2.1 Boolean functions

A Boolean function ${\mathit{f}\mathop{:}\mathbb{B}^{\mathit{n}}\,\to\,\mathbb{B}}$ is a function from ${\mathit{n}}$ Boolean inputs to one Boolean output. We sometimes write ${\mathit{BoolFun}\;\mathit{n}}$ for the type ${\mathbb{B}^{\mathit{n}}\,\to\,\mathbb{B}}$ . The easiest examples of Boolean functions are the functions ${{\mathit{const}_{\mathit{n}}}\;\mathit{b}}$ which ignore the ${\mathit{n}}$ input bits and return ${\mathit{b}}$ . The usual logical gates like ${\mathit{and}_{\mathit{n}}}$ and ${\mathit{or}_{\mathit{n}}}$ are very common Boolean functions. Another example is the dictator function (also known as first projection), which is defined as ${{\mathit{dict}_{\mathit{n}\mathbin{+}\mathrm{1}}}\;[\mskip1.5mu \mathit{x}_{0},\mathbin{...},\mathit{x}_n\mskip1.5mu]\mathrel{=}\mathit{x}_{0}}$ when the dictator is bit ${\mathrm{0}}$ .

A naive representation of a Boolean function could be a pair of an arity and a function ${\mathit{f}}$ : ${[\mskip1.5mu \mathbb{B}\mskip1.5mu]\,\to\,\mathbb{B}}$ , but that turns out to be inefficient when we want to compare and tabulate them (see Section 3.3). Instead we use binary decision diagrams, ${\mathit{BDD}}$ s (Bryant, Reference Bryant1986) as implemented in Masahiro Sakai’s excellent Hackage package ^{Footnote 3} . The package reimplements all the usual Boolean operations on what is semantically expressions in ${\mathit{n}}$ Boolean variables. BDDs are an efficient way of representing Boolean functions, and they can be used for testing, verification, and complexity analysis. For readability, we will present Boolean functions in the naive representation, but the actual code uses the type ${\mathit{BDD}\;\mathit{a}}$ from the ${\mathit{BDD}}$ package (where ${\mathit{a}}$ keeps track of variable ordering). Note that we only use BDDs to represent our Boolean functions, not our decision trees.

In the complexity computation, we only need two operations on Boolean functions which we capture in the following type class interface:

The use of a type class here means we keep the interface to the BDD implementation minimal, which makes proofs easier and gives better feedback from the type system. The first method, ${\mathit{isConst}\;\mathit{f}}$ , returns ${\mathit{Just}\;\mathit{b}}$ iff the function ${\mathit{f}}$ is constant and always returns ${\mathit{b}\mathop{:}\mathbb{B}}$ . The second method, ${\mathit{setBit}\;\mathit{i}\;\mathit{b}\;\mathit{f}}$ , restricts a Boolean function (on ${\mathit{n}\mathbin{+}\mathrm{1}}$ bits) by setting its ${\mathit{i}}$ th bit to ${\mathit{b}}$ . The result is a “subfunction” on the remaining ${\mathit{n}}$ bits, abbreviated $f^{i}_{b}$ , and illustrated in Figure 4.

Fig. 4. The tree of subfunctions of a Boolean function ${\mathit{f}}$ . This tree structure is also the call graph for our generation of decision trees. Note that this tree structure is related to, but not the same as, the decision trees.

