Three messages are not optimal in worst case interactive communication

Let X and Y be two jointly distributed random variables. Suppose person P/sub X/, the informant, knows X, and person P/sub Y/, the recipient, knows Y, and both know the joint probability distribution of the pair (X,Y). Using a predetermined protocol, they communicate over a binary error-free channel in order for P/sub Y/ to learn X, whereas P/sub X/ may or may not learn Y. C/spl circ/m(X/spl verbar/Y) is the minimum number of bits required to be transmitted (by both persons) in the worst case when only m message exchanges are allowed. C/spl circ//spl infin/(X/spl verbar/Y) is the number of bits required when P/sub X/ and P/sub Y/ can communicate back and forth an arbitrary number of times. Orlitsky proved that for all (X,Y) pairs, C/spl circ//sub 2/(X/spl verbar/Y)/spl les/4C/spl circ//spl infin/(X/spl verbar/Y)+3, and that for every positive c and /spl isin/ with /spl isin/<1, there exist (X,Y) pairs with C/spl circ//sub 2/(X/spl verbar/Y)/spl ges/(2/spl minus//spl isin/)C/spl circ//sub 3/(X/spl verbar/Y)/spl ges/(2/spl minus//spl isin/)C/spl circ//spl minus//spl infin/(X/spl verbar/Y)/spl ges/c. These results show that two messages are almost optimal, but not optimal. A natural question, then, is whether three messages are asymptotically optimal. In this work, the authors prove that for any c and /spl isin/ with 00, there exist some (X,Y) pairs for which C/spl circ//sub 3/(X/spl verbar/Y)/spl ges/(2/spl minus//spl isin/)C/spl circ//sub 4/(X/spl verbar/Y)/spl ges/c. That is, three messages are not optimal either.<
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