Discovering Implicit Integrity Constraints in Rule Bases Using ...

Proceedingsof the 28th Annual Hawaii International Conferenceon SystemSciences - I995

Discovering

Implicit

Integrity

Constraints

in Rule Bases Using Metagraphs

Amit Basu and Robert W. BIanning Owen Graduate School of Management Vanderbilt University Nashville, TN 37203

Abstract

A metagraph is a graph formalism that can be used to model the different components of a DSS and to analyre the interactions among them. One such component is a knowledge based system (KBS) in which domain knowledge is stored ss if-then rules. In addition to such rules, a KBS can also contain a set of integrity constraints, which are used to manage the rule base both duringnpdatesand problem solving. In this paper, we show how metagraph representation of a rule base can facilitate discovery of implicit integrity constraints in addition to explicitly stated ones. Introduction

A useful type of information in DSS is expert knowledge, often codified in the form of logic or rules. This knowledge may be used to describe the structure of the DSS in terms of relationships among information sources [Bonceek, Holsapple, and Whinston, 1981, 1982; Dutta and Bssu, 19841,decisionmaking heuristics [Davis and Lenat, 19821,and integrity constraints within and between information sources [Minker and Grant, 19901. Attempts have been made to model rule bases in a comprehensive framework - such as a relational framework [Blanning, 1985,1994] - that can accommodate other information types as well. One such approach that hss proved quite effective is based on metagraphs, which are an extension of digraphs and hypergraphs that can be wed to describe and analyze rules, as well as data bases and model banks pasu and Blanning, 1992; 1994(a,b)J. A metagraph edge describes tbe relationship between two sets of elements, such as the key and content attributes of a file, the input and output attributes of a model, or the antecedent and consequent propositions of a rule.

1060-3425/95 $4.00 Q 1995 IEEE

In the context of rule bases, it can be shown that rule-based inference in a KBS can be modeled as a connectivity problem in the metagraph corresponding to the rule base. This is shown in Section 3 of this paper. Also, certain useful data structures defined on metagraphs can be exploited to operationaliee the solution of this connectivity problem quite elegantly. However, despite the convenience and elegance of this approach, it is not clear that from a computational efficiency standpoint, it has any advantages over existing algorithms for heuristic inference. Nevertheless, we feel that metagraphs are an effective modeling and analysis tool for rule based systems. This is because of two additional capabilities of this approach: 1. The metagraph representation of a rule base can be exploited to discover implicit integrity constraints; 2. Rules represented as metagraph edges can be related to relevant model and data constructs in a DSS, that are also representable as metagraph edges, thus providing a uniform framework for system analysis and design. The focus of this paper is on the first of these issues. An integrity constraint is simply a necessary condition on the knowledge base of a KBS. In other words, any inference made from a knowledge base is valid only if it does not violate any relevant integrity constraint. An example of such a constraint is - “no salaried employee can be paid for overtime”. The major difference between a rule and an integrity constraint is that a rule is a sufficient but not necessary condition for the inference of its consequent. That is, in a rule base containing several integrity constraints with the same antecedent, all these integrity constraints have to be checkedif the antecedent eval-

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Proceedings of the 28th Hawaii International Conference on System Sciences (HICSS '95) 1060-3425/95 $10.00 © 1995 IEEE

Proceedingsof the28th Annual Hawaii International Conferenceon SystemSciences- 1995

uates to true; however, given several rules with identical antecedents, the control mechanism of the KBS may not require all the rules to be fired if the antecedent is true. The paper is organized in five sections. Westart by briefly describing metagraphs and their features in the next section, and the representation of rules as metagraph edgesin Section 3. Then, in Section 4, we discuss how implicit integrity constraints can be discovered from the metagraph corresponding to a rule base. Finally, Section 5 presents some directions for additional research. Metagraphs

