Knowledge representation systems syntactic methods
Reasoning about knowledge and knowledge representation has been an issue of concern in Artificial Intelligence for over two decades. More recently, researchers have realized that these issues also play a crucial role in other subfields of computer science, including cryptography, distributed computation, data base theory, and expert systems.
Any knowledge representation system provides information, usually incomplete, about some parts of perceivable reality, and different authors provide different formal models for the knowledge analysis.
Recently, the notion of rough sets [Pawlak 1982] was introduced, which provides a systematic framework for the study of the problems arising from inprecise and insufficient knowledge.
A rough set, like fuzzy set, is a matematical model used to deal with approximate classification. These concepts have been proved to be independent of one another [Pawlak 1985(2)].
A number of experimental systems has been implemented based on the deterministic rough-set theory. These applications include analysis of medical data of patients with dudendal ulcer [Fibak 1986], control algorithm aquisition in the process of kiln production [Mrózek 1985],linguistic pattern recognition [Wojcik 1986], and approximate reasoning [Rasiowa 1987].
A natural, probabilistic extention of the rough-set model has been proven to be useful mathematical tool for dealing with some problems occuring in machine learning like generation of decision rules from inconsistent training examples [Wong 1986(1)] or database design [Yasdi 1987].
The problem of teaching a student by several imperfect teachers ([Pettorossi,Raś, Zemankova 1987], [Raś, Zemankova 1986]) is handled also within rough-set framework. The authors deal with the case when the teacher and the student understand each other only partially and define a formal system which provides the basis for writing procedures which generate rules of a knowledge base.
The model used in all these investigations assume that in the process of perception one distinguishes entities (objects) and their properties. Properties of objects are perceived through assignment of some characteristics (attributes) and their values of the objects. In this way a universe of discourse (a problem domain) consisting of objects and elementary information items providing characterization of these objects in terms of attributes and attributes values is established. In general, information about objects obtained in this way is not sufficient to characterize objects uniquely; that is, it is not possible to distinguish all the objects by means of the admitted attributes and their values. This means that objects are recognized up to indiscernibility relation determined by elementary information items. Any two objects are indiscernible whehever they assume the same values for all the attributes under consideration.
Next we form concepts; that is, we agregate some objects into sets. Information about a concept is composed from information about objects which are instances of concepts. Since objects are not necessarily distinguishable, information characterizing a concept may be ambigous to some extent. In this case we want to have at least some approximation of our information and we express it in terms of rough sets.
All cited above approaches use purely semantical methods in their investigations, exept [Pettorossi,Raś, Zemankova 1987], [Raś, Zemankova 1986] where mixed semantical and syntactical (formal system) methods are used.
In our paper we focus on purely syntactical methods and problems which can be solved by them. One of the problems is a problem of static learning as defined in [Pawlak 1985(1)].
Its main idea is as follows. Suppose now that we are given a finite subset U of the set of objects O B. Elements of U will be called training examples and O B is called a training set.
Assume futher that O B is classified into non-empty, disjoint subsets O1, O2, ..., O n , (n>2) by a teacher (or expert). The classification represents teacher's knowledge of objects from O B.
Let's now assume that there is another person, a student, who is able to characterize each object from O B in terms of attributes from a set A. Description of objects in terms of attributes from A represents student's knowledge of objects from O B.
Now we want to know if it is possible to describe the classification O1, O2, ..., O n provided by the teacher in terms of attributes from A, or more exactly, to find a classification algorithm which provides teacher's classification on the basis of properties of objects expressed in terms of attributes defined in the student's system S.
We give here a purely syntactical, easy programable method to do so. The method consists of two procedures. First procedure, lets call it Procedure one generates, for a given term of a standard term language for a given system, its equivalent normal form.
The next procedure, called Procedure Two will verify the correctness of proper decision algorithm.
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