t h e m .
As i n d i c a t e d above, descriptors can be compared w i t h
one a n o t h e r on t h e basis of t h e more i n f o r m a t i o n
relation
, w i t h i n the l a t t i c e f o r m e d by a l l possible
subsets of C. ( I t s h o u l d be stressed t h a t this lattice is
a t h e o r e t i c a l e n t i t y a n d need never be e n u m e r a t e d ) . If
a descriptor A
a d e s c r i p t o r B, this is equivalent to
the s t a t e m e n t t h a t A is a B.
We n o t e t h a t A d
A c . T h e r e m a y be m a n y
possibilities b e t w e e n the core and the default of the
t y p e w h i c h f u l f i l l t h e core r e q u i r e m e n t s b u t are i n
v a r i o u s ways d e v i a n t f r o m t h e d e f a u l t . There m a y b e
some characteristics w h i c h seem to belong in the core
d e s c r i p t o r of a t y p e , r a t h e r t h a n in the default
d e s c r i p t o r , w h i c h nevertheless can be absent f r o m
p a r t i c u l a r i n d i v i d u a l s . F o r instance w e w o u l d w a n t t o
say t h a t four-leggedness is a core characteristic for
dogs. H o w e v e r t h e r e are c e r t a i n l y dogs w h o have lost
a leg. O u r p o s i t i o n here is t h a t i n d i v i d u a l s m u s t
always be a l l o w e d to be declared as members of a
class even if t h e y do n o t f u l f i l l the core characteristics
for t h a t class. H o w e v e r no subclass m a y exist w h i c h
does n o t f u l f i l l its p a r e n t class' core characteristics.
T h e f a c t t h a t three-legged dogs exist does n o t
w a r r a n t the c r e a t i o n of a class of such. If in some
a p p l i c a t i o n (e.g. database for a m p u t a t i o n section of a
v e t e r i n a r y h o s p i t a l ) one w a n t e d such a class, t h e n the
f o u r leggedness p r o p e r t y s h o u l d be r e m o v e d f r o m the
t y p e core.
Because we have b o t h core a n d default descriptors for
each t y p e w e can t a l k a b o u t t h e
r e l a t i o n t o and
2.1
Incompatibility
It is also possible to t a l k about i n c o m p a t i b i l i t y
between t w o t y p e descriptors. W h e n w e say t h a t t y p e
A is i n c o m p a t i b l e w i t h t y p e B, w h a t we mean is t h a t
there exists at least one characteristic in the t y p e
descriptor for A, w h i c h is i n c o m p a t i b l e w i t h a
characteristic in the t y p e d e s c r i p t o r for B. (Once
again it is not necessary to p i n p o i n t w h a t the
characteristic is t h a t makes A i n c o m p a t i b l e w i t h B,
t h o u g h of course t h a t w o u l d be e x t r a usable
information.
It
is
enough
to state
that
the
i n c o m p a t i b i l i t y exists). F o r example if one t y p e has
the characteristic ' w e i g h t 5kg.' and another the
characteristic ' w e i g h t 10kg.', then these t w o t y p e
descriptors are i n c o m p a t i b l e w i t h one another in t h a t
no
object
can
belong
to
both
these
types
simultaneously.
The f o r m a l
follows:
definitions
of
incompatibility
are
Definition:
There is a r e l a t i o n
that
holds
iff the
characteristic
i n c o m p a t i b l e w i t h the characteristic c ' .
as
such
c
is
We observe t h a t
is irreflexive and i n t r a n s i t i v e b u t
s y m m e t r i c , i.e. the f o l l o w i n g is satisfied:
Definition:
written
The
descriptor
A
is
It is assumed t h a t no t y p e descriptor
descriptor is in itself i n c o m p a t i b l e .
incompatible,
or
object
We now define the n o t i o n of a c o m p l e m e n t of a t y p e
A, w h i c h i n t u i t i v e l y consistes of the u n i o n of types
whose characteristics are i n c o m p a t i b l e w i t h A. T h i s
complement i s w r i t t e n N O T ( A ) .
