2 “Magic” squares

Experience in teaching FP using a relational approach has led the author to a graphical style of expression and reasoning in which a very simple, geometric construct shines: the (semi) commutative square.

In the classroom this is termed the “magic square” (MS), because much of what we do in logic, FP, database modeling, formal semantics, etc fits in some MS geometry. Each magic square has four sides, say R, P, S, Q, which are binary relations, and each square is a comparison between two paths, of two relations each:

(2.1)

In more detail, each side is an arrow, say , declaring a binary relation R that relates objects of types A and B. Its meaning is as usual: given objects ${{a}\;\mathbin\in \;{A}}$ and ${{b}\;\mathbin\in \;{B}}$ , the proposition ${{a}\;{R}\;{b}}$ tells whether or not a and b are related by R. Take for instance relation . Clearly, ${\mathrm{0}\leq \mathrm{1}}$ holds (it is a true proposition) while ${\mathrm{1}\leq \mathrm{0}}$ does not.

Next, we need to say what a path means, say ${{P} \cdot {R}}$ in (2.1): given some ${{c}\;\mathbin\in \;{C}}$ and some ${{b}\;\mathbin\in \;{B}}$ , ${{c}\;({P} \cdot {R})\;{b}}$ holds whenever there is some mediator ${{a}\;\mathbin\in \;{A}}$ such that both ${{c}\;{P}\;{a}}$ and ${{a}\;{R}\;{b}}$ hold. We say that relation ${{P} \cdot {R}}$ is the composition of relations ${{P}}$ and ${{R}}$ . Relational composition is associative.

Finally, let us say what the comparison ${{R}\; \subseteq \;{S}}$ of two relations means, in general. ( ${{R}\; \subseteq \;{S}}$ should be read: “ ${{R}}$ is at most ${{S}}$ ” or “ ${{R}}$ is included in ${{S}}$ ”.) It means that, for all ${{b}\;\mathbin\in \;{B}}$ and ${{a}\;\mathbin\in \;{A}}$ , if ${{b}\;{R}\;{a}}$ holds then ${{b}\;{S}\;{a}}$ holds too. In summary, relation inclusion “hides” a universal quantifier, while relation composition hides an existential quantifier. In symbols, the logic interpretation of (2.1) is:

(2.2)

The following version of (2.2) makes its connection with (2.1) more explicit:

Comparisons ${{R}\; \subseteq \;{S}}$ form a partial order, and therefore are reflexive, transitive, and antisymmetric.

As will be shown shortly, one can express “a lot” using the “magic” square pattern (2.1), rendered pointwise in (2.2). Some terminology before giving examples: we shall refer to ${{R}\; \subseteq \;{S}}$ as a relational inequality in which ${{R}}$ is the pre (or lower) side and ${{S}}$ is the post (or upper) side. In a composition ${{R} \cdot {S}}$ , relation ${{S}}$ (resp. ${{R}}$ ) will be referred to as the producer (resp. consumer) relation. This terminology bears some relationship to the way information flows in a magic square, cf.

Magic squares compose, not only horizontally

(2.3)

but also vertically:

(2.4)

Every type ${{X}}$ has its own identity relation , which is such that ${{x'}\;{id}\;{x}~\Leftrightarrow~{x'}\mathrel{=}~{x}}$ . Therefore, ${{R} \cdot {id}\mathrel{=}{R}\mathrel{=}{id} \cdot {R}}$ holds for any R and thus squares involving id degenerate into triangles or even “sides”. So, squares in which both pre-consumer and post-producer are identities behave like units of square composition, cf.:

(2.5)

3 Functions

Statements like John loves Mary can be written as Mary is loved by John without changing meaning. The latter version is known as the passive voice of the former. Generalizing, any proposition ${{b}\;{R}\;{a}}$ can be converted to passive voice, written ${{a}\;{{R}}^\circ\;{b}}$ . (In the example, ${{{loves}}^\circ\mathrel{=}{is}\;{loved}\;{by}}$ .) The ${{(_{\unicode{x2014}})}}^\circ$ operator, known as relational converse, will play a major role in our setting.

Converse is involutive ( ${{({{R}^\circ})^\circ}\mathrel{=}{R}}$ ) and contravariant: ${{({R} \cdot {S})^\circ}\mathrel{=}{{S}}^\circ \cdot {{R}}}^\circ$ . So one can take the converse of a square, obtaining an equivalent square where producer and consumer relations swap roles, once conversed:

Some relations, termed functions and denoted in lowercase — see e.g. f in (3.1) below — are special in the sense of fitting in two particular squares involving them and their converses:

(3.1)

By (2.2), the square on the left ( ${{id}\; \subseteq \;{{f}}^\circ \cdot {f}}$ ) converts to:

In words, ${{f}}$ reacts to every possible input (it is “total”). Similarly, the square on the right ( ${{f} \cdot {{f}}^\circ\; \subseteq \;{id}}$ ) converts to:

