Knowledge and Reasoning Supported by Cognitive Maps Alejandro Peña1,2,3, Humberto Sossa3, and Agustin Gutiérrez3 WOLNM1, UPIICSA2 & CIC3 - National Polytechnic Institute2,3 31 Julio 1859, # 1099-B, Leyes Reforma, DF, 09310, México
[email protected],{hsossa,atornes}@cic.ipn.mx
Abstract. A powerful and useful approach for modeling knowledge and qualitative reasoning is the Cognitive Map. The background of Cognitive Maps is the research about learning environments carried out by Cognitive Psychology since the nineteenth century. Along the last thirty years, these underlying findings inspired the development of computational models to deal with causal phenomena. So, a Cognitive Map is a structure of concepts of a specific domain that are related through cause-effect relations with the aim to simulate behavior of dynamic systems. In spite of the short life of the causal Cognitive Maps, nowadays there are several branches of development that focus on qualitative, fuzzy and uncertain issues. With this platform wide spectra of applications have been developing in fields like game theory, information analysis and management sciences. Wherefore, with the purpose to promote the use of this kind of tool, in this work is surveyed three branches of Cognitive Maps; and it is outlined one application of the Cognitive Maps for the student modeling that shows a conceptual design of a project in progress.
1
Introduction
Causal knowledge and reasoning involves many interacting concepts that make them difficult to face, and for which analytical techniques are inadequate [1]. In this case, techniques stemmed from qualitative reasoning, can be used to cope with this kind of knowledge. Thus, a Cognitive Map (CM) is a tool suitable for dealing with interacting concepts. Generally, the underlying elements of the CM are simple. The entities, factors and events of the domain model are outlined as concepts. The causal influences between these concepts are considered as cause-effects relations. So, a CM is graphically depicted as a digraph, where the nodes represent the concepts, the arcs correspond to the causal relations and the direction pictured by the arrow of the arc shows the causation of the target by the source. In general, there are three basic types of causal influences: positive, negative and neutral. The positive means that the source concept stimulates in a direct way the state of the target concept, so when the intensity of the cause concept grows a positive stimulus is trigged to enhance the state of the effect concept, but if the active level of the source concept diminishes then a negative influence is produced on the target concept to decrease its state.
The negative causal influence operates in an inversely way to the positive. So a promotion in the values of the source concept leads to a decrease in the target concept state; and a decrease in the cause concept produces a raise in the effect concept. Finally, the neutral causal influence means that no matter the changes of state that happen in the source concept, they are not going to influence in any way to the target concept; or that there is no a causal relation between this couple of concepts. As regards the kind of CM adopted, the causal influences are outlined by: a set of symbols, a set of crisp values, real values in a continuous range, linguistic variables, probabilistic estimations, or bipolar values. Thus, the basic form to depict the values is by means of the set of symbols {+, -, 0}, which corresponds to positive, negative and neutral causal influences respectively. This set of values is acknowledged by the acronym NPN (negative-positive-neutral). Wherefore, in order to storage and manipulate the causal influence values of a CM it is used an adjacency matrix (w), of size n (the number of concepts), whose entries show the value of the causal relation between the concepts. So, the entry wij contains the value of the causal influence depicted by the arc that comes from the source concept i to the target concept j. Regarding the arcs and its arrows of a CM, two nodes i and j can be linked by a path in three kinds of causal relations: null, direct and indirect. The null causal relation means that there is no a possible path to join the nodes i and j. The direct causal relation corresponds to paths of length equal to one arc. A path with more than one arc, which includes at least one intermediate node different to i and j, depicts an indirect causal relation. So, a propagation of the causal effect is done by the syllogism hypothetic principle. In resume, in a CM two nodes i and j are linked by a null causal relation; or by one direct causal relation and/or at least one indirect causal relation. With this baseline, three versions of a CM are depicted and two applications are outlined next. Thus, the organization of the paper is as follows: in the second section the causal, fuzzy and probabilistic models of a CM are sketched through the underlying formal model of the approach. In the third section it is described the use of two versions of a CM to depict the student modeling process stemmed by the learning experiences in a Web-based Education System. In these cases, the Student Model is the application responsible to fulfill an individual profile of the student in order to provide an adaptive student-centered service. In the conclusions section are presented some comments about the properties of a CM and the CM-based application, besides of identify further work regarding to the automatic generation of a CM.
