Tiled polymorphic temporal media

Abstract

Tiled Polymorphic Temporal Media (tiled PTM) is an algebraic approach to specifying the composition of multimedia values having an inherent temporal quality -- for example sound clips, musical scores, computer animations, and video clips. Mathematically, one can think of a tiled PTM as a tiling in the one dimension of time. A tiled PTM value has two synchronization marks that specify, via an effective notion of tiled product, how the tiled PTM values are positioned in time relative to one another, possibly with overlaps. Together with a pseudo inverse operation, and the related reset and co-reset projection operators, the tiled product is shown to encompass both sequential and parallel products over temporal media. Up to observational equivalence, the resulting algebra of tiled PTM is shown to be an inverse monoid: the pseudo inverse being a semigroup inverse. These and other algebraic properties are explored in detail. In addition, recursively-defined infinite tiles are considered. Ultimately, in order for a tiled PTM to be renderable, we must know its beginning, and how to compute its evolving value over time. Though undecidable in the general case, we define decidable special cases that still permit infinite tilings. Finally, we describe an elegant specification, implementation, and proof of key properties in Haskell, whose lazy evaluation is crucial for assuring the soundness of recursive tiles. Illustrative examples, within the Euterpea framework for musical temporal media, are provided throughout.