Abstract

The Third Homomorphism Theorem is a folk theorem of the constructive algorithmics community. It states that a function on lists that can be computed both from left to right and from right to left is necessarily a list homomorphism---it can be computed according to any parenthesization of the list. We formalize and prove the theorem, and use it to improveanO#n 2 # sorting algorithm to O#n log n#. 1Introduction List homomorphisms are those functions on #nite lists that promote through list concatenation---that is, functions h for which there exists a binary operator # such that, for all #nite lists x and y , h #x ++ y# = hx#hy where `++' denotes list concatenation. Such functions are ubiquitous in functional programming. Some examples of list homomorphisms are: # the identity function id ; # the map function map f , which applies a given function f to every elementof a list; # the function concat , which concatenates a list of lists into a single long list; # the function ...

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