Lazier functional programming, part 2

In Lazier functional programming, part 1, I suggested that some of the standard tools of lazy functional programming are not as lazy as they might be, including even if-then-else. Generalizing from boolean conditionals, I posed the puzzle of how to define a lazier either function, which encapsulates case analysis for sum types. The standard definition:

data Either a b = Left a | Right b

either :: (a → c) → (b → c) → Either a b → c
either f g (Left  x) = f x
either f g (Right y) = g y

The comments to part 1 fairly quickly converged to something close to the laxer definition I had in mind:

eitherL f g = const (f ⊥ ⊓ g ⊥) ⊔ either f g

which is a simple generalization of Luke Palmer‘s laxer if-then-else (reconstructed from memory):

bool c a b = (a ⊓ b) ⊔ (if c then a else b)

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Lazier functional programming, part 1

This post is inspired by a delightful insight from Luke Palmer. I’ll begin with some motivation, and then propose a puzzle.

Consider the following definition of our familiar conditional:

ifThenElse :: Bool → a → a → a
ifThenElse True  t f = t
ifThenElse False t f = f

In a strict language, where there are only two boolean values, these two clauses have a straightforward reading. (The reading is less straightforward when patterns overlap, as mentioned in Lazier function definitions by merging partial values.) In a non-strict language like Haskell, there are three distinct boolean values, not two. Besides True and False, Bool also has a value ⊥, pronounced “bottom” for being at the bottom of the information ordering. For an illustration and explanation of information ordering, see Merging partial values.

Note: In Haskell code, ⊥ is sometimes denoted by “undefined“, which can be confusing, because the meaning is defined precisely. There are many other ways to denote ⊥ in Haskell, and it is impossible to determine whether or not an arbitrary Haskell expression denotes ⊥. I’ll generally use “⊥” in place of “undefined” in this post, as well as for the corresponding semantic value.

The two-clause definition above only addresses two of the three possible boolean values explicitly. What, if anything, does it say indirectly about the meaning of an application like “ifThenElse ⊥ 3 5“?

The Haskell language standard gives an operational answer to this question. Clauses are examined, using pattern matching to select a clause and instantiate that clause’s variables. In case more than one clause matches, the earlier one is chosen.

Pattern matching has three possible outcomes:

  • A single substitution, providing variable bindings that specialize the patterns in a clause’s left-hand side (LHS) to coincide with the actual call. The matching uses semantic, not syntactic, equality and can require forcing evaluation of previously unevaluated thunks (delayed computations).
  • Proof of the nonexistence of such a substitution.
  • Neither conclusion, due to an error or nontermination during evaluation.

In this example, the effort to match True against ⊥ leads to the third outcome. For Haskell as currently defined, the result of the application in such a case is then defined to be ⊥ also. Which is to say that ifThenElse is strict (in its first argument).

So strictness is the Haskell answer, but is it really the answer we want? Are there alternatives that might better fit the spirit of non-strict functional programming?

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“Everything is a function” in Haskell?

There a belief about Haskell that keeps popping up in chat rooms and mailing lists — one that I’ve been puzzling over for a while. One expression of the belief is “everything is a function” in Haskell.

Of course, there are all of these non-functions that need to be accounted for, including integers, booleans, tuples, and lists. What about them? A recurring answer is that such things are “functions of no arguments” or functions of a one-element type or “constant functions”.

I wonder about how beliefs form, spread, and solidify, and so I asked around about how people came to this notion and how they managed to hold onto it. I had a few conjectures in mind, which I kept to myself to avoid biasing people’s responses. Of the responses I got, some were as I’d imagined, and some were quite surprising to me, revealing some of my blind spots about others’ thinking and about conversation dynamics.

My thanks to the many Haskellers, especially newbies, who took the time to help me understand their thought processes. If you’re interested and in a patient mood, you can see the unedited responses on a Haskell reddit thread and on a #haskell IRC log. There were also a few responses on Twitter.


  • 2009-08-04: Added “simplify”: “Would making everything a function really simplify the formal system that is Haskell programming?”. Thanks, SLi.
  • 2009-08-04: Focus on “constant function” story for “It makes things simpler”. I realized that I hadn’t said what I intended there. Thanks, Jonathan Cast.
  • 2011-03-04: Remarks on mutability & dynamic typing, under “Operational thinking”

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Topless data

Functional programming abounds with recursively defined data types. We often draw these structured values as trees with the root at the top and the leaves at the bottom. Lazy functional programming allows values (structures) of these types to be “bottomless”, meaning we can descend forever. There are many examples of how supporting such values gives an enormous boost to modularity. (See, e.g., John Hughes’s classic paper Why Functional Programming Matters.) We usually refer to these values as “infinite”, but I’d like to suggest “bottomless” as a more specific alternative, and to point out a limitation that perhaps is not widely noticed.

Although we can descend infinitely in lazy functional programming, we can only ascend finitely. If I’m traversing a lazy list, there may be infinitely many elements on my right (yet to be visited) but only finitely many on my left (already visited). While traversing a tree, there may be infinite paths below but only a finite one above (leading to my current position).

In other words, our data is bottomless, but not topless. What would it be like to go beyond our usual merely uni-infinite data and program with bi-infinite data instead? With data that is both bottomless and topless?

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Another angle on zippers

The zipper is an efficient and elegant data structure for purely functional editing of tree-like data structures, first published by Gérard Huet. Zippers maintain a location of focus in a tree and support navigation operations (up, down, left, right) and editing (replace current focus).

