The Tractability of Path-Based Inheritance - Semantic Scholar

The Tractability of Path-Based Inheritance B a r t S e l m a n a n d H e c t o r J . Levesque* Dept. of Computer Science University of Toronto Toronto, Canada M5S 1A4

Abstract T o u r e t z k y (1984) proposed a f o r m a l i s m for nonmonotonic m u l t i p l e inheritance reasoning which is sound in the presence of ambiguities and redundant links. We show t h a t Touretzky's inheritance n o t i o n is N P - h a r d , and thus, provided P # N P , c o m p u t a t i o n a l l y intractable. T h i s result holds even when one only considers unambiguous, t o t a l l y acyclic inheritance net­ works. A direct consequence of this result is t h a t the c o n d i t i o n i n g strategy proposed by T o u r e t z k y to allow for fast parallel inference is also intractable. Therefore, it follows t h a t nonmonotonic m u l t i p l e inheritance hierarchies, a l t h o u g h compact representations, may not al­ low for efficient retrieval of i n f o r m a t i o n as has been suggested in a t t e m p t s to use such hierar­ chies, e.g., in NETL (Fahlman 1979). We also analyze the influence of various design choices made by Touretzky. We show t h a t all ver­ sions of downward (coupled) inheritance, i.e., on-path or off-path preemption and skeptical or credulous reasoning, are intractable. How­ ever, t r a c t a b i l i t y can be achieved when using upward (decoupled) inheritance.

1

Introduction

Since the early semantic networks of Q u i l l i a n (1968), i n ­ heritance hierarchies have been used to provide a com­ pact representation and efficient reasoning mechanism for certain kinds of taxonomic i n f o r m a t i o n . One prob­ lem t h a t has plagued these systems is t h a t of exceptions (or cancellation) in non-tree hierarchies. Early a t t e m p t s to systematically deal w i t h this issue, such as t h a t of NETL (Fahlman 1979), were later shown to be unsound in the presence of redundant links and ambiguities (Reiter and Criscuolo 1983; Touretzky 1984). T h e first com­ prehensive definition t h a t appeared to solve these prob­ lems was t h a t of Touretzky (1984). Since t h e n , other equally sound schemes have been proposed (Sandewall 1986; I l o r t y et ai 1987). B u t despite a decade of study, w i t h increasingly subtle examples and counter-examples being considered, consensus has yet to emerge regarding *

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Fellow o f T h e C a n a d i a n I n s t i t u t e for A d v a n c e d Research.

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the proper t r e a t m e n t of m u l t i p l e inheritance w i t h can­ cellations. F r o m a knowledge representation s t a n d p o i n t , p a r t of the p r o b l e m is t h a t there are tremendous subtleties in reasoning w i t h propositions t h a t a d m i t exceptions, such as " B i r d s fly" ( B r a c h m a n 1985). N o t surprisingly, the research on inheritance systems mentioned above has not a t t e m p t e d to settle the larger logical and semanti­ cal issues associated w i t h defeasible reasoning. Rather, the argument has been t h a t inheritance systems need to conform to a set of special i n t u i t i o n s i n v o l v i n g paths t h r o u g h hierarchies. We therefore call such systems pathbased inheritance systems and contrast t h e m w i t h the more general nonmonotonic reasoning systems (Reiter 1987). T h e latter approach a t t e m p t s to establish a log­ ical account of defeasible reasoning (using, for example, autoepistemic, circumscriptive, c o n d i t i o n a l , or default logic), and somehow absorb hierarchies and inheritance as a special case. W h i l e the nonmonotonic systems tend to be more principled and semantically m o t i v a t e d on the whole, they have yet to be applied successfully to prob­ lems as intricate as those considered by the path-based approaches. B u t if there are indeed irreducible i n t u i t i o n s about i n ­ heritance and paths t h r o u g h hierarchies, these i n t u i t i o n s are sometimes in conflict and can give rise to different inheritance systems (Touretzky et ai 1987). How then to choose among the competing systems, especially since there is no independent semantic characterization t h a t adequately covers the phenomena in question? W h i l e we do not c l a i m to have an answer to this question, we do propose here a criterion t h a t should be taken into account when comparing systems. W h a t we w i l l show is t h a t there can be a significant difference in the compuiational tractability of inheritance depending on fine points of the definition used. In other words, t w o ac­ counts of inheritance t h a t cover by and large the same t e r r i t o r y , differing only in certain complex cases, may nonetheless be quite different in their overall c o m p u t a ­ t i o n a l demands. T h e m a i n technical result of this paper is t h a t the def­ i n i t i o n of inheritance proposed by T o u r e t z k y 1 is inher­ ently N P - h a r d , and remains so even for t o t a l l y acyclic 1

