Artificial Intelligence and Soft Computing: Behavioral ... - Arteimi.info

Cognitive maps are generally used for describing soft knowledge [6]. For
example, political and sociological problems, where a very clear formulation
of the mathematical models is difficult, can be realized with cognitive maps.
The cognitive map describing the political relationship to hold the middle east
peace is presented in fig. 16.1 [13]. It may be noted from this figure that the
arcs here are labeled with + or – signs. A positive arc from the node A to F
denotes an increase in A will cause a further increase in F. Similarly, a
negative arc from A to B represents an increase in A will cause a decrease in
B. In the cognitive map of fig. 16.1, we do not have weights with the arcs.
Now, suppose the arcs in this figure have weights between –1 to +1. For
instance, assume that the arc from A to B has a weight –0.7. It means a unit
increase in A will cause a 0.7 unit decrease in B. There exist two types of
model of the cognitive map. One type includes both positive and negative
weighted arcs and the events are all positive. The other type considers only
positive arcs, but the events at the nodes can be negated. In this chapter, we
will use the second type representation in our models.
The principles of cognitive mapping [12] for describing relationships
among facts were pioneered by Axelord [1] and later extended by Kosko [5].
Kosko introduced the notion of fuzzy logic for approximate reasoning with
cognitive maps. According to him, a cognitive map first undergoes a training
phase for adaptation of the weights. Once the training phase is completed, the
same network may be used for generating inferences from the known beliefs
of the starting nodes. Kosko’s model is applicable in systems, where there
exists a single cause for a single effect. Further, the process of encoding of
weights in Kosko’s model is stable for acyclic networks and exhibits limit
cycle behavior for cyclic nets. The first difficulty of Kosko’s model has been
overcome by Pal and Konar [8], who used fuzzy Petri nets to represent the
cognitive map. The second difficulty, which refers to limit cycles in a cyclic
cognitive map, has also been avoided in [3] by controlling one parameter of
the model. In the forgoing sections, we will present these models with their
analysis for stability. Such analysis is useful for applications of the models in
practical systems.
16.2 Learning by a Cognitive Map
A cognitive map can encode its weights by unsupervised learning [7] like the
Long Term Memory (LTM) in the biological brain. In this chapter, we employ
the Hebbian learning [5] following Kosko [6], which may be stated as
follows.
The weight Wij connected between neuron Ni and Nj increases when the
signal strength of the neurons also increase over time.