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Joined 2026-09-21 Message
Incoming links: Haskell
Bottom value 2026-10-24
An undefined or nonterminating computation in the denotational account of Haskell. A structured value may contain bottom values while retaining a defined outer constructor:
undefined : [] is distinguishable from an entirely undefined list.[ref]Haskell 2010 Language Report. §3.1, Errors
Haskell Ord class 2026-10-24
The Haskell class providing ordering operations, including
compare. A comparison-based sorting algorithm requires the associated constraint to inspect the order of arbitrary keys. Its correctness additionally assumes that the supplied operations obey the intended ordering laws. Irrefutable pattern 2026-10-24
A pattern that does not force its matched value merely to establish the match. A Haskell pattern written with
~ can defer evaluation of a recursive pair until one component is demanded. This can preserve productivity in definitions involving lazy evaluation. It does not make every later use of the components terminate. Leaf-labelled binary tree 2026-10-24
A binary tree with each data label stored at a leaf of a tree. One useful Haskell datatype has constructors
One a for a leaf and Two (Tree a) (Tree a) for an internal node. Every finite inhabitant has at least one leaf, so this datatype cannot encode an empty leaf sequence. Its natural tree fold has a leaf operation a -> b and an internal operation b -> b -> b; replacing the two constructors by those operations defines the structural recursion. Optional value 2026-10-24
Partial-list interleaving obstruction 2026-10-25
No generic sequential pure Haskell operation can guarantee emission of every available input occurrence when either of two input spines may be the bottom value. To obtain an arbitrary element, it must eventually demand one input. That input can be undefined while the other is a defined singleton, so the demand blocks the promised singleton output. Choosing the other input first reverses the counterexample. Lazy evaluation cannot determine which computation will diverge; the positive two-stream fairness result needs proper lazy lists or a stronger concurrent computational model.
Source release: Oxford examination archive, member
2018/T/a10097w1.pdf. The cover dates this paper to 18 June 2019; the archive member's directory is misleading. The solutions cover every option. In the code, different definitions offered in different parts are alternatives, and the Haskell Prelude operations have their usual meanings unless redefined locally.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.Fix
s and put . The right fold equations giveThese are exactly the left fold equations. Structural induction, with the induction hypothesis quantified over all accumulators, proves equality on finite terminating lists.To include the other input spines, first observe that both applied expressions give on an undefined spine. The same induction proves equality on every finite prefix ending in : the step
k s x f a demands the continuation f when its result is demanded. Finally use that the two applied operations are Scott continuous maps, taking the limit over prefixes interpreted as a finite approximation of a lazy list. This covers infinite spines, on which both applied expressions are , and proves the identity for all lists under the stated Haskell semantics. The intermediate right fold builds an accumulator-transforming function; it does not provide a terminating left fold of an infinite list.No universally correct sequential Haskell implementation has that guarantee for partial lists. Compare
undefined paired with [v], and [v] paired with undefined, where the element type is arbitrary. A generic program cannot fabricate v; it must eventually inspect an input spine to obtain it. Whichever input it first demands can be chosen to be the undefined one, blocking access to the available element in the other input.In particular, the implementation in part b can emit a defined prefix and then block when alternation reaches an undefined tail. Lazy evaluation postpones demands; it cannot recognize that a demanded computation will never finish. The result is the partial-list interleaving obstruction, not an objection to fairness under the proper-input hypothesis. Additional concurrent primitives would change the computational model.[ref]
Haskell 2010 Language Report. §3.1, errors and nontermination; §3.17, demands made by pattern matching
Source paper: released examination archive, member 2020/L/a10097w1.pdf.
The held cover identifies the Long Vacation sitting of 8 September 2021. All eight questions and all 37 printed letter leaves are answered here, including the unnumbered cases within 5(a) and 5(b). Code is written for Haskell 2010 unless marked as pseudocode. Sorting assumes a lawful terminating ordering on finite inputs; the partial-value qualifications in Question 1 are stated where they matter. No native Haskell compiler is installed in the checking environment, so executable checks use independent Python models and are not claimed as native compilation.
For finite lists, use structural induction: establish
P []; then, for arbitrary x and xs, assume P xs and prove P (x:xs). The induction hypothesis must quantify over any other parameters that need to change in the step.For all Haskell list values, finite terminating spines are insufficient: also establish
P undefined, and allow arbitrary, possibly undefined, heads in the cons step. This proves the equation for finite partial lists ending in the bottom value. An infinite lazy list is the limit of these finite approximations. Equations between pure definable computable functions are preserved by those limits, so the equation extends to infinite inputs. This last argument requires that admissibility property; an arbitrary logical property of lists need not have it. The distinction between [] and the bottom value is essential.[ref]A Haskell representation is:Every tree stores a value at its root, including internal nodes. Interpret the forest by root-left-right preorder traversal, concatenating the results in forest order. Valid values have correct cached sizes, perfect binary trees, nondecreasing sizes and only the permitted initial duplicate. The raw constructors can express invalid representations, so the operations below assume valid values and should form the public interface. Sizes are assumed to fit in
import Prelude hiding (null, head, tail)
data RTree a = Leaf a | Node a (RTree a) (RTree a)
newtype List a = List [(Int, RTree a)]Int; the stated time complexities use constant-time size arithmetic.A strict function satisfies , where is the bottom value representing an undefined or nonterminating computation. This is a semantic implication: obtaining a defined result requires enough definedness of the argument. It does not prescribe an evaluation order or say that every component of a structured argument is evaluated.[ref]
For example, in Haskell,is strict, since every result is , but never needs to inspect its argument. Moreover, lazy evaluation may leave an entire application unevaluated when its result is unused. Strictness is therefore different from the operational statement that an argument is evaluated on every call.