As an example, for the function ${\mathit{and}_{2}}$ we have that ${\mathit{setBit}\;\mathit{i}\;{\mathbf{0}}{}\;\mathit{and}_{2}\mathrel{=}{\mathit{const}_{\mathrm{1}}}\;{\mathbf{0}}{}}$ and ${\mathit{setBit}\;\mathit{i}\;{\mathbf{1}}{}\;\mathit{and}_{2}\mathrel{=}\mathit{id}}$ . For ${\mathit{and}_{2}}$ we get the same result for ${\mathit{i}\mathrel{=}\mathrm{0}}$ , or ${\mathrm{1}}$ but for the dictator function it depends if we pick the dictator index ( ${\mathrm{0}}$ ) or not. We get ${\mathit{setBit}\;\mathrm{0}\;\mathit{b}\;{\mathit{dict}_{\mathit{n}\mathbin{+}\mathrm{1}}}\mathrel{=}{\mathit{const}_{\mathit{n}}}\;\mathit{b}}$ , because the result is dictated by bit ${\mathrm{0}}$ . Otherwise, we get ${\mathit{setBit}\;(\mathit{i}\mathbin{+}\mathrm{1})\;\mathit{b}\;{\mathit{dict}_{\mathit{n}\mathbin{+}\mathrm{1}}}\mathrel{=}{\mathit{dict}_{\mathit{n}}}}$ irrespective of the value of ${\mathit{b}}$ since only the value of the dictator bit matters. This behavior is shown in Figure 5.

Fig. 5. The tree of subfunctions of the ${{\mathit{dict}_{\mathit{n}\mathbin{+}\mathrm{1}}}}$ function.

2.2 Decision trees

Consider a decision tree that picks the ${\mathit{n}}$ bits of a Boolean function ${\mathit{f}}$ in a deterministic way depending on the values of the bits picked further up the tree. Decision trees are referred to as algorithms in Landau et al. (Reference Landau, Nachmias, Peres and Vanniasegaram2006); Garban & Steif (Reference Garban and Steif2014); Jansson (Reference Jansson2022). Given a Boolean function ${\mathit{f}}$ , a decision tree ${\mathit{t}}$ describes one way to evaluate the function ${\mathit{f}}$ . The Haskell datatype is as follows:

Parts of the “rules of the game” in the mathematical literature are that you must return a ${\mathit{Res}}$ ult if the function is constant and you may only ${\mathit{Pick}}$ an index once. We can capture most of these rules with a type family version of the ${\mathit{DecTree}}$ datatype (here expressed in ${\mathit{Agda}}$ syntax). Here we use two type indices: ${\mathit{t}\mathop{:}\mathit{DecTree}\;\mathit{n}\;\mathit{f}}$ is a decision tree for the Boolean function ${\mathit{f}}$ , of arity ${\mathit{n}}$ . The ${\mathit{Res}}$ constructor may only be used for constant functions (but for any arity), while ${\mathit{Pick}\;\mathit{i}}$ takes two subtrees for Boolean functions of arity ${\mathit{n}}$ to a tree of arity ${\mathit{suc}\;\mathit{n}\mathrel{=}\mathit{n}\mathbin{+}\mathrm{1}}$ .

Note that the dependently typed version of ${\mathit{setBit}}$ clearly indicates that the resulting function ${\mathit{g}\mathrel{=}(\mathit{setBit}\;\mathit{i}\;\mathit{b}\;\mathit{f})\mathop{:}\mathit{BoolFun}\;\mathit{n}}$ has arity one less that of ${\mathit{f}\mathop{:}\mathit{BoolFun}\;(\mathit{suc}\;\mathit{n})}$ . This helps maintaining the invariant that each input bit may only be picked once. ^{Footnote 4} We use the Haskell versions, but the Agda versions capture the invariants better.

We can use these rules backward to generate all possible decision trees for a certain function. If the function is constant, returning ${\mathit{b}\mathop{:}\mathbb{B}}$ , we immediately know that the only decision tree allowed is ${\mathit{Res}\;\mathit{b}}$ . If it is not constant, we pick any index ${\mathit{i}}$ , any decision tree ${\mathit{t}_{0}}$ for the subfunction ${\mathit{setBit}\;\mathit{i}\;{\mathbf{0}}{}\;\mathit{f}}$ and ${\mathit{t}_{1}}$ for the subfunction ${\mathit{setBit}\;\mathit{i}\;{\mathbf{1}}{}\;\mathit{f}}$ recursively. We get back to this in Section 3.1 after some preparation.

Note that we do not use binary decision diagrams (BDDs) to represent our decision trees. An example of a decision tree for the majority function ${\mathit{maj}_{3}}$ on three bits is defined by the expression ${\mathit{ex1}}$ visualised in Figure 6.

