Matthias is known for many contributions to the field of Programming Languages. He received the SIGPLAN Programming Languages Achievement Award in 2012, the citation for which states:

He introduced evaluation contexts as a notation for specifying operational semantics, and progress-and-preservation proofs of type safety, both of which are used in scores of research papers each year, often without citation. His other contributions include small-step operational semantics for control and state, A-normal form, delimited continuations, mixin classes and mixin modules, a fully-abstract semantics for Sequential PCF, web programming techniques, higher-order contracts with blame, and static typing for dynamic languages.

Absent from this list, perhaps because it wasn’t brought back into the collective memory until 2013, is a very early presentation of the idea of using effects and handlers for extensible language design.

However, perhaps most prominent among Matthias’s contributions to our field is a long series of projects on teaching introductory programming, from TeachScheme! through Program By Design to a latest incarnation in the form of the How to Design Programs textbook (“HtDP”), co-authored with Robby Findler, Matthew Flatt, and Shriram Krishnamurthi, and now in its second edition. The HtDP book is my focus in this post; I have access only the First Edition, but the text of the Second Edition is online. I make no apologies for using Haskell syntax in my examples, but at least that gave Matthias something to shout at!

One key aspect of HtDP is the emphasis on *design recipes* for solving programming tasks. A design recipe is a template for the solution to a problem: a contract for the function, analogous to a type signature (but HtDP takes an untyped approach, so this signature is informal); a statement of purpose; a function header; example inputs and outputs; and a skeleton of the function body. Following the design recipe entails completing the template—filling in a particular contract, etc—then fleshing out the function body from its skeleton, and finally testing the resulting program against the initial examples.

The primary strategy for problem solving in the book is via analysis of the structure of the input. When the input is composite, like a record, the skeleton should name the available fields as likely ingredients of the solution. When the input has “mixed data”, such as a union type, the skeleton should enumerate the alternatives, leading to a case analysis in the solution. When the input is of a recursive type, the skeleton encapsulates *structural recursion*—a case analysis between the base case and the inductive case, the latter case entailing recursive calls.

So, the design recipe for structural recursion looks like this:

PhaseGoalActivityData Analysis and Design to formulate a data definition develop a data definition for mixed data with at least two alternatives; one alternative must not refer to the definition; explicitly identify all self-references in the data definition Contract Purpose and Header to name the function; to specify its classes of input data and its class of output data; to describe its purpose; to formulate a header name the function, the classes of input data, the class of output data, and specify its purpose: ;; name : in1 in2 … –> out

;; to compute … from x1 …

(define (name x1 x2 …) …)

Examples to characterize the input-output relationship via examples create examples of the input-output relationship; make sure there is at least one example per subclass Template to formulate an outline develop a cond-expression with one clause per alternative; add selector expressions to each clause; annotate the body with natural recursions; Test: the self-references in this template and the data definition match! Body to define the function formulate a Scheme expression for each simple cond-line; explain for all other cond-clauses what each natural recursion computes according to the purpose statement Test to discover mistakes (“typos” and logic) apply the function to the inputs of the examples; check that the outputs are as predicted

The motivating example for structural recursion is Insertion Sort: recursing on the tail of a non-empty list and inserting the head into the sorted subresult.

A secondary, more advanced, strategy is to use *generative recursion*, otherwise known as divide-and-conquer. The skeleton in this design recipe incorporates a test for triviality; in the non-trivial cases, it splits the problem into subproblems, recursively solves the subproblems, and assembles the subresults into an overall result. The motivating example for generative recursion is QuickSort (but not the fast in-place version): dividing a non-empty input list into two parts using the head as the pivot, recursively sorting both parts, and concatenating the results with the pivot in the middle.

As far as I can see, no other program structures than structural recursion and generative recursion are considered in HtDP. (Other design recipes are considered, in particular accumulating parameters and imperative features. But these do not determine the gross structure of the resulting program. In fact, I believe that the imperative recipe has been dropped in the Second Edition.)

My thesis is that HtDP has missed an opportunity to reinforce its core message, that **data structure determines program structure**. Specifically, I believe that the next design recipe to consider after structural recursion, in which the shape of the program is determined by the shape of the input, should be *structural corecursion*, in which the shape of the program is determined instead by the shape of the output.

More concretely, a function that generates “mixed output”—whether that is a union type, or simply a boolean—might be defined by case analysis over the output. A function that generates a record might be composed of subprograms that generate each of the fields of that record. A function that generates a recursive data structure from some input data might be defined with a case analysis as to whether the result is trivial, and for non-trivial cases with recursive calls to generate substructures of the result. HtDP should present explicit design recipes to address these possibilities, as it does for program structures determined by the input data.

For an example of mixed output, consider a program that may fail, such as division, guarded so as to return an alternative value in that case:

The program performs a case analysis, and of course the analysis depends on the input data; but the analysis is not determined by the *structure* of the input, only its *value*. So a better explanation of the program structure is that it is determined by the structure of the output data.

For an example of generating composite output, consider the problem of extracting a date, represented as a record:

from a formatted string. The function is naturally structured to match the output type:

For an example of corecursion, consider the problem of “zipping” together two input lists to a list of pairs—taking and to , and for simplicity let’s say pruning the result to the length of the shorter input. One can again solve the problem by case analysis on the input, but the fact that there are two inputs makes that a bit awkward—whether to do case analysis on one list in favour of the other, or to analyse both in parallel. For example, here is the outcome of case analysis on the first input, followed if that is non-empty by a case analysis on the second input:

Case analysis on both inputs would lead to four cases rather than three, which would not be an improvement. One can instead solve the problem by case analysis on the output—and it is arguably more natural to do so, because there is only one output rather than two. When is the output empty? (When either input is empty.) If it isn’t empty, what is the head of the output? (The pair of input heads.) And from what data is the tail of the output recursively constructed? (The pair of input tails.)

And whereas Insertion Sort is a structural recursion over the input list, inserting elements one by one into a sorted intermediate result, *Selection Sort* is a structural corecursion towards the output list, repeatedly extracting the minimum remaining element as the next element of the output: When is the output empty? (When the input is empty.) If the output isn’t empty, what is its head? (The minimum of the input.) And from what data is the tail recursively generated? (The input without this minimum element.)

(here, denotes list with the elements of list removed).

I’m not alone in making this assertion. Norman Ramsey wrote a nice paper On Teaching HtDP; his experience led him to the lesson that

Last, and rarely, you could design a function’s template around the introduction form for the result type. When I teach [HtDP] again, I will make my students aware of this decision point in the construction of a function’s template: should they use elimination forms, function composition, or an introduction form? They should use elimination forms usually, function composition sometimes, and an introduction form rarely.

He also elaborates on test coverage:

Check functional examples to be sure every choice of input is represented. Check functional examples to be sure every choice of

outputis represented. This activity is especially valuable for functions returning Booleans.

(his emphasis). We should pay attention to output data structure as well as to input data structure.

Only once this dual form of program structure has been explored should students be encouraged to move on to generative recursion, because this exploits both structural recursion and structural corecursion. For example, the QuickSort algorithm that is used as the main motivating example is really structured as a corecursion to construct an intermediate tree, followed by structural recursion over that tree to produce the resulting list:

The structure of both functions and is determined by the structure of the intermediate datatype, the first as structural corecursion and the second as structural recursion. A similar explanation applies to any divide-and-conquer algorithm; for example, MergeSort is another divide-and-conquer sorting algorithm, with the same intermediate tree shape, but this time with a simple splitting phase and all the comparisons in the recombining phase:

(Choosing this particular tree type is a bit clunky for Merge Sort, because of the two calls to required in . It would be neater to use non-empty externally labelled binary trees, with elements at the leaves and none at the branches:

Then you define the main function to work only for non-empty lists, and provide a separate case for sorting the empty list. This clunkiness is a learning opportunity: to realise the problem, come up with a fix (no node labels in the tree), rearrange the furniture accordingly, then replay the development and compare the results.)

Having identified the two parts, structural recursion and structural corecursion, they may be studied separately; separation of concerns is a crucial lesson in introductory programming. Moreover, the parts may be put together in different ways. The divide-and-conquer pattern is known in the MPC community as a hylomorphism, an unfold to generate a call tree followed by a fold to consume that tree. As the QuickSort example suggests, the tree can always be deforested—it is a virtual data structure. But the converse pattern, of a fold from some structured input to some intermediate value, followed by an unfold to a different structured output, is also interesting—you can see this as a change of structured representation, so I called it a metamorphism. One simple application is to convert a number from an input base (a sequence of digits in that base), via an intermediate representation (the represented number), to an output base (a different sequence of digits). More interesting applications include encoding and data compression algorithms, such as arithmetic coding.

Although I have used Haskell as a notation, nothing above depends on laziness; it would all work as well in ML or Scheme. It is true that the mathematical structures underlying structural recursion and structural corecursion are prettier when you admit infinite data structures—the final coalgebra of the base functor for lists is the datatype of finite *and infinite* lists, and without admitting the infinite ones some recursive definitions have no solution. But that sophistication is beyond the scope of introductory programming, and it suffices to restrict attention to finite data structures.