In words: ${{f}}$ is univocal. Because ${{f}}$ in (3.1) always yields some output for any possible input, and such an output is unique, one can interpret the proposition ${{b}\;{f}\;{a}}$ as ${{b}\mathrel{=}{f}\;{a}}$ , that is, ${{b}}$ is unequivocally the output of ${{f}}$ once ${{a}}$ is passed as input. This saves one from the two existential quantifiers of a very frequent and useful pattern involving functions and function converses:Footnote ³

(3.2)

We shall refer to (3.2) as the “nice” rule, where ${{f}}$ and ${{g}}$ are functions and ${{R}}$ is not constrained.Footnote ⁴

It can be shown that functions — i.e. relations ${{f}}$ fitting in (3.1) — are precisely those that satisfy the following square equivalences,

(3.3)

(3.4)

cf.

— similarly for (3.4). These equivalences, known as “shunting rules,” are very useful for reasoning about functions. One particular follow-up is that comparing functions amounts to equating them,

(3.5) \begin{eqnarray} {{f}\; \subseteq \;{g}~\Leftrightarrow~{f}\mathrel{=}{g}} \end{eqnarray}

as can be easily shown:

4 Reynolds squares

In the sequel, we shall be particularly interested in squares (2.1) in which both pre-consumers and post-producers are functions, say ${{f}}$ and ${{g}}$ in (4.1) given next:

(4.1)

By (3.3) followed by the “nice rule” (3.2), square (4.1) captures the situation in which, for ${{R}}$ -related inputs, the two functions ${{f}}$ and ${{g}}$ always yield ${{S}}$ -related outputs: if ${{a}\;{R}\;{b}}$ holds then ${({f}\;{a})\;{S}\;({g}\;{b})}$ holds, for any ${{a}}$ and ${{b}}$ .

We shall refer to squares of pattern (4.1) as Reynolds squares because of the role they will play in expressing free theorems of parametric polymorphic functions, due to John Reynolds (Reference Reynolds1983) and made popular by Philip Wadler (Reference Wadler1989). Note however that pattern (4.1) can be found in far more mundane settings, for instance relational database modeling.

Take for instance the very simple data model below of a small grocery in which customers can use coupons to obtain discounts when purchasing goods:

The “polygons” of the data model can be regarded as “opportunities” to dig up business rules, i.e. data invariants. Indeed, the triangle on the right can be oriented as the square

(4.2)

which is indeed meaningful: coupons cannot be used beyond their expiry date. And the lefthand triangle can be oriented as

(4.3)

bearing in it another relevant business rule: coupons can only be used by the customers who own them. Footnote ⁵

Squares in general (2.1) arise naturally in formal modeling in various guises, involving data relations (e.g. key-value maps), object attributes etc, and capturing not only domain-specific invariants such as those illustrated above but also structural ones. Particularly frequent squares are those that capture referential integrity,

(4.4)

where ${\top }$ is the largest relation of its type, that is, for all ${{b}}$ and ${{k}}$ . Think of ${{R}}$ and ${{S}}$ as key-value stores with keys ${{K}}$ and ${{K'}}$ , respectively, the latter playing the role of foreign key referred to by attribute ${{f}}$ of ${{A}}$ (a function).Footnote ⁶

Reynolds arrow Back to the genericity of Reynolds squares, fix the pre-producer ${{R}}$ and the post-consumer ${{S}}$ of (4.1). Then (4.1) expresses a (higher-order) relation on functions ${{f}\;({R}\to {S})\;{g}}$ defined by:

(4.5) \begin{eqnarray}{ {{f}\;({R}\to {S})\;{g}} \equiv f \cdot R \subseteq S \cdot g} \end{eqnarray}

Usinghe exponential notation ${{{Y}}^{{X}}}$ to denote the type of all functions from type ${{X}}$ to type ${{Y}}$ , and extending it to relations, ${{{S}}^{{R}}\mathrel{=}({R}\to {S})}$ , one has:

Thus we obtain the Reynolds arrow, a higher-order relational operator that builds, from relations ${{R}}$ and ${{S}}$ , the (higher-order) relation ${{{S}}^{{R}}}$ on functions identified by Backhouse (Reference Backhouse1990) and defined above by (4.5). We will use the notations ${{{S}}^{{R}}}$ , ${{R}\to {S}}$ and ${{S}\leftarrow {R}}$ interchangeably, depending on how convenient they are. (E.g. we avoid ${{({{S}}^{{R}})}^{{Q}}}$ and write ${{Q}\to {{S}}^{\textit{R}}}$ ^R instead.)

For instance, the ${{Used}\to (\leq )}$ higher-order relation relates attributes ${{date}}$ and ${{expiry}}$ in (4.2), expressing the semantic constraint that the use of a coupon can only occur on a date prior ( ${\leq }$ ) to its expiry date.

