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Lifting Theorems for Equality

Authors Bruno Loff , Sagnik Mukhopadhyay

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Bruno Loff
  • INESC-TEC and University of Porto, Porto, Portugal
Sagnik Mukhopadhyay
  • Computer Science Institute of Charles University, Prague, Czech Republic


We are thankful to Suhail Sherif, Mark Vinyals, and Susanna de Rezende for many helpful discussions, and Or Meir for pointing out an important bug in an earlier draft of the paper. We also thank the anonymous referees whose insights improved the paper by a substantial amount. We owe an extraordinary debt to Arkadev Chattopadhyay, an outstanding companion of many tea-break conversations on the subject of this paper.

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Bruno Loff and Sagnik Mukhopadhyay. Lifting Theorems for Equality. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 50:1-50:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)


We show a deterministic simulation (or lifting) theorem for composed problems f o Eq_n where the inner function (the gadget) is Equality on n bits. When f is a total function on p bits, it is easy to show via a rank argument that the communication complexity of f o Eq_n is Omega(deg(f) * n). However, there is a surprising counter-example of a partial function f on p bits, such that any completion f' of f has deg(f') = Omega(p), and yet f o Eq_n has communication complexity O(n). Nonetheless, we are able to show that the communication complexity of f o Eq_n is at least D(f) * n for a complexity measure D(f) which is closely related to the AND-query complexity of f and is lower-bounded by the logarithm of the leaf complexity of f. As a corollary, we also obtain lifting theorems for the set-disjointness gadget, and a lifting theorem in the context of parity decision-trees, for the NOR gadget. As an application, we prove a tight lower-bound for the deterministic communication complexity of the communication problem, where Alice and Bob are each given p-many n-bit strings, with the promise that either all of the strings are distinct, or all-but-one of the strings are distinct, and they wish to know which is the case. We show that the complexity of this problem is Theta(p * n).

Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
  • Theory of computation → Communication complexity
  • Theory of computation → Oracles and decision trees
  • Communication complexity
  • Query complexity
  • Simulation theorem
  • Equality function


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