Testing Oz Facilities in Distributed Problem Solving. Werner Dilger, Rainer Staudte, Kay Anke. groups, teachers, and rooms. lessons.

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1 Testing Oz Facilities in Distributed Problem Solving Werner Dilger, Rainer Staudte, Kay Anke Abstract The Oz programming language incorporates several features that have become important for developing solutions for complex problems. Among these features are constraint programming, object orientation, and concurrent programming. This paper describes a special solution to a standard problem, the time table problem. The solution proceeds in two phases: the construction of a time table that generally is not optimal, and a subsequent improvement of that time table. The rst phase was realized using constraint programming, the second phase is being realized as a multi-agent system based on the idea of the contract net protocol. The implementation done so far is sketched. Keywords Multi-agent systems, contract net model, constraints, time table problem, Oz 1. Introduction For a few years the AI research group at Chemnitz University has been dealing with multi-agent systems. For this reason, the Oz language became interesting for us and we began to study this language. Our contribution aims to gain a deeper insight into basic Oz facilities. We selected the well-known time table problem as an example. The main purpose of this choice is not to develop a special solution for this problem, but to check several Oz features in a real situation. In section 2 the time table problem is introduced and illustrated with an example. Section 3 gives a formalization of the problem which is the basis of the implementation. The principles of the implementation are described in section 4. It consists of a constraint system and a multi-agent system. The latter simulates the behavior of teachers in those times when time table programs did not yet exist, but includes the student groups as agents too. Section 5 sketches the implementation done so far and section 6 summarizes the results. 2. The Problem The model of a time table introduced in this section differs somewhat from the problem in [1]. For given sets of subjects (that consist of several lessons, student groups, teachers, rooms, and times) assignments are to be found such that a well dened set of constraints holds and several quality criteria of the time table are obeyed as far as possible. Typical constraints are for instance: For each subject there is given a set of student groups, teachers, rooms, and times to chose from. Werner Dilger, Fakultat fur Informatik, TU Chemnitz-Zwickau. dilgerinformatik.tu-chemnitz.de Rainer Staudte, Fakultat fur Informatik, TU Chemnitz-Zwickau. staudteinformatik.tu-chemnitz.de Kay Anke, Fakultat fur Informatik, TU Chemnitz-Zwickau. k.ankeinformatik.tu-chemnitz.de There are times when subjects, student groups, teachers, and rooms are unavailable. Not more than one lesson at any time for student groups, teachers, and rooms. Rooms of sucient size have to be assigned to the lessons. Quality criteria for a time table can be: As few as possible time gaps for teachers and students between lessons. Minimal moves between rooms in remote parts of the building. The length of a working day for teachers and students should not exceed a given limit. The following example will be used in the next section to illustrate the denition of the time table problem. Example 1: A time table for ve days per week (Monday till Friday) and six time slots per day is to be found. The last slot on Wednesday and the last two slots on Friday are closed for planning. There are 12 subjects as follows: Trainer Sparwasser gives two swimming courses in the pool. The group 7901 has three exercises, the groups 7902 and 7903 have to train together four exercises. Dr Bitner gives a computer science lesson for all three groups once a week. Beside this he gives programming courses, twice a week for the group 7901, and three times for each of the other groups. Miss Hurtig gives Spanish courses ve times a week for each group separately. Prof Fuchs has an algebra seminar four times a week for each group separately. The swimming pool is open only on Monday and Wednesday. The last two times of each day are excluded for group Dr Bitner is accessable only on Wednesday and Thursday. The time table in this example should have the following features: All subjects should be planned as demanded. Groups and teachers should not have gaps in their time tables. The times should be distributed over the week as good as possible. The number of room changes for groups should be minimal.