Fig. 6. An example of a decision tree for ${\mathit{maj}_{3}}$ . The root node branches on the value of bit 0. If it is ${\mathbf{0}}$ , it picks bit 2, while if it is ${\mathbf{1}}$ , it picks bit 1. It then picks the last remaining bit if necessary.

We will define several functions as folds over ${\mathit{DecTree}}$ and to do that we introduce a type class ${\mathit{TreeAlg}}$ (for “Tree Algebra”) which collects the two methods ${\mathit{res}}$ and ${\mathit{pic}}$ which are then used in the fold to replace the constructors ${\mathit{Res}}$ and ${\mathit{Pick}}$ .

The ${\mathit{TreeAlg}}$ class is used to define our decision trees but also for several other purposes. (In the implementation, we additionally require some total order on ${\mathit{a}}$ to enable efficient set computations.) We see that our decision tree type is the initial algebra of ${\mathit{TreeAlg}}$ and that we can reimplement a generic version of ${\mathit{ex1}}$ which can be instantiated to any ${\mathit{TreeAlg}}$ instance:

2.3 Expected cost

For a function ${\mathit{f}}$ and a specific input ${\mathit{xs}\mathop{:}\mathbb{B}^{\mathit{n}}}$ , the cost of evaluating ${\mathit{f}}$ according to a decision tree ${\mathit{t}}$ is the length of the path from root to leaf dictated by the bits in ${\mathit{xs}}$ . We then let the bits be independent and identically distributed with probability $p \in [0,1]$ for ${{\mathbf{1}}{}}$ and compute the expected cost (averaging over all $2^n$ inputs). Expected cost can be implemented as an instance of ${\mathit{TreeAlg}}$ .

Note that the expected cost of any decision tree for a Boolean function of ${\mathit{n}}$ bits will always be a polynomial. We represent polynomials as lists of coefficients: ${\mathit{P}\;[\mskip1.5mu \mathrm{1},\mathrm{2},\mathrm{3}\mskip1.5mu]}$ represents ${\lambda \mathit{p}\,\to\,\mathrm{1}\mathbin{+}\mathrm{2}\times\mathit{p}\mathbin{+}\mathrm{3}\times{\mathit{p}^{\mathrm{2}}}}$ and use ${\mathit{evalP}\mathop{:}\mathit{Ring}\;\mathit{a}\Rightarrow \mathit{Poly}\;\mathit{a}\,\to\,(\mathit{a}\,\to\,\mathit{a})}$ to evaluate polynomials. The polynomial implementation borrowed from Jansson et al. (Reference Jansson, Ionescu and Bernardy2022) includes the polynomial ring operations ( ${(\mathbin{+})}$ , ${(\mathbin{-})}$ , ${(\times)}$ ), ${\mathit{gcd}}$ , ${\mathit{divMod}}$ , symbolic derivative, and ordering. The ${\mathit{res}}$ and ${\mathit{pic}}$ functions are as follows:

Here ${\mathit{zero}\mathrel{=}\mathit{P}\;[\mskip1.5mu \mskip1.5mu]} and {\mathit{one}\mathrel{=}\mathit{P}\;[\mskip1.5mu \mathrm{1}\mskip1.5mu]} represent {\mathit{const}\;\mathrm{0}}$ and ${\mathit{const}\;\mathrm{1}}$ , respectively, while ${\mathit{xP}\mathrel{=}\mathit{P}\;[\mskip1.5mu \mathrm{0},\mathrm{1}\mskip1.5mu]}$ is “the polynomial ${\mathit{x}}$ ”. For ${\mathit{pickPoly}\;- \;\mathit{q}_{0}\;\mathit{q}_{1}}$ , we first have to pick one bit and then if this bit is ${\mathbf{0}}$ (with probability $\mathbb{P}(x_{i} = {\mathbf{0}}$ ) = (1-p)) we get $q_0$ which is the polynomial for this case. If the bit is instead ${\mathbf{1}}$ (with probability $\mathbb{P}(x_{i} = {\mathbf{1}}) = p$ ) we get $q_1$ . The expected cost of the decision tree ${\mathit{ex1}}$ is $2 + 2p - 2p^2$ . From now on we will use Haskell’s overloading to write ${\mathrm{0}}$ and ${\mathrm{1}}$ for ${\mathit{zero}}$ and ${\mathit{one}}$ even when working with polynomials.

