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In the automata-theoretic approach to model checking we check the emptiness of the product of a system S with an automaton A-ψ for the complemented specification. This gives rise to two automata-theoretic problems: complementation of word automata, which is used in order to generate A-ψ, and the emptiness problem, to which model checking is reduced. Both problems have numerous other applications, and have been extensively studied for nondeterministic Büchi word automata (NBW). Nondeterministic generalized Büchi word automata (NGBW) have become popular in specification and verification and are now used in applications traditionally assigned to NBW. This is due to their richer acceptance condition, which leads to automata with fewer states and a simpler underlying structure.In this paper we analyze runs of NGBW and use the analysis in order to describe a new complementation construction and a symbolic emptiness algorithm for NGBW. The complementation construction exponentially improves the best known construction for NGBW and is easy to implement. The emptiness algorithm is almost identical to a known variant of the Emerson-Lei algorithm, and our contribution is the strong relation we draw between the complementation construction and the emptiness algorithm--both naturally follow from the analysis of the runs, which easily implies their correctness. This relation leads to a new certified model-checking procedure, where a positive answer to the model-checking query is accompanied by a certificate whose correctness can be checked by methods independent of the model checker. Unlike certificates generated in previous works on certified model checking, our analysis enables us to generate a certificate that can be checked automatically and symbolically.