# Centre for Discrete and Applicable Mathematics

## CDAM Research Report, LSE-CDAM-2006-19

December 2006

The Common Knowledge of Formula Exclusion

Robert Samuel Simon

A multi-partition with evaluations is defined by two sets S and X, a collection P1, . . . ,Pn of partitions of S and a function ψ: 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 ∈ Pj and P ∩ C ≠ ∅ imply that P ⊆ C. Construct a sequence R0,R1, . . . of partitions of S by R0 = { ψ-1(a) | a ∈ {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 R be the limit of the Ri, namely x and y belong to the same member of R 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 Ω such that from any multi-partition with evaluations (using the same X and n) there is a canonical map to Ω with the property that x and y are mapped to the same point of Ω if and only if x and y share the same member of R. We define a cell of Ω 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 ∈ Pj in the cell has finitely many elements. All cells of Ω with finite fanout are surjective, but the converse does not hold.

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