2
Profile of Cognitive Maps
The research on spatial learning begins in the nineteenth century focus on orientation task in animals and human beings. However, was Tolman [2] in 1948, who called CM to the mental structure that storages and recalls the spatial knowledge. Next in 1955, Kelly introduces the Personal Construct Theory to depict an individual’s multiple perspectives [3]. Afterwards, in 1976 Axelroad [3] states the computational version of a CM. Nowadays new work has been doing to face imprecise knowledge, uncertainty and fuzzy views of domain. So, along this section it is resumed the underlying concepts of three versions of a CM: causal, fuzzy and probabilistic.
2.1 Causal Cognitive Maps The baseline of the Causal Cognitive Map (CCM) rests in the relational theory outlined by Axelroad [4] and Nakamura et al. [5], who work out in the fields of the international relations and the decisions support respectively. Thus, a CCM is a directed graph that represents an individual’s beliefs with respect to the model domain that is defined as: CM:= (C, A). Where C is the set of concepts pictured like vertices, and A is the set of causal relations, depicted like arcs, between the concepts. The arrows are labeled by elements of the set δ:={+, -. 0, ⊕, Ө, ±, a, ?} that means respectively: positive, negative, neutral, positive or neutral, negative or neutral, positive or negative, conflict, and positive, negative or neutral causal effect. Four operators are defined on the set δ of causal relations. They are union (U), intersection (∩), sum (|) and multiplication (*). The laws of union and intersection are derived from: +, -, 0, ⊕, Ө, ±, a, ?; when they are considered as shorthands for: {+}, {-}, {0}, {0, +}, {0, -}, {+, -}, {}, {+, 0, -} respectively. The guidelines of these operators are outlined in the Table 1, where C is the set of concepts. Table 1. Laws for the causal operators. Union (U), intersection (∩), sum (|) and multiplication (*), with do meaning distributes over
U (union) and ∩ (intersection) (1a) (1b) (1c) (1d) (1e)
⊕ = 0 U+ Ө = 0 U± = + U? = 0 U+Ua=+∩0=+∩-=0∩-
| (sum) For any x, y Є C (2a) 0 | y = y (2b) a | y = a (2c) y | y = y (2d) + | - = ? (2e) | do U (2f) x | y = y | x
* (multiplication) For any x, y Є C (3a) + * y = y (3b) 0 * y = 0, if y ≠ a (3c) a * y = a (3d) - * - = + (3e) * do U (3f) x * y = y * x
The multiplication (*) operator estimates indirect causal effects, e.g., if a path from node i to node j has an intermediate node k, with the effects (i) -Æ (k)-Æ(j); so it produces a positive indirect effect according to (3d). Whereas the sum (|) operator computes direct causal effects from different paths that link two nodes i and j; e.g., there is one path from i to j with negative total indirect effect and other path with positive total indirect effect, then the total direct effect is ?, according to law (2d). The operators * and | can be lifted to matrices, as follows. Consider A and B as square valency matrices of size n. The addition and multiplication operators are defined by equations (1) and (2). The nth power of a square matrix A, for n > 0 is defined in (3). Thus, the total effect of one concept on another is estimated by the total effect matrix At whose entry Aij owns the total effect of i on j, according as (4). Due to the sum (|) operator is monotonic, there is a k such that represents the total causal effect from one concept on another, depicted by (5). This model for a CCM is based on an intuitive perspective with ad hoc rules, and lacks of a formal treatment of relations. Wherefore, it is advisable to review the proposal stated by Chaib-draa [6] to deal with this issues; his model has a precise semantics based on relation algebra and it has been used for qualitative decision-making and agent reasoning.