f :: a -> b
f _ = undefinedUse these structural recursion definitions in Haskell:The first definition returns a known outer constructor even when
tails :: [a] -> [[a]]
tails xs = xs : ptails xs
ptails :: [a] -> [[a]]
ptails [] = []
ptails (_:xs) = tails xs
ntails :: [a] -> [[a]]
ntails [] = []
ntails xs@(_:rest) = xs : ntails restxs is undefined, so null (tails undefined) is False, as required. The pattern in ntails must first determine whether the input is empty. Each suffix of a list is shared with the original list; on a finite input of length , constructing the outer lists takes work and additional outer-list space. No suffix is copied or measured repeatedly.For prefixes of a list, the corresponding Haskell definitions areThe empty prefix is available without inspecting the input in
inits :: [a] -> [[a]]
inits xs = [] : ninits xs
pinits :: [a] -> [[a]]
pinits [] = []
pinits (x:xs) = [] : map (x:) (pinits xs)
ninits :: [a] -> [[a]]
ninits [] = []
ninits (x:xs) = map (x:) (inits xs)inits. Every nonempty prefix of a list begins with its input's head, explaining the last equation. For pinits (x:xs), prepend x only to proper prefixes of xs, and include the empty prefix.On finite inputs,
inits [] = [[]], whereas both other functions return []. On an infinite input, all finite prefixes are proper and these definitions produce them incrementally. Fully constructing all elements of all prefixes takes work for an input of length , matching the size of that output; there are no repeated end-append operations to add another factor.For finite, fully defined representations, yes; for arbitrary infinite ones in Haskell, no. At the level of mathematical occurrence sets, splitting by parity, dividing by two, and then applying the inverse index transformations recovers every original position exactly once. Thus the equality holds on finite total position lists.
For a proper infinite representation, take the alternating Boolean sequence, represented byThis is a total representation: both queries return productive infinite occurrence lists. Butfilters for odd indices in uses
alternating :: Seq Bool
alternating False = [0,2..]
alternating True = [1,3..]odds alternating False[0,2..], so it produces no outer constructor and denotes . The callilv (evens alternating) (odds alternating) FalsemergePos to inspect both transformed lists before emitting its first result; it therefore also diverges. In contrast, alternating False starts with 0. Applying head distinguishes the two. The productivity failure is real even though the mathematical position-set identity is correct; the hypothesis that the input properly represents an infinite sequence does not remove it.A tree fold can compute insertion while retaining the unchanged children by carrying a pair. The following fold-derived implementation preserves the lazy evaluation behaviour of part (f):The invariant of the folded pair is
insertViaFold :: Ord a => a -> Tree a -> Tree a
insertViaFold q t = snd (foldTree node base t)
where
base = (Empty, Tree Empty q Empty)
node ~(l,li) x ~(r,ri) = (original,updated)
where
original = Tree l x r
updated = case compare q x of
LT -> Tree li x r
EQ -> original
GT -> Tree l x ri(original subtree, subtree with q inserted). The empty case establishes it. At a node, the first components reconstruct the original tree, and the second components supply whichever child insertion is needed. The irrefutable pair patterns ~(l,li) and ~(r,ri) avoid demanding an unused child's fold. Induction on finite approximations therefore proves equivalence even for partial trees, and only the selected search route needs insertion work.There is a distinction if “written as a fold” is required to mean a direct same-result-type expression Here
insert q = foldTree node base, with no richer result and final projection. On finite fully defined binary search trees, such an expression is possible by undoing insertion of a new extreme key in the child that should remain unchanged. For completeness, a literal direct fold on that finite, fully defined domain is:eraseFresh :: Ord a => a -> Tree a -> Tree a
eraseFresh _ Empty = Empty
eraseFresh q (Tree l x r) = case compare q x of
LT -> Tree (eraseFresh q l) x r
EQ -> Empty
GT -> Tree l x (eraseFresh q r)
insertDirectFold :: Ord a => a -> Tree a -> Tree a
insertDirectFold q = foldTree node (Tree Empty q Empty)
where
node left x right = case compare q x of
LT -> Tree left x (eraseFresh q right)
EQ -> Tree (eraseFresh q left) x (eraseFresh q right)
GT -> Tree (eraseFresh q left) x righteraseFresh is used only to undo a newly inserted extreme key, which is a leaf; it is not a general deletion operation. The binary search tree bound at each root guarantees exactly that precondition on every child being restored.On all partial Haskell trees, however, a direct expression with result type
Tree a cannot implement part (f).For that obstruction, fix Both can be completed to valid right subtrees of a root labelled
q = 0, and consider the two right subtreesr1 = Tree Empty undefined Empty
r2 = Tree Empty undefined (Tree Empty 3 Empty)1. Yet insert 0 r1 and insert 0 r2 are both bottom, because insertion must compare 0 with their undefined root label. A direct fold equal to insertion would therefore receive the same folded left value, root 1, and bottom folded right value on both larger trees Tree Empty 1 r1 and Tree Empty 1 r2. It must return the same result. Part (f) instead preserves r1 or r2, which are distinguishable by inspecting their right-child constructors without inspecting the undefined labels. This proves the direct-fold obstruction. The pair-valued implementation avoids losing that information and states precisely which sense of fold implementation is being used. Pattern matching 2026-10-24
Standard Prelude 2026-10-24
The automatically available standard module of Haskell, supplying basic types, classes and operations such as list folds, arithmetic and list comprehension support. A program can explicitly hide an imported name when providing an alternative definition. The reference definitions specify behavior, rather than requiring a particular implementation strategy.[ref]
Haskell 2010 Language Report. Chapter 9, introductory paragraphs and module Prelude exports