  How can interactions with A.I. systems be designed? This paper explores the
design space for A.I. interaction to develop tools for designers to think about
tangible and physical A.I. interactions. Our proposed framework consists of two
dimensions: initiative (human, mixed, or machine) and materiality (physical,
combined, or digital form). A particularly interesting area of interactions we
identify is the quadrant of physical, machine-initiated interactions. With our
framework, we examine calculator interactions and attempt to expand these to
the tangible, mixed-initiative space. We illustrate each area in our proposed
framework with one representative example of a calculator -- a common and
well-known example of a computing device. We discuss existing examples of
calculators and speculative future interactions with mixed-initiative and
physical calculator systems. We reflect on the implications of our framework
for the larger task of designing human-A.I. collaborative systems. Designers
can also apply this framework as a guideline for analogous solutions to
problems in the same domain.