Reynolds arrows are central to the inference of free theorems, as will be shown later, in particular in the following situation, in which ${{R}}$ is the converse of a function ( ${R}\mathbin{:=}{{h}}^\circ$ ) and ${{S}}$ is a function ( ${{S}\mathbin{:=}{k}}$ ):

(4.6)

By shunting ${{{h}}}^\circ$ in (4.6) via (3.4), ${{{h}}^\circ\to {k}}$ becomes the higher-order function

(4.7) \begin{eqnarray} {({{h}^{\circ}}\to {k})\;{g}} &=& {{k} \cdot {g} \cdot {h}} \end{eqnarray}

as can be easily checked:

The special cases of (4.7),

(4.8) \begin{eqnarray}{({id}\to {k})\;{g}\mathrel{=}{k} \cdot {g}} \end{eqnarray}

(for A=B) and

(4.9) \begin{eqnarray}{({{h}^{\circ}}\to {id})\;{g}\mathrel{=}{g} \cdot {h}}\end{eqnarray}

(for ${{C}\mathrel{=}{D}}$ ) capture post-and pre-composition, that is:

(4.10)

(4.11)

By taking converses of both sides of (4.5) and swapping ${{f}}$ and ${{g}}$ via shunting rules (3.3, 3.4), one gets

(4.12)

Thus:

(4.13)

(4.14)

Higher-order Reynolds squares Exponential relations ${{{S}}^{{R}}}$ can involve other exponentials, for instance ${{({{S}}^{{Q}})}^{{R}}}$ i.e. ${{R}\to {{S}}^Q}$ . This happens frequently with functional programmers, who tend to use functions curried rather than uncurried. Such is the case in

Let us unfold this, assuming all fresh variables universally quantified:

(4.15)

(4.16)

Perfect squares Allelations in square (4.6) but the pre-producer are functions. Suppose now that all relations are functions in a magic square (2.1). Then, by (3.5) path comparison becomes path equality and producer/consumer paths become interchangeable. In this case, we drop the ${ \subseteq }$ symbol from the square:

Clearly:

(4.17) \begin{eqnarray} {{f}\;{{k}}^{{h}}\;{g}} &\equiv& {{f} \cdot {h}\mathrel{=}{k} \cdot {g}} \end{eqnarray}

5 More on Reynolds square expressiveness

Let the pre-producer ${{R}}$ of a Reynolds square (4.1) be the identity:

(5.1)

In this case, ${{f}}$ and ${{g}}$ have the same input type and are said to be pointwise ${{S}}$ -related. Using the dotted notation ${\stackrel.{{S}}}$ for ${{{S}}^{{id}}}$ , one may write ${{f}\;\stackrel.{{S}}\;{g}}$ to mean square (5.1), that is: . Thus, the following (useful) “lifting” rule:

(5.2)

A typical situation arises in (5.1) when the post-consumer ${{S}}$ is an ordering ${(\leq )}$ .

Then ${{f}\;\stackrel.{\leq }\;{g}}$ (i.e. ${{{f}}\subseteq{(\leq ) \cdot {g}}}$ ) means that ${{f}\;{a}\leq {g}\;{a}}$ for every input ${{a}}$ , i.e. ${{f}}$ is pointwise-smaller than ${{g}}$ with respect to ${(\leq )}$ .

Relational types. The intersection of ${{{S}}^{{R}}}$ and (higher-order) id captures all Reynolds squares (4.1) in which ${{f}\mathrel{=}{g}}$ :

(5.3)

In this case, we often abbreviate ${{f}\;({R}\to {S})\;{f}}$ to ${{f}\mathbin{:}{R}\to {S}}$ , meaning that ${{f}}$ has relational type ${{R}\to {S}}$ .Footnote ⁷ As seen before, this means that ${{f}}$ maps ${{R}}$ -related inputs to ${{S}}$ -related outputs.

We can also write to express (5.3), stressing the fact that ${{f}}$ can be regarded as a morphism of a category in which relations are the objects, mapped by functions that preserve them in the sense of (5.3). Indeed, by vertical composition (2.4) such morphisms compose and composition has identities (2.5).Footnote ⁸

For ${{R}}$ and ${{S}}$ instantiated to preorders, say ${{R},{S}\mathbin{:=}(\sqsubseteq ),(\leq )}$ , the square ${{f}\mathbin{:}(\sqsubseteq )\to (\leq )}$ is a concise way of saying that ${{f}}$ is a monotonic function: .

Predicates as types A special case of square (5.3) pops up wherever ${{R}}$ and ${{S}}$ are partial identities, also called coreflexive relations ( ${{R}\; \subseteq \;{id}}$ and ${{S}\; \subseteq \;{id}}$ ). These relations are one-to-one correspondent to predicates: given a predicate , its associated coreflexive is the relation ${{{p}}?\mathbin{:}{A}\to {A}}$ defined by

\begin{eqnarray*} {{a'}\;({{p}}?)\;{a}~\Leftrightarrow~{p}\;{a}\mathrel{\wedge}{a'}\mathrel{=}{a}}\end{eqnarray*}

Let ${{f}}$ have relational type ${{{p}}?\to {{q}}?}$ , as shown next:

(5.4)

