Sensitivity and block sensitivity of specific problem, is there an errata? Arora-Barak exercise 12.4 There's an exercise in Arora-Barak on sensitivity and block sensitivity. The statement of the problem is:


  Let $f$ be a function on $n = k^2$ variables that is an OR of $k$ applications of $g: \{0, 1\}^n \rightarrow \{0,1\}$, all $k$ blocks of variables are disjoint. $g(x_1, \ldots, x_k) = 1$ if there exists $i \in [k-1]$ s.t. $x_i = x_{i=1} = 1$ and $x_j = 0$ for all $j \neq i$. Prove that $s(f) = \sqrt{n}, bs(f) = n/2$.


I am confused about $x_{i=1} = 1$. Is there a typo and should it be $x_{i-1} = 1?$ If it is the case, what about $i=1$? Or is it simply $x_1 = 1?$

Another question is whether $g = 1$ iff the condition holds, or if it is only a sufficient condition.
The definition of $g$ should be: $g(x_1,\ldots,x_k) = 1$ if there exists $i \in [k-1]$ such that $x_i=x_{i+1}=1$ and $x_j = 0$ for $j \neq i,i+1$.

As for your question, whether $g=1$ iff the condition holds, or whether it is just a sufficient condition: in mathematics, when we define something, we use the word "if" to mean "iff". Check your favorite textbook, monograph or paper for many examples.