Exploration with a generative formalism must necessarily account for the nature of interaction between humans and the design space explorer. Established accounts of design interaction are made complicated by two propositions in Woodbury and Burrow's Keynote on design space exploration. First, the emphasis on the primacy of the design space as an ordered collection of partial designs (version, alternatives, extensions). Few studies exist in the design interaction literature on working with multiple threads simultaneously. Second, the need to situate, aid, and amplify human design intentions using computational tools. Although specific research and practice tools on amplification (sketching, generation, variation) have had success, there is a lack of generic, flexible, interoperable, and extensible representation to support amplification. This paper addresses the above, working with design threads and computer-assisted design amplification through a theoretical model of dialogue based on Grice's model of rational conversation. Using the concept of mixed initiative, the paper presents a visual notation for representing dialogue between designer and design space formalism through abstract examples of exploration tasks and dialogue integration.