In this master thesis the hierarchy of automata and related languages is presented. First, the concepts, associated with recognizable and unrecognizable languages are introduced, followed by further introduction of a subclass of recognized languages; these are decidable languages. By means of decidable and undecidable languages the concept of decidable and undecidable problems was translated and strictly defined. The following examples of undecidable problems are described in detail: the Turing's halting problem, the Post correspondence problem, the busy beaver problem and the Hilbert's tenth problem. In the frame of the master thesis, a web application which is searching for concrete solutions of the Post correspondence problem, was developed. The programming environment of the application and the interpretation of the source code with instructions for its using are described in detail. An example for the usage of the application at teaching, for example in a computer club, is presented.
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