2 3. A Time Table Model In the time table model of [2], a time table is viewed as a state in a search process. Such a time table may be complete or incomplete, i. e. not all possible assignments have been made yet. Denition 1: The quadruple T T P = (C; M; Q; S 0 ) is called a time table problem if the following holds: 1. C = (L; G; T; R; U) collects the sets of subjects L = fl 1 ; l 2 ; ; l n Lg, student groups G = fg 1 ; g 2 ; ; g n Gg, teachers T = ft 1 ; t 2 ; ; t n T g, rooms R = fr 1 ; r 2 ; ; r n Rg, and times U = fu 1 ; u 2 ; ; u n Ug. 2. M denotes a set of mappings fm G ; m T ; m R ; m U g, where m G : L! N G species the number of student groups to be selected and the set of student groups to select from, m T : L! N T species the number of teachers to be selected and the set of teachers to select from, m R : L! N R species the number of rooms to be selected and the set of rooms to select from, m U : L! N U species the number of times to be selected and the set of times to select from for each subject, respectively. 3. The set Q = fq 1 ; q 2 ; ; q n g includes several quality criteria q k of a time table usually formulated as functions q k : fs i ji = 0(1)pg! N from the set of all (complete or incomplete) time tables into the set of natural numbers N. The value 0 for a certain q k expresses the ideal (possibly utopic) value. Especially q 1 denotes the number of student groups, teachers, rooms, and times that are not yet assigned to any subjects. 4. S 0 is an initial time table, where the values q k (S 0 ) do not t certain objective or subjective demands. Example 2: Let's transform example 1 of the previous chapter: T T P = (C; M; Q; S 0 ) C = (L; G; T; R; U) L = fswimming 1 ; swimming 2 ; computer science; programming 1 ; programming 2 ; programming 3 ; spanish 1 ; spanish 2 ; spanish 3 ; algebra 1 ; algebra 2 ; algebra 3 g G = f7901; 7902; 7903g T = fbitner; F uchs; Hurtig; Sparwasserg m G : L! N G R = f216; 301; swimming poolg U = f1; ; 17; 19; ; 28g swimming 1! (2; f7902; 7903g) swimming 2! (1; f7901g) computer science! (3; f7901; 7902; 7903g) programming! (1; f7901g) m T : L! N T algebra! (1; f7903g) swimming 1! (1; fsparwasserg) swimming 2! (1; fsparwasserg) computer science! (1; fbitnerg) programming! (1; fbitnerg) m R : L! N R algebra! (1; ff uchsg) swimming 1! (1; fswimming poolg) swimming 2! (1; fswimming poolg) computer science! (1; f216; 301g) programming! (1; f216; 301g) m U : L! N U algebra! (1; f216; 301g) swimming 1! (4; f1 6; 13 17g) swimming 1! (3; f1 4; 13 16g) computer science! (1; f13 16; 19 22g) programming! (2; f13 16; 19 22g) algebra! (4; f1 17; 19 28g) q 1 : S i! N denotes the number of unassigned objects. q 2 : S i! N denotes the number of gaps in teacher plans. q 3 : S i! N denotes the number of gaps in group plans. As a rule, the numerous quality criteria of a time table cannot be satised simultaneously. Instead one has to nd a compromise between the criteria. There are dierent ways to cope with this problem. For instance modeling the problem by means of soft constraints has gained much attraction in the last few years. Unlike in this approach we model Q from the standpoint of multiple objective decision making. The special criterion q 1 (S ) = 0 must be satised, but for all other q k with k > 1 the complete time table should have values that are in the Pareto set or at least close to it.

