We extend the automata-theoretic framework for reasoning about infinite-state sequential systems to handle also the global model-checking problem. Our framework is based on the observation that states of such systems, which carry a finite but unbounded amount of information, can be viewed as nodes in an infinite tree, and transitions between states can be simulated by finite-state automata. Checking that the system satisfies a temporal property can then be done by a two-way automaton that navigates through the tree. The framework is known for local model checking. For branching time properties, the framework uses two-way alternating automata. For linear time properties, the framework uses two-way path automata. In order to solve the global model-checking problem we show that for both types of automata, given a regular tree, we can construct a nondeterministic word automaton that accepts all the nodes in the tree from which an accepting run of the automaton can start.