Definition:
N O T ( A ) is defined
according to the f o l l o w i n g e q u a t i o n :
to
be
a
set
f r o m b o t h t h e core and t h e d e f a u l t points for each
t y p e . T h i s allows us to m a k e the s t a t e m e n t t h a t A is
a B w i t h t h e 4 f o l l o w i n g nuances:
*
A ' s are a l w a y s t y p i c a l B's
Note t h a t if A is inconsistent t h e n N O T ( A ) is the
e m p t y set.
Padgham
1087
P r o p o s i t i o n 1:
That is
to say incompatibility in a descriptor is necessarily
inherited by all more specific descriptors.
The incompatibility relation can of course be between
any combination of core and default type pairs,
giving similar nuances in negative statements as in
positive. Thus we can say:
incompatible w i t h a consistent descriptor B, then all
consistent descriptors more specific than A are
NOT(B).
1088
Commonsense Reasoning
T h e positive and negative relationships described here
t h e n make it possible to draw a d i a g r a m such as fig.
1, where we can follow paths t h r o u g h the graph to
collect i n f o r m a t i o n . By defining a default assumption
operation E w h i c h takes us f r o m a core point to its
corresponding default we can come to a new point in
the g r a p h , enabling us to collect f u r t h e r i n f o r m a t i o n .
These j u m p s u p w a r d in i n f o r m a t i o n content, are
essentially the reasoning step 'we know t h a t we have
an A, in the absence of i n f o r m a t i o n to the contrary,
let us assume t h a t we have a t y p i c a l A . '
Definition:
A default assumption, w r i t t e n
is
the step w h i c h allows us to add the relation 'object
T d ' to our set of relations.
We also allow
to add in a relationship ' o b j e c t _
T d -δ', where T d - δ is
T c b u t has some of the T d
i n f o r m a t i o n removed ( i n order to deal w i t h p a r t i a l
defaults). F o r f u r t h e r explanation of p a r t i a l defaults
see [Padgham 89].
The logical necessity of the conclusions obtained by
strict negative reasoning relies, of course, on
acceptance of the basic tenets of the model, namely
that:
*
types can be defined (theoretically) by their core
and default type descriptor each of w h i c h is a set
of characteristics.
*
these sets of characteristics can be compared
with
each
other
according
to
the
more
i n f o r m a t i o n relation
*
incompat ability between types A and B can be
explained as incomp at a b i l i t y between at least
one
characteristic
of
type
A
and
one
characteristic of type B.
Given t h a t we accept the model, proof for the
correctness of strict negative reasoning is as follows:
P r o p o s i t i o n 3:
I f X , Y , W , Z are type descriptors
such t h a t Y
X, W
Z. and Y
W is
incompatible, then X € N O T ( Z ) .
T h u s f r o m fig. 1 we can say w i t h certainty t h a t A is
a B, and is n o t a C. A f t e r m a k i n g the default
assumption t h a t we have a t y p i c a l B, we can also add
the i n f o r m a t i o n t h a t A is a D.
3.
Negative Reasoning
If we now add e x t r a i n f o r m a t i o n to fig 1, to obtain
the d i a g r a m shown in f i g . 2, w h a t more can we say
about A? O u r c l a i m is t h a t we can say t h a t if we
accept the previous conclusions about A, then A is
also definitely not a P, and probably not a Q. The
reasoning t h a t A is definitely not a P, we refer to as
strict negative reasoning because given t h a t we
believe A is not a C, the extension to believing that
A is also not a P does not require any default
assumptions. T h e reasoning t h a t A is also probably
not a Q relies on the assumption that if A is
definitely not a t y p i c a l Q, then A is probably not a Q
at all. T h u s we call this default negative reasoning.
More f o r m a l j u s t i f i c a t i o n of the reasoning used to
o b t a i n these e x t r a negative conclusions is given in the
f o l l o w i n g t w o subsections.