( A | B) ij = Aij | Bij .
(1)
( A * B ) ij = ( Ai1 * B1 j ) | ... | ( Ain * Bnj ) .
(2)
A1 := A; and An := A * An−1 .
(3)
At = A1 | A2 | A3 | A4 | .... .
(4)
At = A1 | A2 | A3 | A4 | .... | Ak .
(5)
2.2 Fuzzy Cognitive Maps Kosko [7] in 1986 proposes the Fuzzy Cognitive Map (FCM) as a CM whose causal relations and concept values are defined by fuzzy knowledge. The arcs and nodes values are depicted by fuzzy membership functions that are associated to fuzzy sets. These functions translate real world values to qualitative measures of the concepts presence in a conceptual domain, by mean of crisp values of a set, as {0, 1} or {-1, 0, 1}, or a real values in a range, as [-1, 1]. Concepts with positive values indicate that the concept is strongly present. Values around zero mean the concept is practically inactive in the conceptual domain. Negative values outline negative states of presence of the concept. Whereas, positive, zero and negative arc values depict different gray levels of the causal influence from the source concept on the target concept. Thus, besides of the adjacency matrix for the fuzzy values of the causal relations, there is a vector concept used to describe along the time the values state of the concepts. Once the FCM is depicted, a simulation process is activated to predict causal behavior. This process is carried out along discrete steps, where the values of the concept vector change, according to the fuzzy causal influences; whereas the values of the valency matrix remain fixed, unless the FCM has an adaptive behavior. So, with the aims to produce the initial values of the concept vector, real world domain values are estimated for feeding the fuzzy membership functions. Once it is depicted the initial concept vector, an iterative process begins at step time t = 0. In each cycle a new state for the concepts is computed by taking the normalize result of the sum of the inputs. At step t, the inputs to the concept i are estimated by the state values, at step t= t-1, of the nodes j with edges coming into i, multiplied by the corresponding weights wij. Due to a FCM is a qualitative model; a threshold function is applied to the result of the sum of the product of the inputs by the weights to normalize the concept values according to the set or range associated to the concept. The formal representation of the state of a FCM is defined in formula (6), where C is the state concept vector, t is the iteration, u is the threshold function, s is the result of the sum of the inputs, and wij is the entry with the fuzzy value of the arc from concept j to concept i. Also, the equations (7) to (9) picture the threshold functions used to achieve respectively the sets {0, 1} and {-1, 0, 1}, and the range [-1, 1].
Ci (t ) = u ( s ); where : s = (∑ j =1 wij * C j (t − 1)) .
(6)
u ( s ) = 0, s ≤ 0; u ( s ) = 1, s > 0 .
(7)
u ( s ) = −1, s ≤ −0.5; u ( s) = 0, s > −0.5 ∧ s < 0.5; u ( s) = 1, s ≥ 0.5 .
(8)
u = 1 /(1 + e − cs ) .