It is easy to see that (5.4) converts to the pointwise , meaning that ${{p}}$ is a sufficient condition on the input of ${{f}}$ (pre-condition) for ${{q}}$ to hold on the output (post-condition). By a not so dangerous abuse of notation one might abbreviate ${{f}\mathbin{:}{{p}}?\to {{q}}?}$ to ${{f}\mathbin{:}{p}\to {q}}$ and regard predicates ${{p}}$ and ${{q}}$ as types.Footnote ⁹

Another perspective is to look at (5.4) as a Hoare-triple ${{p}\;\{\mskip1.5mu {f}\mskip1.5mu\}\;{q}}$ where ${{f}}$ is a functional program that satisfies such pre/post conditions.Footnote ¹⁰

Algebraic squares A quite interesting situation arises in a Reynolds square (4.1) when ${{f},{g}\mathbin{:=}\alpha ,\beta }$ , where ${\alpha }$ and ${\beta }$ are algebras of a relator :Footnote ¹¹

(5.5)

The pointwise equivalent of (5.5) is , for all ${{x}}$ and ${{y}}$ . For ${\alpha \mathrel{=}\beta }$ and endo-relation ${{R}}$ , says that ${{R}}$ is compatible with algebra ${\alpha }$ . For ${{R}}$ an equivalence relation, it further says that ${{R}}$ is a congruence relation with respect to ${\alpha }$ .Footnote ¹² In case ${{R}}$ is a function, say ${{R}\mathbin{:=}{h}}$ in (5.5), the square becomes perfect

meaning that ${{h}}$ is an -homomorphism (Bird & de Moor, Reference Bird and de Moor1997).

Coalgebraic squares Let ${{f},{g}\mathbin{:=}\gamma ,\phi }$ in a Reynolds square, where ${\gamma }$ and ${\phi }$ are -coalgebras:

(5.6)

Then ${{R}}$ is said to be a bisimulation (Sangiorgi, Reference Sangiorgi2004) between the two coalgebras, meaning:

In case ${{R}}$ is coreflexive, it is known as a coalgebraic invariant (Barbosa et al., Reference Barbosa, Oliveira and Silva2008). Squares (5.5) and (5.6) show that, in a sense, a bisimulation is a kind of co-logical-relation.

Galois connections. To complete this review of concepts that magic squares are able to easily capture, let ${{R}}$ and ${{S}}$ in (4.1) be two preorders, ${(\sqsubseteq )}$ and ${(\leq )}$ respectively. Moreover, reverse the direction of ${{f}}$ and let the square be perfect:

(5.7)

This very special kind of square is known as a Galois connection (GC) between two so-called adjoint functions, ${{f}}$ (the lower or left adjoint) and ${{g}}$ (the upper or right adjoint). Going pointwise in (5.7) via (2.2, 3.2) etc one gets:

(5.8) \begin{eqnarray} {{f}\;{b}\sqsubseteq {a}~\Leftrightarrow~{b}\leq {g}\;{a}} \end{eqnarray}

Two cancellation laws are easily obtained from (5.8):

These inequalities tell that f and g are inverses of each other in an imperfect way: the round-trip ${{f} \cdot {g}}$ loses information while ${{g} \cdot {f}}$ yields an over-approximation of its input. This is captured by the diagram

and often abbreviated by simply writing ${{f}\mathbin\dashv {g}}$ .

Examples of the usefulness of GCs in both mathematics and computing are abundant, see e.g. von Karger (Reference von Karger1998), Backhouse & Backhouse (Reference Backhouse and Backhouse2004), Mu & Oliveira (Reference Mu and Oliveira2011). The following GC between multiplication and whole division in the natural numbers

(5.9) \begin{eqnarray} {{a} \times {y}\leq {x}} & {~\Leftrightarrow~} & {{a}\leq {{x}}\div{{y}}} \end{eqnarray}

— ${( \times {y})\mathbin\dashv (\div {y})}$ in short, for ${{y}\not=\mathrm{0}}$ — will be of interest in the sequel.

6 Free theorem squares

In his landmark paper Theorems for free!, Philip Wadler (Reference Wadler1989) wrote: “From the type of a polymorphic function we can derive a theorem that it satisfies. (…) How useful are the theorems so generated? Only time and experience will tell (…)”. Under this “theorems for free” catchy phrase, Wadler made popular an important result on parametric polymorphism due to John Reynolds (Reference Reynolds1983). Four decades later there is ample evidence that such free theorems are indeed very useful, see for instance the work by Backhouse & Backhouse (Reference Backhouse and Backhouse2004), Voigtländer (Reference Voigtländer, Shao and Pierce2009) and Hackett & Hutton (Reference Hackett and Hutton2015).