3.1
Strict Negative Reasoning
We define strict negative reasoning as the reasoning
w h i c h takes a negative conclusion regarding the
i n c o m p a t i b i l i t y of the object being reasoned about
w i t h some t y p e , and deduces all logically necessary
negative conclusions based on this i n c o m p a t i b i l i t y .
For example, if we have
C) (i.e. we
believe N O T ( C ) ) and we know t h a t P
Since Y _ W is incompatible and _ has been
shown to preserve i n c o m p a t i b i l i t y , it follows t h a t
X
Z is incompatible which implies X €
NOT(Z).
We note t h a t in the situation where we have C
P d , we are not logically bound to add in N O T ( P ) , as
we do not have a definite i n c o m p a t i b i l i t y w i t h P c .
I.e. C is-not-a P means only C E N O T ( P c ) .
If there have been default assumptions made on the
way to the conclusion on w h i c h negative reasoning is
based, then the negative conclusions w h i c h follow are
of course default conclusions rather t h a n certain
conclusions. However this reasoning is referred to as
strict
(or
monotonic),
because
if
the
initial
i n c o m p a t i b i l i t y is believed then we are logically
bound to also believe the consequences of strict
negative reasoning based on the i n c o m p a t i b i l i t y . To
not do so results in a set of conclusions which is
either incomplete or inconsistent.
C, then we
are logically b o u n d to believe N O T ( P ) .
Padgham
1089
3.2
Default Negative Reasoning
Default negative reasoning is the step t h a t can be
i n t u i t i v e l y described as 'if we k n o w t h a t w h a t we
have does not f i t the description for a t y p i c a l X, t h e n
let us assume t h a t it is not an X . ' W h a t we do
f o r m a l l y is to define an operation N w h i c h allows us
given
to assume
thus adding
the i n f o r m a t i o n A € N O T ( X ) . F o l l o w i n g such an
operation we can then continue our strict negative
reasoning.
Definition:
A
negative
default
assumption,
w r i t t e n N(T) is the step w h i c h allows us t o , given
the conclusion
add the r e l a t i o n
T h a t is if we believe t h a t the object about w h i c h we
are reasoning is an element of N O T ( T d ) t h e n we
assume t h a t it is also an element of N O T ( T c ) .
This default negative reasoning is n o t p r o v a b l y
correct in the w a y t h a t the strict negative reasoning
is, j u s t because it is default reasoning, and as such
requires an assumption. However the nature of the
assumption required is essentially s y m m e t r i c a l to t h a t
required for positive default reasoning. It also seems
i n t u i t i v e l y reasonable.
3.3
Preference for Specificity
W h e n doing positive reasoning one can obtain
conclusions t h a t conflict w i t h one another, by v i r t u e
of the fact t h a t positive reasoning can result in a final
negative conclusion. W h e n such conflicts occur, the
decision as to w h a t should be preferred is based on a
n o t i o n of specificity, preferring conclusions w h i c h rely
on default assumptions at the most specific level
possible.[Padgham 88]
In negative reasoning it is n o t possible to arrive at
conclusions w h i c h conflict w i t h other conclusions
derived v i a negative reasoning. T h i s is because the
negative reasoning methods can only lead to negative
conclusions, m a k i n g it impossible to o b t a i n b o t h X
and N O T X in the negative reasoning phase.
Consequently conclusions based on negative reasoning
methods do not have to be ordered according to the
specificity of the default assumption on w h i c h they
are based. However conclusions obtained during
negative reasoning can conflict w i t h conclusions
obtained d u r i n g positive reasoning.
Conclusions obtained by strict negative reasoning
f r o m a conclusion (e.g. N O T ( X ) ) t h a t was obtained
by deduction using S should clearly be resolved using
the specificity preference. The conclusions of the
strict negative reasoning depend on the same default
assumption as the conclusion N O T ( X ) itself depends
on. T h i s must then be compared for specificity w i t h
the default assumption on w h i c h the conflicting
conclusion depends.