HtDP already stipulates a termination argument in the design recipe for generative recursion; the same kind of argument should be required for structural corecursion (and is easy to make for the sorting examples given above). Of course, structural recursion over finite data structures is necessarily terminating. Laziness is unnecessary for co-programming.

HtDP actually draws on a long tradition on relating data structure and program structure, which is a theme dear to my own heart (and indeed, the motto of this blog). Tony Hoare wrote in his Notes on Data Structuring in 1972:

There are certain close analogies between the methods used for structuring data and the methods for structuring a program which processes that data.

Fred Brooks put it pithily in The Mythical Man-Month in 1975:

Show me your flowcharts and conceal your tables, and I shall continue to be mystified. Show me your tables, and I won’t usually need your flowcharts; they’ll be obvious.

which was modernized by Eric Raymond in his 1997 essay The Cathedral and the Bazaar to:

Show me your code and conceal your data structures, and I shall continue to be mystified. Show me your data structures, and I won’t usually need your code; it’ll be obvious.

HtDP credits Jackson Structured Programming as partial inspiration for the design recipe approach. As Jackson wrote, also in 1975:

The central theme of this book has been the relationship between data and program structures. The data provides a model of the problem environment, and by basing our program structures on data structures we ensure that our programs will be intelligible and easy to maintain.

and

The structure of a program must be based on the structures of

allof the data it processes.

(my emphasis). In a retrospective lecture in 2001, he clarified:

program structure should be dictated by the structure of its input and output data streams

—the JSP approach was designed for processing sequential streams of input records to similar streams of output records, and the essence of the approach is to identify the structures of each of the data files (input and output) in terms of sequences, selections, and iterations (that is, as regular languages), to refine them all to a common structure that matches all of them simultaneously, and to use that common data structure as the program structure. So even way back in 1975 it was clear that we need to pay attention to the structure of output data as well as to that of input data.

HtDP apologizes that generative recursion (which it identifies with the field of algorithm design)

is much more of an ad hoc activity than the data-driven design of structurally recursive functions. Indeed, it is almost better to call it inventing an algorithm than designing one. Inventing an algorithm requires a new insight—a “eureka”.

It goes on to suggest that mere programmers cannot generally be expected to have such algorithmic insights:

In practice, new complex algorithms are often developed by mathematicians and mathematical computer scientists; programmers, though, must th[o]roughly understand the underlying ideas so that they can invent the simple algorithms on their own and communicate with scientists about the others.

I say that this defeatism is a consequence of not following through on the core message, that data structure determines program structure. QuickSort and MergeSort are not ad hoc. Admittedly, they do require some insight in order to identify structure that is not present in either the input or the output. But having identified that structure, there is no further mystery, and no ad-hockery required.

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Recall that is defined as follows:

It satisfies the following *universal property*, which gives necessary and sufficient conditions for a function to be expressible as a :

As one consequence of the universal property, we get a *fusion theorem*, which states sufficient conditions for fusing a following function into a :

(on finite lists—infinite lists require an additional strictness condition). Fusion is an equivalence if is surjective. If is not surjective, it’s an equivalence on the range of that fold: the left-hand side implies

only for of the form for some , not for all .

As a second consequence of the universal property (or alternatively, as a consequence of the free theorem of the type of ), we have *map–fold fusion*:

The code golf “” is a concise if opaque way of writing the binary operator pointfree: .

Similarly, is defined as follows:

( is tail recursive, so it is unproductive on an infinite list.)

also enjoys a universal property, although it’s not so well known as that for . Because of the varying accumulating parameter , the universal property entails abstracting that argument on the left in favour of a universal quantification on the right:

(For the proof, see Exercise 16 in my paper with Richard Bird on arithmetic coding.)

From the universal property, it is straightforward to prove a map– fusion theorem:

Note that . There is also a fusion theorem:

This is easy to prove by induction. (I would expect it also to be a consequence of the universal property, but I don’t see how to make that go through.)

Of course, and are closely related. §3.5.1 of Bird and Wadler’s classic text presents a *First Duality Theorem*:

when is associative with neutral element ; a more general *Second Duality Theorem*:

when associates with (that is, ) and (for all , but fixed ); and a *Third Duality Theorem*:

(again, all three only for finite ).

The First Duality Theorem is a specialization of the Second, when (but curiously, also a slight strengthening: apparently all we require is that , not that both equal ; for example, we still have , even though is not the neutral element for ).

“When is a function a fold?” Evidently, if function on lists is injective, then it can be written as a : in that case (at least, classically speaking), has a post-inverse—a function such that —so:

and so, letting , we have

But also evidently, injectivity is not necessary for a function to be a fold; after all, is a fold, but not the least bit injective. Is there a simple condition that is both sufficient and necessary for a function to be a fold? There is! The condition is that lists equivalent under remain equivalent when extended by an element:

We say that is *leftwards*. (The *kernel* of a function is a relation on its domain, namely the set of pairs such that ; this condition is equivalent to for every .) Clearly, leftwardsness is necessary: if and , then

Moreover, leftwardsness is sufficient. For, suppose that is leftwards; then pick a function such that, when is in the range of , we get for some such that (and it doesn’t matter what value returns for outside the range of ), and then define

This is a proper definition of , on account of leftwardsness, in the sense that it doesn’t matter which value we pick for , as long as indeed : any other value that also satisfies entails the same outcome for . Intuitively, it is not necessary to completely invert (as we did in the injective case), provided that preserves enough distinctions. For example, for (for which indeed implies ), we could pick . In particular, is obviously in the range of ; then is chosen to be some such that , and so by construction —in other words, acts as a kind of partial inverse of . So we get:

and therefore where .

A *list homomorphism* is a fold after a map in a monoid. That is, define

provided that and form a monoid. (One may verify that this condition is sufficient for the equations to completely define the function; moreover, they are almost necessary— and should form a monoid on the range of .) The Haskell library defines an analogous method, whose list instance is

—the binary operator and initial value are determined implicitly by the instance rather than being passed explicitly.

Richard Bird’s Introduction to the Theory of Lists states implicitly what might be called the *First Homomorphism Theorem*, that any homomorphism consists of a *reduction* after a map (in fact, a consequence of the free theorem of the type of ):

Explicitly as Lemma 4 (“Specialization”) the same paper states a *Second Homomorphism Theorem*, that any homomorphism can be evaluated right-to-left or left-to-right, among other orders:

I wrote up the Third Homomorphism Theorem, which is the converse of the Second Homomorphism Theorem: any function that can be written both as a and as a is a homomorphism. I learned this theorem from David Skillicorn, but it was conjectured earlier by Richard Bird and proved by Lambert Meertens during a train journey in the Netherlands—as the story goes, Lambert returned from a bathroom break with the idea for the proof. Formally, for given , if there exists such that

then there also exists and associative such that

Recall that is “leftwards” if

and that the leftwards functions are precisely the s. Dually, is *rightwards* if

and (as one can show) the rightwards functions are precisely the s. So the Specialization Theorem states that every homomorphism is both leftwards and rightwards, and the Third Homomorphism Theorem states that every function that is both leftwards and rightwards is necessarily a homomorphism. To prove the latter, suppose that is both leftwards and rightwards; then pick a function such that, for in the range of , we get for some such that , and then define

As before, this is a proper definition of : assuming leftwardsness and rightwardsness, the result does not depend on the representatives chosen for . By construction, again satisfies for any , and so we have:

Moreover, one can show that is associative, and its neutral element (at least on the range of ).

For example, sorting a list can obviously be done as a , using insertion sort; by the Third Duality Theorem, it can therefore also be done as a on the reverse of the list; and because the order of the input is irrelevant, the reverse can be omitted. The Third Homomorphism Theorem then implies that there exists an associative binary operator such that . It also gives a specification of , given a suitable partial inverse of —in this case, suffices, because is idempotent. The only characterization of arising directly from the proof involves repeatedly inserting the elements of one argument into the other, which does not exploit sortedness of the first argument. But from this inefficient characterization, one may derive the more efficient implementation that merges two sorted sequences, and hence obtain merge sort overall.

Here is another relationship between the directed folds ( and ) and list homomorphisms—any directed fold can be implemented as a homomorphism, followed by a final function application to a starting value:

Roughly speaking, this is because application and composition are related:

(here, is right-associating, loose-binding function application). To be more precise:

This result has many applications. For example, it’s the essence of parallel recognizers for regular languages. Language recognition looks at first like an inherently sequential process; a recognizer over an alphabet can be represented as a finite state machine over states , with a state transition function of type , and such a function is clearly not associative. But by mapping this function over the sequence of symbols, we get a sequence of functions, which should be subject to composition. Composition is of course associative, so can be computed in parallel, in steps on processors. Better yet, each such function can be represented as an array (since is finite), and the composition of any number of such functions takes a fixed amount of space to represent, and a fixed amount of time to apply, so the steps take time.

Similarly for carry-lookahead addition circuits. Binary addition of two -bit numbers with an initial carry-in proceeds from right to left, adding bit by bit and taking care of carries, producing an -bit result and a carry-out; this too appears inherently sequential. But there aren’t many options for the pair of input bits at a particular position: two s will always *generate* an outgoing carry, two s will always *kill* an incoming carry, and a and in either order will *propagate* an incoming carry to an outgoing one. Whether there is a carry-in at a particular bit position is computed by a of the bits to the right of that position, zipped together, starting from the initial carry-in, using the binary operator defined by

Again, applying to a bit-pair makes a bit-to-bit function; these functions are to be composed, and function composition is associative. Better, such functions have small finite representations as arrays (indeed, we need a domain of only three elements, ; and are both left zeroes of composition, and is neutral). Better still, we can compute the carry-in at all positions using a , which for an associative binary operator can also be performed in parallel in steps on processors.