The rest of this paper will be devoted to one particular application of such free theorems. Before that, the question arises: what is a free theorem and how does one derive it (for free)? It turns out that free theorems are magic squares. To put it simply, let a parametric function be given. Then its free theorem states that ${{f}}$ has relational type

(6.1) \begin{eqnarray} {{f}\mathbin{:}\mathsf{F} \;{R}\to \mathsf{G} \;{R}} \end{eqnarray}

for any ${{R}}$ relating its parameters, as shown in the corresponding square:

This extends to multiparametric functions, as shown next with a simple example: consider the Haskell constant function ${\mathsf{const}\mathbin{:}{a}\to {b}\to {a}}$ . In exponential notation (and following our uppercase notation for type parameters), we may write ${\mathsf{const}\mathbin{:}{A}\to {{A}}^{{B}}}$ . Then, by (6.1), ${\mathsf{const}}$ has relational type ${{R}\to {{R}}^{\textit{S}}}$ , that is,

(6.2) \begin{eqnarray}{\mathsf{const} \cdot {R}\; \subseteq \;{{R}}^{{S}} \cdot \mathsf{const}} \end{eqnarray}

holds, cf.

As already mentioned, this square is the free theorem of ${\mathsf{const}}$ . Recalling (4.15, 4.16), its pointwise version is, for all ${{a}}$ , ${{b}}$ , ${{c}}$ , ${{d}}$ :

\begin{eqnarray*}{{a}\;{R}\;{c}\Rightarrow {b}\;{S}\;{d}\Rightarrow (\mathsf{const}\;{a}\;{b})\;{R}\;(\mathsf{const}\;{c}\;{d})}\end{eqnarray*}

As a foretaste of what is to come, let us see how productive (6.2) is. Select, for instance, ${{R},{S}\mathbin{:=}{id},{k}}$ . Then (6.2) becomes ${\mathsf{const}\; \subseteq \;{{id}}^{{k}} \cdot \mathsf{const}}$ that, by (4.13,3.3) and (3.5) becomes ${( \cdot {k}) \cdot \mathsf{const}\mathrel{=}\mathsf{const}}$ , which is equivalent to ${\mathsf{const}\;{a} \cdot {k}\mathrel{=}\mathsf{const}\;{a}}$ — a well-known fusion law for constant functions. Swapping the roles of ${{R}}$ and ${{S}}$ to ${{R},{S}\mathbin{:=}{k},{id}}$ , one gets ${\mathsf{const} \cdot {k}\; \subseteq \;{{k}}^{{id}} \cdot \mathsf{const}}$ which, by (4.8) etc, rewrites to ${\mathsf{const} \cdot {k}\mathrel{=}({k} \cdot ) \cdot \mathsf{const}}$ yielding another well-known law: ${{k} \cdot \mathsf{const}\;{a}\mathrel{=}\mathsf{const}\;({k}\;{a})}$ .

Flipping. As another example, consider the well-known function in Haskell that swaps the arguments of a curried function:

(6.3)

Its free theorem states that has relational type ${{{{Q}}^{{S}}}^{{R}}\to {{{Q}}^{{R}}}^{{S}}}$ meaning

that is,

(6.4)

where (as in the sequel) we use the tilde notation to abbreviate , as in the right hand square above. For ${{Q}\mathbin{:=}{id},{S}\mathbin{:=}{id}\;{and}\;{R}\mathbin{:=}{r}}$ (a function), one gets:

This is known as the fusion-law of flipping (Oliveira, Reference Oliveira2020).

Flipping will be relevant in the rest of the paper, in which we shall investigate the theory that stems from the free theorems of two very popular functional programming combinators, ${\mathsf{foldl}{}{}}$ and ${\mathsf{foldr}{}{}}$ , heading toward discussing under what conditions they have the same behaviour.

7 foldl and foldr

In Haskell, there is a class ${{Foldable}}$ that, according to the standard documentation, “represents data structures that can be reduced to a summary value one element at a time”.Footnote ¹³ The class offers, in particular, two standard ways of accessing foldable data structures:

(7.1)

Lists are a prominent instance of the class — and, indeed, its main inspiration — whose standard definitions are:

The free theorems of ${\mathsf{foldl}{}{}}$ and ${\mathsf{foldr}{}{}}$ — valid for any instance of class ${{Foldable}}$ — can be written in the form of relational types:Footnote ¹⁴

(7.2)

(7.3)

There are two squares involved in each type. The one on the left will play the role of a side-condition for the one on the right to hold.

Foldl. Let us see (7.2) first:

(7.4)

For ${{R},{S}\mathbin{:=}{id},{h}}$ (hence ${{A}\mathbin{:=}{X}}$ ), both and ${{{S}}^{{R}}}$ reduce to ${({h} \cdot )}$ by Footnote ¹⁵ and (4.8). So the squares become perfect and one has:

(7.5)

Going pointwise via the usual laws, from (7.5) one gets:

This is the fusion law of ${\mathsf{foldl}{}{}}$ given by Bird & Gibbons (Reference Bird and Gibbons2020), who prove it (for lists) by finite-list induction. Note however that, as a corollary of a free theorem, it needs no dedicated proof and it holds for all instances of class ${{Foldable}}$ , not just for lists. Moreover, the scope of (7.4) is far wider than perfect squares involving only functions, as shown next.

