 Lowercase Roman letters \(f,g,h,\ldots\) denote functions of type \(\mathbb{N}^k \rightarrow \mathbb{N}\) (for some \(k\))—i.e., the class of functions with domain \(\mathbb{N}^k\) and range \(\mathbb{N}\). For a fixed \(j\), \(f:\mathbb{N}^j \rightarrow \mathbb{N}\) expresses that \(f\) is a \(j\)-ary function (or has arity \(j\))—i.e., \(f\) has domain \(\mathbb{N}^j\) and range \(\mathbb{N}\).