The Trick seems to be related to Cayley’s Theorem, on monoids rather than groups: any monoid is equivalent to a monoid of endomorphisms. That is, corresponding to every monoid , with a set, and an associative binary operator with neutral element , there is another monoid on the carrier of endomorphisms of the form ; the mappings and are both monoid homomorphisms, and are each other’s inverse. (Cayley’s Theorem is what happens to the Yoneda Embedding when specialized to the one-object category representing a monoid.) So any list homomorphism corresponds to a list homomorphism with endomorphisms as the carrier, followed by a single final function application:

Note that takes a list element the endomorphism . The change of representation from to is the same as in The Trick, and is also what underlies Hughes’s novel representation of lists; but I can’t quite put my finger on what these both have to do with Cayley and Yoneda.

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Here is a simplified version of arithmetic coding; compared to the previous posts, we use a fixed rather than adaptive model, and rather than ing some arbitrary element of the final interval, we pick explicitly the lower bound of the interval (by , when it is represented as a pair). We suppose a function

that returns a positive count for every symbol, representing predicted frequencies of occurrence, from which it is straightforward to derive functions

that work on a fixed global model, satisfying the central property

For example, an alphabet of three symbols `a’, `b’, `c’, with counts 2, 3, and 5 respectively, could be encoded by the intervals . We have operations on intervals:

that satisfy

Then we have:

For example, with the alphabet and intervals above, the input text “abc” gets encoded symbol by symbol, from left to right (because of the ), to the narrowing sequence of intervals from which we select the lower bound of the final interval. (This version doesn’t stream well, because of the decision to pick the lower bound of the final interval as the result. But it will suffice for our purposes.)

Note now that the symbol encoding can be “fissioned” out of , using map fusion for backwards; moreover, is associative (and its neutral element), so we can replace the with a ; and finally, now using fusion forwards, we can fuse the with the . That is,

and we can fuse the back into the ; so we have , where

Now encoding is a simple and decoding a simple . This means that we can prove that decoding inverts encoding, using the “Unfoldr–Foldr Theorem” stated in the Haskell documentation for (in fact, the only property of stated there!). The theorem states that if

for all and , then

for all finite lists . Actually, that’s too strong for us here, because our decoding always generates an infinite list; a weaker variation (hereby called the “Unfoldr–Foldr Theorem With Junk”) states that if

for all and , then

for all finite lists —that is, the inverts the except for appending some junk to the output. It’s a nice exercise to prove this weaker theorem, by induction.

We check the condition:

Now, we hope to recover the first symbol, ie we hope that :

Fortunately, it is an invariant of the encoding process that —the result of —is in the unit interval, so indeed . Continuing:

as required. Therefore decoding inverts encoding, apart from the fact that it continues to produce junk having decoded all the input; so

for all finite .

Whereas AC encodes longer and longer messages as more and more precise fractions, ANS encodes them as larger and larger integers. Given the function on symbols, we can easily derive definitions

such that gives the cumulative counts of all symbols preceding (say, in alphabetical order), the total of all symbol counts, and looks up an integer :

(for example, we might tabulate the function using a scan).

We encode a text as an integer , containing bits of information (all logs in this blog are base 2). The next symbol to encode has probability , and so requires an additional bits; in total, that’s bits. So entropy considerations tell us that, roughly speaking, to incorporate symbol into state we want to map to . Of course, in order to decode, we need to be able to invert this transformation, to extract and from ; this suggests that we should do the division by first:

so that the multiplication by the known value can be undone first:

How do we reconstruct ? Well, there is enough headroom in to add any non-negative value less than without affecting the division; in particular, we can add to , and then we can use on the remainder:

But we have still lost some information from the lower end of through the division, namely ; so we can’t yet reconstruct . Happily, for any , so there is still precisely enough headroom in to add this lost information too without affecting the , allowing us also to reconstruct :

We therefore define

Using similar reasoning as for AC, it is straightforward to show that a decoding step inverts an encoding step:

Therefore decoding inverts encoding, modulo final junk, by the Unfoldr–Foldr Theorem With Junk:

for all finite . For example, with the same alphabet and symbol weights as before, encoding the text “abc” (now from right to left, because of the ) proceeds through the steps , and the last element is the encoding.

In fact, the inversion property holds whatever value we use to start encoding with (since it isn’t used in the proof); in the next section, we start encoding with a certain lower bound rather than . Moreover, is strictly increasing on states strictly greater than zero, and strictly decreasing; which means that the decoding process can stop when it returns to the lower bound. That is, if we pick some and define

then the stronger version of the Unfoldr–Foldr Theorem holds, and we have

for all finite , without junk.

The previous versions all used arbitrary-precision arithmetic, which is expensive. We now change the approach slightly to use only bounded-precision arithmetic. As usual, there is a trade-off between effectiveness (a bigger bound means more accurate approximations to ideal entropies) and efficiency (a smaller bound generally means faster operations). Fortunately, the reasoning does not depend on much about the actual bounds. We will represent the integer accumulator as a pair which we call a :

such that is a list of digits in a given base , and is an integer with (where we define for the upper bound of the range), under the abstraction relation

where

We call the “window” and the “remainder”. For example, with and , the pair represents ; and indeed, for any there is a unique representation satisfying the invariants. According to Duda, should divide into evenly. On a real implementation, and should be powers of two, such that arithmetic on values up to fits within a single machine word.

The encoding step acts on the window in the accumulator using , which risks making it overflow the range; we therefore renormalize with by shifting digits from the window to the remainder until this overflow would no longer happen, before consuming the symbol.

Some calculation, using the fact that divides , allows us to rewrite the guard in :

For example, again encoding the text “abc”, again from right to left, with and , the accumulator starts with , and goes through the states and on consuming the `c’ then the `b’; directly consuming the `a’ would make the window overflow, so we renormalize to ; then it is safe to consume the `a’, leading to the final state .

Decoding is an unfold using the accumulator as state. We repeatedly output a symbol from the window; this may make the window underflow the range, in which case we renormalize if possible by injecting digits from the remainder (and if this is not possible, because there are no more digits to inject, it means that we have decoded the entire text).

Note that decoding is of course symmetric to encoding; in particular, when encoding we renormalize before consuming a symbol; therefore when decoding we renormalize *after* emitting a symbol. For example, decoding the final encoding starts by computing ; the window value 68 has underflowed, so renormalization consumes the remaining digit 3, leading to the accumulator ; then decoding proceeds to extract the `b’ and `c’ in turn, returning the accumulator to .

We can again prove that decoding inverts encoding, although we need to use yet another variation of the Unfoldr–Foldr Theorem, hereby called the “Unfoldr–Foldr Theorem With Invariant”. We say that predicate is an invariant of and if

The theorem is that if is such an invariant, and the conditions

hold for all and , then

for all finite lists , provided that the initial state satisfies the invariant . In our case, the invariant is that the window is in range (), which is indeed maintained by and . Then it is straightforward to verify that

and

when is in range, making use of a lemma that

when is in range. Therefore decoding inverts encoding:

for all finite .

The version of encoding in the previous section yields a , that is, a pair consisting of an integer window and a digit sequence remainder. It would be more conventional for encoding to take a sequence of symbols to a sequence of digits alone, and decoding to take the sequence of digits back to a sequence of symbols. For encoding, we have to flush the remaining digits out of the window at the end of the process, reducing the window to zero:

For example, . Then we can define

Correspondingly, decoding should start by populating an initially-zero window from the sequence of digits:

For example, . Then we can define

One can show that

provided that is initially in range, and therefore again decoding inverts encoding:

for all finite .

Note that this is not just a data refinement—it is not the case that yields the digits of the integer computed by . For example, , whereas . We have really changed the effectiveness of the encoding in return for the increased efficiency.

We would now like to stream encoding and decoding as metamorphisms: encoding should start delivering digits before consuming all the symbols, and conversely decoding should start delivering symbols before consuming all the digits.

For encoding, we have

with a fold; can we turn into an unfold? Yes, we can! The remainder in the accumulator should act as a queue of digits: digits get enqueued at the most significant end, so we need to dequeue them from the least significant end. So we define

to “deal out” the elements of a list one by one, starting with the last. Of course, this reverses the list; so we have the slightly awkward

Now we can use *unfold fusion* to fuse and into a single unfold, deriving the coalgebra

by considering each of the three cases, getting where

As for the input part, we have a where we need a . Happily, that is easy to achieve, at the cost of an additional , since:

on finite . So we have , where

It turns out that the streaming condition does not hold for and —although we can stream digits out of the remainder:

we cannot stream the last few digits out of the window:

We have to resort to *flushing streaming*, which starts from an apomorphism

rather than an unfold. Of course, any unfold is trivially an apo, with the trivial flusher that always yields the empty list; but more interestingly, any unfold can be factored into a “cautious” phase (delivering elements only while a predicate holds) followed by a “reckless” phase (discarding the predicate, and delivering the elements anyway)

where

In particular, the streaming condition may hold for the cautious coalgebra when it doesn’t hold for the reckless coalgebra itself. We’ll use the cautious coalgebra

which carefully avoids the problematic case when the remainder is empty. It is now straightforward to verify that the streaming condition holds for and , and therefore , where

and where streams its output. On the downside, this has to read the input text in reverse, and also write the output digit sequence in reverse.