When speaking of "Touretzky's definition of inheritance" or "Touretzky's inheritance system," we are referring to the system defined in Touretzky (1984).

unambiguous networks. T h u s there cannot be an algo­ r i t h m t h a t correctly determines if one node is inheritable f r o m another t h a t runs in t i m e t h a t is p o l y n o m i a l in the size of the network. 2 An immediate consequence of this is t h a t the c o n d i t i o n i n g of a network, which Touretzky proposes to allow for fast parallel inference, is itself com­ p u t a t i o n a l l y intractable. T h i s suggests t h a t a Touretzky inheritance procedure cannot r u n unsupervised, unless the network can be restricted in f o r m or in size. B u t the news is n o t all bad. We also show that there are plausible variants of the Touretzky definition t h a t are indeed tractable. A m o n g these is the definition proposed by H o r t y et al. (1987), who first exhibited a polynomial a l g o r i t h m for c o m p u t i n g inheritance. We extend this work and demonstrate which parts of the definition are responsible for m a k i n g inheritance tractable. A g a i n , we do not wish to claim t h a t the t r a c t a b i l i t y of inheritance implies the correctness of the definition; b u t it is certainly one issue among many that needs to be taken i n t o account in resolving differences among com­ peting accounts. In the next section, we consider the precise definition of several forms of path-based inheritance. In the subse­ quent section, we examine the complexity of inheritance, and which c o m b i n a t i o n of features in the definition affect the t r a c t a b i l i t y of the associated reasoning. In the final section, we summarize our results and discuss directions for f u t u r e research.

2

P a t h - B a s e d I n h e r i t a n c e Systems

For our purposes, an inheritance network consists of a fi­ nite set TV of objects called nodes denoted w i t h lower-case letters x, y, z, and a set T of edges, defined as follows: D e f i n i t i o n : Edge An edge is an element of TV x { 0 , 1 } x TV, t h a t is, any ordered pair of nodes and a 0-1 value called its polarity. Edges w i t h a 0 are called negative and those w i t h a 1 are called positive. We draw positive edges as x—>y and negative edges as x+y. I n t u i t i v e l y , the nodes of a network are intended to stand for concepts or properties such as " b i r d , " "pen­ g u i n , " "Tweety," or "flies." Edges, on the other hand, stand for statements: A positive edge x—>y stands for the statement "an x is n o r m a l l y a y," while the corre­ sponding negative edge represents the statement "an x is n o r m a l l y not a y . " 3 Path-based inheritance is concerned w i t h the logic of statements of this type only. 4 T h e goal is to define w h a t it means for a new edge x—>y to be 2

For the purpose of this paper, and to keep the provisos to a minimum, we assume that P # N P . 3 I n the case where x is a property, instead of 'an x," this should read "something with property x." This also applies to y as a property. When x denotes an individual concept, instead of "an x is normally," the phrase "x is" should be used. The case w i t h y as an individual concept never arises. 4 Touretzky also defines "no-conclusion" edges. Such edges are not often used in practice, and therefore we will ignore them here in order to simplify our definitions. Note that these edges cannot decrease the complexity of the inheritance reasoning.

