Centre for Discrete and Applicable Mathematics 

CDAM Research Report, LSECDAM200619December 2006 
The Common Knowledge of Formula Exclusion
Robert Samuel Simon
A multipartition with evaluations is defined by two sets S and X, a collection P^{1}, . . . ,P^{n} of partitions of S and a function ψ: S → {0, 1}^{X}. To each partition P^{j} corresponds a person j who cannot distinguish between any two points belonging to the same member of P^{j} but can distinguish between different members of P^{j}. A cell of a multipartition is a minimal subset C such that for all j the properties P ∈ P^{j} and P ∩ C ≠ ∅ imply that P ⊆ C. Construct a sequence R_{0},R_{1}, . . . of partitions of S by R_{0} = { ψ^{1}(a)  a ∈ {0, 1}^{X}} and x and y belong to the same member of R_{i} if and only if x and y belong to the same member of R_{i1} and for every person i the members P_{x} and P_{y} of P^{j} containing x and y respectively intersect the same members of R_{i1}. Let R_{∞} be the limit of the R_{i}, namely x and y belong to the same member of R_{∞} if and only if x and y belong to the same member of R_{i} for every i. For any set X and number n of persons there is a canonical multipartition with evaluations defined on a set Ω such that from any multipartition 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 multipartition with evaluations that maps to it does so surjectively. A cell of a multipartition with evaluations has finite fanout if every P ∈ P^{j} in the cell has finitely many elements. All cells of Ω with finite fanout are surjective, but the converse does not hold.A PDF file (121 kB) with the full contents of this report can be downloaded by clicking here.
Alternatively, if you would like to get a free hard copy of this report, please send the number of this report, LSECDAM200619, together with your name and postal address to:
CDAM Research Reports Series Centre for Discrete and Applicable Mathematics London School of Economics Houghton Street London WC2A 2AE, U.K. 

Phone: +44(0)207955 7494. Fax: +44(0)207955 6877. Email: info@maths.lse.ac.uk 
Introduction to the CDAM Research Report Series.  
CDAM Homepage. 