To give a simple example, let ${{S}}$ be an ordering ${(\leq )}$ and keep ${{R}\mathbin{:=}{id}}$ . Since both squares will feature the relational type ${(\leq )\to {(\leq )}^{{id}}}$ . Then, if the left square holds, i.e. for all ${{b}}$ , ${{y}}$

then the right square

will hold too.

Perhaps more interesting is to observe that, if types a and b are replaced by predicates p and q, under the interpretation given by (5.4), then

is a corollary of the free theorem, saying:

NB: ${\mathsf{all}\;{p}}$ , available for all instances of class ${{Foldable}}$ , is the predicate associated to the coreflexive that relates a -structure to itself if and only if all data in it satisfy predicate ${{p}}$ .

Foldr. Repeating the above exercise for (7.3), on the right we get the same square as in (7.4), but the side condition on the left is different:

(7.6)

For ${{R},{S}\mathbin{:=}{id},{h}}$ , such a side-condition square unfolds to:

Altogether, one gets:

(7.7)

Going fully pointwise,

(7.8)

one obtains the fusion law of ${\mathsf{foldr}{}{}}$ proved for finite lists (by induction) by Bird & Gibbons (Reference Bird and Gibbons2020). Again we stress that (7.7) is a corollary of a free theorem and needs no proof, holding for all instances of class ${{Foldable}}$ .

Furthermore, by swapping the roles of ${{R}}$ and ${{S}}$ and making ${{S},{R}\mathbin{:=}{id},{h}}$ , the side-condition square of (7.6) becomes ${{g}\;({{id}}^{{h}})\;{f}}$ which, by (4.17) is ${{f}\mathrel{=}{g} \cdot {h}}$ . Replacing ${{f}}$ in the right square we get

(7.9)

Law (7.9) is often referred to as the absorption law of “fmap” () by ${\mathsf{foldr}{}{}}$ . It is another corollary of free theorem (7.6) and therefore none of its instances in the ${{Foldable}}$ class needs a proof. The default implementation of ${{foldMap}}$ in Data.Foldable arises from (7.9).

Maybe Just to give an example of (7.8) holding for instances of class ${{Foldable}}$ other than lists, let us check the ${{Maybe}}$ instance defined in ${{{Data}.Foldable}}$ , with respect to (7.7):

We have two cases: for ${{xs}\mathbin{:=}\mathsf{Nothing}}$ , the right square trivially reduces to ${{h}\;{e}\mathrel{=}{h}\;{e}}$ and we are done. In the case ${{xs}\mathbin{:=}\mathsf{Just}\;{x}}$ , we have ${\mathsf{foldr}{{f}}{{e}}}$ ${(\mathsf{Just}\;{x})} = {{f}\;{x}\;{e}}$ in (7.8) and ${\mathsf{foldr}{{g}}{({h}\;{e})}\;(\mathsf{Just}\;{x})\mathrel{=}{g}\;{x}\;({h}\;{e})}$ . Then: ${{g}\;{x}\;({h}\;{y})\mathrel{=}{h}\;({f}\;{x}\;{y})}$ ${\Rightarrow }$ ${{h}\;({f}\;{x}\;{e})\mathrel{=}{g}\;{x}\;({h}\;{e})}$ trivially holds too.

Permutativity Let ${{f}}$ and g be the same function in (7.7), say s, and ${{h}\mathbin{:=}{s}\;{a}}$ :

(7.10)

Property ${({s}\;{x}) \cdot ({s}\;{a})\mathrel{=}({s}\;{a}) \cdot ({s}\;{x})}$ — i.e. the fully pointwise ${{s}\;{x}\;({s}\;{a}\;{y})\mathrel{=}{s}\;{a}\;({s}\;{x}\;{y})}$ — is called (left) permutativity by Danvy (Reference Danvy2023).Footnote ¹⁶ It can be easily shown that if s is associative and commutative then it is permutative:

That is:

(7.11)

On the other hand, if s is permutative and has unit e, then s is commutative: ${{s}\;{x}\;({s}\;{a}\;{e})\mathrel{=}{s}\;{a}\;({s}\;{x}\;{e})}$ and thus ${{s}\;{x}\;{a}\mathrel{=}{s}\;{a}\;{x}}$ .

8 Universal properties

Suppose a particular instance of ${{Foldable}}$ such that

(8.1) \begin{eqnarray} {\mathsf{foldr} \, {\alpha } \, {\gamma }\mathrel{=}{id}} \end{eqnarray}

holds, for some ${\alpha }$ and ${\gamma }$ . By (7.6) the types are and . That is, ${\alpha }$ and ${\gamma }$ are constructors of type . Such is the case of lists, in which ${\mathsf{foldr}{(\mathbin{:})}{[\mskip1.5mu \mskip1.5mu]}\mathrel{=}{id}}$ , that is, ${\alpha \mathrel{=}(\mathbin{:})}$ and ${\gamma \mathrel{=}[\mskip1.5mu \mskip1.5mu]}$ in (8.1). Then one has the following corollary of (7.8):

\begin{eqnarray*}{{g}\;{x}\;({h}\;{y})\mathrel{=}{h}\;(\alpha \;{x}\;{y})}& {\Rightarrow } &{{h}\;{xs}\mathrel{=}\mathsf{foldr}{{g}}{({h}\;\gamma )}\;{xs}}\end{eqnarray*}

which, by introducing ${{z}\mathrel{=}{h}\;\gamma }$ and dropping ${{xs}}$ , becomes:

(8.2)

Thiseans that, if ${\gamma }$ and ${\alpha }$ exist such that (8.1) holds, by the free-theorem of ${\mathsf{foldr}{}{}}$ the system of equations (in ${{h}}$ )

has a unique solution, ${{h}\mathrel{=}\mathsf{foldr}{{g}}{{z}}}$ . Since ${\mathsf{foldr}{{g}}{{z}}}$ satisfies the equations for which it is a solution, then

(8.3)

hold, unveiling the definition of ${\mathsf{foldr}{}{}}$ itself. Moreover, this definition is mathematically equivalent to (just replace h by ${\mathsf{foldr}{{g}}{{z}}}$ and simplify):

(8.4)

Putting (8.2,8.4) together, we get the universal property of ${\mathsf{foldr}{}{}}$ ,

(8.5)

which, for lists, is

(8.6)

cf. e.g. Hutton (Reference Hutton1999).

9 Is foldl equal to foldr?

Looking at (7.1), the type-wise distance between ${\mathsf{foldr}{}{}}$ and ${\mathsf{foldl}{}{}}$ is the flip (6.3) of the first parameter.Footnote ¹⁷ So the “best fit” one can aim at here is

(9.1)

possibly valid for (as wide as possible) a class of functions f and instances of class ${{Foldable}}$ .

Our first step is to conjecture a definition for ${\mathsf{foldl}{}{}}$ that is type-wise consistent (over the ${{Foldable}}$ class) with (8.3) which — recall — was derived from the free-theorem of ${\mathsf{foldr}{}{}}$ . A possibility is to involve ${\mathsf{foldr}{}{}}$ itself in the definition. Looking at the flipped ${{f}}$ in (9.1), which is of type ${{A}\to {{B}}^{\textit{B}}}$ , one can think of expressing by a higher-order fold:

(9.2)

Note the type ${\theta\;{f}\mathbin{:}{A}\to {({{B}}^{{B}})}^{({{B}}^{{B}})}}$ for ${{f}\mathbin{:}{B}\to {{B}}^{{A}}}$ .

Next, we unfold (9.2) via universal property (8.5). For easier reference, we instantiate (with no loss of generality) ${\gamma }$ and ${\alpha }$ for lists, as these are very well known to functional programmers:

This confirms the standard definition of ${\mathsf{foldl}{}{}}$ for lists.

Universal- ${\mathsf{foldl}{}{}}$ An advantage of defining ${\mathsf{foldl}{}{}}$ “as a ${\mathsf{foldr}{}{}}$ ” (9.2) is that the universal property of the latter induces the universal property of the former:Footnote ¹⁸

Thus we get the universal-property of ${\mathsf{foldl}{}{}}$ :

(9.3)

Equating ${\mathsf{foldl}{}{}}$ and ${\mathsf{foldr}{}{}}$ Next we address the question (9.1) that opened this section: under what conditions does hold? We can use ${\mathsf{foldl}{}{}}$ -universal (9.3) to find an answer:

We conclude that holds for the instances of class ${{Foldable}}$ such that (8.1) holds, provided that is a permutative operation (recall Section 7).

The usual assumption that ${\mathsf{foldl}{{f}}{{e}}}$ and are the same for ${{f}}$ associative and ${{e}}$ its unitFootnote ¹⁹ is therefore too strong. By (7.11) we know that associativity and commutativity ensure permutativity.Footnote ²⁰ However, the converse implication does not hold, take e.g. ${{f}\mathrel{=}(\div )}$ (5.9) as counter-example: neither ${(\div }$ ) nor are associative or commutative, and yet is permutative. Why? Let us check it by indirect equality (Dijkstra, Reference Dijkstra2001):

Also to be noted, the second parameter e remains unconstrained in both folds. For example:

Of course, the previous calculation can generalize to any Galois connection ${{f}\mathbin\dashv {g}}$ .

Altogether, we have that is granted for those g that, being neither associative nor commutative, participate in an adjunction ${{f}\mathbin\dashv {g}}$ whose other adjoint is so.

10 Summary and discussion

This article addresses the use of relational, pointfree techniques in reasoning about functional programs. In particular, it shows the role of relational semi-commutative squares (suggestively referred to as “magic” squares) in expressing (and reasoning about) formal concepts relevant to computing and, in particular, to FP. By switching to relational types one gets a quite rich setting in which programming and expressing properties of programs blend naturally.Footnote ²¹ The calculation of “theorems-for-free” is given as an application of this, in particular used to infer conditions for the combinators ${\mathsf{foldl}{{f}}{{e}}}$ and to compute the same output for any ${{e}}$ .