Metagraphs are graphical structures consisting of a set of elements and a set of edges that contain some of the elements [Basu and Blanning, 1994(a,b)]. However, unlike a digraph where each edge is an ordered pair of elements [Berge, 19851,or a hypergraph where each edge is a non-n,ull subset of the set of elements [Berge, 19891,in a metagraph, each edge is an ordered pair of eubsetg of the set of elements. The first subset is called the invertexof the edge, the second subset is tbe outvertex, and at least one of these vertices must be non-null. A formal definition is as follows: I: Given a finite generating set X = { zi, i = l...I}, a metagraph is an ordered pair S = ,whereEisasetofedgesE={er,,k= l...K}, each of which is an ordered pair ck = < v,,B$>WithVk,Wk~XandVkUWk##.Each vk is the invertex of the edge ek, and wk is the putvertex.’ An edge in a metagraph may represent any DSS component viewed ss a black box. The component may be a file, a model, or a situation-action rule. If the edge represents a file, then the elements are data attributes, the invertex is the key of the file (i.e., the set of key elements), and the outvertex is the content. If the edge represents a decision model, the elements are variables, the invertex is the input to the model, and the outvertex is the output. If the edge represents a rule, then the elements are propceitions that may be either true or false, the invertex represents the antecedent to the rule, and

the outvertex represents the consequent. For example, consider a rule base consisting of five propositions 21 . ..25(~ldtwo~lt8(z~h2~)~ (2s I\ $4) and 24 + 25. The first rule would be rep resented by an edge ei = < V’, WI > with invertex and v-1 = {z~,Q} and outvertex WI = {~,24}, the second rule would be represented by the edge e2 = < (24), { ~5) >. This is illustrated in Figure 1. An important property of metagraphs, as with any graph structure, is connectivity. In digraphs and hypergraphs connectivity exists between pairs of elements whenever there is a path from the first element to the second element. This is also the case in metagraphs. For example, in Figure 1 there is a path from 21 to 25, consisting of the sequence of edges < ei, ez >. There is no path from t5 to 21, nor for that matter from t5 to z5 nor from 21 to 2~. We formalize this concept as follows: DEFINITION II: Given a generating set X, a metagraph S =C X, E >, and two elements a E X from u to b is a sequence and b E X, a ~ of edges h(a, b) = < ei . . .ei > such that: 1. aEV/,bE

2. W~nV,‘+,#~forZ=l...L-1 3. {e;,I=l...L}EE

DEFINITION

lh thin paper, we um curled brackets ( . . . ) to denote a ret and angle bracketa < . . . > to denote a sequence.

Wt

The coinola_tof a in h(a, b) ia [~4’\~W]\I.) 1x1

I=1

and the cooutm& of b in h(o, b) is

A type of path that is of special interest is a &, which is a path from any element to itself. That is, a cycle is a path h(a,a) for some a E X. A metagraph that contains no cycles is said to be acyclic. There are two difficulties with the concept of a simple path. The first is that a simple path is limited to a sequenceof edges, but we may wish to consider edges that do not form a sequence. The second is that a simple path is defined between a pair of elements, but we may wish to examine connectivity from one set of elements to another set of elements.

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Proceedings of the 28th Hawaii International Conference on System Sciences (HICSS '95) 1060-3425/95 $10.00 © 1995 IEEE

Proceedingsof the 28th Annual Hawaii International Conferenceon SystemSciences- 1995