Fortunately, decoding is rather easier to stream. We have

with an unfold; can we turn into a fold? Yes, we can! In fact, we have , where

That is, starting with for the accumulator, digits are injected one by one into the window until this is in range, and thereafter appended to the remainder.

Now we have decoding as an after a , and it is straightforward to verify that the streaming condition holds for and . Therefore on finite inputs, where

and streams. It is perhaps a little more symmetric to move the from the final step of encoding to the initial step of decoding—that is, to have the least significant digit first in the encoded text, rather than the most significant—and then to view both the encoding and the decoding process as reading their inputs from right to left.

Inlining definitions and simplifying, we have ended up with encoding as a flushing stream:

and decoding as an ordinary stream:

Note that the two occurrences of can be taken to mean that encoding and decoding both process their input from last to first. The remainder acts as a queue, with digits being added at one end and removed from the other, so should be represented to make that efficient. But in fact, the remainder can be eliminated altogether, yielding simply the window for the accumulator—for encoding:

and for decoding:

This gives rather simple programs, corresponding to those in Duda’s paper; however, the calculation of this last version from the previous steps currently eludes me.

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Recall that

Encoding and decoding work together. But they work only in batch mode: encoding computes a fraction, and yields nothing until the last step, and so decoding cannot start until encoding has finished. We really want encoding to yield as the encoded text a list of bits representing the fraction, rather than the fraction itself, so that we can stream the encoded text and the decoding process. To this end, we replace by , where

The obvious definitions have yield the shortest binary expansion of any fraction within , and evaluate this binary expansion. However, we don’t do quite this—it turns out to prevent the streaming condition from holding—and instead arrange for to yield the bit sequence that when extended with a 1 yields the shortest expansion of any fraction within (and indeed, the shortest binary expansion necessarily ends with a 1), and compute the value with this 1 appended.

Thus, if then the binary expansion of any fraction within starts with 0; and similarly, if , the binary expansion starts with 1. Otherwise, the interval straddles ; the shortest binary expansion within is it the expansion of , so we yield the empty bit sequence.

Note that is a hylomorphism, so we have

Moreover, it is clear that yields a finite bit sequence for any non-empty interval (since the interval doubles in width at each step, and the process stops when it includes ); so this equation serves to uniquely define . In other words, is a *recursive coalgebra*. Then it is a straightforward exercise to prove that ; so although and differ, they are sufficiently similar for our purposes.

Now we redefine encoding to yield a bit sequence rather than a fraction, and decoding correspondingly to consume that bit sequence:

That is, we move the part of from the encoding stage to the decoding stage.

Just like , the new version of encoding consumes all of its input before producing any output, so does not work for encoding infinite inputs, nor for streaming execution even on finite inputs. However, it is nearly in the right form to be a *metamorphism*—a change of representation from lists of symbols to lists of bits. In particular, is associative, and is its unit, so we can replace the with a :

Now that is in the right form, we must check the streaming condition for and . We consider one of the two cases in which is productive, and leave the other as an exercise. When , and assuming , we have:

as required. The last step is a kind of associativity property:

whose proof is left as another exercise. Therefore the streaming condition holds, and we may fuse the with the , defining

which streams the encoding process: the initial bits are output as soon as they are fully determined, even before all the input has been read. Note that and differ, in particular on infinite inputs (the former diverges, whereas the latter does not); but they coincide on finite symbol sequences.

Similarly, we want to be able to stream decoding, so that we don’t have to wait for the entire encoded text to arrive before starting decoding. Recall that we have so far

where is an and a . The first obstacle to streaming is that , which we need to be a instead. We have

Of course, is not associative—it doesn’t even have the right type for that. But we can view each bit in the input as a function on the unit interval: bit~0 is represented by the function that focusses into the lower half of the unit interval, and bit~1 by the function that focusses into the upper half. The fold itself composes a sequence of such functions; and since function composition is associative, this can be written equally well as a or a . Having assembled the individual focussers into one composite function, we finally apply it to . (This is in fact an instance of a general trick for turning a into a , or vice versa.) Thus, we have:

where yields either the lower or the upper half of the unit interval:

In fact, not only may the individual bits be seen as focussing functions and on the unit interval, so too may compositions of such functions:

So any such composition is of the form for some interval , and we may represent it concretely by itself, and retrieve the function via :

So we now have

This is almost in the form of a metamorphism, except for the occurrence of the adapter in between the unfold and the fold. It is not straightforward to fuse that adapter with either the fold or the unfold; fortunately, however, we can split it into the composition

of two parts, where

in such a way that the first part fuses with the fold and the second part fuses with the unfold. For fusing the first half of the adapter with the fold, we just need to carry around the additional value with the interval being focussed:

where

For fusing the second half of the adapter with the unfold, let us check the fusion condition. We have (exercise!):

where the is the functorial action for the base functor of the datatype, applying just to the second component of the optional pair. We therefore define

and have

and therefore

Note that the right-hand side will eventually lead to intervals that exceed the unit interval. When , it follows that ; but the unfolding process keeps widening the interval without bound, so it will necessarily eventually exceed the unit bounds. We return to this point shortly.

We have therefore concluded that

Now we need to check the streaming condition for and . Unfortunately, this is never going to hold: is always productive, so will only take production steps and never consume any input. The problem is that is too aggressive, and we need to use the more cautious flushing version of streaming instead. Informally, the streaming process should be productive from a given state only when the whole of interval maps to the same symbol in model , so that however is focussed by subsequent inputs, that symbol cannot be invalidated.

More formally, note that

where

and

That is, the interval is “safe” for model if it is fully included in the encoding of some symbol ; then all elements of decode to . Then, and only then, we may commit to outputting , because no further input bits could lead to a different first output symbol.

Note now that the interval remains bounded by unit interval during the streaming phase, because of the safety check in , although it will still exceed the unit interval during the flushing phase. However, at this point we can undo the fusion we performed earlier, “fissioning” into again: this manipulates rationals rather than intervals, so there is no problem with intervals getting too wide. We therefore have:

Now let us check the streaming condition for and the more cautious . Suppose that is a productive state, so that holds, that is, all of interval is mapped to the same symbol in , and let

so that . Consuming the next input leads to state . This too is a productive state, because for any , and so the whole of the focussed interval is also mapped to the same symbol in the model. In particular, the midpoint of is within interval , and so the first symbol produced from the state after consumption coincides with the symbol produced from the state before consumption. That is,

as required. We can therefore rewrite decoding as a flushing stream computation:

That is, initial symbols are output as soon as they are completely determined, even before all the input bits have been read. This agrees with on finite bit sequences.

We will leave arithmetic coding at this point. There is actually still quite a bit more arithmetic required—in particular, for competitive performance it is important to use only *fixed-precision* arithmetic, restricting attention to rationals within the unit interval with denominator for some fixed~. In order to be able to multiply two numerators using 32-bit integer arithmetic without the risk of overflow, we can have at most . Interval narrowing now needs to be *approximate*, rounding down both endpoints to integer multiples of . Care needs to be taken so that this rounding never makes the two endpoints of an interval coincide. Still, encoding can be written as an instance of . Decoding appears to be more difficult: the approximate arithmetic means that we no longer have interval widening as an exact inverse of narrowing, so the approach above no longer works. Instead, our 2002 lecture notes introduce a “destreaming” operator that simulates and inverts streaming: the decoder works in sympathy with the encoder, performing essentially the same interval arithmetic but doing the opposite conversions. Perhaps I will return to complete that story some time…

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The basic idea behind arithmetic coding is essentially to encode an input text as a subinterval of the unit interval, based on a *model* of the text symbols that assigns them to a partition of the unit interval into non-empty subintervals. For the purposes of this post, we will deal mostly with half-open intervals, so that the interval contains values such that , where are rationals.

For example, with just two symbols “a” and “b”, and a static model partitioning the unit interval into for “a” and for “b”, the symbols in the input text “aba” successively narrow the unit interval to , and the latter interval is the encoding of the whole input. And in fact, it suffices to pick any single value in this final interval, as long as there is some other way to determine the end of the encoded text (such as the length, or a special end-of-text symbol).

We introduce the following basic definitions for intervals:

We’ll write “” for , and “” for .

A crucial operation on intervals is *narrowing* of one interval by another, where is to as is to the unit interval:

We’ll write “” for . Thus, is “proportionately of the way between and “, and we have

Conversely, we can *widen* one interval by another:

We’ll write “” for . Note that is inverse to , in the sense

and consequently widening is inverse to narrowing:

We work with inputs consisting of sequences of symbols, which might be characters or some higher-level tokens:

The type then must provide the following operations:

- a way to look up a symbol, obtaining the corresponding interval:

- conversely, a way to decode a value, retrieving a symbol:

- an initial model:

- a means to
*adapt*the model on seeing a new symbol:

The central property is that encoding and decoding are inverses, in the following sense:

There are no requirements on and , beyond the latter being a total function.

