Genetic Algorithms to allow Intelligent Tutoring Systems to be efficient. We consider supervised teaching where the teacher assigns a initial profile and a finale ...
GAITS : Fuzzy Sets-Based Algorithms for Computing Strategies Using Genetic Algorithms Mohamed Quafafou*, Mohammed Naf'm** (*) (**)
IRIN, Universit6 de Sciences et des Techniques de Nantes, France. Digital Equipment Corporation, Sophia-Antipolis, France.
Abstract. This paper introduces a new method combining Fuzzy Logic and
Genetic Algorithms to allow Intelligent Tutoring Systems to be efficient. We consider supervised teaching where the teacher assigns a initial profile and a finale profile to each student before starting the teaching. The system computes an optimal strategy which represents a way of evolving the student's knowledge from the initial profile. So~ the student's knowledge will change progressively to reach the final profile where the teaching objective is judged to be reached. The paper deals with an example presenting a simulation of the result's of a good student and a bad one.
1 Introduction The architecture of Intelligent Tutoring System (ITS) is based on several interacting components. The following four components are usually used [8] : 9 The expert module; 9 The instructional module; 9 The student model; 9 Interfaces. But these modules became complex, because research and development of an ITS to be used in real world is an extremely lenghty process which requires appropriate people and funding. Besides, when we develop an industrial educational system we have to cope with different kind of problems because phenomena are complex to formulate and generaly imperfectly known [6]. Symbolic tools was developed allowing to ITSs to be powerful and well organized educational systems [1, 2]. We propose an alternative numeric method, based on fuzzy logic and genetic algorithms which allows to ITSs to be efficient and to provide a maximum of competence in a minimum of time. The main body of the paper comprises four sections. In the section 2 we discuss the interest of fuzzy logic for ITS and we propose a model based on fuzzy sets. In the first part of the section 3 we give a brief introduction to genetic algorithms (GAs). Then, we describe algorithms which allow to compute the refrence strategy and simulate student's feedback and answers. Finally, we deal with an example comparing the reference strategy to results obtained by both a good and a bad student.
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2 Fuzzy logic & ITS Recently, fuzzy logic was applied to ITS [7] in order to improve the speed in decision support over rule based inference engines and to have a more experienced control of learning sequence. Fuzzy logic was also used to improve a supervised customized teaching using pedagogical rules which are defined by the author [5]. In this paper, we assume that the tutor manages basic elements (knowledge elements). Each one defines the subject to be teached. Besides, we have a set of dialogues which can be proposed to the student considering his level. They have a normalized description characterized by two sets of worthed knowledge elements : 9 prerequisite knowledge elements (LP) : a dialogue is candidate to be proposed to the student (admissible dialogue) if and only if its prerequisite elements are satisfied; 9 teached knowledge elements (LT) : the dialogue execution allows to enrich the knowledge the student has.
For example, let us consider four knowledge elements el, e2, e3 and e4 and the dialogue L1 defined by LP1 = (0 0 .I .3) and LT1 (.2 0.4.7). Note that el and e2 are not prerequisite elements for L1 (their levels are equal to 0 in LP1) when only e2 is not teached by this dialogue (its level is equal to 0 in LT1). The dialogue L1 is considered an admissible dialogue iff the level of e3 and e4 in the student model are respectively greater then .1 and .3. Let us consider supervised teaching where the teacher assigns the following profiles to each student before starting the teaching : 9 initial profile I : is the student knowledge presupposed model at the time he starts his training and corresponds to the class of the student belongs to; 9final profile F : is a final state which must be reached by the student and where the teaching objective is judged reached.
Let us consider the precedent four knowledge elements and define the "teaching interval" by I = (0.1 0.2) and F = (.6.7.3 .5). So, the teaching session allows to change progressively the student's profile from the initial profile to the final one. We define a strategy as a way of evolving the student's profile from I to F. In the following section we show how to compute the strategy taking account all useful knowledge. We can view the LP i, LTi, I and F as fuzzy sets. They correspond to fit vectors and each fit value measures partial set membership or degrees of elementhood [4] For example, the fit value mI(e2) equal to .1 indicates that the knowledge element e2 belongs only slightly to the fuzzy set I. Besides, the fit value mF(e4) equal to .5 indicates that the knowledge element e4 belongs to F as much as it does not. The value 0 indicates the abscence of the ith knowledge element ei in the fuzzy set.
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3 Student's and expert's strategies This section outlines the operation of a basic genetic algorithm (GA) and describes the GA-based method adopted in this study to compute the expert's strategy (reference strategy). For more details on GAs, see [3].