To see this, consider the metagraph in Figure 2. In thii metagraph there are two simple patha from 22 to ts. The first is the path < el, es > with coinput {t~,zd) and eooutput (zs), and the second is the path < e2, es >, with coinput (23) and cooutput {Q}. However, if neither tg nor t4 is available, then neither path can be ueed. But there is a third poesibility, to use all three edgee. Tbia will change both the required input and the resulting output. That is, the elementa 21 and 22 must be available, and the elemente ts and 24, along with 86, are generated as outputs. However, the ee.t(cl, es, es) ie not a simple path, since it ia not a sequenceof edges. To overcome thie problem, we define a metapath from one mt of elements to another set of elementa - for example, from (21,22) to (25). Henceforth we will m+ethe operator denoted Set( ) to convert a eequence into a set: Set(< a, b,c.. . >) = {a,b,c.. .}. DEFINITION III: Given a generating set X, a metagraph S =< X, E >, and two disjoint subsets BEXandCEX,awfromBtoCisa set of edges M(B, C) = (ef, 1 = 1.. . L) C E such that: 1. For each I = 1.. . L there is a simple path h,(b, c) from some b E B to some c E C such that ei E Set(hl(b,c)) C M(B,C). 2. IJ;,

v,‘\U;“=,

W,’ E B

3. c E u;“t, w;. Any metagraph can be described by an adj& cency matrix, which contains one row and one column for each element in the generating set [Baeu and Blanning, 1994(b)]. The adjacency matrix, A, of the metagraph in Figure 2 appears in Figure 3. Each member, aij , of A containe a triple for each edge connecting the t7h element in the generating set to the Jamelement. For example, all = asI = 4, since there are no edgee connecting 21 to itself or connecting 23 to 21. On the other hand, a13 contains one triple, since there is one edge connecting 21 to x3. The first component of the triple is the coinput of x1 for thie edge, the second component is the cooutput of 23, and the third component is the edge. Since 21 has coinput 22, while 25 haa no cooutput, and the edge is e1, we have a13 = {< {ta), 4, < el >>}. Similarly, since e1 connects 22 to 24 with no coinputs or cooutpute and no other edges connect 22 to 24 we have 424 = {< +,d, < e2 >>}.

There is also a multiplication operation for metagraph adjacency matrices that can be used to calculate the power of such a matrix [Basu and Blanning, 1994(b)]. The nt* power of the adjacency matrix dieclosesall simple paths of length n connecting any two elements. For example, A2 con&&u mostly of null members, since there are only three paths of length two in our example. One of these connects t1 to 25. The triple 4!5 = (< {tp,24}, (x3}, c e1,es>>}discloaesthatthepathis,that z1 has coinputs {22,24} and z5 baa cooutput (23). The situation with regard to ai5 is more complex, since there are two paths of length 2 connecting 22 to 25: < el, es > and < e2, ea >. These paths, along with the corresponding coinputs of 22 and cooutputs of 25, are disclowd by the two triples that make up al5 : < {21,24},d23},< el,e2 >> and < (23}, {,24}, c es, e3 >>. All members of A2 except for 4f5 and ai are null. Since there are no paths of length 3 or greater in the example, all members of A” are null for n 2 3. More generally, for an acyclic metagraph all members of A” will be null for any n that exceeds the number of edges. In addition, all diagonal elements of A” for any power of n will be null. Adjacency matrices can not only be multiplied, they can be added. For example, each member of the sum of A and A2 consists of the union of the triples in the corresponding members of A and of A2. This will disclose all paths of length 1 or 2 along with their coinputs and cooutputs. More generally, the sum of the matrices A” for n = 1,2.. . until all higher powers of A contain only null elements will disclose all paths of any length between any two elements in the metagraph. This is &led the closure of the adjacency matrix and is denoted A*. In the example, A* = A + A2 (Figure 4). Rule

Representation

Using

Metagraphs

In Section 2 we formalized the concepts of metagraphs and connectivity. We now discuss how these ideas relate to the organization and processing of rule bases. We begin by defining a rule base. A rule base consists of a collection of rules such as “IF (the shipment bas not arrived AND the inventory is less than 500 units) THEN (a purchase order should be prepared AND the shipping department should be notified).” Each rule can be represented as an expression of the form

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Proceedings of the 28th Hawaii International Conference on System Sciences (HICSS '95) 1060-3425/95 $10.00 © 1995 IEEE