The original zipper type and operations are customized for a single type, but it’s not hard to see how to adapt to other tree-like types, and hence to regular data types. There have been many follow-up papers to The Zipper, including a polytypic version in the paper Type-indexed data types.

All of the zipper adaptations and generalizations I’ve seen so far maintain the original navigation interface. In this post, I propose an alternative interface that appears to significantly simplify matters. There are only two navigation functions instead of four, and each of the two is specified and implemented via a fairly simple one-liner.

I haven’t used this new zipper formulation in an application yet, so I do not know whether some usefulness has been lost in simplifying the interface.

The code in this blog post is taken from the Haskell library functor-combo and completes the Holey type class introduced in Differentiation of higher-order types.


  • 2010-07-29: Removed some stray Just applications in up definitions. (Thanks, illissius.)
  • 2010-07-29: Augmented my complicated definition of tweak2 with a much simpler version from Sjoerd Visscher.
  • 2010-07-29: Replaced fmap (first (:ds')) with (fmap.first) (:ds') in down definitions. (Thanks, Sjoerd.)

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Differentiation of higher-order types

A “one-hole context” is a data structure with one piece missing. Conor McBride pointed out that the derivative of a regular type is its type of one-hole contexts. When a data structure is assembled out of common functor combinators, a corresponding type of one-hole contexts can be derived mechanically by rules that mirror the standard derivative rules learned in beginning differential calculus.

I’ve been playing with functor combinators lately. I was delighted to find that the data-structure derivatives can be expressed directly using the standard functor combinators and type families.

The code in this blog post is taken from the Haskell library functor-combo.

See also the Haskell Wikibooks page on zippers, especially the section called “Differentiation of data types”.

I mean this post not as new research, but rather as a tidy, concrete presentation of some of Conor’s delightful insight.

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Details for non-strict memoization, part 1

In Non-strict memoization, I sketched out a means of memoizing non-strict functions. I gave the essential insight but did not show the details of how a nonstrict memoization library comes together. In this new post, I give details, which are a bit delicate, in terms of the implementation described in Elegant memoization with higher-order types.

Near the end, I run into some trouble with regular data types, which I don’t know how to resolve cleanly and efficiently.


  • 2010-09-10: Fixed minor typos.

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Memoizing higher-order functions

Memoization incrementally converts functions into data structures. It pays off when a function is repeatedly applied to the same arguments and applying the function is more expensive than accessing the corresponding data structure.

In lazy functional memoization, the conversion from function to data structure happens all at once from a denotational perspective, and incrementally from an operational perspective. See Elegant memoization with functional memo tries and Elegant memoization with higher-order types.

As Ralf Hinze presented in Generalizing Generalized Tries, trie-based memoization follows from three simple isomorphisms involving functions types:

1 → a ≅ a

(a + b) → c ≅ (a → c) × (b → c)

(a × b) → c ≅ a → (b → c)

which correspond to the familiar laws of exponents

a ^ 1 = a

ca + b = ca × cb

ca × b = (cb)a

When applied as a transformation from left to right, each law simplifies the domain part of a function type. Repeated application of the rules then eliminate all function types or reduce them to functions of atomic types. These atomic domains are eliminated as well by additional mappings, such as between a natural number and a list of bits (as in patricia trees). Algebraic data types corresponding to sums of products and so are eliminated by the sum and product rules. Recursive algebraic data types (lists, trees, etc) give rise to correspondingly recursive trie types.

So, with a few simple and familiar rules, we can memoize functions over an infinite variety of common types. Have we missed any?

Yes. What about functions over functions?


  • 2010-07-22: Made the memoization example polymorphic and switched from pairs to lists. The old example accidentally coincided with a specialized version of trie itself.
  • 2011-02-27: updated some notation

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Elegant memoization with higher-order types

A while back, I got interested in functional memoization, especially after seeing some code from Spencer Janssen using the essential idea of Ralf Hinze’s paper Generalizing Generalized Tries. The blog post Elegant memoization with functional memo tries describes a library, MemoTrie, based on both of these sources, and using associated data types. I would have rather used associated type synonyms and standard types, but I couldn’t see how to get the details to work out. Recently, while playing with functor combinators, I realized that they might work for memoization, which they do quite nicely.

This blog post shows how functor combinators lead to an even more elegant formulation of functional memoization. The code is available as part of the functor-combo package.

The techniques in this post are not so much new as they are ones that have recently been sinking in for me. See Generalizing Generalized Tries, as well as Generic programming with fixed points for mutually recursive datatypes.


  • 2011-01-28: Fixed small typo: “b^^a^^” ⟼ “ba
  • 2010-09-10: Corrected Const definition to use newtype instead of data.
  • 2010-09-10: Added missing Unit type definition (as Const ()).

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Non-strict memoization

I’ve written a few posts about functional memoization. In one of them, Luke Palmer commented that the memoization methods are correct only for strict functions, which I had not noticed before. In this note, I correct this flaw, extending correct memoization to non-strict functions as well. The semantic notion of least upper bound (which can be built of unambiguous choice) plays a crucial role.


  • 2010-07-13: Fixed the non-strict memoization example to use an argument of undefined (⊥) as intended.
  • 2010-07-23: Changed spelling from “nonstrict” to the much more popular “non-strict”.
  • 2011-02-16: Fixed minor typo. (“constraint on result” → “constraint on the result type”)

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