For example, we might support adaptive coding via a model that counts the occurrences seen so far of each of the symbols, represented as a histogram:

This naive implementation works well enough for small alphabets. One might maintain the histogram in decreasing order of counts, so that the most likely symbols are at the front and are therefore found quickest. For larger alphabets, it is better to maintain the histogram as a binary search tree, ordered alphabetically by symbol, and caching the total counts of every subtree.

Now encoding is straightforward to define. The function takes an initial model and a list of symbols, and returns the list of intervals obtained by looking up each symbol in turn, adapting the model at each step:

That is,

We then narrow the unit interval by each of these subintervals, and pick a single value from the resulting interval:

All we require of is that ; then yields a fraction in the unit interval. For example, we might set , where

So much for encoding; how do we retrieve the input text? In fact, we can retrieve the first symbol simply by using . Expanding the encoding of a non-empty text, we have:

The proof obligation, left as an exercise, is to show that

which holds when is of the form for some .

Now

and indeed, encoding yields a fraction in the unit interval, so this recovers the first symbol correctly. This is the foothold that allows the decoding process to make progress; having obtained the first symbol using , it can adapt the model in precisely the same way that the encoding process does, then retrieve the second symbol using that adapted model, and so on. The only slightly tricky part is that when decoding an initial value , having obtained the first symbol , decoding should continue on some modified value ; what should the modification be? It turns out that the right thing to do is to scale by the interval associated in the model with symbol , since scaling is the inverse operation to the s that take place during encoding. That is, we define:

(Of course, , by the inverse requirement on models, and so the new scaled value is again within the unit interval.)

Note that decoding yields an infinite list of symbols; the function is always productive. Nevertheless, that infinite list starts with the encoded text, as we shall now verify. Define the round-trip function

Then we have:

From this it follows that indeed the round-trip recovers the initial text, in the sense that yields an infinite sequence that starts with ; in fact,

yielding the original input followed by some junk, the latter obtained by decoding the fraction (the encoding of ) from the final model that results from adapting the initial model to each symbol in in turn. To actually retrieve the input text with no junk suffix, one could transmit the length separately (although that doesn’t sit well with streaming), or append a distinguished end-of-text symbol.

So far we have an encoder and a decoder, and a proof that the decoder successfully decodes the encoded text. In the next post, we’ll see how to reimplement both as streaming processes.

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So for example, we can convert an infinite fraction in base 3 to one in base 7 with

where

In this post, we’ll see another number conversion problem, which will deliver the digits of . For more details, see my paper—although the presentation here is simpler now.

Leibniz showed that

From this, using Euler’s convergence-accelerating transformation, one may derive

or equivalently

This can be seen as the number in a funny mixed-radix base , just as the usual decimal expansion

is represented by the number in the fixed-radix base . Computing the decimal digits of is then a matter of conversion from the mixed-radix base to the fixed-radix base.

Let’s remind ourselves of how it should work, using a simpler example: conversion from one fixed base to another. We are given an infinite-precision fraction in the unit interval

in base , in which for each digit . We are to convert it to a similar representation

in base , in which for each output digit . The streaming process maintains a state , a pair of rationals; the invariant is that after consuming input digits and producing output digits, we have

so that represents a linear function that should be applied to the value represented by the remaining input.

We can initialize the process with . At each step, we first try to produce another output digit. The remaining input digits represent a value in the unit interval; so if and have the same integer part, then that must be the next output digit, whatever the remaining input digits are. Let be that integer. Now we need to find such that

for any remainder ; then we can increment and set to and the invariant is maintained. A little algebra shows that we should take and .

If and have different integer parts, we cannot yet tell what the next output digit should be, so we must consume the next input digit instead. Now we need to find such that

for any remainder ; then we can increment and set to and the invariant is again maintained. Again, algebraic manipulation leads us to and .

For example, , and the conversion starts as follows:

That is, the initial state is . This state does not yet determine the first output digit, so we consume the first input digit 0 to yield the next state . This state still does not determine the first output, and nor will the next; so we consume the next two input digits 2 and 0, yielding state . This state does determine the next digit: and both start with a 1 in base 7. So we can produce a 1 as the first output digit, yielding state . And so on.

The process tends to converge. Each production step widens the non-empty window by a factor of , so it will eventually contain multiple integers; therefore we cannot produce indefinitely. Each consumption step narrows the window by a factor of , so it will tend towards eventually producing the next output digit. However, this doesn’t always work. For example, consider converting to base 3:

The first output digit is never determined: if the first non-3 in the input is less than 3, the value is less than a third, and the first output digit should be a 0; if the first non-3 is greater than 3, then the value is definitely greater than a third, and it is safe to produce a 1 as the first output digit; but because the input is all 3s, we never get to make this decision. This problem will happen whenever the value being represented has a finite representation in the output base.

Let’s return now to computing the digits of . We have the input

which we want to convert to decimal. The streaming process maintains a pair of rationals—but this time representing the linear function , since this time our expression starts with a sum rather than a product. The invariant is similar: after consuming input digits and producing output digits, we have

Note that the output base is fixed at 10; but more importantly, the input *digits* are all fixed at 2, and it is the input *base* that varies from digit to digit.

We can initialize the process with . At each step, we first try to produce an output digit. What value might the remaining input

represent? Each of the bases is at least , so it is clear that , where

which has unique solution . Similarly, each of the bases is less than , so it is clear that , where

which has unique solution . So we consider the bounds and ; if these have the same integer part , then that is the next output digit. Now we need to find such that

for any remainder , so we pick and . Then we can increment and set to , and the invariant is maintained.

If the two bounds have different integer parts, we must consume the next input digit instead. Now we need to find such that

for all , so we pick and . Then we can increment and set to , and again the invariant is maintained.

The conversion starts as follows:

Happily, non-termination ceases to be a problem: the value being represented does not have a finite representation in the output base, being irrational.

We can plug these definitions straight into the function above:

where

and

(The s make rational numbers in Haskell, and force the ambiguous fractional type to be rather than .)

In fact, this program can be considerably simplified, by inlining the definitions. In particular, the input digits are all 2, so we need not supply them. Moreover, the component of the state is never used, because we treat each output digit in the same way (in contrast to the input digits); so that may be eliminated. Finally, we can eliminate some of the numeric coercions if we represent the component as a rational in the first place:

Then we have

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I don’t think I’ve written about them yet in this series—another story, for another day—but *hylomorphisms* consist of a fold after an unfold. One very simple example is the factorial function: is the product of the predecessors of . The predecessors can be computed with an unfold:

and the product as a fold:

and then factorial is their composition:

Another example is a tree-based sorting algorithm that resembles Hoare’s quicksort: from the input list, grow a binary search tree, as an unfold, and then flatten that tree back to a sorted list, as a fold. This is a divide-and-conquer algorithm; in general, these can be modelled as unfolding a tree of subproblems by repeatedly dividing the problem, then collecting the solution to the original problem by folding together the solutions to subproblems.

This post is about the opposite composition, an unfold after a fold. Some examples:

- to reformat a list of lists to a given length;
- to sort a list;
- to convert a fraction from base to base ;
- to encode a text in binary by “arithmetic coding”.

In each of these cases, the first phase is a fold, which consumes some structured representation of a value into an intermediate unstructured format, and the second phase is an unfold, which generates a new structured representation. Their composition effects a change of representation, so we call them metamorphisms.

Hylomorphisms always *fuse*, and one can deforest the intermediate *virtual data structure*. For example, one need not construct the intermediate list in the factorial function; since each cell gets constructed in the unfold only to be immediately deconstructed in the fold, one can cut to the chase and go straight to the familiar recursive definition. For the base case, we have:

and for non-zero argument , we have:

In contrast, metamorphisms only fuse under certain conditions. However, when they do fuse, they also allow infinite representations to be processed, as we shall see.

Fusion seems to depend on the fold being tail-recursive; that is, a :

For the unfold phase, we will use the usual list unfold:

We define a metamorphism as their composition:

This transforms input of type to output of type : in the first phase, , it consumes all the input into an intermediate value of type ; in the second phase, , it produces all the output.

Under certain conditions, it is possible to fuse these two phases—this time, not in order to eliminate an intermediate data structure (after all, the intermediate type need not be structured), but rather in order to allow some production steps to happen before all the consumption steps are complete.

To that end, we define the function as follows:

This takes the same arguments as . It maintains a current state , and produces an output element when it can; and when it can’t produce, it consumes an input element instead. In more detail, it examines the current state using function , which is like the body of an unfold; this may produce a first element of the result and a new state ; when it yields no element, the next element of the input is consumed using function , which is like the body of a ; and when no input remains either, we are done.

The *streaming condition* for and is that

Consider a state from which the body of the unfold is productive, yielding some . From here we have two choices: we can either produce the output , move to intermediate state , then consume the next input to yield a final state ; or we can consume first to get the intermediate state , and again try to produce. The streaming condition says that this intermediate state will again be productive, and will yield the same output and the same final state . That is, instead of consuming all the inputs first, and then producing all the outputs, it is possible to produce some of the outputs early, without jeopardizing the overall result. Provided that the streaming condition holds for and , then

for all finite lists .