2.4 Complexity

Now that we have introduced expected cost, we can introduce the level- ${\mathit{p}}$ -complexity ${{D_p(\mathit{f})}}$ as the pointwise minimum of the expected cost over all of ${\mathit{f}}$ ’s decision trees:

where the generation of decision trees is explained in Section 3.1. When minimizing we do not necessarily get a polynomial, but a piecewise polynomial function. For simplicity, we represent a piecewise polynomial function as a set of polynomials:

This representation will be inefficient if the set is big, but as a specification it works fine and we will later use thinning to keep the set small (see Sections 3.2 and 3.4). We say that one polynomial ${\mathit{q}}$ is “uniformly worse” than another polynomial ${\mathit{p}}$ when ${\mathit{p}\;\mathit{x}{\leqslant} \mathit{q}\;\mathit{x}}$ for all ${\mathrm{0}{\leqslant} \mathit{x}{\leqslant} \mathrm{1}}$ and ${\mathit{p}\;\mathit{x}\mathbin{<}\mathit{q}\;\mathit{x}}$ for some ${\mathrm{0}\mathbin{<}\mathit{x}\mathbin{<}\mathrm{1}}$ . For some polynomials, we cannot determine which is worse, see Figure 1 where four polynomials all intersect. In this case, they are incomparable.

When computing the level- ${\mathit{p}}$ -complexity, it would be possible to take both ${\mathit{f}}$ and the probability ${\mathit{p}}$ as arguments and return the smallest expected cost for that probability, but we prefer to just take ${\mathit{f}}$ as an argument and compute a piecewise polynomial function representation. In this way, we can analyze the result symbolically to find minima, maxima, number of polynomial pieces, etc.

2.5 Examples of Boolean functions and their costs

Now that we have introduced expected cost and level- ${\mathit{p}}$ -complexity, we give a few examples of Boolean functions and their costs to give a feeling of how the computations work. The impatient reader can skip forward to Section 3. As mentioned earlier (in Section 2.1), we present the Boolean functions as Haskell functions for readability, but every example has a BDD counterpart.

For the constant functions ( ${{\mathit{const}_{\mathit{n}}}\;\mathit{b}}$ ), there is just one legal decision tree ${\mathit{t}\mathrel{=}\mathit{Res}\;\mathit{b}}$ and thus ${\mathit{expCost}\;\mathit{t}\mathrel{=}\mathrm{0}}$ which gives $D_p({{\mathit{const}_{\mathit{n}}}\;\mathit{b}}) = 0$ . For the dictator function, there are many decision trees, but as we can see in Figure 5, picking bit 0 first is optimal and gets us to the constant case just covered. Thus, the optimal tree is ${\mathit{optTree}\mathrel{=}\mathit{Pick}\;\mathrm{0}\;(\mathit{Res}\;{\mathbf{0}}{})\;(\mathit{Res}\;{\mathbf{1}}{})}$ , and we can compute the expected cost as follows.

