Abstraction Techniques for Connguration Systems

ion Techniques for Con guration Systems Rainer Weigel and Boi Faltings DRAFT Laboratoire d'Intelligence Arti cielle Ecole Polytechnique F ed erale de Lausanne (EPFL) IN-Ecublens, CH-1015 Lausanne Switzerland fweigel,[email protected] .ch Abstract In this paper, we address abstraction methods for conguration. Con guration is a design activity where the set of available components and their allowed combinations are known a priori and the goal of the conguration process is to nd the sets of components ful lling the customers wishes and respecting all the compatibility constraints. Abstraction techniques for con guration are becoming more and more important when dealing with the complexity of real world systems. We have developed abstraction methods for nite and discrete constraint satisfaction problems (CSPs) and will describe herein the advantages of using these methods for con guration system. In particular the elaboration of abstraction hierarchies and compacting knowledge using the concept of interchangeability and substitutability on the di erent levels will be described. By using interchangeability we can simply merge objects that are indistinguishable in an abstract space and thereby simplify the problem representation. Other advantages of nding interchangeability and substitutability are that (1) we do not have to bother the user with redundant information on different abstraction levels, and (2) it provides us ways to compact huge amount of data in a way that it can be overlooked by humans and (3) it gives us a possibility for integrating optimization procedures. With clustering methods we can transform a constraint based representation on a very ne level of granularity into a more coarse one. This enables us to transform a representation based on parts or components into a representation based on assemblies. Introduction A simple illustrative example will be given rst in order to demonstrate the importance of the concept of interchangeability and substitutability on the di erent abstraction levels. In a telecommunication con guration application we will have several analog and ISDN telecommunication facilities. For the ISDN facility a ISDN box and a 220V power supply are necessary. Provide telecommunication faciliy etc. etc. ISDN connection with fax provide ISDN connection provide analog telephone connection Figure 1: A function abstraction hierarchy Each ISDN telephone needs a compatible ISDN box which we will call then a ISDN system. These systems might be di erent in that some can provide additional functionality that other will not provide i.e. with or without a fax for example. However if a customer requires simply a ISDN system without specifying more details, we can see that all the possible systems are interchangeable for such a query. If a customer simply wants a telephone we see that on the next higher level that the analog and the ISDN installations are not interchangeable because the ISDN installations need a power supply. However, if there is a power supply available at the customers place then the analog and the ISDN installations would be interchangeable. This kind of interchangeability is called context dependent interchangeable (0). Similarly if several abstraction levels are considered at the same time the IDSN installations are substitutable for the analog installation since every functionality the analog system provides can also, beside other, be provided by the ISDN installation. Since we have to assume that a power supply is available we should call it more precisely context dependent substitutability. Thus we can leave the decision about the equipment to come after the choice of the system type (see Fig.: 1). Nevertheless the decision could be force by by decisions on the lower level in the hierarchy. The following points are important: 55 A ISDN telephone can not provide functionality by its own; it must be clustered or combined with a ISDN box. In general, a certain functionality of a product can only be achieved after single parts are clustered into assemblies. As soon as a ISDN system fed throught the telephone line is invented we have to recalculate interchangeability. The new systems will be interchangeable respectivly substitutable with analog systems in general and not only in the context where 220V power supply is available. Thus the decision hierarchy must have to be changed ; there is no longer the need of deciding on the type of connection so early in the decision process. Automatically generating such a hierarchy based on the concepts of interchangeability and substitutability is of major importance when changes of the knowledgebase occur frequently. Abstraction methods have been used successfully in various AI systems (0; 0; 0; 0). While abstractions were mainly used in those systems to improve search, we will exploit abstractions to facilitate also the interaction with the user . Our abstraction method combines the concept of interchangeability and substitutability with clustering methods. The goal of this paper is to demonstrate that 1.) con guration problems should be represented on different levels of abstraction and 2.) that the exploitation of interchangeability is in particularly useful for con guration problems. We furthermore claim that due to the internal structure of many con guration problems one can often avoid problems concerning the combinatorics that often appears in random CSPs. We present in the next sections a abstraction technique which is based on variable abstraction and value abstraction. Variable abstraction clusters variables into a new meta-variable and results after adjusting the constraints in a new CSP which is called meta-CSP in (0). Similarly value abstraction replaces interchangeable values into a single meta-value. The concept of k-interchangeability (0) plays a major role in the value abstraction procedure and can be seen as a basis for algorithms to calculate partial-interchangeability or meta-interchangeability also mentioned both in (0). The complete abstraction procedure can be described by applying recursively variableand valueabstraction procedures until a certain abstraction level is reached. Lets rst de ne the CSPs and describe then the two abstraction methods. A binary CSP is de ned by P = (X;D;C;R), where X = fX1; : : : ; Xng is a set of variables, D = fD1; : : : ; Dng a set of nite domains associated with the variables, C = fC1; : : : ; Cmg a set of constraints, and R =fRij Di Dj for Cij applicable to Xi and Xjg a set of relations that determine the de nition of the constraints. Solving a discrete CSP amounts to nding value assignments to variables subject to constraints. The constraint graph of a CSP is a graph G = (V;E) where V = fX1; : : : ; Xng and E = fe1; : : : ; emg with ei is the edge between the nodes related by Ci. Variable Abstraction Variables, inducing subproblems of the CSP, are clustered together and treated as a single meta-variable. Each solution to a subproblem becomes a structured value of this new meta-variable. Adjusting the constraints must in turn be done for all the values of each new variable. In con guration problems components can be put together to establish assembly variables. These new assembly variables itself might be subassemblies of even bigger assemblies and so on. By applying the clustering algorithm several times we will generate what we will call the \abstract graph" of a CSP. For con guration problems, it might turn out that the nodes on the di erent abstraction levels will also appear in the structural decomposition of the product. Fig 1 shows four constraint graphs of the single constraint problem on di erent levels of abstraction. Constraint graphs of problems that arise from physical interactions often resemble clustered graphs. This is in particular true for constraint graphs of product con guration problems with their modular and hierarchical structure. In those graphs we can nd high interactions between parts within a assembly and weaker interactions in between di erent assemblies. These structures are said to exhibit a ultrametric topology (0; 0). var1292 var1293 var1291 var1290 var1288 var1289 var1287 var1278 var1271 var1286 var1284 var1279 var1268 var1285var1280 var1195 var1282 var1283 var1194 var1281 var1276 var1277 var1275 var1274 var1273 var1272 var1270 var1269 var1238 var1267 var1266 var1262 var1265 var1264 var1263 var1256 var1250 var1260 var1261 var1259 var1258