Before revision:  Mathematically, the biology-inspired concept of canalization means that a single input can determine the output of a function , irrespective of all the other inputs. Here, we relax this definition and explore the broader class of collectively canalizing Boolean functions as well as the k-set canalizing proportion, which is the proportion of sets of k inputs which canalize a Boolean function. We exhibit how the number and type of these sets influence a function's behavior and define a new measure for the canalizing strength of any Boolean function.
After revision:  Mathematically, the biology-inspired concept of canalization means that a single input can determine the output of a function , and the class of functions studied here captures a key feature of biological network dynamics, namely that a subset of one or more variables, under certain conditions, can dominate the value of a Boolean function, to the exclusion of all others. These functions have rich mathematical properties to be explored. The paper shows how the number and type of these sets influence a function's behavior and define a new measure for the canalizing strength of any Boolean function.