Augmenting the Process

Abstract

In section 4, we considered “application of less than n inputs” in terms of the coproduct $\coprod\limits_{j = 0}^{n - 1} {{\text{IX}}^{\text{k}} }$, and then required X to preserve coproducts. Unfortunately, this requirement rules out tree automata—we know from [3] that the only X: Set → Set which preserves coproducts is the sequential machine process X = − × X_o. section, we show how to achieve the goals of Section 4 in a less restrictive setting, by augmenting X to obtain a new functor X̄

$$Q\bar X = QX + Q$$

(where + is the coproduct) whose n-fold application directly generates “application of at most n inputs”. Consider the case X = − × X_O. It is easy to check that QX̄ⁿ is then essentially $\coprod\limits_{j \leqslant n} {(Q \times X\frac{j} {o})}$, where the “essentially” reminds us that $Q \times X\frac{j} {o}$ is actually present as ($(\frac{n} {j})$) disjoint copies. With this motivation, we turn to the general development