inferrable f r o m a given set of edges T. To do so, we use the notion of a p a t h . D e f i n i t i o n : Path A path is a sequence of edges f r o m nodes x 0 to x1 to . .. to where and the first n — 1 edges are positive. 5 T h e polarity of a path is the polarity of the final edge. T h e nodes x 0 and x n are called the start point and end point of the p a t h . The edge formed by t a k i n g the start and end points and the polarity of a p a t h we call the conclusion supported by the p a t h . We w i l l let lower-case Greek letters stand for paths and draw them It is t e m p t i n g to define the set of edges t h a t are i n ­ ferrable f r o m a network T directly as the set of all con­ clusions supported by at least one path formed by edges in T. The complication is t h a t a path may be invali­ dated in one of two ways: it may be contradicted by other paths (in which case neither path wins) or it may be preempted by a more specific path (in which case the more specific p a t h wins). For the following definitions, we w i l l let be any set of paths, be any path and x be any node. D e f i n i t i o n : Contradiction a is contradicted in $ iff there is path in w i t h the same start and end points as a, b u t of opposite polarity. So for example, the path xo—>x\ —+X2 is contradicted by the path since the start and end points are the same b u t the final edge is of opposite polarity. For preemption, the idea is t h a t a path is preempted by an edge of opposite polarity f r o m an intermediate of the p a t h . We w i l l consider two definitions of intermedi­ ate. D e f i n i t i o n : On-path intermediate x is an on-path intermediate of a

some j < m. to node £ ; _ i . )

in $ iff for some a positive p a t h w i t h x = yj for (Note t h a t a and 7 must be identical up

D e f i n i t i o n : Off-path intermediate x is an off-path intermediate of a m iff there is a posi­ tive path 7 in $ f r o m XQ to x n _ i that contains x. (Note that cr and 7 can be completely disjoint except for the nodes xo and x „ _ i . ) D e f i n i t i o n : Preemption (off-path or on-path) a is preempted m _ ifT there is a node x t h a t is an i n ­ termediate of a in , and an edge in of the opposite polarity of a f r o m x to x „ . For

consider the set of paths = where C , r e , e , and g respec­ tively denote "Clyde," "royal elephant," "elephant," and "gray." Figure 1(a) gives the underlying inheritance net­ work. (The figure contains an additional node Ae, which w i l l be discussed below.) The p a t h < would be on-path preempted in by the since 5

example,

S o every edge in a n e t w o r k is a p a t h .

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speaks of decoupled inheritance. B u t w i t h downward p a t h concatenation the p a t h , cannot be ob­ tained f r o m the paths in To do so, one w o u l d also need as->a—+e w h i c h , in fact, is contradicted in by So, in this case, there is a coupling between the properties t h a t J i l l can i n h e r i t and those inherited by the class of adult students. We now define the inheritable paths. D e f i n i t i o n : Inheritable p a t h (on-path and off-path, downward and upward) is inheritable in iff Figure 1: Examples of preemption and p a t h concatena­ t i o n . Arrows w i t h a cross bar denote negative edges, the other arrows denote positive edges.

re is an o n - p a t h intermediate of the p a t h . 6 It w o u l d also be off-path preempted by the same edge. On the other h a n d , the p a t h where At denotes " A f r i c a n elephant," w o u l d o n l y be off-path preempted

