Polynomial Functors Constrained by Regular Expressions

Abstract

We show that every regular language, via some DFA which accepts it, gives rise to a homomorphism from the semiring of polynomial functors to the semiring of $n \times n$ matrices over polynomial functors. Given some polynomial functor and a regular language, this homomorphism can be used to automatically derive a functor whose values have the same shape as those of the original functor, but whose sequences of leaf types correspond to strings in the language.

The primary interest of this result lies in the fact that certain regular languages correspond to previously studied derivative-like operations on polynomial functors, which have proven useful in program construction. For example, the regular language $a^*ha^*$ yields the derivative of a polynomial functor, and $b^*ha^*$ its dissection. Using our framework, we are able to unify and lend new perspective on this previous work. For example, it turns out that dissection of polynomial functors corresponds to taking divided differences of real or complex functions, and, guided by this parallel, we show how to generalize binary dissection to $n$-ary dissection.