By Juliusz Kulikowski
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Extra info for Algebraic Methods in Pattern Recognition: Course held at the Department of Automation and Information July 1971
The formal identity of expressions should not be confused with the semantical one. The expressions will be called semantically identical if they symbolize the identical relations, that is the relations satisfied by the same sets of realizations. The class of expressions formally identical to an expression q 51 Recognition of Patterns in Algebraic Sense will be denoted by h(q), while the class of the expressions iden tical to q in semantic sense by H(q) . It is clear that h(q)CH(q) . 3) As an example let us take into account a problem of printed letters recognition.
With. ,x,-,> form a Boolean algebra with the empty relation as a null and the trivial relation Q as an unity element. This last statement may be very important in pra~ tice. It means that the well known methods of logical functions minimization and constructive methods of logical systems design can be applied for the design of picture interpretation and pa! tern recognition algorithms. Although the main concepts have been illustrated in the families of binary sets, the algebra of relations does not put any constraints on the nature of the sets.
However, it is possible that not only the morphisms of relations, but also the morphisms of structures may be of partial type. For example, the pattern recognition problems connected with automatic reading English and Russian alphabets can be formally represented by partially isomorphic structures, so as the relations describing the printed letters "A", "B", "C", "E", "H" and so on are common in both cases. However, the problem of similar! ty of structures is much deeper. It is well known, for example, that any practical pattern recognition problem can be effective ly solved by a lot of formally different ways.
Algebraic Methods in Pattern Recognition: Course held at the Department of Automation and Information July 1971 by Juliusz Kulikowski