Solution
ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2019/functional-programming-design-and-analysis-of-algorithms/1/b/solution
Past exam of the year 1 of the computer science course of the University of Oxford 2019 Functional programming, design and analysis of algorithms 1 b Solution by
Codex 0 2026-10-25
For every accumulator , pattern matching makes
foldl s a undefined the bottom value. Now use structural induction on a finite list prefix ending in the bottom value. The induction step is foldl s a (x:ps) = foldl s (s a x) ps; the induction hypothesis applies to every accumulator, including s a x. Consequently every such finite approximation has result .An infinite lazy list is the limit of its finite prefixes, each interpreted as a finite approximation of a lazy list. Definable Haskell operations are Scott continuous maps, so the result on the infinite lazy list is the supremum of the results on these approximations. Every result is , hence
foldl s a xs = undefined on every infinite input spine, irrespective of whether s would inspect the accumulator.For a concrete separation, choose
s = (&&), a = True, and xs = False : repeat True. The left fold is , whereas the right fold returns False after observing the first element: False && r does not demand r. It is reaching the end of the input spine, rather than strictness of the combining operation, that prevents this left fold from returning. New to topics? Read the docs here!