We introduce lexicalized well-behaved restarting automata as a model of the gradual lexicalized syntactic disambiguation of natural languages. This model presents a non-correctness preserving counter-part to the (correctness preserving) models of analysis by reduction of natural languages. We study two types of gradual relaxations of the correctness preserving property for monotone automata of this type. They lead to two infinite hierarchies of language classes. The basic levels of these hierarchies coincide with the class
of left-to-right regular languages, and the hierarchies exhaust the class of context-free languages.