In the last 40 years, complexity theory has grown to a rich and powerful field in theoretical computer science. The main task of complexity theory is the classification of problems with respect to their consumption of resources (e.g., running time or required memory). To study the computational complexity (i.e., consumption of resources) of problems, similar problems are grouped into so called complexity classes. During the systematic study of numerous problems of practical relevance, no efficient algorithm for a great number of studied problems was found. Moreover, it was unclear whether such algorithms exist. A major breakthrough in this situation was the introduction of the complexity classes P and NP and the identification of hardest problems in NP. These hardest problems of NP are nowadays known as NP-complete problems. One prominent example of an NP-complete problem is the satisfiability problem of propositional formulas (SAT). Here we get a propositional formula as an input and it must be decided whether an assignment for the propositional variables exists, such that this assignment satisfies the given formula. The intensive study of NP led to numerous related classes, e.g., the classes of the polynomial-time hierarchy PH, P, #P, PP, NL, L and #L. During the study of these classes, problems related to propositional formulas were often identified to be complete problems for these classes. Hence some questions arise: Why is SAT so hard to solve? Are there modifications of SAT which are complete for other well-known complexity classes? In the context of these questions a result by E. Post is extremely useful. He identified and characterized all classes of Boolean functions being closed under superposition. It is possible to study problems which are connected to generalized propositional logic by using this result, which was done in this thesis. Hence, many different problems connected to propositional logic were studied and classified with respect to their computational complexity, clearing the borderline between easy and hard problems.
Komplexitätstheorie, Boolesche Funktionen, Isomorphie, abgeschlossene Klassen, Zählprobleme, Computational complexity, Boolean functions, Boolean isomorphism, Boolean equivalence, Dichotomy, counting problems
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