Complexidade descritiva de classes de complexidade probabilísticas de tempo polinomial e das classes ⊕P e NP∩coNP através de lógicas com quantificadores de segunda ordem

Abstract

Many computable problems can be solved more efficiently or in a more natural way through probabilistic algorithms, which shows that the use of such algorithms is quite relevant in Computer Science. However, probabilistic algorithms may return a wrong answer with a certain probability. Also, the use of probabilistic algorithms does not solve problems that are not computable. In Computational Complexity, the complexity of a problem is characterized based on the amount of computational resources, such as space and time, needed to solve it. Problems that have the same complexity compose the same class. The computational complexity classes are related by a hierarchy. In Descriptive Complexity, a logic is used to express problems and capture computational complexity classes in order to express all and only the problems of this class. Thus, the complexity of a problem does not depend on physical factors, such as time and space, but only on the expressiveness of the logic that defines it. Important results of the area states that several classes of computational complexity can be characterized by a logic. For example, the class NP has been shown equivalent to the class of problems expressed by the existential fragment of Second-Order Logic. This close relationship between these areas allows some results about Logics to be transferred to Computational Complexity and vice versa. Despite of the importance of probabilistic algorithms and of Descriptive Complexity, there are few results on the characterization, by a logic, of probabilistic computational complexity classes. In this work, we show characterizations for each of the polinomial time probabilistic complexity classes. In our results, we use second-order generalized quantifiers to simulate the acceptance of the nondeterministic machines of these classes. We found Logical characterizations in the literature only for classes PP and BPP. In the first case, the logic employed was the first-order added by a quantifier most of second-order. With the approach established in this work, we obtain an alternative proof for the characterization of PP. With the same methodology, we also characterize the class ⊕P through a logic with a second-order parity quantifier. In the case of BPP , there was a result that used a logic with probabilistic semantics. Using our approach of generalized quantifiers, we obtain an alternative characterization for this class. With the same method, we were able to characterize the probabilistic semantic classes RP, coRP, ZPP and the semantic class NP ∩ coNP. Finally, we show an application of Descriptive Complexity results in the creation of algorithms from a logic specification.