Proceedings of the 28th Annual Hawaii International Conference on System Sciences - 1995

lb-+/b

PEY

qEZ

where Y, 2 E P, a set of propositions, “A” denotes conjunction and “+” denotes implication. For example, the above rule could be restated as

In previous work, we have shown that the metagraph corresponding to a rule base can be used to determine whether a set of literals is a logical consequence of another set [Bssu and Blanning, 1994(a)]. We begin by establishing the .correspondence between metapaths and valid inferences. Theorem I: Let X = {zi, i = 1. . . N} be a generating set and 5 =< X,E > be au acyclic metagraph on X corresponding to a rule base T =< P, R >. Given two nonempty disjoint sets of elements Xi, Xs c X corresponding to two sets of propositions 4, Ps C P, there is a metagraph M from X1 to X2 in S iff the following implication is valid in T:

where pl = n the shipment has not arrived”, pz = “the inventory is less than 500 units”, ps = n a purchaseorder should be prepared”, and p4 = n the ehipping department abould be notified”. Formally, a rule base can be defined as follows: DEfXNITION IV: A rule base is an ordered pair T =< P, R > in which P is a set of propositions {pj,i= l...I}andRisasetof&R={rs,k= QES PEP% l...X} with each rule fk being an expression of the form Proven in [Barn and Blanning, 1994(a)].

AP-+/\'I

b’lb ,‘EYk

YEZk

with Yk C_P and Zs C P. Yk is the &p&$&& of rk and Zk is the mauent.. A rule base can be represented as a metagraph in which P is the generating set with each pi an element in the set, R is the set of edges with each fk an edge, and Yk and Zk are the invertex and Outvertex of fk . DEFINITION V: Given a generating set X and a metagraph S =< X,E > on X with E = {ek,k = l,... K}andarulebaseT=,S corresoonds to T if there are bijective mappings between X and P and between E and R such that for anyzEXandpEPandforanyl_ be a metagraph on X. Given two nonempty disjoint sets of elements X1,X2 z X and the set of all simple paths 0 = {hl , hs . . . h,) from any element 2 E Xi to (y} where y E X2, for every metapath M from X1 to y E X2,3H E 2e, where 2’ is the power set of 0, such that M = Set(H). Proven in [Basu and Blanning,

1994(a)]

Theorem II implies that the search for a metapath from Xi to Xs can be limited to combinations of simple paths from elements in Xi to elements in X2. This is useful, since we know from our earlier discussion that the set of triples in each cell sb of the A* matrix represent all of the simple paths from si to sj. It follows then that the set of all simple paths from elements in X1 to elements in Xs is given by:

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Proceedings of the 28th Hawaii International Conference on System Sciences (HICSS '95) 1060-3425/95 $10.00 © 1995 IEEE

Proceedings of the 28th Annual Hawaii International Conference on System Sciences - 1995

In other words, given an interpretation in which the elements in Xr are known to be true, we can infer that the elements of the set X2 are also true if some combination of triples in the intemection of the X1 tow8 and the X2 columns of A’ comprise a metapath from X1 to X2. The meet eignificant aspect of this ia that the search for a metauath can be limited to combinations of triples in a 8ubset of the columns of the A* matrix. It is possible to simplify the tesk of finding a metapath still further. We note that given q path8 hr,... h, from a set of elements XI to some set of elements X2, let ai = @vertices on ith path), and pi = U(outvettices on i’h path). If IjQi\ljPi i=l

Exl

a

Discovering

:

x2

+

e3

:

x3

A 24

24 -+

26

Now consider the question “Can 26 be inferred from 21 and 22? That is, L the formula21 A 22 4 26 a logical consequenceof the tde base? n To answer this question, we consider the triples in the cells ai6 and ag6 in Figure 4, and find that there are three such triples:



>




(21922,241 from

{x1,22)

to

{23,2&d

to

(26)

In the previous sections, we have aesumed that all the rules in the rule baserepresented by a metagraph contain only positive propositions. In this section, we discuss how information about complementary literals can be used to make certain useful ttansfotmations in the metagraph representation of a rule base that can reveal relevant integrity constraints. The issue of integrity maintenance is an impottant one in the context of knowledge baaed eystems and information By&ems in general. One way in which integrity is enforced in a rule based KBS is through the use of integrity constraints. While any sentence can be used aa an integrity con&Ant, a common and fairly widely applicable form for integrity constraints is a statement of the form: ‘21

v

‘522..

