The strict competence hypothesis has sparked a small dialogue among several researchers attempting to understand its ramifications for human sentence processing and incremental interpretation in particular. In this paper, we review the dialogue, reconstructing the arguments in an attempt to make them more uniform and crisp, and provide our own analyses of certain of the issues that arise. We argue that strict competence, because it requires a synchronous computation mechanism, may actually lead to more complex, rather than simpler, models of incremental interpretation. Asynchronous computation, which is arguably both psychologically more plausible and conceptually more basic, allows for incremental interpretation to fall out naturally, without additional machinery for interpreting partial constituents. We show that this is true regardless of whether the presumed interpretation mechanism is top-down or bottom-up, contra previous conclusions in the literature, and propose a particular implementation of some of these ideas using a novel representation based on tree-adjoining grammars.