As a simple example, consider the `buffering’ process , where

Note that , so is just a complicated way of writing as a . But the streaming condition holds for and (as you may check), so may be streamed. Operationally, the streaming version of consumes one list from the input list of lists, then peels off and produces its elements one by one; when they have all been delivered, it consumes the next input list, and so on.

The streaming version of is actually rather special, because the production steps can always completely exhaust the intermediate state. In contrast, consider the `regrouping’ example where

from the introduction (here, yields where , with when and otherwise). This transforms an input list of lists into an output list of lists, where each output `chunk’ except perhaps the last has length —if the content doesn’t divide up evenly, then the last chunk is short. One might hope to be able to stream , but it doesn’t quite work with the formulation so far. The problem is that is too aggressive, and will produce short chunks when there is still some input to consume. (Indeed, the streaming condition does not hold for and —why not?) One might try the more cautious producer :

But this never produces a short chunk, and so if the content doesn’t divide up evenly then the last few elements will not be extracted from the intermediate state and will be lost.

We need to combine these two producers somehow: the streaming process should behave cautiously while there is still remaining input, which might influence the next output; but it should then switch to a more aggressive strategy once the input is finished, in order to flush out the contents of the intermediate state. To achieve this, we define a more general *flushing stream* operator:

This takes an additional argument ; when the cautious producer is unproductive, and there is no remaining input to consume, it uses to flush out the remaining output elements from the state. Clearly, specializing to retrieves the original operator.

The corresponding metamorphism uses an *apomorphism* in place of the unfold. Define

Then behaves like , except that if and when stops being productive it finishes up by applying to the final state. Similarly, define flushing metamorphisms:

Then we have

for all finite lists if the streaming condition holds for and . In particular,

on finite inputs : the streaming condition does hold for and the more cautious , and once the input has been exhausted, the process can switch to the more aggressive .

The main advantage of streaming is that it can allow the change-of-representation process also to work on infinite inputs. With the plain metamorphism, this is not possible: the will yield no result on an infinite input, and so the will never get started, but the may be able to produce some outputs before having consumed all the inputs. For example, the streaming version of also works for infinite lists, providing that the input does not end with an infinite tail of empty lists. And of course, if the input never runs out, then there is no need ever to switch to the more aggressive flushing phase.

As a more interesting example, consider converting a fraction from base 3 to base 7:

We assume that the input digits are all either 0, 1 or 2, so that the number being represented is in the unit interval.

The fold in is of the wrong kind; but we have also

Here, the intermediate state can be seen as a defunctionalized representation of the function , and applies this function to :

Now there is an extra function between the and the ; but that’s no obstacle, because it fuses with the :

However, the streaming condition does not hold for and . For example,

That is, , but , so it is premature to produce the first digit 2 in base 7 having consumed only the first digit 1 in base 3. The producer is too aggressive; it should be more cautious while input remains that might invalidate a produced digit.

Fortunately, on the assumption that the input digits are all 0, 1, or 2, the unconsumed input—a tail of the original input—again represents a number in the unit interval; so from the state the range of possible unproduced outputs represents a number between and . If these both start with the same digit in base 7, then (and only then) is it safe to produce that digit. So we define

and we have

Now, the streaming condition holds for and (as you may check), and therefore

on finite digit sequences in base 3. Moreover, the streaming program works also on *infinite* digit sequences, where the original does not.

(Actually, the only way this could possibly produce a finite output in base 7 would be for the input to be all zeroes. Why? If we are happy to rule out this case, we could consider only the case of taking infinite input to infinite output, and not have to worry about reaching the end of the input or flushing the state.)

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What is the correct way to write breadth first traversal of a ?

He’s thinking of “traversal” in the sense of the class, and gave a concrete declaration of rose trees:

It’s an excellent question.

First, let’s think about breadth-first enumeration of the elements of a tree. This isn’t compositional (a fold); but the related “level-order enumeration”, which gives a list of lists of elements, one list per level, is compositional:

Here, is “long zip with”; it’s similar to , but returns a list as long as its longer argument:

(It’s a nice exercise to define a notion of folds for , and to write as a fold.)

Given , breadth-first enumeration is obtained by concatenation:

Incidentally, this allows trees to be foldable, breadth-first:

Level-order enumeration is invertible, in the sense that you can reconstruct the tree given its shape and its level-order enumeration.

One way to define this is to pass the level-order enumeration around the tree, snipping bits off it as you go. Here is a mutually recursive pair of functions to relabel a tree with a given list of lists, returning also the unused bits of the lists of lists.

Assuming that the given list of lists is “big enough”—ie each list has enough elements for that level of the tree—then the result is well-defined. Then is determined by the equivalence

Here, the of a tree is obtained by discarding its elements:

In particular, if the given list of lists is the level-order of the tree, and so is exactly the right size, then will have no remaining elements, consisting entirely of empty levels:

So we can take a tree apart into its shape and contents, and reconstruct the tree from such data.

This lets us traverse a tree in breadth-first order, by performing the traversal just on the contents. We separate the tree into shape and contents, perform a list-based traversal, and reconstruct the tree.

This trick of traversal by factoring into shape and contents is explored in my paper Understanding Idiomatic Traversals Backwards and Forwards from Haskell 2013.

We’ve seen that level-order enumeration is invertible in a certain sense, and that this means that we perform traversal by factoring into shape and contents then traversing the contents independently of the shape. But the contents we’ve chosen is the level-order enumeration, which is a list of lists. Normally, one thinks of the contents of a data structure as simply being a list, ie obtained by breadth-first enumeration rather than by level-order enumeration. Can we do relabelling from the breadth-first enumeration too? Yes, we can!

There’s a very clever cyclic program for breadth-first relabelling of a tree given only a list, not a list of lists; in particular, breadth-first relabelling a tree with its own breadth-first enumeration gives back the tree you first thought of. In fact, the relabelling function is precisely the same as before! The trick comes in constructing the necessary list of lists:

Note that variable is defined cyclically; informally, the output leftovers on one level also form the input elements to be used for relabelling all the lower levels. Given this definition, we have

for any . This program is essentially due to Geraint Jones, and is derived in an unpublished paper Linear-Time Breadth-First Tree Algorithms: An Exercise in the Arithmetic of Folds and Zips that we wrote together in 1993.

We can use this instead in the definition of breadth-first traversal:

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The idea goes back to the adjunction between extension and intension in set theory—you can define a set by its *extension*, that is by listing its elements:

or by its *intension*, that is by characterizing those elements:

Expressions in the latter form are called *set comprehensions*. They inspired a programming notation in the SETL language from NYU, and have become widely known through list comprehensions in languages like Haskell. The structure needed of sets or of lists to make this work is roughly that of a *monad*, and Phil Wadler showed how to generalize comprehensions to arbitrary monads, which led to the “**do**” notation in Haskell. Around the same time, Phil Trinder showed that comprehensions make a convenient database query language. The comprehension notation has been extended to cover other important aspects of database queries, particularly aggregation and grouping. Monads and aggregations have very nice algebraic structure, which leads to a useful body of laws to support database query optimization.

Just as a warm-up, here is a reminder about Haskell’s list comprehensions.

This (rather concocted) example yields the list of all values of the expression as is drawn from and from and such that is divisible by , namely .

To the left of the vertical bar is the *term* (an expression). To the right is a comma-separated sequence of *qualifiers*, each of which is either a *generator* (of the form , with a variable and a list expression ) or a *filter* (a boolean expression). The scope of a variable introduced by a generator extends to all subsequent generators and to the term. Note that, in contrast to the mathematical inspiration, bound variables need to be generated from some existing list.

The semantics of list comprehensions is defined by translation; see for example Phil Wadler’s Chapter 7 of The Implementation of Functional Programming Languages. It can be expressed equationally as follows:

(Here, denotes the empty sequence of qualifiers. It’s not allowed in Haskell, but it is helpful in simplifying the translation.)

Applying this translation to the example at the start of the section gives

More generally, a generator may match against a pattern rather than just a variable. In that case, it may bind multiple (or indeed no) variables at once; moreover, the match may fail, in which case it is discarded. This is handled by modifying the translation for generators to use a function defined by pattern-matching, rather than a straight lambda-abstraction:

or, more perspicuously,

It is clear from the above translation that the necessary ingredients for list comprehensions are , singletons, , and the empty list. The first three are the operations arising from lists as a functor and a monad, which suggests that the same translation might be applicable to other monads too. But the fourth ingredient, the empty list, does not come from the functor and monad structures; that requires an extra assumption:

Then the translation for list comprehensions can be generalized to other monads:

(so ). The actual monad to be used is implicit; if we want to be explicit, we could use a subscript, as in ““.

This translation is different from the one used in the Haskell language specification, which to my mind is a little awkward: the empty list crops up in two different ways in the translation of list comprehensions—for filters, and for generators with patterns—and these are generalized in two different ways to other monads (to the method of the class in the first case, and the method of the class in the second). I think it is neater to have a monad subclass with a single method subsuming both these operators. Of course, this does mean that the translation forces a monad comprehension with filters or possibly failing generators to be interpreted in a monad in the subclass rather than just —the type class constraints that are generated depend on the features used in the comprehension. (Perhaps this translation was tried in earlier versions of the language specification, and found wanting?)