in

C o n t r a d i c t i o n and preemption tell us how a p a t h in a network may be i n v a l i d a t e d . B u t once a p a t h is i n v a l ­ i d a t e d , other paths t h a t contain it may be invalidated also. So in d e t e r m i n i n g inheritable paths, we must be sure to concatenate only those paths t h a t have n o t been ruled o u t . As it turns o u t , there are t w o ways to con­ catenate paths. D e f i n i t i o n : U p w a r d concatenation a is an upward concatenation of paths in iff the last edge of a is in and the p a t h consisting of all b u t the last edge of a is also in D e f i n i t i o n : D o w n w a r d concatenation a is a downward concatenation of paths in iff the p a t h consisting of all b u t the first edge of a is in and the path consisting of all b u t the last edge of a is also in ■ . T h e latter f o r m of concatenation was o r i g i n a l l y called double chaining in Touretzky (1984). Here we o p t for the more recent terminology used in T o u r e t z k y et al. (1987). T h e definition of inheritance based on u p w a r d concate­ nation t h a t we w i l l consider leads to so-called decoupled inheritance, as opposed to the coupled inheritance when using downward p a t h concatenation. To illustrate the difference between the t w o forms of concatenation, consider the f o l l o w i n g set of paths where J , a s , s , a , and e respectively denote " J i l l , " " a d u l t s t u d e n t , " " s t u d e n t , " " a d u l t , " and "employed." T h e u n d e r l y i n g inheritance network is given in figure 1(b). T h e p a t h can now be formed by u p w a r d p a t h concatenation. T h i s p a t h supports the conclusion while in supports as—e. So, in this case, J i l l and the class of adult students differ w . r . t . the p r o p e r t y "employed." In general, when there is no coupling between the proper­ ties of a class and the properties of its superclasses, one Royal elephant is a subclass of elephant; therefore, i n ­ formation associated with it should override information as­ sociated with the elephant class. This is precisely what is captured by preemption.

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I n t u i t i v e l y , the paths t h a t are inheritable in are those t h a t are inferrable f r o m b u t not invalidated by B u t where does this set come from? There are different ways of choosing a set We first consider Touretzky's definition of so called credulous inheritance reasoning. In this approach, is chosen to be the least set of paths whose edges are those of T and closed under inheritance. We call such sets credulous grounded extensions. 7 D e f i n i t i o n : Credulous grounded extension is a credulous grounded extension of a set of edges T

iff

U n f o r t u n a t e l y , there need not be a unique credulous grounded extension for a given set F, t h a t is a network can be ambiguous. Inheritance reasoners allowing for m u l t i p l e extensions, such as Touretzky's, are called cred­ ulous reasoners because they explore the various alter­ natives in different extensions. Instead of allowing for m u l t i p l e extensions, H o r t y et al. (1987) propose a f o r m of skeptical inheritance in which at most one grounded extension is generated. In this f o r m of inheritance a unique extension is inductively constructed by i n c l u d i n g paths t h a t are inheritable only if a p a t h w i t h the same start and end p o i n t b u t of opposite p o l a r i t y could not be inherited. T h e i n d u c t i o n is based on the degree of a path: D e f i n i t i o n : Degree Given a set of edges F, the degree of a path w i t h start p o i n t x and end point y is the length of the longest p a t h in T f r o m x to y (ignoring the p o l a r i t y of the edges). H o r t y et al. restrict their definition of skeptical inher­ itance to acyclic networks: D e f i n i t i o n : Acyclic A set of edges T is acyclic iff the graph formed by the elements of Y is acyclic. 8 We can now state the definition of skeptical inheri­ tance as follows: 7

These were called grounded expansions in T o u r e t z k y (1984). 8 T o u r e t z k y (1984) speaks of totally acyclic, as d i s t i n ­ guished f r o m IS-A acyclic networks in which the graph f o r m e d by only the positive edges is required to be acyclic.

D e f i n i t i o n : Skeptical grounded extension is a skeptical grounded extension of a set of edges T iff where  i is defined as follows:

T;

contains the paths in ø i and each p a t h a of degree i + 1 w i t h the following properties: • a can be obtained by concatenation of two paths in • there neither exists an edge in øi, nor a path inheritable in ø i w i t h the same start and end p o i n t as a b u t of opposite polarity. W h a t we u l t i m a t e l y care about is the conclusions (that is, the edges defined by the start and end points) sup­ ported by the paths in a grounded extension of a net­ w o r k . So, we define: D e f i n i t i o n : Conclusion set A set of edges is a conclusion set for T iff for some grounded extension ø, the edges are all the conclusions supported by the elements of . T h e inheritance p r o b l e m , t h e n , is this: D e f i n i t i o n : Inheritance problem Given an acyclic network T, find a conclusion set of T. Note t h a t we are really t a l k i n g about 8 inheritance problems here according to whether we consider offp a t h or on-path preemption, upward or downward path concatenation, and skeptical or credulous reasoning. Touretzky's d e f i n i t i o n , for example, would be classified as o n - p a t h , d o w n w a r d , and credulous, while Horty's ver­ sion is off-path, u p w a r d , and skeptical. We now consider the c o m p u t a t i o n a l difficulty of the inheritance problem.

3

Computational Complexity

T h e f o l l o w i n g theorem shows t h a t for Touretzky's inheri­ tance n o t i o n there is no p o l y n o m i a l a l g o r i t h m t h a t takes as i n p u t an acyclic network F and finds a conclusion set

of T. T h e o r e m 1 The inheritance problem for downward, credulous inheritance (Touretzky

on-path, 1984) is

NP-hard. T h e p r o o f of this theorem is based on a reduction from the NP-complete decision problem " p a t h w i t h forbidden pairs" (or P W F P ) defined by Gabow, Maheshwari, and Osterweil (1976). An instance of P W F P consists of a directed graph G = ( V , E ) , specified vertices s,t V, and a collection C = { ( a 1 , b 1 ) , . . . , ( a n , bn)} of pairs of vertices f r o m V. T h e question is: does there exist a path f r o m s to t in G t h a t contains at most one vertex from each pair in C? T h i s problem remains NP-complete even if we only consider acyclic graphs and all pairs in C are 9

The perhaps more natural condition "a is inheritable in øi" leads to a different notion of skeptical inheritance. We use the condition given above for compatibility with Horty et al. (1987). Under this definition, the credulous grounded extension of an unambiguous inheritance network need not be identical to its skeptical grounded extension (Selrnan and Levesque 1989).

disjoint. Consider an instance of this restricted version of P W F P . W i t h o u t loss of generality, we may assume that each forbidden pair (a,, bi) is such t h a t there does not exist a path f r o m bi, to ai in C. We now construct an acyclic network T f r o m this i n ­ stance of P W F P . First, we include every edge of G as a positive edge of T. Then for every node a t h a t is the first element of a forbidden pair in C, we replace a in T by the network shown in figure 2. Figure 2(a) shows a node a f r o m the forbidden pair (a, b,) w i t h all of its its neighbors in G, and 2(b) shows the structure in T t h a t replaces a: two additional nodes (for a t o t a l of three) and n + 4 additional edges must be included, where n is the number of edges p o i n t i n g to a in G. Note t h a t of the three new nodes, the middle one a* is linked to 6 by a negative edge. Since G is acyclic and the forbidden pairs are such t h a t there is no path f r o m bi to ai for any pair in C, it follows that T is an acyclic network. Moreover, F is such that given an arbitrary on-path, downward, cred­ ulous conclusion set C of T, there is a p a t h in G f r o m s to t containing at most one vertex f r o m each forbidden pair iff the conclusion s—>t is in C . 1 0 Now, consider an algorithm that takes as i n p u t an instance of P W F P , constructs (in p o l y n o m i a l time) an acyclic network T as outlined above, then finds an onp a t h , downward, credulous conclusion set C of T, and, finally, returns "yes" if s—>tøG, and and " n o " other­ wise. Such an a l g o r i t h m returns "yes" iff there exists a path f r o m s to t in G containing at most one node f r o m each forbidden pair; but if finding a conclusion set can be done in polynomial t i m e , the overall a l g o r i t h m w i l l run in polynomial t i m e . Since P W F P is N P - h a r d , the inheritance problem for on-path, downward, credulous reasoning must be NP-hard too. The construction shown in figure 2 exploits vari­ ous properties of inheritance reasoning in general, and Touretzky's notion specifically. Firstly, we rely on pre­ emption since preemption prevents inheritance of a p a t h f r o m s to t going through a"and b (corresponding to a 10

We prove this property of T in Selman and Levesque (1989).