For those negative conclusions w h i c h depend on a
negative default assumption, it is less clear how they
relate to conflicting conclusions based on a positive
default
assumption.
It
seems d i f f i c u l t
to use
specificity in an i n t u i t i v e and w e l l defined way in
t h a t it is not entirely clear how the specificity of
N O T A ' s should be compared to the specificity of
B's. An i n i t i a l i n t u i t i o n is t h a t positive default
assumptions m a y always be preferred over negative
default assumptions.
Using the example in f i g . 3 default negative reasoning
gives the effect t h a t if we have an object X w h i c h is a
Grey
Thing,
then
we
know
that
R o y a l E l e p h a n t d ) (because X
Grey T h i n g
and
I(GreyThing
RoyalElephantd))
N(RoyalElephant) then allows us to add the r e l a t i o n
R o y a l E l e p h a n t c ) , thus giving the conclusion
t h a t X is not a R o y a l Elephant. T h i s seems
i n t u i t i v e l y reasonable, and in keeping w i t h h u m a n
reasoning.
4.
E x a m p l e U s i n g N e g a t i v e Reasoning
Let us look now at a concrete example where the k i n d
of negative reasoning described could be used. Let us
suppose t h a t we have a decision support system for
general medical practitioners w h i c h includes, amongst
other things, a knowledge base of various types of
drugs, organised according to our lattice based model.
One of the things w h i c h doctors m u s t be very
1090
Commonsense Reasoning
concerned about when prescribing drugs is to be
aware of contra-indications for giving a particular
drug. Contra-indications are often other drugs t h a t
the patient may be taking (or t h a t the doctor is
a b o u t to prescribe). T h u s t h e doctor must consider
the entire drug group t h a t the patient will take and
ascertain t h a t it is compatible.
Let us imagine now t h a t a patient comes to the
doctor and is diagnosed as having high blood pressure
which needs to be t r e a t e d by medication. The patient
is also a diabetic and is currently taking medication
for diabetes. T h e doctor must ensure t h a t the blood
pressure medication he prescribes does not conflict
with the diabetes medication.
A p a r t of the knowledge base of drugs is shown in fig.
4. If we know t h a t the patient is currently taking
Sulphonylurea Glibencamide for diabetes, we can use
the inheritance reasoning strategies described in this
paper to ascertain t h a t we should not prescribe any
of t h e thiazides for treating the p a t i e n t ' s high blood
pressure, as this will lead to an incompatibility
regarding w h a t is happening with the blood glucose
level.
If we refer to the group of drugs which the patient is
going to take as D, then we can use ordinary
default / i n h e r i t a n c e reasoning (following links and
using t h e € operation), to collect the information
Sulphonylurea
Glibencamide, Diabetic treatment drug,
Blood glucose
lowerer,
NOTfblood glucose
raiser).
We can t h e n continue further using negative
reasoning, s t a r t i n g from the negative conclusion
NOT(Blood glucose raiser). By following the links in
a backward direction, and using the default operation
J\f to go from default to core of diuretics, we collect
the
additional
information
NOTfdiuretic),
NOT (thiazides).
Given that thiazides are in fact one of the groups of
drugs which are high blood pressure drugs, it is very
useful to be able to obtain the information t h a t a
consistent group of drugs D, which contains
Sulphonylurea
Glibencamide
may
not
contain
thiazides. In this case it is easily possible to choose
one of the other groups of high blood pressure drugs
(Calcium antagonists or Beta blockers), which do not
lead to any incompatibility. If there were no
alternative drug groups then there are a number of
other reasonable possibilities, e.g. try to find a
thiazide t h a t is an atypical diuretic and is not a blood
glucose raiser, or alter the drug used for diabetes
treatment. However these questions are outside the
scope of the present paper.