This research shares much with the work of Backhouse & Backhouse (Reference Backhouse and Backhouse2004), who give perhaps the most impressive account of the power of free-theorems of all literature on the subject, contributing with sharp results on the role of Galois connections in (higher-order) abstract interpretation. Such a paper is, however, not an easy read for the average functional programmer. The (less ambitious) approach proposed in the current paper bears in mind the need to make such fantastic results more and more accessible to the programming community.

There is a sharp contrast between the effectiveness of relational reasoning in the style of e.g. Bird & de Moor (Reference Bird and de Moor1997) and Backhouse & Backhouse (Reference Backhouse and Backhouse2004) and the low popularity of the methodology. This is rather unfortunate. Having decided to teach such relational methods to computer science students two decades ago, the author has invested into widening scope and applying such a reasoning style to areas such as database programming, formal modeling and, of course, functional programming (Oliveira, Reference Oliveira2024). In this process, diagrams have proved very effective as a device enhancing the perception of the information flow and of how the overall reasoning is conducted. Thus magic squares (MS) emerged as a simple, yet very powerful design unit of a pointfree, relational approach to formal reasoning.

Relational exponentiation ${{{S}}^{{R}}}$ plays a major role in the way higher-order functions are handled. By MS vertical composition (2.4), one immediately infers that, as expected, it is monotone on the base and anti-monotonic on the exponent:

We also know that ${{{id}}^{{id}}\mathrel{=}{id}}$ (4.14). By horizontal composition (2.3), we get

(10.1) \begin{eqnarray}{{{{S}}^{{R}} \cdot {{S'}}^{{R'}}}\subseteq{{({S} \cdot {S'})}^{({R} \cdot {R'})}}} \end{eqnarray}

However, the converse inclusion does not hold and so relational exponentiation is not in general a (bi)relator.Footnote ²² Backhouse & Backhouse (Reference Backhouse and Backhouse2004) give conditions for strengthening (10.1) to an equality that include the cases involving functions and converses of functions used in this paper.

Functional programmers have realized that many abstract concepts they love have a categorical nature. This has led to a strong focus on explaining the whole paradigm in categorial terms, see e.g. Hinze et al. (Reference Hinze, Wu and Gibbons2015) — a trend that of course includes the today so important concept of a monad (Wadler, Reference Wadler1990; Moggi, Reference Moggi1991; Gibbons & Hinze, Reference Gibbons and Hinze2011). It all happened as if, in a sense, FP eventually came to rescue category theory from being considered by the programming community as mere abstract nonsense, with no practical application. In this setting, theorems-for-free have been formalized via (lax) dinatural transformations (Hackett &Hutton, Reference Hackett and Hutton2015; Voigtländer, Reference Voigtländer2019) in order-enriched categories. We believe that the category of relations, which is “naturally” ordered by relation inclusion, provides an overall simpler approach, much in the spirit of the quote by Freyd & Scedrov (Reference Freyd and Scedrov1990) that opens this paper.

Framed in this trend, the ${\mathsf{foldl}{}{}\mathbin{/}\mathsf{foldr}{}{}}$ case study could have been given as a pointfree calculation carried out via the adjoint-fold theorem (Hinze, Reference Hinze2013; Oliveira, Reference Oliveira, Madeira and Martins2023) and enabled by the self-adjunction witnessed by ${{flip}}$ Footnote ²³ — as in a similar calculation concerning left and right-iteration (Oliveira, Reference Oliveira2020). It should be noted that knowing that permutativity is enough for foldr/foldl “equality” is not new — see e.g. Danvy (Reference Danvy2023). Danvy’s reasoning is, however, quite different: he postulates permutativity as a side condition and then proves it in Coq by induction on lists. In this paper, permutativity arises by free-theorem calculation. Moreover, it shows that Danvy’s result can be extended via Galois connections.Footnote ²⁴

The generic approach to this case study given in the current paper throws attention to the ${{Foldable}}$ class of the Haskell standard library system. The permutativity of ${{f}}$ that grants the outcome “for-free” is not alone: a reflection constraint (8.1) is also needed. This constraint is quite strong in the sense of identifying the type constructors ${\gamma }$ and ${\alpha }$ of the particular ${{Foldable}}$ instance in hands. The types of ${\alpha }$ and ${\gamma }$ in (8.1) clearly point to lists as being the prototypical instance of the class.

It is not difficult to find instances of class ${{Foldable}}$ that do not meet reflection constraint (8.1) and for which the ${\mathsf{foldl}{}{}\mathbin{/}\mathsf{foldr}{}{}}$ result cannot be obtained as in the current paper. Interestingly enough, seems to be granted across the class as an axiom, that is, is the default definition of ${\mathsf{foldl}{}{}}$ , assuming ${\mathsf{foldr}{}{}}$ defined. Studying this better and applying the same kind of reasoning to other classes of the Haskell standard libraries is a topic for future research.