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p a t h t h r o u g h a and 6 in G) because of the negative edge between a* and 6. 1 1 Secondly, we use quasi-redundant edges (Touretzky 1984, p. 10; H o r t y et ai 1987, Techn. R e p o r t , p. 21). T h e edge a'→a" is an example of a quasiredundant edge, since this edge is s t r i c t l y redundant for concluding a ' → a " ; however, to be able to conclude, for example, p 1 —>a", the edge is essential (so it allows us to have a p a t h f r o m s to t t h r o u g h p1 and a" in a grounded extension, corresponding to a p a t h f r o m s to t v i a p1 and a in G). T h i r d l y , the reduction relies on coupled inheritance; we w i l l see below t h a t decoupled (upward) reasoning allows for p o l y n o m i a l t i m e inheritance. A direct consequence of theorem 1 is t h a t the condi­ t i o n i n g of a network as proposed in Touretzky (1984), to enable fast parallel inference, is also intractable. C o n ­ d i t i o n i n g is a process of adding edges to an inheritance network in such a way t h a t a parallel marker-passing procedure can subsequently be used to draw conclusions in t i m e p r o p o r t i o n a l to the height of the inheritance net­ work. We show in Selman and Levesque (1989) t h a t the i n ­ heritance network T used in the above reduction is u n a m ­ biguous. T h u s , it follows t h a t even when we restrict our­ selves to unambiguous acyclic networks the inheritance p r o b l e m based on Touretzky's n o t i o n of inheritance is i n t r a c t a b l e . Note t h a t it also follows t h a t d e t e r m i n i n g whether an unambiguous acyclic network supports a par­ ticular conclusion is N P - h a r d . Recently, Geffner and V e r m a (1989) used a v a r i a t i o n of our reduction to show t h a t reasoning based on their definition of defeasible inheritance is N P - h a r d . T h u s , the technique given above may prove to be useful in deter­ m i n i n g the complexity of f u t u r e proposals for path-based inheritance reasoners. We w i l l now consider the influence of the various de­ sign choices made in Touretzky (1984; H o r t y et ai 1987) on the c o m p u t a t i o n a l complexity of the inheritance rea­ soning. Our results are summarized in the f o l l o w i n g the­ orem: Theorem 2 The computational complexity of the inher­ itance problem for the various choices between off-path or on-path, upward or downward path concatenation, and skeptical or credulous reasoning is given in the following table (P stands for doable in polynomial time):

Clearly the choice between off-path or o n - p a t h pre­ e m p t i o n and between skeptical or credulous inheritance does not change the complexity of inheritance reason­ i n g . However, when we consider u p w a r d (decoupled) i n ­ heritance, we do o b t a i n t r a c t a b i l i t y . T h e l a t t e r f i n d i n g generalizes the t r a c t a b i l i t y result obtained by H o r t y et al. (1987) for their off-path, u p w a r d , skeptical inheri­ tance reasoner. T h e NP-hardness results are proved by showing t h a t the various design choices do n o t affect the correctness of the above reduction when dealing w i t h Downward concatenation is also relevant here.

