A note on the tight example in On the randomised query complexity of composition

TL;DR

This paper discusses a recent example in randomized query complexity composition, highlighting its implications for query-to-communication lifting and the necessity of using relations in such examples.

Contribution

It reveals limitations of randomized query-to-communication lifting for relations and emphasizes the importance of relations in tight examples.

Findings

Lifting does not always hold for relations in randomized query complexity.

Relations are essential in constructing tight examples for composition theorems.

The example clarifies boundaries of existing theoretical frameworks.

Abstract

We make two observations regarding a recent tight example for a composition theorem for randomized query complexity: (1) it implies general randomized query-to-communication lifting is not always true if one allows relations, (2) it is in a certain sense essential that a relation is used in constructing the example.