Composition and Simulation Theorems via Pseudo-random Properties

We prove a randomized communication-complexity lower bound for a composed function — OrderedSearch ◦ IP — by lifting the randomized query-complexity lower-bound of OrderedSearch to the communication-complexity setting. We do this by extending ideas from a paper of Raz and Wigderson [RW89]. We think that the techniques we develop will be useful in proving a randomized simulation theorem. We also generalize the deterministic simulation theorem of Raz and McKenzie [RM99], to any inner-function which satisfies certain hitting property. We prove that IP satisfies this property, and as a corollary we obtain deterministic simulation theorem for an inner-function gadget with logarithmic block-size with respect to the arity of the outer function. This answers an open question posed by Göös, Pitassi and Watson [GPW15]. Our result also implies the previous results for the Indexing inner-function. Tata Institute of Fundamental Research, Mumbai, [email protected] Charles University, Prague, [email protected] Charles University, Prague, [email protected] Tata Institute of Fundamental Research, Mumbai, [email protected] ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 14 (2017)