. v -xn

which can also be istated in the form 2lA

22

...AXk-l/\2k+l...A2N-,~21

for any k between 1 and N. Typically, aa part of the definition of the rule base, a number of such integrity constraints may be included. These conetraints can be used not only during the problem solving proceeg to eliminate infeasible solutions, but also ae integrity checks during any update8 to the rule base [Minker and Grant, 19901.

Combining these triples and reducing the input aeta gives us the following additional metapaths: A& :

btegrity

hpkit

i=l

then UizISet(hi) is metapath from X1 to X2. In other word& in testing each combination of triples in the candidate set, we can take the unions of their inputs and outputs respectively, and can remoYe any elements from the input set that alao appear in the output set. If the resulting input set is contained in X1, and the output set contains X2, the union of the edge8in the triples comprises a valid metapath from X1 to X2. Example: Consider again the metagraph in Figure 2 which corresponds to the following rule base el : 21 A22 423 e2

Since hf2 i8 a metapath from (21~22) t0 (x6}, we can conclude that 26 can be inferred from tl and 22-thatis,2lh22+25. Note that in the above example, tbe search for a relevant metapath could be limited to just two of the cells in the A* matrix. In other words, even in the case of a large rule base defined on a large generating set, if the A’ matrix is precompiled and stored, it can he used to quickly infer desired results from given inputs. The complexity of the run time task is clearly a function of tbe number of triples in the intersection cells, and is thus most effective when this number if relatively small. We contend that even in many large rule b-, the number of paths between two elemellts may not be very large, and in such ewes, the metagraph representation of the rule base provides a useful precompilation and knowledge management tool.

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Proceedings of the 28th Hawaii International Conference on System Sciences (HICSS '95) 1060-3425/95 $10.00 © 1995 IEEE --,-..1

_,I..

----,--/

-.-_

--.

-~

__._-_,,_,,___,.._

-

.,-

.--

---

---

-.

-~_

Proceedingsof the 28th Annual Hawaii International Conferenceon SystemSciences- 1995

In addition to the explicitly stated integrity constraints in the rule base, implicit constraints may exist in the knowleae base, or may rmult from determinations that some propoeitions are either complementary or mutually contradictory. Consider the following rule base.: ((a + ~),(a -+ q)}, where a is come conjunction of proposi+ns containing neither p nor q. In this rule base, if we impose an explicit integrity con&r&t (7pV -q), we find that there is an implicit integrity constraint -a which the eystem designer or UoGr may not be aware of. cl Discovery of such implicit integrity constrainta can be very useful during problem solving. Methods for the discovery of such implicit constraints are valuable, since they facilitate rule base management. In this section, we show that metagraph teptesentation of rules and the corresponding A* matrix can facilitate tL-. ;L;,.vz~~ of ikilplicit iptogtity COIl&&iKlt8. For thirs purpose, we need to define an augmented form of the triples in A*. Given a triple < {a}, {a), < h >> in a~j (where a and P are sets of elements and h is a eequenceof edger), the augmented triple corresponding to this is given by < {a,Zi}, {Pt2j), < h >>*

that the antecedent of the rule is never true, and thus the rule is uselessand can be deleted. Transformation 2: Triple of thie type correspond to the implication a A 2i + B A 2j. which can be simplified to the two clauses