In this case, all bits have to be picked to determine the parity, regardless of input. We prove that for all decision trees ${\mathit{t}}$ of ${\mathit{par}_{\!\mathit{n}}}$ or ${\neg \;\mathit{par}_{\!\mathit{n}}}$ , we have that ${\mathit{expCost}\;\mathit{t}\mathrel{=}\mathit{n}}$ using induction over ${\mathit{n}}$ . For the base case, ${\mathit{n}\mathrel{=}\mathrm{0}}$ we have that ${\mathit{par}_{\!\mathrm{0}}\mathrel{=}{\mathit{const}_{\mathrm{0}}}\;{\mathbf{0}}{}}$ and ${\neg \;\mathit{par}_{\!\mathrm{0}}\mathrel{=}{\mathit{const}_{\mathrm{0}}}\;{\mathbf{1}}{}}$ so that ${\mathit{expCost}\;\mathit{t}\mathrel{=}\mathrm{0}}$ for all decision trees ${\mathit{t}}$ as shown above. For the induction step we assume that for all decision trees ${\mathit{t}}$ of ${\mathit{par}_{\!\mathit{n}}}$ or ${\neg \;\mathit{par}_{\!\mathit{n}}}$ we have that ${\mathit{expCost}\;\mathit{t}\mathrel{=}\mathit{n}}$ and show that for all decision trees ${\mathit{t}}$ of ${\mathit{par}_{\!\mathit{n}\mathbin{+}\mathrm{1}}}$ or ${\neg \;\mathit{par}_{\!\mathit{n}\mathbin{+}\mathrm{1}}}$ we have that ${\mathit{expCost}\;\mathit{t}\mathrel{=}\mathit{n}\mathbin{+}\mathrm{1}}$ . Any decision tree for ${\mathit{par}_{\!\mathit{n}\mathbin{+}\mathrm{1}}}$ or ${\neg \;\mathit{par}_{\!\mathit{n}\mathbin{+}\mathrm{1}}}$ is of the form ${\mathit{Pick}\;\mathit{i}\;\mathit{t}_{0}\;\mathit{t}_{1}}$ where ${\mathit{t}_{0}}$ and ${\mathit{t}_{1}}$ are decision trees for ${\mathit{par}_{\!\mathit{n}}}$ or ${\neg \;\mathit{par}_{\!\mathit{n}}}$ as seen in Figure 7.

Fig. 7. The recursive structure of the parity function ( ${\mathit{par}_{\!\mathit{n}}}$ ). The pattern repeats all the way down to ${\mathit{par}_{\!\mathrm{0}}\mathrel{=}{\mathit{const}_{\mathrm{0}}}\;{\mathbf{0}}{}}$ .

To calculate the expected cost, we get

Thus, the induction proof is complete and as ${\mathit{expCost}\;\mathit{t}\mathrel{=}\mathit{n}}$ for all decision trees then also the minimum is n, thus $D_p({\mathit{par}_{\!\mathit{n}}}) = n$ . Comparing Figure 5 with Figure 7, we see that the minimum depth of the dictator tree is 1, while the minimum depth of the parity tree is n. The parity function and the constant function are interesting extreme cases of Boolean functions as they have highest and lowest possible level- ${\mathit{p}}$ -complexity ${\mathit{n}}$ and 0. Either all bits have to be picked to determine the parity or none of them need to be picked to determine the constant function.

We now introduce the Boolean function ${\mathit{same}}$ which checks if all bits are equal:

Using ${\mathit{same}}$ we construct the example ${\mathit{sim}_{5}}$ from the introduction. We first split the bits into two groups, one with the first three bits and the second with the last two bits. On the first group, called ${\mathit{as}}$ , we check if the bits are not the same, and on the second group, called ${\mathit{cs}}$ we check if the bits are the same.

The point of this function is to illustrate a special case where the best decision tree depends on ${\mathit{p}}$ so that the level- ${\mathit{p}}$ -complexity consists of more than one polynomial piece. This computation is shown in Section 4.1.

One of the major goals of this paper was to calculate the level- ${\mathit{p}}$ -complexity of 9 bit iterated majority called ${\mathit{maj}_{3}^2}$ . When extending the majority function to ${\mathit{maj}_{3}^2}$ , we use ${\mathit{maj}_{3}}$ inside ${\mathit{maj}_{3}}$ .

It is hard to calculate $D_p({\mathit{maj}_{3}^2})$ by hand because there are very many different decision trees, and this motivated our Haskell implementation.