(9)
n
where, according to Mohr [8], c is critical in determining the degree of fuzzification of the function, due to at large values, the logistic signal function approaches discrete threshold functions, so a c=5 value is advisable. Stability in dynamic systems, as a FCM, is typically analyzed through the use of Lyapunov functions. Thus, a FCM with discrete threshold functions, as (7) or (8), will either converge to a limit cycle or reach an equilibrium state, due to these functions force fuzzy state vectors to non-fuzzy values. Whereas, a FCM using the logistic signal threshold function, as (9), may become nonlinear under some conditions of feedback. Since the state vector of the map at time t is determined by its values at time t-1, the equilibrium state of a FCM may be easily detected by comparing two successive patterns of states concepts, composed by one or more state vectors. If they are identical, then the map has reached an equilibrium state and the execution ends. In despite of the single inference mechanism of a FCM, the outcomes achieved by the FCM can be non-linear, and the problem of finding whether a state is reachable in the FCM simulation is nondeterministic polynomial (NP) hard. Wherefore, it is advisable to review the study carried out by Miao and Liu [9], that focuses on the causal inference mechanism of a FCM with crisp binary concept states {0, 1}. They stated that given initial conditions, a FCM is able to reach only certain states in its state space. So, they show that splitting the whole FCM in several basic FCM modules, it is possible to study their inference patterns in a hierarchical fashion. 2.3 Probabilistic Cognitive Maps Wellman, in 1994 [10], carries out a Probabilistic version of a Cognitive Map (PCM) focuses on the assurance of the soundness of the inference for the sign relations of a CCM. This sign relation is depicted by (1) and (2), but now they are integrated by equation (10), where Pa,b is the set of paths in the PCM from a to b, and δ is the causal sign of the arc between the nodes c, and c’. In this version, it is considered that: if the signs denote a causal correlation and the concepts random variables, then the path tracing is not sound. So, for instance, if i is negatively correlated with j, and j negatively correlated with k, it is still possible that i and k be negatively correlated, instead of positively supported by the law (3d). Thus, correlation is not a good interpretation for the sign of causal relations. Other issues that Wellman addressed were: the effect of blocking the path by instantiated evidence and the evidential reasoning produced by the effect concept on the cause concept.
In any of those cases, it would be possible to conditionalize the conclusion on partial information about the concepts, assuming that the values of some of them may have been observed or revealed. With the aim to determine the validity of inference rules, such the depicted in (10), the definition of the rule should be local as far as possible. Thus, in assessing the validity of a signed edge (c, c’, δ), where δ is a causal sign, the attention is limited to the neighborhood of concepts c and c’. The rule should be unambiguously determined by the precise causal relation among the concepts, so that the sign relation, depicted by (10), is an abstraction of the precise relation. Thus, if the precise relation were a functional dependency, the sign would be an abstraction of the function relating the concepts. So, if the causal relation were probabilistic, the sign would be an abstraction of the probabilistic dependence, defined in terms of conditional probability (Pr) With this baseline, the PCM is depicted as follows: The PCM is an acyclic digraph, with nodes (a, b) regarding to concepts and signed edges picturing abstract causal relations. The concepts are interpreted as random variables, although the variables domains need not be explicitly specified, what matters is: the relative ordering among values. The edges denote the sign of probabilistic dependence. So an edge (c, c’, +) means that for all values c1 > c2 of c, c’0 of c’, and all assignments x to other predecessors of c’ in the PCM, applies the correlation stated by equation (11).
⎛ ⎞ ⎜ | δ ⎟. p∈Pa ,b ⎝ ( c ,c ',δ )∈ p ⎠
(10)
Pr(c' ≥ c'0 | c1 x) ≥ Pr(c' ≥ c'0 | c2 x) .
(11)
*
where symbols *, | correspond to sum and multiplication operators, P depicts a path and Pr a conditional probability. An edge (c, c’, -) is defined analogously with ≤ to substitute the central inequality in (11). If there is no edge from c to c’ and no path from c’ to c, then the left and right hand sides of (11) are equal; so, c and c’ are conditionally independent given the predecessors of c’. If none of these cases hold, and there is no path from c’ to c, then there is an ambiguous edge (c, c’, ?). The path analysis formula (10) applies to direct paths from a to b that corresponds to pure causal inference, but in a CM there may be undirected pathways between two variables that not all of them are purely causal paths. Wherefore, sometimes appear situations where the values of the variables have been observed, so that these variables have the effect of blocking the path where they are. Thus, if e is observed evidence, and e is in some of the paths between the concepts a to b, then all the paths that includes e have to be removed from Pa,b. Other type of inference is the evidential reasoning produced by the effect concept on the cause concept. The sign of the probabilistic dependence from c’ to c is the same as that from c to c’, as a result of applying the Baye’s rule to (11). Other issues considered in the PCM are: the intuitive relations among target concepts of the same source concept, the causes of the same effect, and the relation between two causes given their common effect depend on how they interact in producing the effect.