The model is only relevant for cases where there are
no complex interactions between the two drugs which
cannot be expressed as the union of their combined
characteristics. Also it may be t h a t the inconsistency
detected is not always relevant. For instance if 'blood
glucose lowerer' had been an u n i m p o r t a n t side effect
of the diabetes medication, r a t h e r than the desired
effect, then it would not have mattered t h a t this is
blocked by some other medication. Reasoning beyond
the detection of the inconsistency can either be done
by the doctor or by some other reasoning module, but
is not part of the mechanisms described here.
Padgham
1091
5.
Conclusions
T h e lattice based m o d e l of t y p i n g schemas presented
previously [Padgham 88] extends n a t u r a l l y to include
reasoning about w h a t an object is N O T . A subset of
the results of this extension are similar to results
reported by H o r t y a n d T h o m a s o n [88] 1 . However this
model provides a proof of correctness for t h e results,
w h i c h are obtained simply by d e f i n i t i o n w i t h i n their
model. T h e e x t r a results w h i c h come f r o m the default
negative reasoning are
also considered
to be
important.
T h e results of strict negative reasoning are available
in any m o d e l based reasoning system based on
classical logic (e.g. c i r c u m s c r i p t i o n [ M c C a r t h y 1986],
default
logic
[Reiter 1980] etc.), as following
backwards along t h e links f r o m a negative conclusion
amounts s i m p l y t o t a k i n g the contrapositive f o r m o f
an
implication.
Those
approaches
such
as
c i r c u m s c r i p t i o n where the logic is classical, and the
default behaviour comes f r o m m i n i m i s i n g some sort of
a b n o r m a l i t y predicate w i l l also o b t a i n the conclusions
obtained by our N operator. However such systems
cannot distinguish between conclusions obtained by
contraposing (or use of modus tolens) and conclusions
obtained by other mechanisms equivalent in our
system to use of the E operator. This lack of
d i s t i n c t i o n can lead to m a n y extensions w i t h no
s t r u c t u r a l mechanism available to choose between
t h e m . We believe t h a t having A/' as a separate
operator, allows b o t h the power of d r a w i n g e x t r a
conclusions as w e l l as the a b i l i t y to discriminate and
make distinctions. T h i s a b i l i t y to discriminate is
needed when one has m a n y possible conclusions b u t
wishes to prefer some conclusion sets over others.
The
model
provides
for
clean
and
simple
d e t e r m i n a t i o n of such things as correct pre-emption
behaviour and w h i c h graphs are inconsistent in the
i n f o r m a t i o n they c o n t a i n , and can therefore not be
expected to lead to c o r r e c t / m e a n i n g f u l results. M o r e
work is needed in the area of default negative
reasoning, to determine w h a t p r i o r i t y conclusions
obtained f r o m this reasoning should have w i t h respect
to positive conclusions. These issues have n o t been
explored in the literature because (at least as far as
this author is aware) there have n o t been systems
w h i c h are able to b o t h use contraposition and
distinguish it from other reasoning methods. One
1 Based on personal c o m m u n i c a t i o n ( H o r t y ,
Feb.
'89) the
beliefs s u p p o r t e d by s t r i c t negative reasoning appear to be
essentially equivalent to those o b t a i n e d by H o r t y ' s d e f i n i t i o n of
a skeptical reasoner using m i x e d l i n k types. We do not have
any proof of equivalence.
1092
Commonsense Reasoning
possibility is to compare the specificity of the level at
w h i c h the N operation is used, w i t h t h e specificity of
the level at w h i c h S is used to o b t a i n the competing
positive conclusion. However our i n t u i t i o n is t h a t we
should instead prefer a l l positive conclusions over
negative conclusions obtained using default negative
reasoning. It m a y be however, t h a t preference is
determined by w h a t one wishes to use the reasoner
for. It is a d i s t i n c t advantage of the presented model
t h a t such changes are simple and clean to make in
the algorithms w h i c h m a n i p u l a t e the model, and do
not require a change w i t h i n the m o d e l itself.
Acknowledgements
I t h a n k R a l p h Ronnquist for help w i t h the f o r m a l
expressions, and Toomas T i m p k a for help w i t h the
medical example.
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