Generic Accumulations

Abstract

Accumulations are recursive functions that keep intermediate results in additional parameters which are eventually used in later stages of the computation . We present a generic definition of accumulations obtained by the introduction of a new recursive operator on inductive types. We also show that the notion of downwards accumulation developed by Gibbons is subsumed by our notion of accumulation.

Unifying structured recursion schemes

“…(a) catamorphism [10,19] Id Id depth (b) anamorphism [10,19] Id Id mutumorphism [8] ∆ (×) even/odd (mutumorphism) (+) ∆ special case: zygomorphism [18] perfect -special case: paramorphism [21] wc apomorphism [31] fold with a parameter [25] − × P (−) P cat, depths -generalised fold [4] (− • P) Ran P total (generalised unfold) Lan P (− • P) recursion scheme from a comonad [30] U N Cofree N λ -coiteration [2] Free M U M special case: histomorphism [26] knapsack special case: futumorphism [26] This paper shows again the importance of adjunctions. They have played a pivotal role in the categorical analysis of logic; we believe that will prove just as important in the theory of programming.…”

Shortcut fusion rules for the derivation of circular and higher-order monadic programs

“…In this work, we consider a third alternative consisting of defining f :: (µF, z ) → a in terms of a program scheme called pfold (a fold with parameters) [19]. The definition of pfold relies on the concept of strength of a functor F , a polymorphic function τF :: (F a, z ) → F (a, z ) that satisfies certain coherence axioms (see [19,6] for details).…”

“…The definition of pfold relies on the concept of strength of a functor F , a polymorphic function τF :: (F a, z ) → F (a, z ) that satisfies certain coherence axioms (see [19,6] for details). The strength distributes the value of type z to the variable positions (of type a) of the functor.…”

“…Given a functor F and a function h :: (F a, z ) → a, pfold [19] is defined as the least function such that:…”

The Essence of Dataflow Programming

“…Lewis et al [25] must have contemplated employing the product comonad to handle implicit parameters (see the conclusion of their paper), but did not carry out the project. Comonads have also been used in the semantics of intuitionistic linear logic and modal logics [5,7], with their applications in staged computation and elsewhere, see e.g., [15], and to analyse structured recursion schemes, see e.g., [39,28,9]. In the semantics of intuitionistic linear and modal logics, comonads are strong symmetric monoidal.…”