3. A Case of Use of Cognitive Maps This section shows an example of the use of a CM to support the student modeling in Web-Based Education Systems (WBES) stemmed from the currently work done by the authors [11, 12]. Thus, among the trends of the WBES is the provision of studentcentered education with the support of the artificial intelligence. The aim is that the WBES carries out an adaptive behavior to depict the plans, the content and the learning experiences according to the dynamic student needs. So, the student model depicts a belief-based student profile, with regard to his/her cognitive skills, learning preferences, behavior, outcomes and knowledge domain acquired. Due to the WBES works out a teaching-learning process, the CM was selected as the underlying approach to achieve the student model. Wherefore, the teaching task can be see as the cause concept, and the learning activity as the effect concept. So a logical analogy arises between the teaching-learning application domain and the CMbased student model. Thus in this section are introduced a couple of student models depicted by causal and fuzzy Cognitive Map versions. 3.1 Causal Cognitive Map-based Student Model Before delivering a teaching-learning experience, it is necessary to consider the causal effects that the subjects of the knowledge domain, produce on the student’s cognitive performance. So, a small version of CCM is sketched in the Figure 1, is able to depict with them and to simulate the causal impact along the further iterations, as follows: The CCM pictures four concepts regarding to the development of reusable computer programs. These concepts are sketched as the nodes (a) to (d), whereas their causal relations are show as labeled arcs, with positive (+) or negative (-) values. As regards the arrows, it is possible to identify and to follow the causal flow among the concepts. Thus, several paths are appreciated; some of them are direct paths as (a) + Æ(b); others are indirect paths as (a)+ Æ(b)- Æ(d). The CCM is a cyclic map due there are two paths, (a)+ Æ(b)- Æ(d) and (a)+ Æ(c)+ Æ(d), that arrive to concept (d); and one link, (d) + Æ(a), that points to the concept (a) in order to trigger a new cycle. (a) understanding OO philosophy + + (b) manage of the (c) design of classes structured paradigm + (d) reusable applications Fig. 1. Student Model depicted by a Causal Cognitive Map
The behavior of the CCM is computed through the use of the equations (1) to (5) along several iterations. As a consequence of the activation, the adjacency matrix of the CCM is transformed to depict the causal effects in the way showed in the Table 2.
Table 2. Evolution of the Adjacency Matrix of the Causal Cognitive Map through 4 states
A1 Initial State, i=1 a b c d A3 After 2 iterations, i=1 a b c d
a
b
c
-
+
D +
+ a
b
c
+ -
+
D
+
+
A2 After 1 iteration, i=2 a b c d A4 After 3 iteration, i=4 a b c d
a
B
c
d +
+
a
-
+
b
c
d +
+ -
+
where A1 corresponds to the initial state, before begin the simulation process. For this reason, the values in the matrix are the direct causal relations among each couple of concepts linked by only one arc. A2 shows the indirect causal effects among couples of concepts linked by a path with two arcs. For instance, in the relation between a and d there are two paths, the first path is (a)- Æ(b)- Æ(d) with the indirect causal value of + (positive); and the second path is (a)+ Æ(c)+ Æ(d) with the indirect causal value of +, as a result of apply the laws of * (multiplication) and the laws of | (sum). Thus, in the entry A2a,d appears + as the indirect causal value between a and d. Matrix A3 corresponds to the values achieved by paths with lengths of 3 arcs (e.g., (a)- Æ(b)Æ(d) + Æ(a) the result produced is – (negative). Finally, in matrix A4 it is possible to identify that it has been reached a matrix with equilibrium states, due to the resulting values are the same that those in matrix A2, therefore the process ends. 3.2 Fuzzy Cognitive Map-based Student Model In this section is introduced a FCM to analyze the cognitive skills of the student model. Through the activation of a simulation process, it is possible to predict the fuzzy evolution of the states of the concepts involved in the model. So, a brief example of this approach is sketched in the Figure 2 with four concepts. They represent cognitive skills elicited by the tutor of the student. According to the fuzzy causal relations, which are labeled by real values in the range [-1, 1], is appreciated that: the concentration enhances abstraction; abstraction promotes logic reasoning; logic contributes problems solution; problems solution stimulates concentration; but concentration feedbacks negatively to problems solution as a result of the work done. The fuzzy causal values for the concepts and their relations are represented in the Table 3, where the entries of the 2nd to the 6th rows correspond to the values of the relations between cause concepts, stated as rows, and effect concepts, identified in the column headers; whereas, in the last row appears initial concept vector.
