1. Languages with Bounded Multiparty Communication Complexity Arkadev Chattopadhyay (McGill) Joint work with: Andreas Krebs (Tubingen) Michal Koucky (Czech Acad. Sciences) Mario Szegedy (Rutgers) Pascal Tesson (Laval) Denis Therien (McGill) TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA A A A

2. ‘Number on Forehead’ Model 010001110 Player 1 110111001 Player 2 111000001 Player 3 1101 0111 1101 Cost of protocol is worst case cost. D k (f) is the cost of best protocol for f for the worst partition.

3. A Theorem for k =2 Question. What functions can be computed in constant communication for the worst partition by two players (denoted by CCC 2 )? Remark: A priori there is no reason to believe that CCC 2 should have any relationship to space-time complexity classes! Theorem (Szegedy93). Every function in CCC 2 can be computed by linear sized ACC 0 circuits.

4. Three Players Question: What can we say about CCC k, for k ¸ 3 ? We show, Theorem 1. CCC 3 contains functions that have arbitrarily large circuit complexity. We use a coding trick to show this.

5. The Coding Idea Let C : { 0,1 } * ! { 0,1 } * be an encoding function. Definition. For a L, define C(L) as follows: y 2 C(L) if there exists x 2 L s.t. C(x)=y. Observation. If C has efficient encoding and decoding algorithms, then L and C(L) have comparable complexity. Fact. If the relative distance of C is more than 2/3, then C(L) 2 CCC 3.

6. Proof of Fact 0100011101001 Player X 110111001 Player Y 1 1 0 111000001 Player Z Find w ² x Find w ² x w ² y ? w ² z ? w 2 C(L)?w 2 C(L)?

7. Such Codes Exist Fact: Reed-Solomon codes concatenated with unary codes can be used to carry out this idea! Remark: Picking a hard L (that is guaranteed to exist by a counting argument), proves our Theorem. Question. What makes 3 players so powerful?

8. Key Features of Three Players Every pair of input bits is looked at by some player. At least a third of the input bits overlap the view of two players. Each player knows the precise position in the input word of every input bit that he sees. Question. How useful is the third feature? Answer. We obtain insight into this question by considering two simple classes of functions.

9. Neutral Letter and Symmetricity Definition. Boolean function f :  * ! { 0, 1 } has neutral letter e if for every x, y 2  *, f(xey) = f(xy). Theorem 2. Every language with a neutral letter that is in CCC k for some fixed k, is regular. We can also give a decidable algebraic characterization of such languages. Definition. f over  is symmetric if for any permutation  and any input string x, f(x)=f(  (x)). Theorem 3. A symmetric function f is in CCC k for some fixed k iff it is in CCC 2.

10. Promised Partition Let A be a 0-1 matrix of dimension k £ n. Let there be a promise that each column of A has at most one 1. Definition. PPart n k (A) is 1 if each column of A has a 1. Theorem. PPart n k cannot be computed by k players using c bits of communication for the row-wise partition of inputs if n ¸ HJ(k, 2 c ). The above Theorem will give us a handle on languages with a neutral letter in CCC k

11. Partition to Neutral Letter Let f, g be two functions that have alphabet  = { a,b,e }, where e is neutral. Let w 2 { a,b } * be the minimal word s.t. f(w)  g(w), with | w |= m. Let D 3 (f) = D 3 (g) = c. Claim: PPart m 3 can be computed by three players using 2c bits of communication for the row-wise partition of input bits.

12. Proof of the Claim 0100 Player X 1000 Player Y 0001 Player Z Let w = abab, m =| w |= 4 x = ebee z = eeeb y = aeee u = x § y § z = eae bee eee eeb Compute f(u) =f(abb) Compute g(u) = g(abb) Output 1 iff f(u)  g(u)

13. Consequence Corollary: If f and g are any two functions over a given alphabet with a neutral letter, can be computed by k players using c bits of communication and they agree on all inputs of length at most HJ(k, 2 2c ), then they must be identical. Remark: There are only a finite number of such functions over a given alphabet that can be computed by k players communicating c bits, for each fixed k and c. Fact: This observation can be used to show that languages in CCC k having a neutral letter, are regular.

14. Conclusion We omit the characterization for symmetric functions. If your are interested, please check out the full version on ECCC at http://eccc.hpi-web.de/eccc-reports/2006/TR06- 117/Paper.pdf THANK YOU!