3 q 1 q 3 6 S 3 (2) S 2 S 4 (2) (2) / S 5 BB (3) (3) (1) BN S 6 Q Qs S - S XXXXXXXXXz 0 (1) S 1 Fig. 1. Multiple objective decision making in the time table model On the basis of this model, the planning process divides into three parts which are illustrated in gure In the rst, the monotone phase (S 0! S 1! S 2 ) the value of q 1 (S i ) is to be decreased to zero. 2. In the second, the cooperative phase (S 2! S 3! S 4! S 5 ) at least one value q j (S i ) with j > 1 decreases in each step, while all other values q k (S i ) with j 6= k do not increase. 3. The third, the contradictory phase (S 5! S 6! S 7 S ) is a search in the Pareto set, i. e. decreasing one value q j (S i ) causes increasing of at least one other value q k (S i ), j > 1, k > 1, j 6= k, and still q 1 (S i ) = Solution Strategy We want to implement a time table problem solver (T T P S) in Oz, thereby testing dierent facilities of this language. For the rst, i. e. the monotone phase of the planning process, the use of constraints seems to be the adequate means of implementation. For the other two phases, the cooperative and the contradictory one, we prefer a multi-agent system designed according to the principles of Reid Smith's contract net model [3], [4]. The rst part of the T T P S, the constraint system, generates a plan for each object occuring in the time table domain (teachers, student groups, rooms) taking into account the hard constraints of the objects. There may also be soft constraints, but these are not regarded, therefore the plan produced by the constraint system is generally not optimal. When the plan generation has been nished, the second part of the T T P S, the multi-agent system starts to improve the plan. According to the contract net model, an agent can take on two roles in the problem solving process: As a manager it subdivides a task into subtasks and announces them to other agents. If there are agents that are ready to execute q 2 the subtasks (so-called "contractors") it supervises the execution of the subtasks. As a contractor it is responsible for the actual execution of a task, either by doing it itself or by playing the role of a manager with respect to that task. Obviously, an agent can take on both roles simultaneously. In the time table domain there are three kinds of objects: teachers, student groups, and rooms. The rooms are regarded as mere objects, the teachers and student groups are taken as agents. They are able to improve their respective time tables by negotiating with each other. We distinguish between the two groups of agents, the teachers and the student groups, as dierent agent groups. If during the planning process an agent of one group has some contacts with an agent of the other group, the latter is called the complementary agent. If A is an agent, each entry in its time table consists of a pair (K; R), with K a complementary agent of A and R a room. There are four basic operations that an agent can apply to improve its plan: 1. Moving the assignments of a certain time (agent, complementary agent, room) to time gap. 2. Replacing a time gap with the assignments to another time. 3. Exchanging two lessons. 4. Exchanging three or more lessons. For reasons of eciency, the rst three operations should be preferred. The negotiation process starts when one agent, say A, becomes active and proceeds in two phases: (1) Identication of potential partners, (2) Realization of the exchange. For the rst phase there are three possible alternatives. Identication of a partner A determines a time gap t 1 with a high valuation. 2. A determines all complementary agents K 1 ; ; K n and rooms R 1 ; ; R m which have a time gap at t 1. To reduce the amount of communication, only complementary agents are taken into regard that occur in A's plan. 3. For each K i A determines the times at which it occurs in its time table. These times are provided with priorities; times with low priorities are not taken into regard. 4. As soon as a time has been found that is suited for an exchange, A sends a lock command to all objects involved. If all receivers accept the command, A enters the time in a list, called active list. If at least one receiver rejects the command (because the time has already been locked), the time will be unlocked by all receivers and A enters it in the alternative list. Identication of a partner A determines a pair (K; R) that it wants to move to some other time slot.

4 2. A determines all free time slots that can be occupied according to the priority values, say t 1 ; ; t k. 3. A asks K, which of the t 1 ; ; t k are free for it. 4. For those times that are free for K A sends a lock command to K. If K accepts the command, A looks for non occupied rooms and enters the time in a list, called active list. If K rejects the command (because the time has already been locked), the time will be unlocked and A enters it in the alternative list. Identication of a partner A selects a time t 1 with respect to a valuation function q i which it wants to change and determines the entry (K; R) at t 1 in its time table. 