2) abstraction facility
+ 0.808 1) concentration
+ 0.703
- 0.506
3) likeness logic
+ 0.603
+ 0.802
4) problems solution
Fig. 2. Fuzzy Cognitive Map. Depicts some cognitive skills for student modeling
Table 3. Valency Matrix and State Vector of the Fuzzy Cognitive Map Adjacency Matrix 1) concentration 2) abstraction facility 3) likeness logic 4) problems solution
(1) concentration
(2) abstraction +0.808
(3) logic
(4) solution -0.506
+0.703 +0.603 +0.802
Initial State Vector
+0.5
+0.6
-0.2
0
The causal simulation of the FCM is sketched in the Table 4. So, in this table appear the results estimated for the four concepts along several iterations, according to the equations (6) and (9). In the entries of a specific column, it is possible to appreciate the behavior of a particular cognitive skill; e.g., the right column, shows the state evolution of the concept solution, which begins with 0, grows to 0.80, and drops to 0.63, where achieves an equilibrium state. In the same way, the behavior of the whole FCM is represented by the state vector depicted in each row. Thus, the simulation begins with the initial state, evolves through successively iterations until arrive to a stable situation produced in the eighteen cycle. So that, the interpretation is that the skills concentration, abstraction and logic reasoning achieve a high active state, but the solution skill develops a lightly positive increase in its state. This interpretation is stemmed from the sigmoid threshold function (9), where values close to 1.0 mean high positive activation, values around 0.5 represent lightly activation, and values close to 0.0, outline high negative activation of presence in the model. Table 4. Evolution of the State Vector. As a result of the activation of the fuzzy Cognitive Map Iteration 1 2 3 4 10 16 18
(1) concentration 0.5 0.5 0.630904 0.962182 0.928551 0.92876 0.928759
(2) abstraction 0.6 0.883013 0.883013 0.927605 0.977212 0.977124 0.977125
(3) logic -0.2 0.891978 0.957176 0.957176 0.968884 0.968872 0.968872
(4) solution 0 0.133612 0.806615 0.784781 0.639364 0.639974 0.639971
Conclusions In this work it has been presented the baseline of Cognitive Maps and three versions oriented to deal with causality, fuzziness and non-deterministic situations. All of them are related to qualitative knowledge representation and causal reasoning. The CM is suitable to deal with dynamic systems modeling, where there are significant feedback and a nonlinear behavior along the simulation of the problem domain. Wherefore, the CM represents a qualitative approach for modeling a wide range of situations in fields as the political, social, economical, education, and engineering. As an instance of a CM application, in this paper were outlined two student models by causal and fuzzy versions of a CM. In these cases, the preferences and the mental skills of the student were depicted and two simulations of behavior were achieved. As a further work is the development of an approach oriented to automatically generate CM through the use of evolutionary strategies and ontologies of the domain, with the aim to generate a student model in an adaptive fashion.
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Acknowledgments The first author states that: this work was inspired in a special way for my Father, my brother Jesus and my Helper as part of the research projects of World Outreach Light to the Nations Ministries (WOLNM). Also this work was partially supported by the IPN, CONACYT, and Microsoft México.
