Mathematics-Based Models

3.2 Mathematics-Based Models

As we have mentioned in Chap. 1 , models defined on the basis of various mathemat- ical theories play a fundamental role in Artificial Intelligence. The first methods used for solving one of the key problems of AI, namely the auto- matic recognition of objects, appeared at the beginning of computer science. This is the field of pattern recognition. Objects (phenomena, processes, etc.) are represented by sets of features (attributes). Recognition of an unknown object/phenomenon is performed by a classifier, which ascribes the object to one of a number of predefined

categories. 6 In order to construct a classifier a set of example objects with their cor- rect classifications should be available. 7 Such a set, called a learning (training) set, is a kind of knowledge base of a classifier. The main idea of a classification process can be defined as the task of finding the object X in the learning set which is “similar” most to the unknown object. If the classifier finds such an object X, it ascribes the unknown object to the class that the object X belongs to. In fact, this (simplified here) general idea of the classification is implemented with the help of advanced math- ematical models such as the Bayesian probability model, discriminating functions,

minimum-distance models, etc. These models are presented in detail in Chap. 10 . The complementary issue of cluster analysis, which consists of grouping a set of objects/phenomena into classes (categories), is discussed in Chap. 10 as well. The second important group of mathematics-based methods relates to the crucial issue of the possibility of formally specifying:

5 James Lloyd “Jay” McClelland—a professor of psychology at Carnegie Mellon University and Stanford University. His work concerns psycholinguistics and applications of connectionist models

in pattern recognition, speech understanding, machine learning, etc. 6 The categories, also called classes, should be defined earlier, i.e., when we formulate the problem.

For example, if a classifier is constructed to support medical diagnosis, then disease entities are the categories.

7 Such a set corresponds to human experience in solving a given classification problem. For example, in medical diagnosis it corresponds to the diagnostic experience of a physician.

26 3 Computational Intelligence

• vague notions that are used for a description of the world, and • an inference process, when only imperfect knowledge is available.

Imperfect knowledge can result from various causes. For example, input informa- tion can be uncertain (uncertainty of knowledge), measurements of signals received by an AI system can be imprecise (imprecision of knowledge) or the system may not know all required facts (incompleteness of knowledge).

The model of Bayes networks is used for inference that is based on propositions to which the probability of an event occurring is added. The probability measure expresses our uncertainty related to the knowledge rather than the degree of truth- fulness of a specific proposition. There are usually a lot of possible factors which influence the result of such probabilistic reasoning. An assessment of probabilities of these factors as well as their combinations is often impossible in real-world appli- cations. Therefore, in this model we construct a graph which represents connections between only those factors which are essential for reasoning.

If our knowledge is incomplete, we can use Dempster-Shafer theory for reasoning. In this theory we use specific measures to express the degree of our ignorance. If we acquire new knowledge, these measures are modified to express the fact that our ignorance is reduced.

Knowledge is continuously acquired by intelligent systems. In a classical logic we assume that after adding new propositions to a model the set of its consequences does not decrease. However, this assumption is not true in the case of AI systems which reason over the real-world environment. To put it simply, new facts can cause old assumptions not to be true any more. To solve this problem we use non-monotonic logics such as default logic, autoepistemic logic, or circumscription, or specific models like the Closed-World Assumption model.

Bayes networks, Dempster-Shafer Theory, and non-monotonic logics are pre- sented in Chap. 12 . The problem of defining formal specifications of concepts which are used for

a description of the world seems to be even more difficult. On one hand, we have vague notions, which are used in everyday life. On the other hand, mathematics-based models require notions which should be precise and unambiguous.

The vagueness of notions can be considered in two ways. First of all, a notion can

be ambiguous, which usually results from its subjective nature. Notions relating to the height of a human being (e.g., tall, short) are good examples of such notions. In this case we should take into account the subjective nature of a notion by introducing

a measure, which grades “being tall (short)”. This is the main idea of fuzzy set theory. The vagueness of a notion can relate to the degree of precision (detail, accuracy) which is required during a reasoning process. This degree should be adequate with respect to the problem to be solved, i.e., it should be determined in such a way that our AI system should distinguish between objects which are considered to belong to different categories and it should not distinguish between objects which are treated as belonging to the same category. This is the main idea of rough set theory.

Both theories which are used to solve the problem of vagueness of notions are introduced in Chap. 13 .

3.3 Biology-Based Models 27