2. A determines all times t 2 ; :::; t k that are suited for an exchange with respect to q i and for which the following condition holds: 9K 0 9R 0 : (K; R 0 ) is the entry at t i or (K 0 ; R 0 ) is the entry at t i and K 0 is free at t 1 and K is free at t i where K 6= K 0 and i = 2; ; k 3. As soon as a time has been found that is suited for an exchange, A sends a lock command to all objects involved. If all receivers accept the command, A enters the time in a list, called active list. If at least one receiver rejects the command (because the time has already been locked), the time will be unlocked by all receivers and A enters it in the alternative list. Realization of the exchange 1. If A's active list is not empty, it chooses the entry with the highest valuation, say (K; R), removes it from the list, and negotiates with K about a possible move of the entry. If the active list is empty, it continues with step If the negotiation is successful, A updates its plan and informs the other objects about the result such that they can update their respective plans. It also sends an unlock command to all objects involved in the identication process. The realization procedure stops with success. 3. If the negotiation does not succeed, A sends an unlock command to all objects involved in the identication process and continues with step If the alternative list is not empty, A selects the now unlocked entries in this list, puts them into the active list, and continues with step If the alternative list is empty, the realization procedure stops with failure. When A selects the entry with the highest valuation from its active list, it assumes that the complementary agent K is willing to negotiate, according to the identication procedure. The negotiation can proceed in four dierent forms: Negotiating!!!!!!!! (Reject) (Report) (Report) Request a aaaaaaa Commanding Oering (Reject) Fig. 2. The negotiating protocol Negotiating (Reject) 1. K rejects negotiation because time t 1 has been locked in the meantime. 2. K agrees with an exchange at time t 1 because this is advantageous for it, too. 3. K suggests an additional exchange with A at time t K tries to nd another partner, say B, for an exchange. In this case B is a complementary agent for K, i. e. it belongs to the same agent group as A. In case 3 the negotiation process becomes recursive. For reasons of eciency it should be avoided that too many agents are involved in the process, therefore the depth of the recursion is restricted. Instead of negotiating with too many partners, a sequence of negotiations between restricted numbers of partners will take place. Using the generic congurable cooperation protocols of [5] the negotiation process can be illustrated by gure 2. The change between the agent groups in the recursive negotiation is similar to the negotiation according to the contract net protocol, where each agent can take on the role of a manager or a contractor. However, our approach diers from the contract net protocol insofar as no bids are delivered, rather an agent selects a complementary agent according to its own valuation function. 5. Implementation The mappings m G ; m T ; m R ; m U as the most important input dates for a time table are almost all given explicitly. For the sake of simplicity several sets to select from can be collected in extra variables. For instance, in our example the rooms 216 and 301 are arbitrarily available for all subjects except for swimming, and it is more convenient to group together both rooms into one set. The time constraints of student groups, teachers and

5 rooms are used to calculate the availability of times for subjects. This is managed by means of the FD.get-constraint. Up to now only the time lists of such student groups, teachers, and rooms are included, where no alternatives are possible. The time constraints of the other objects are considered only at the moment of their actual selection. A second way can be an exact formulation of these times by means of FD.OR-connected constraints. The exclusive use of a time for subjects with common student groups, teachers or rooms can be enforced by the FD.allDierent-constraint. The question, whether this will be a good idea for large time tables is open. Obviously this constraint doesn't reduce the problem solving space considerably. For the implementation of the multi-agent approach we cannot yet make a statement about Oz properties, but the object oriented paradigm promises to be a solid basis. 6. Results The work introduced here is still under development. The constraint solver in Oz is well suited to implement the monotone phase of the time table problem in a smart way. It works well for small examples (cf. the example in this article). But up to now it is not clear whether the inexperienced users did use all Oz facilities eciently. A more realistic time table problem is very complex, and we have to gure out whether the Oz constraint facilities and our techniques are sucient to solve problems of the above formulated classes. The multi-agent approach combines the cooperative and the contradictory phases. At the moment the implementation is not yet completed. Therefore we can not give nal results about our experiences with Oz in this area. References [1] Martin Henz and Jorg Wurtz, Using Oz for College Time Tabling, Extended Abstract, DFKI [2] Klaus Philipp und Rainer Staudte, Das n-dimensionale Zuweisungsproblem bei mehrfacher Zielsetzung, Wissenschaftliche Zeitschrift der Technischen Hochschule Karl-Marx-Stadt, 24(1982), pp [3] Reid G. Smith, The Contract Net Protocol: High-Level Communication and Control in a Distributed Problem Solver, in: Alan H. Bond and Less Gasser (Eds.), Readings in Distributed Articial Intelligence Morgan Kaufmann, 1988, pp [4] Randall Davis and Reid G. Smith, Negotiation as a Metaphor for Distributed Problem Solving, Articial Intelligence, 20(1983), pp [5] B. Burmeister, A. Haddadi, K. Sundermeyer, Generic congurable cooperation protocols for multi-agent systems, Technical report, Research and Technology, Daimler-Benz AG, Berlin 1992