Improved Lower Bounds for Testing Triangle-freeness in Boolean Functions via Fast Matrix Multiplication

Understanding the query complexity for testing linear-invariant properties has been a central open problem in the study of algebraic property testing. Triangle-freeness in Boolean functions is a simple property whose testing complexity is unknown. Three Boolean functions f1, f2 and f3 : F2 → {0, 1} are said to be triangle free if there is no x, y ∈ F2 such that f1(x) = f2(y) = f3(x + y) = 1. This property is known to be strongly testable [16], but the number of queries needed is upper-bounded only by a tower of twos whose height is polynomial in 1/ , where is the distance between the tested function triple and triangle-freeness, i. e., the minimum fraction of function values that need to be modified to make the triple triangle free. A lower bound of ( 1 ) 2.423 for any one-sided tester was given by Bhattacharyya and Xie (2010). In this work we improve this bound to ( 1 ) 6.619. Interestingly, we prove this by way of a combinatorial construction called uniquely solvable puzzles that was at the heart of Coppersmith and Winograd (1990)’s renowned matrix multiplication algorithm. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems