Synthesis of Algorithms on Sets Represented as Monotone Lists

ABSTRACT. We choose to represent sets as sorted lists without duplications, which we call monotone lists. Without loss of generality, we assume that the elements of the sets are from an ordered domain, and we take increasingly sorted lists. This approach leads to more ecient algorithms then the ones for operating on sets represented as non-sorted lists. We use two functions: the function R which applied to a set returns its representation as a monotone list, and the function S which applied to a (monotone) list returns its corresponding set. These two functions are bijective and reverse to each other and represent the isomorphism between the domain of nite sets (with the usual signature) and the domain of monotone lists with the appropriate signature induced by this isomorphism. The synthesis problem consists in producing the algorithms implementing this signature.

Each synthesis starts as an inductive constructive proof which applies specic strategies and inference rules. These strategies and inference rules are based on the corresponding properties of sets (as monotone lists). We synthesize (by extracting from proofs) ve algorithms among with two predicates: Member and Inclusion; and three functions: Union, Intersection, and Difference.

In parallel with the process of algorithm synthesis we explore the theory of monotone lists.