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downward p a t h concatenation. Details are given in Sel­ m a n and Levesque (1989) which also contains polyno­ m i a l a l g o r i t h m s for the tractable cases. To summarize, theorem 2 clearly shows t h a t the type of p a t h concatenation (upward or downward) is the de­ t e r m i n i n g factor in the complexity of the reasoning. T h i s d i s t i n c t i o n corresponds to the difference between decou­ pled and coupled reasoning. It should be noted t h o u g h , t h a t in formalisms w i t h substantially different definitions of p r e e m p t i o n or grounded extension, other factors may also influence the complexity of the reasoning. A f u r t h e r understanding of the complexity issues u n ­ d e r l y i n g inheritance reasoning can be obtained by con­ sidering the p o l y n o m i a l algorithms for u p w a r d inheri­ tance. These algorithms iteratively construct a conclu­ sion set. T h e m a i n difficulty in adding a new conclusion to the set arises f r o m having to determine whether the p a t h t h a t supports this conclusion is preempted or n o t . So, we have to search for intermediates, which requires access to the set of paths in the underlying grounded ex­ tension. At worst, one would have to keep track of the f u l l extension obtained so far (which can be of exponen­ t i a l size). B u t for upward inheritance it is sufficient to keep track of the current conclusion set, since one can rederive in p o l y n o m i a l t i m e any particular p a t h in the u n d e r l y i n g extension, and thus, search for intermediates efficiently. As a final topic, we w i l l consider goal-directed inheri­ tance. Note t h a t so far we have considered the complex­ i t y of finding any a r b i t r a r y conclusion set given an inher­ itance hierarchy. In some situations, however, it m i g h t be wasteful to generate the entire conclusion set, and moreover, one m i g h t be interested in an extension t h a t supports some particular conclusion (or set of conclu­ sions). We therefore define the problem of goal-directed inheritance reasoning: given an inheritance network T and a conclusion x→y (or x+y), does there exists an extension of T s u p p o r t i n g x→y (or x-f+y)! W h e n a network is unambiguous or when skeptical rea­ soning is desired, the c o m p u t a t i o n a l complexity of this inference task is essentially the same as t h a t of search­ ing for a conclusion set, since, on the one h a n d , after finding the unique conclusion set, it a t r i v i a l to deter­ mine whether it contains a certain conclusion, a n d , on the other h a n d , after at most a p o l y n o m i a l number of queries, one can determine the conclusion set (for a net­ work containing n concepts one has to consider at most 2 n ( n — 1) possible conclusions). Moreover, our reduc­ t i o n f r o m P W F P shows t h a t goal-directed inheritance for o n - p a t h and off-path, credulous, d o w n w a r d inheri­ tance is N P - h a r d , j u s t like the p r o b l e m of finding an extension. (Consider the query: does the constructed network have an extension t h a t supports the conclusion s—>t?) T h e f o l l o w i n g theorem, however, shows t h a t goaldirected reasoning is s t r i c t l y more difficult t h a n search­ ing for an a r b i t r a r y conclusion set: 1 2 12

Kautz and Selman (1989) show that goal-directed rea­ soning for default logic is also strictly harder than generating an arbitrary extension.

T h e o r e m 3 Goal-directed, inheritance is NP-hard.

on-path,

upward,

credulous

T h e p r o o f of t h i s t h e o r e m is again based on a reduc­ t i o n f r o m P W F P ( S e l m a n a n d Levesque 1989). For this r e d u c t i o n , it is essential for t h e constructed network to be a m b i g u o u s . T h i s t h e o r e m reveals some of the e x t r a difficulties i n i n h e r i t a n c e reasoning due t o a m b i g u i t i e s , a l t h o u g h these difficulties do n o t arise w h e n searching for an a r b i t r a r y conclusion set, as shown by theorem 2. T h e c o m p l e x i t y o f g o a l - d i r e c t e d , o f f - p a t h , u p w a r d , credulous i n h e r i t a n c e remains a n open p r o b l e m .