Taking this approach gives basically the monad comprehension notation from Wadler’s Comprehending Monads paper; it loosely corresponds to Haskell’s **do** notation, except that the term is to the left of a vertical bar rather than at the end, and that filters are just boolean expressions rather than introduced using .

We might impose the law that is a “left” zero of composition, in the sense

or, in terms of comprehensions,

Informally, this means that any failing steps of the computation cleanly cut off subsequent branches. Conversely, we do not require that is a “right” zero too:

This would have the consequence that a failing step also cleanly erases any effects from earlier parts of the computation, which is too strong a requirement for many monads—particularly those of the “launch missiles now” variety. (The names “left-” and “right zero” make more sense when the equations are expressed in terms of the usual Haskell bind operator , which is a kind of sequential composition.)

One more ingredient is needed in order to characterize monads that correspond to “collection classes” such as sets and lists, and that is an analogue of set union or list append. It’s not difficult to see that this is inexpressible in terms of the operations introduced so far: given only collections of at most one element, any comprehension using generators of the form will only yield another such collection, whereas the union of two one-element collections will in general have two elements.

To allow any finite collection to be expressed, it suffices to introduce a binary union operator :

We require composition to distribute over union, in the following sense:

or, in terms of comprehensions,

For the remainder of this post, we will assume a monad in both and . Moreover, we will assume that is the unit of , and is both a left- and a right zero of composition. To stress the additional constraints, we will write “” for “” from now on. The intention is that such monads exactly capture collection classes; Phil Wadler has called these structures *ringads*. (He seems to have done so in an unpublished note *Notes on Monads and Ringads* from 1990, which is cited by some papers from the early 1990s. But Phil no longer has a copy of this note, and it’s not online anywhere… I’d love to see a copy, if anyone has one!)

(There are no additional methods; the class is the intersection of the two parent classes and , with the union of the two interfaces, together with the laws above.) I used roughly the same construction already in the post on Horner’s Rule.

As well as (finite) sets and lists, ringad instances include (finite) bags and a funny kind of binary tree (externally labelled, possibly empty, in which the empty tree is a unit of the binary tree constructor). These are all members of the so-called Boom Hierarchy of types—a name coined by Richard Bird, for an idea due to Hendrik Boom, who by happy coincidence is named for one of these structures in his native language. All members of the Boom Hierarchy are generated from the empty, singleton, and union operators, the difference being whether union is associative, commutative, and idempotent. Another ringad instance, but not a member of the Boom Hierarchy, is the type of probability distributions—either normalized, with a weight-indexed family of union operators, or unnormalized, with an additional scaling operator.

The well-behaved operations over monadic values are called the *algebras* for that monad—functions such that and . In particular, is itself a monad algebra. When the monad is also a ringad, necessarily distributes also over —there is a binary operator such that (exercise!). Without loss of generality, we write for ; these are the “reductions” of the Bird–Meertens Formalism. In that case, is a ringad algebra.

The algebras for a ringad amount to aggregation functions for a collection: the sum of a bag of integers, the maximum of a set of naturals, and so on. We could extend the comprehension notation to encompass aggregations too, for example by adding an optional annotation, writing say ““; although this doesn’t add much, because we could just have written “” instead. We could generalize from reductions to collection homomorphisms ; but this doesn’t add much either, because the map is easily combined with the comprehension—it’s easy to show the “map over comprehension” property

Leonidas Fegaras and David Maier develop a monoid comprehension calculus around such aggregations; but I think their name is inappropriate, because nothing forces the binary aggregating operator to be associative.

Note that, for to be well-defined, must satisfy all the laws that does— must be associative if is associative, and so on. It is not hard to show, for instance, that there is no on sets of numbers for which ; such an would have to be idempotent, which is inconsistent with its relationship with . (So, although denotes the sum of the squares of the odd elements of bag , the expression (with now a set) is not defined, because is not idempotent.) In particular, must be the unit of , which we write .

We can derive translation rules for aggregations from the definition

For empty aggregations, we have:

For filters, we have:

For generators, we have:

And for sequences of qualifiers, we have:

Putting all this together, we have:

We have seen that comprehensions can be interpreted in an arbitrary ringad; for example, denotes (the set of) the squares of the odd elements of (the set) , whereas denotes the bag of such elements, with a bag. Can we make sense of “heterogeneous comprehensions”, involving several different ringads?

Let’s introduced the notion of a *ringad morphism*, extending the familiar analogue on monads. For monads and , a monad morphism is a natural transformation —that is, a family of arrows, coherent in the sense that for —that also preserves the monad structure:

A ringad morphism for ringads is a monad morphism that also respects the ringad structure:

Then a ringad morphism behaves nicely with respect to ringad comprehensions—a comprehension interpreted in ringad , using existing collections of type , with the result transformed via a ringad morphism to ringad , is equivalent to the comprehension interpreted in ringad in the first place, but with the initial collections transformed to type . Informally, there will be no surprises arising from when ringad coercions take place, because the results are the same whenever this happens. This property is straightforward to show by induction over the structure of the comprehension. For the empty comprehension, we have:

For filters, we have:

For generators:

And for sequences of qualifiers:

For example, if is the obvious ringad morphism from bags to sets, discarding information about the multiplicity of repeated elements, and a bag of numbers, then

and both yield the set of squares of the odd members of . As a notational convenience, we might elide use of the ringad morphism when it is “obvious from context”—we might write just even when is a bag, relying on the “obvious” morphism . This would allow us to write, for example,

(writing for the extension of a bag), instead of the more pedantic

There is a forgetful function from any poorer member of the Boom hierarchy to a richer one, flattening some distinctions by imposing additional laws—for example, from bags to sets, flattening distinctions concerning multiplicity—and I would class these forgetful functions as “obvious” morphisms. On the other hand, any morphisms in the opposite direction—such as sorting, from bags to lists, and one-of-each, from sets to bags—are not “obvious”, and so should not be elided; and similarly, I’m not sure that I could justify as “obvious” any morphisms involving non-members of the Boom Hierarchy, such as probability distributions.

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*Accumulations* or *scans* distribute information contained in a data structure across that data structure in a given direction. The paradigmatic example is computing the running totals of a list of numbers, which can be thought of as distributing the numbers rightwards across the list, summing them as you go. In Haskell, this is an instance of the operator:

A special case of this pattern is to distribute the elements of a list rightwards across the list, simply collecting them as you go, rather than summing them. That’s the function, and it too is an instance of :

It’s particularly special, in the sense that it is the most basic ; any other instance can be expressed in terms of it:

This is called the *Scan Lemma* for . Roughly speaking, it states that a replaces every node of a list with a applied to that node’s predecessors. Read from right to left, the scan lemma is an efficiency-improving transformation, eliminating duplicate computations; but note that this only works on expressions where is a , because only then are there duplicate computations to eliminate. It’s an important result, because it relates a clear and simple specification on the right to a more efficient implementation on the left.

However, the left-to-right operators , , and are a little awkward in Haskell, because they go against the grain of the cons-based (ie, right-to-left) structure of lists. I leave as a simple exercise for the reader the task of writing the more natural , , and , and identifying the relationships between them. Conversely, one can view etc as the natural operators for snoc-based lists, which are constructed from nil and snoc rather than from nil and cons.

What would , , , etc look like on different—and in particular, non-linear—datatypes? Let’s consider a simple instance, for homogeneous binary trees; that is, trees with a label at both internal and external nodes.

for which the obvious fold operator is

I’m taking the view that the appropriate generalization is to distribute data “upwards” and “downwards” through such a tree—from the leaves towards the root, and vice versa. This does indeed specialize to the definitions we had on lists when you view them vertically in terms of their “cons” structure: they’re long thin trees, in which every parent has exactly one child. (An alternative view would be to look at distributing data horizontally through a tree, from left to right and vice versa. Perhaps I’ll come back to that another time.)

The upwards direction is the easier one to deal with. An upwards accumulation labels every node of the tree with some function of its *descendants*; moreover, the descendants of a node themselves form a tree, so can be easily represented, and folded. So we can quite straightforwardly define:

where yields the root of a tree:

As with lists, the most basic upwards scan uses the constructors themselves as arguments:

and any other scan can be expressed, albeit less efficiently, in terms of this:

The downwards direction is more difficult, though. A downwards accumulation should label every node with some function of its *ancestors*; but these do not form another tree. For example, in the homogeneous binary tree

the ancestors of the node labelled are the nodes labelled . One could represent those ancestors simply as a list, ; but that rules out the possibility of a downwards accumulation treating left children differently from right children, which is essential in a number of algorithms (such as the parallel prefix and tree drawing algorithms in my thesis). A more faithful rendering is to define a new datatype of *paths* that captures the left and right turns—a kind of non-empty cons list, but with both a “left cons” and a “right cons” constructor:

(I called them “threads” in my thesis.) Then we can capture the data structure representing the ancestors of the node labelled

by the expression . I leave it as an exercise for the more energetic reader to work out a definition for

to compute the tree giving the ancestors of every node, and for a corresponding .

