Unate function

A unate function is a type of boolean function which has monotonic properties. They have been studied extensively in switching theory. A function f(x_1,x_2,\ldots,x_n) is said to be positive unate in x_i if for all possible values of x_j , j\neq i : f(x_1,x_2,\ldots,x_{i-1},1,x_{i+1},\ldots,x_n) \ge f(x_1,x_2,\ldots,x_{i-1},0,x_{i+1},\ldots,x_n).\, Likewise, it is negative unate in x_i if : f(x_1,x_2,\ldots,x_{i-1},0,x_{i+1},\ldots,x_n) \ge f(x_1,x_2,\ldots,x_{i-1},1,x_{i+1},\ldots,x_n).\, If for every x_i f is either positive or negative unate in the variable x_i then it is said to be unate (note that some x_i may be positive unate and some negative unate to satisfy the definition of unate function). A function is binate if it is not unate (i.e., is neither positive unate nor negative unate in at least one of its variables). For example, the logical disjunction function or with boolean values used for true (1) and false (0) is positive unate. Conversely, Exclusive or is non-unate, because the transition from 0 to 1 on input x0 is both positive unate and negative unate, depending on the input value on x1.

A unate function is a type of boolean function which has monotonic properties. They have been studied extensively in switching theory. A function f(x_1,x_2,\ldots,x_n) is said to be positive unate in x_i if for all possible values of x_j , j\neq i f(x_1,x_2,\ldots,x_{i-1},1,x_{i+1},\ldots,x_n) \ge f(x_1,x_2,\ldots,x_{i-1},0,x_{i+1},\ldots,x_n).\, Likewise, it is negative unate in x_i if f(x_1,x_2,\ldots,x_{i-1},0,x_{i+1},\ldots,x_n) \ge f(x_1,x_2,\ldots,x_{i-1},1,x_{i+1},\ldots,x_n).\, If for every x_i f is either positive or negative unate in the variable x_i then it is said to be unate (note that some x_i may be positive unate and some negative unate to satisfy the definition of unate function). A function is binate if it is not unate (i.e., is neither positive unate nor negative unate in at least one of its variables). For example, the logical disjunction function or with boolean values used for true (1) and false (0) is positive unate. Conversely, Exclusive or is non-unate, because the transition from 0 to 1 on input x0 is both positive unate and negative unate, depending on the input value on x1.

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