The common knowledge of formula exclusion

Abstract

A multi-partition with evaluations is defined by two sets S and X, a collection P1, . . . ,Pn of partitions of S and a function : S ! {0, 1}X. To each partition Pj corresponds a person j who cannot distinguish between any two points belonging to the same member of Pj but can distinguish between different members of Pj . A cell of a multi-partition is a minimal subset C such that for all j the properties P 2 Pj and P \ C 6= ; imply that P C. Construct a sequence R0,R1, . . . of partitions of S by R0 = { −1(a) | a 2 {0, 1}X} and x and y belong to the same member of Ri if and only if x and y belong to the same member of Ri−1 and for every person i the members Px and Py of Pj containing x and y respectively intersect the same members of Ri−1. Let R1 be the limit of the Ri, namely x and y belong to the same member of R1 if and only if x and y belong to the same member of Ri for every i. For any set X and number n of persons there is a canonical multi-partition with evaluations defined on a set such that from any multi-partition with evaluations (using the same X and n) there is a canonical map to with the property that x and y are mapped to the same point of if and only if x and y share the same member of R1. We define a cell of to be surjective if every multi-partition with evaluations that maps to it does so surjectively. A cell of a multi-partition with evaluations has finite fanout if every P 2 Pj in the cell has finitely many elements. All cells of with finite fanout are surjective, but the converse does not hold