Having seen ad-hoc constructions for a particular kind of binary tree, we should consider what the datatype-generic construction looks like. I discussed datatype-generic upwards accumulations already, in the post on Horner’s Rule; the construction was given in the paper Generic functional programming with types and relations by Richard Bird, Oege de Moor and Paul Hoogendijk. As with homogeneous binary trees, it’s still the case that the generic version of labels every node of a data structure of type with the descendants of that node, and still the case that the descendants form a data structure also of type . However, in general, the datatype does not allow for a label at every node, so we need the *labelled variant* where . Then we can define

where returns the root label of a labelled data structure—by construction, every labelled data structure has a root label—and is the unique arrow to the unit type. Moreover, we get a datatype-generic operator, and a Scan Lemma:

The best part of a decade after my thesis work, inspired by the paper by Richard Bird & co, I set out to try to define datatype-generic versions of downward accumulations too. I wrote a paper about it for MPC 1998, and then came up with a new construction for the journal version of that paper in SCP in 2000. I now think these constructions are rather clunky, and I have a better one; if you don’t care to explore the culs-de-sac, skip this section and the next and go straight to the section on derivatives.

The MPC construction was based around a datatype-generic version of the datatype above, to represent the “ancestors” of a node in an inductive datatype. The tricky bit is that data structures in general are non-linear—a node may have many children—whereas paths are linear—every node has exactly one child, except the last which has none; how can we define a “linear version” of ? Technically, we might say that a functor is linear (actually, “affine” would be a better word) if it distributes over sum.

The construction in the paper assumed that was a sum of products of literals

where each is either , , or some constant type such as or . For example, for leaf-labelled binary trees

the shape functor is , so (there are two variants), (the first variant has a single literal, ) and (the second variant has two literals, and ), and:

Then for each we define a -ary functor , where is the “degree of branching” of variant (ie, the number of s occurring in , which is the number of for which ), in such a way that

and is linear in each argument except perhaps the first. It’s a bit messy explicitly to give a construction for , but roughly speaking,

where is “the next unused ” when , and just otherwise. For example, for leaf-labelled binary trees, we have:

Having defined the linear variant of , we can construct the datatype of paths, as the inductive datatype of shape where

That is, paths are a kind of non-empty cons list. The path ends at some node of the original data structure; so the last element of the path is of type , which records the “local content” of a node (its shape and labels, but without any of its children). Every other element of the path consists of the local content of a node together with an indication of which direction to go next; this amounts to the choice of a variant , followed by the choice of one of identical copies of the local contents of variant , where is the degree of branching of variant . We model this as a base constructor and a family of “cons” constructors for and .

For example, for leaf-labelled binary trees, the “local content” for the last element of the path is either a single label (for tips) or void (for bins), and for the other path elements, there are zero copies of the local content for a tip (because a tip has zero children), and two copies of the void local information for bins (because a bin has two children). Therefore, the path datatype for such trees is

which is isomorphic to the definition that you might have written yourself:

For homogeneous binary trees, the construction gives

which is almost the ad-hoc definition we had two sections ago, except that it distinguishes singleton paths that terminate at an external node from those that terminate at an internal one.

Now, analogous to the function which labels every node with its descendants, we can define a function to label every node with its ancestors, in the form of the path to that node. One definition is as a fold; informally, at each stage we construct a singleton path to the root, and map the appropriate “cons” over the paths to each node in each of the children (see the paper for a concrete definition). This is inefficient, because of the repeated maps; it’s analogous to defining by

A second definition is as an unfold, maintaining as an accumulating parameter of type the “path so far”; this avoids the maps, but it is still quadratic because there are no common subexpressions among the various paths. (This is analogous to an accumulating-parameter definition of :

Even with an accumulating “Hughes list” parameter, it still takes quadratic time.)

The downwards accumulation itself is defined as a path fold mapped over the paths, giving a Scan Lemma for downwards accumulations. With either the fold or the unfold definition of paths, this is still quadratic, again because of the lack of common subexpressions in a result of quadratic size. However, in some circumstances the path fold can be reassociated (analogous to turning a into a ), leading finally to a linear-time computation; see the paper for the details of how.

I was dissatisfied with the “…”s in the MPC construction of datatype-generic paths, but couldn’t see a good way of avoiding them. So in the subsequent SCP version of the paper, I presented an alternative construction of downwards accumulations, which does not go via a definition of paths; instead, it goes directly to the accumulation itself.

As with the efficient version of the MPC construction, it is coinductive, and uses an accumulating parameter to carry in to each node the seed from higher up in the tree; so the downwards accumulation is of type . It is defined as an unfold, with a body of type

The result of applying the body will be constructed from two components, of types and : the first gives the root label of the accumulation and the seeds for processing the children, and the second gives the children themselves.

These two components get combined to make the whole result via a function

This will be partial in general, defined only for pairs of -structures of the same shape.

The second component of is the easier to define; given input , it unpacks the to , and discards the and the (recall that is the labelled variant of , where ).

For the first component, we enforce the constraint that all output labels are dependent only on their ancestors by unpacking the and pruning off the children, giving input . We then suppose as a parameter to the accumulation a function of type to complete the construction of the first component. In order that the two components can be zipped together, we require that is shape-preserving in its second argument:

where is the unique function to the unit type. Then, although the built from these two components depends on the partial function , it will still itself be total.

The SCP construction gets rid of the “…”s in the MPC construction. It is also inherently efficient, in the sense that if the core operation takes constant time then the whole accumulation takes linear time. However, use of the partial function to define a total accumulation is a bit unsatisfactory, taking us outside the domain of sets and total functions. Moreover, there’s now only half an explanation in terms of paths: accumulations in which the label attached to each node depends only on the *list* of its ancestors, and not on the left-to-right ordering of siblings, can be factored into a list function (in fact, a ) mapped over the “paths”, which is now a tree of lists; but accumulations in which left children are treated differently from right children, such as the parallel prefix and tree drawing algorithms mentioned earlier, can not.

After another interlude of about a decade, and with the benefit of new results to exploit, I had a “eureka” moment: the linearization of a shape functor is closely related to the beautiful notion of the *derivative* of a datatype, as promoted by Conor McBride. The crucial observation Conor made is that the “one-hole contexts” of a datatype—that is, for a container datatype, the datatype of data structures with precisely one element missing—can be neatly formalized using an analogue of the rules of differential calculus. The one-hole contexts are precisely what you need to identify which particular child you’re talking about out of a collection of children. (If you’re going to follow along with some coding, I recommend that you also read Conor’s paper Clowns to the left of me, jokers to the right. This gives the more general construction of *dissecting* a datatype, identifying a unique hole, but also allowing the “clowns” to the left of the hole to have a different type from the “jokers” to the right. I think the explanation of the relationship with the differential calculus is much better explained here; the original notion of derivative can be retrieved by specializing the clowns and jokers to the same type.)

The essence of the construction is the notion of a *derivative* of a functor . For our purposes, we want the derivative in the second argument only of a bifunctor; informally, is like , but with precisely one missing. Given such a one-hole context, and an element with which to plug the hole, one can reconstruct the whole structure:

That’s how to consume one-hole contexts; how can we produce them? We could envisage some kind of inverse of , which breaks an -structure into an element and a context; but this requires us to invent a language for specifying which particular element we mean— is not injective, so needs an extra argument. A simpler approach is to provide an operator that annotates every position at once with the one-hole context for that position:

One property of is that it really is an annotation—if you throw away the annotations, you get back what you started with:

A second property relates it to —each of elements in a hole position plugs into its associated one-hole context to yield the same whole structure back again:

(I believe that those two properties completely determine and .)

Incidentally, the derivative of a bifunctor can be elegantly represented as an *associated type synonym* in Haskell, in a type class of bifunctors differentiable in their second argument, along with and :

Conor’s papers show how to define instances of for all polynomial functors —anything made out of constants, projections, sums, and products.

The path to a node in a data structure is simply a list of one-hole contexts—let’s say, innermost context first, although it doesn’t make much difference—but with all the data off the path (that is, the other children) stripped away:

This is a projection of Huet’s zipper, which preserves the off-path children, and records also the subtree in focus at the end of the path:

Since the contexts are listed innermost-first in the path, closing up a zipper to reconstruct a tree is a over the path:

Now, let’s develop the function , which turns a tree into a labelled tree of paths. We will write it with an accumulating parameter, representing the “path so far”:

Given the components of a tree and a path to its root, must construct the corresponding labelled tree of paths. Since and , this amounts to constructing a value of type . For the first component of this pair we will use , the path so far. The second component can be constructed from by identifying all children via , discarding some information with judicious s, consing each one-hole context onto to make a longer path, then making recursive calls on each child:

That is,

Downwards accumulations are then path functions mapped over the result of . However, we restrict ourselves to path functions that are instances of , because only then are there common subexpressions to be shared between a parent and its children (remember that paths are innermost-first, so related nodes share a tail of their ancestors).

Moreover, it is straightforward to fuse the with , to obtain

which takes time linear in the size of the tree, assuming that and take constant time.

Finally, in the case that the function being mapped over the paths is a as well as a , then we can apply the Third Homomorphism Theorem to conclude that it is also an associative fold over lists. From this (I believe) we get a very efficient parallel algorithm for computing the accumulation, taking time logarithmic in the size of the tree—even if the tree has greater than logarithmic depth.

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