3.1 Genetic Algorithm The basic idea of a genetic algorithm is to model a chromosome for a particular species and what happens to the chromosome during successive generations, when the underlying rule of nature is the survival of the fittest. A simple GA is composed of three operators : reproduction, crossover, mutation. But, before using this operators and for each particular problem we have to define a scheme for coding of the parameter space which corresponds, in effect, to a descritisation of the parameter space. There must be a mapping between the points of the descretised space and a chromosome, which is just a finite length binary string in the basic algorithm. Each chromosome has an associated fitness value which is calculated by the fitness function. At the start of the algorithm an initial population is generated randomly. Once the members of the population have been generated their fitness values are calculated via the predefined fitness function. Each string is a representation of a solution to the problem and its fitness value is a measure of the goodness of the solution that it represents. The initial population will be transformed by the genetic operators (reproduction, crossover, mutation) into populations with higher fitness values. Using the reproduction operator, individual strings are copied according to their fitness function. The crossover operator chooses pairs of strings at random and produces new pairs. The crossover probability is the probability that the offspring will be different. If crossover is to take place then an integer j is selected at random to determine the crossover position. For example consider strings A and B and suppose that j = 1 then the two new individuals A', B' will be : B=01010111
A'=100111~I B'=010101
0
After crossover, the mutation operation is performed for each bit in each chromosome and if a mutation occurs then the particular bit just flip it's value. The basic genetic algorithm consists of repeating the above three operations of reproduction, crossover and mutation to repeatedly obtain new generations of individuals with, one hopes, higher levels of fitness.
3.2 The reference strategy Before starting the teaching session, the tutor must be able to compute a strategy which allows to reach the final profil. This strategy is a way of changing progressively the student's profile from the initial profile to the final one. We call it the reference strategy, because it allows to judge student's results. In order to define
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this strategy, we compute at each teaching step T the reference profile TIT taking into account the following considerations : 9 the student knowledge must evolve progressively; 9 the teaching objective must be satisfied with the maximum degree at each step; 9 all useful knowledge at the step T must be considered. These considerations can be stated in numerical terms as follows : 9 minimize d01T, Current profile); 9 minimize d('qT, Final profile); ~ minimize the average of d(r W, Li) for all admissible dialogues Li at the step T. So, the objective function to minimize is :
where
1 k ST(TIT) = d01T,C) + d(TIT,F) + k * ~ wi*d01T,Li) Y.wi i=l i=l k : the number of admissible dialogues at the step T; 11T : the reference profile at the step T; C : the current profile; F : the final profile; Li: the ith admissible dialogue at the step T; wi: the weighting coefficient for Li. d(X,Y) : the distance between the two fuzzy sets X, Y.
(1)
wi express the importance of the dialogue Li. For example, wi is higher than wj if the dialogue Li is more important than Lj, that is, if LTi is more near to the final profile F than LTj (d(LTj, F) < d(LTi, F)). The task of the optimization module is to determine the fuzzy set TIT by computing n values mrlT(ei), so as to minimise ST(rW). The GA module contains the following procedures : 9 Initialization : This procedure randomly generates the starting strings. Each
one is of the form Xl x2 ... Xm., where xi = mllT(ei) and are themselves binary strings; 9 F i t n e s s : This procedure computes the fitness of each solution string. Fitness scaling [6] is adopted; 9 Selection : It implements the gene reproduction function; 9 Crossover : It performs the gene crossover operation; 9 M u t a t i o n : It carries out the gene mutation operation. The computing r e f e r e n c e s t r a t e g y algorithm is R e f e r e n c e S t r a t e g y shown in the following. Input to ReferenceStrategy is a set of dialogues, the initial profile and the final profile. Ouput of ReferenceStrategy is the reference strategy ~expert composed by fuzzy sets 111, r12 ..... 11f. Besides, the satisfaction of the teaching objective degree fiT at the step T are given as results.
63 So, before starting the teaching session, the tutor computes the reference strategy using the ReferenceStrategy procedure. Firstly, admissible dialogues are selected taking into account the current profile (I). Then the profile reference 111 is computed minimizing the function S 1. Considering 1] 1 we compute the satisfaction of the teaching objective degree 81 = Degree(rl 1 ~ F). Then we update the current profile and we start a teaching cycle by selecting a new admissible dialogue. The teaching objective is considered satisfied at the step p if 819 -- 1. So, the reference strategy is defined by 111,112..... rip. Procedure ReferenceStrategy; 9 a initial profile I. 9 a final profile F. 9 a set of dialogues { Li, i = 1 .... n } Output : 9 ~ : reference strategy. 9 8i : satisfaction of the teaching objective degree at the step i. begin CurrentProfile the ith Reference profile vii } ~i } { Added Li to q~ }