4

Conclusions

We have shown t h a t path-based inheritance reasoning as defined in T o u r e t z k y (1984) is N P - h a r d , even when restricted t o acyclic u n a m b i g u o u s n e t w o r k s . Moreover, the versions of t h i s f o r m of i n h e r i t a n c e t h a t use skeptical reasoning and o f f - p a t h p r e e m p t i o n are also i n t r a c t a b l e . T h u s , w h i l e T o u r e t z k y (1984) showed t h a t inheritance networks can be c o n d i t i o n e d to allow for correct and efficient retrieval of i n f o r m a t i o n ( t i m e 0(log(n)) for n concepts) such as in NETL ( F a h l m a n 1979), our results d e m o n s t r a t e t h a t t h i s c o n d i t i o n i n g procedure itself is i n t r a c t a b l e w h e n based on d o w n w a r d (coupled) inher­ itance. However, o u r other c o m p l e x i t y results, gener­ a l i z i n g the t r a c t a b i l i t y result o b t a i n e d by H o r t y ct ai (1987), also suggest t h a t t h e various f o r m s of u p w a r d (decoupled) inheritance can be used to achieve t r a c t a b i l ­ ity. One possible d i r e c t i o n for f u t u r e research is to consider f u r t h e r restrictions o n the f o r m o f inheritance networks t h a t w o u l d allow for a p o l y n o m i a l inference mechanism based on d o w n w a r d i n h e r i t a n c e . One candidate we have already begun to explore is i n h e r i t a n c e restricted to com­ pletely balanced hierarchies. Such hierarchies have a m a x i m u m d e p t h o f 0 ( l o g ( n ) ) , where n i s the t o t a l n u m ­ ber of concepts in the hierarchy. Such a r e s t r i c t i o n seems q u i t e reasonable, given t h a t t a x o n o m i c hierarchies w i l l often be " s h a l l o w . " However, we have f o u n d a reduc­ t i o n f r o m the s m a l l - c l i q u e p r o b l e m 1 3 t o d o w n w a r d i n ­ heritance reasoning w i t h such n e t w o r k s . T h i s result i n ­ dicates t h a t T o u r e t z k y ' s i n h e r i t a n c e n o t i o n restricted t o balanced hierarchies is m o s t l i k e l y s t i l l i n t r a c t a b l e ( i . e . , n o t p o l y n o m i a l ) . See Selman and Levesque (1989) for a m o r e detailed discussion of these issues.

Acknowledgments This work was supported in part by a Government of Ontario Scholarship to the first author, and a grant to the second from the Natural Sciences and Engineering Research Council of Canada. We would like to thank David Etherington for point­ ing out to us that the complexity of inferential distance was unknown. We also would like to thank Sue Becker, Jim des Rivieres, Russ Greiner, Jeff Horty, Gerhard Lakemeyer, Peter Patel-Schneider, Rich Thomason, David Touretzky, Wlodek Zadrozny, and the anonymous referees for providing useful comments.

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Annual Re­

Reiter, R. and Criscuolo, G. (1983). Some Representational Issues in Default Reasoning, Computers & Mathemat­ ics with Applications, (Special Issue on Computational Linguistics), 9 (1), 1983, 1-13. Sandewall, E. (1986). Nonmonotonic Inference Rules for Multiple Inheritance with Exceptions. Proceedings of the IEEE, vol. 74, 1986, 81-132. Selman, B. and Levesque, H. J. (1989). The Tractability of Path-Based Inheritance. Technical Report, Dept. of Computer Science, University of Toronto, Toronto, Ont., Canada, forthcoming. Touretzky, D.S. (1984). The Mathematics of Inheritance Systems. Report CMU-CS-84-136, Dept. of Computer Science, Carnegie-Mellon University, Pittsburgh, PA, 1984; also London, U K : Pitman, 1986. Touretzky, D.S., Horty, J.F., and Thomason, R.H.(1987). A Clash of Intuitions: The Current State of Nonmono­ tonic Multiple Inheritance Systems. Proceedings IJCAI87, Milan, Italy, 1987, 476-482.

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Given a graph G w i t h n vertices, does G contain a clique of size log(n)? (Karchmer 1989; Megiddo and Vishkin 1988)

Selman and Levesque

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