Examples

Theorem II& Given a rule baee and its corresponding metagraph s, if the constraint TXi V -2j is added, then the following rules of transformation can be used to simplify the A’ matrix of S (where a,/3 C X, ad Ei,Sj E X ad

{zi,zj}n(aU@)

= 4,

and where h is a sequenceof edge8forming a simple path): 1. Any triple of the form < can be deleted;

{2i,2j9a),{/3),
>

The second clause can be resolved with the integrity constraint to infer the integrity con&taint -ra V y2ir which is the desired result. Tranefotmation 3: Triples of this type corte8pond to the implication a -+ p A2i A 2j. However, the given integrity conetraint is equivalent to the expression y(Ei A Zj). Hence the consequent of the rule must alwaye be false. This in turn impliee that a must always be false (i.e., la), which is the desired result. QED. The transformations in the above theorem can be used to eliminate some triples (?‘ransformation 1) and to extract implicit integrity constraints (lknsformations 2 and 3). We illustrate thie in the following example: Example: Gonrrideting again the metagtaph in Figure 2, and its A* matrix in Figure 4. The following separate examples illustrate each of the three rules in Theorem III: If we introduce the conetraint 7~1 V 724, then the triple < {zl,z4},{23},< el,es >> in a& can be deleted, by Rule 1. If we introduce the constraint 1%~ V -23, then the integrity constraint 1x1 V 1x2 V -24 can be inferred by Rule 2. If we introduce the constraint 724 V 726, this lead8 to the constraint -22 V -23 by Rule 3. El Furthermore, if we coneidet not only the simple paths repteeented by the triples in A’, but also metapaths, then we get the following additional transformation:

2. From any augmented triple of the form < {zi,a},{xj,/3},< h >> the integrity constraint ~a V -Xi can be inferred; Theorem IV: Given a rule base, if the constraint 3. From any augmented triple of the form < ‘xi V -2j is added, then for any set of elements a ~?MG, 2;:~ iierz the iW@r con- such that there is an acyclic metapath in the cortespending metagraph from a to both 2i and tj, the 7 integrity constraint -a can be inferred. Proof: From the previous section, we know that if Proof: We consider each rule in turn: Transformation 1: Given the integrity constraint, there is a metapath from a to both 2i and 2j, then clearly both zi and xj can never be both true, 80 it follows that

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Proceedings of the 28th Hawaii International Conference on System Sciences (HICSS '95) 1060-3425/95 $10.00 © 1995 IEEE

Pmeedings

0 +

of the 28th Annual Hawaii International Conference on System Sciences -

1995

which would require consideration of more complex edge types, including embedded edges. These research efforts, and others that may be spawned from them, may be quite helpful in developing a general view of DSS that incorporates a wide variety of information types.

2i A 2j

Bowever, this corresponds to the case where Transformation 3 applies in the previoua theorem, and the redt follow& QED. Example: Consider yet again the metagraph in Figure 2, and its corresponding A and A’ matrices. The metapath {ei, cr} connects the set (~1~~2) to the set {ZQ, 24}. If we add the constraint 1~3 V -7~4, then by Theorem IV, we get the integrity constraint -21 V -22 (it ie worth noting that in this case, we also find that by Transformation 1, the edge es can be deleted).

ACKNOWLEDGEMENT: This research was trupported by the Dean’s Fund for Faculty Research of the Owen Graduate School of Management of Vanderbilt University.

REFERENCES

0

Thus, we have shown that the A’ matrix for a metagraph corresponding to a rule base provides a useful basis for not only determining valid inferences (as described in the previous section), but also for detecting implicit integrity constrainta. Furthermor+since-t-set of integrik --nn+++.ir+.only changes when the rule base is updated, the proctss of extracting the constraints can be viewed as part of the compilation of the rule base in terms of the A* matrix.

