On ∃ ∀ ∃! solving: A case study on automated synthesis of magic card tricks

In formal synthesis, the goal is to find a composition of components from a finite library such that the composition satisfies a given logical specification. In this paper, we consider the problem of synthesizing magic card tricks from component actions, where some of the actions depend on non-deterministic choices made by the audience. This problem can be naturally represented as a quantified logical formula of the form: Exists a composition, Forall nondeterministic choices, Uniquely-Exists intermediate and final outputs satisfying a logical specification, that is, an (∃∀∃!) satisfiability problem. We present a novel approach to solve this problem that exploits the unique-existence of intermediate and final outputs for any given composition and choice values. We illustrate how several popular magic card tricks can be recovered using this approach. These tricks evolved through human ingenuity over decades, but we demonstrate that formal synthesis can generate a number of novel variants of these tricks within minutes. In contrast, a direct encoding to quantified SMT problem fails to find a solution in hours.