Minority Report (film) composed and conducted by John Williams and orchestrated by John Neufeld, with vocals by Deborah Dietrich. Williams normally enters Spielberg productions at an early stage, well before the movie starts shooting. For "Minority Report" however, his entry was delayed due to his work on "", and he joined the film when it was nearly completed, leaving him scant production time. The soundtrack takes inspiration from Bernard Herrmann's work. Williams decided not to focus on the science fiction elements, and made a score suitable for film noir. He included traditional noir elements such as a female singer in the Anne Lively
Directed set While the definition of a directed set is for an "upward-directed" set (every pair of elements has an upper bound), it is also possible to define a downward-directed set in which every pair of elements has a common lower bound. A subset of a poset is downward-directed if and only if its upper closure is a filter. Directed subsets are used in domain theory, which studies directed complete partial orders. These are posets in which every upward-directed set is required to have a least upper bound. In this context, directed subsets again provide a generalization of convergent sequences. Directed set