1. Basu, Amit and Blanning, Robert W., “Enterprise Modeling Using Metagrapha,” in Decision Support Systems:Experiences and Expectations, ed. by Tawflk Jelaesi, Michel

R. Klein, and W.M. Mayon-White, NorthBolland, Amsterdam, 1992, pp. 183-199. 2. Basu, Amit and Blanning, Robert W., “A Graph-Theoretic Approach to the Analysis of Knowledge Bases Containing Rules, Models and Data”, Working Paper, Owen Graduate School of Management, Vanderbilt University, 1994(a).

Conclusion

In this paper we have shown that the closure of the adjacency matrix (A’) ia a useful way of organieing information about a rule base not only to determine valid inferences but also to detect implicit integrity constraints. In this paper, we have restricted our attention to integrity constraints that have a specific form, namely disjunctions of literals. However, as mentioned earlier, any logical sentence can be used as an integrity corurtraint. We feel that metagraphs might be useful in analyzing the impact of such conetrainta as well, by examining the consequencesof forcing edges corresponding to such constraints to be valid in any interpretation. However, this requires further study. The existence of a single framework, metagraphs, for the integration of these types of information suggests two possible area8 for further research. One ia an extension of this work to the case in which the corresponding metagraph contains cycles - that is, when the rule base ia recursive. The other area is the inclusion of more complex knowledge structures, such as semantic nets and frames,

3. Beau, Amit and Blanning, Robert W., “Metagraphs:A Tool for Modeling Decision Support Systems,“, to appear in Management Science, 1994(b). 4. Basu, Amit and Blanning, Robert W., “Model Integration Using Metagraphs”, to appear in Information Systems Research, 1994(c). 5. Berge, Claude, Hypergraphs: Combinations of Finite Sets, North-Holland, Amsterdam, 1989. 6. Berge, Claude, Graphs, (2nd ed.) Holland, Amsterdam, 1985.

North-

7. Blanning, Robert W., “A Relational Algebra for Propositional Logic”, Decision Suppori Systems, Vol. 11, No. 2, February 1994, pp. 211218. 8. Blanning, Robert W., “A Relational Framework for Assertion Management,” Decision

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Proceedings of the 28th Annual Hawaii International Conference on System Sciences -

Support Systems, Vol. pp 167-172.

1, No.

2, April

1985,

9. Boncsek, Robert H., Holsapple, Clyde W., and Whinston, Andrew D., “The Evolution from MIS to DSS: Extension of Data Management to Model Management,” in Decision Support Systems, ed. by Michael J. Ginzberg, Walter Reitman, and Edward A. Stohr, North-Holland, Amsterdam 1982. 10. Bonczek, Robert H. Holsapple, Clyde W., and Whinston, Andrew B., Foundations of Decision Support Systems, Academic Press, New York, 1981.

1995

12. Dutta, Amitava and Basu, Amit, “An Artificial Intelligence Approach to Model Management in Decision Support Systems,” IEEE Computer, Vol. 17, No. 9, September 1984, pp. 89-97. 13. Maier, David, The Theory of Relational Databases, Computer Science Press, Rockville, 1983. 14. Minker, J. and Grant, J., “Integrity Constraints in Knowledge Based Systems,” in Knowledge Engineering, Vol. II, Applications, ed. by H. Adeli, McGraw-Hill, New York, 1990,. pp. l-25.

11. Davis, Randall and Lenat, Douglas B., Knowledge-Based Systems in Artificial Intelligence, McGraw-Hill, New York, 1982.

FIGURE 1: A Metagraph with a Simple Path

FIGURE 2: A Metapath

Proceedings of the 28th Hawaii International Conference on System Sciences (HICSS '95) 1060-3425/95 $10.00 © 1995 IEEE

Proceedings of the 28th Annual Hawaii International Conference on System Sciences -

(4x

I I,= 1’

>>I 1

(J

1995

x2

0

0

0

x3

0

0

0

0

(I

3 =I

0

FIGURE 3: The Adjacency Matrix (A)

I 0.e

0

x4 I b3L