On the Structure of a Common Knowledge Created by Correlated Observations and Transmission over Helping Channels

Abstract

Suppose that two individuals, person X and person Y, communicate with each other in such a way that X sends one of M _X messages to Y and, simultaneously, Y sends one of M _Y messages to X. The messages are numbered by the integers 1, ..., M _X and 1, ..., M _Y . Assuming the numbers to be the identifiers for the corresponding messages, we consider the pairs of the exchanged messages (i, j) ∈ {1, ..., M _X } × {1, ..., M _Y } as possible common values of X and Y which describe their common knowledge. Suppose also that there is another person, called the source, who gives the same binary vector x of length n to the individuals. Then X and Y update their knowledge by including this vector, which means that now they have a triple (i, j, x) in common and if 2ⁿ is much greater than M _X M _Y , then the total number of possible common values is also much greater. However, if the source changes the rules in such a way that x is given to X and y is given to Y, where the vectors x and y do not coincide, but correlated, then this updating of the transmitted pair of messages is not possible any more, and the individuals can revert the situation in which they may agree on M _X M _Y common values. An alternative algorithm can be fixed as follows: X and Y compute their messages using deterministic functions of the observations and each individual, based on the vector given by the source and the message received from the other person, constructs a value belonging to some “virtual” space, which is assumed to be common to both of them and can be formally presented as a finite set Ω. The algorithm should be assigned in such a way that the values are also common. We will investigate this possibility and demonstrate the example in which one of 20 pairs of messages is exchanged, one of 60 pairs of vectors is given by the source, while X and Y construct one of 50 common values.

on leave from the Data Security Association “Confident”, 193060 St.-Petersburg, Russia

The work was supported by the University of Bielefeld (Germany) and the Eindhoven University of Technology (the Netherlands). The author is grateful to Professor Rudolf Ahlswede and Professor Imre Csiszár for helpful and stimulating discussions, which essentially affected this research and presentation of the results. The help of Dr. Roger Bultitude in the preparing of the manuscript is also highly appreciated.