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.
References ETH87 E t h e r i n g t o n , D . W . F o r m a l i z i n g Nonmonotonic Reasoning Systems. Artificial Intelligence, v o l . 3 1 , 1987, p p . 41-85. FAH79 F a h l m a n , S.E. NETL: A System for Representing and Using Real- World Knowledge. T h e M I T Press, C a m b r i d g e , M A , 1979. HOR88 H o r t y , J. F., T h o m a s o n , R. H. M i x i n g Strict and Defeasible Inheritance. Proceedings of AAAI '88 St P a u l , Minnesota, A u g u s t 20-26, 1988, v o l 2, pp. es427-432 PAD88 P a d g h a m , L. A M o d e l and Representation for T y p e I n f o r m a t i o n and its Use in Reasoning w i t h Defaults. Proceedings of AAAI '88 St Paul, Minnesota, August 20-26, 1988, v o l 2, pp.409-414 REI80 Reiter, R. A Logic for Default reasoning, Artificial Intelligence 13, N o r t h - H o l l a n d , 81-132 SAN86 Sandewall, E. Non-monotonic Inference Rules for M u l t i p l e Inheritance w i t h Exceptions. Proceedings of the IEEE, v o l 74, 1986, p p . 1345-1353. TOU86 T o u r e t z k y , D.S. The Mathematics of Inheritance Systems. M o r g a n K a u f m a n n Publishers, Los Altos, C A , 1986. MCC86 Applications of C i r c u m s c r i p t i o n to Formalizing C o m m o n Sense Knowledge, Artificial Intelligence Journal, v o l 28, p p . 89-116