3.1 Generating decision trees and other tree algebras

The complexity computation starts from a Boolean function ${\mathit{f}\mathop{:}\mathit{BoolFun}\;\mathit{n}}$ and generates all decision trees for it. There are two top level cases: either the function ${\mathit{f}}$ is constant (and returns ${\mathit{b}\mathop{:}\mathbb{B}}$ ), in which case there is only one decision tree: ${\mathit{res}\;\mathit{b}}$ ; or the function ${\mathit{f}}$ still depends on some of the input bits (and thus the arity is at least 1). In the latter case, for each index ${\mathit{i}\mathop{:}\mathit{Index}}$ , we can generate two subfunctions $f^{i}_{{\mathbf{0}}} = {\mathit{setBit}\;\mathit{i}\;{\mathbf{0}}{}\;\mathit{f}}$ and $f^{i}_{{\mathbf{1}}} = {\mathit{setBit}\;\mathit{i}\;{\mathbf{1}}{}\;\mathit{f}}$ . We then recursively generate a decision tree ${\mathit{t}_{0}}$ for $f^{i}_{{\mathbf{0}}}$ and ${\mathit{t}_{1}}$ for $f^{i}_{{\mathbf{1}}}$ and combine them to a bigger decision tree using ${\mathit{pic}\;\mathit{i}\;\mathit{t}_{0}\;\mathit{t}_{1}}$ . This is done for all combinations of ${\mathit{i}}$ , ${\mathit{t}_{0}}$ , and ${\mathit{t}_{1}}$ in a set comprehension. To make it easier to later extend the definition (for thinning and memoization), we make the recursive step explicit.

We would like to enumerate the cost polynomials of all the decision trees of a particular Boolean function ( ${\mathit{n}\mathrel{=}\mathrm{9}}$ , ${\mathit{f}\mathrel{=}\mathit{maj}_{3}^2}$ is our main goal). Without taking symmetries into account, there are ${\mathrm{2}\times\mathit{n}}$ immediate subfunctions $f^{i}_{b}$ and if $T_g$ is the cardinality of the enumeration for subfunction g we have that

$T_{{\mathit{f}}} = \sum_{i=0}^{n-1} T_{f^{i}_{{\mathbf{0}}}}\times T_{f^{i}_{{\mathbf{1}}}}$

These numbers can be really big if we count all decision trees, but if we only care about their cost polynomials, many decision trees will collapse to the same polynomial, making the counts more manageable (but still possibly really big). Even the total number of subfunctions encountered (the number of recursive calls) can be quite big. If all the ${\mathrm{2}\times\mathit{n}}$ immediate subfunctions are different, and if all of them would generate ${\mathrm{2}\times(\mathit{n}\mathbin{-}\mathrm{1})}$ different subfunctions in turn, the number of subfunctions would be $2^n \times n!$ . But in practice many subfunctions will be the same. When computing the polynomials for the 9-bit function ${\mathit{maj}_{3}^2}$ , for example, only ${\mathrm{215}}$ distinct subfunctions are encountered.

As a smaller example, for the 3-bit majority function ${\mathit{maj}_{3}}$ , choosing ${\mathit{i}\mathrel{=}\mathrm{0},\mathrm{1},} or {\mathrm{2}}$ gives exactly the same subfunctions. Figure 8 illustrates a simplified call graph of ${{\mathit{genAlg}_{\mathrm{3}}}\;\mathit{maj}_{3}}$ and the results (the expected cost polynomials) for the different subfunctions. In this case, all the sets are singletons, but that is very unusual for more realistic Boolean functions. It would take too long to compute all polynomials for the 9-bit function ${\mathit{maj}_{3}^2}$ , but there are 21 distinct 7-bit subfunctions, and the first one of them already has ${\mathrm{18021}}$ polynomials. Thus, we can expect billions of polynomials for ${\mathit{maj}_{3}^2}$ , and this means we need to look at ways to keep only the most promising candidates at each level. This leads us to the algorithmic design technique of thinning.

Fig. 8. A simplified computation tree of ${genAlg}_{\mathrm{3}}\;{maj}_{3}$ . In each node, ${f}\mapsto {ps}$ shows the input f and output ${ps}={genAlg}_{n}\;{f}$ of each local call. As all the functions involved are “symmetric” in the index ( ${setBit}\;{i}\;{b}\;{f} == {setBit}\;{j}\;{b}\;{f}$ for all i and j), we only show edges for 0 and 1 from each level.

3.2 Thinning

The general shape of the specification has two phases: “generate all candidates” followed by “pick the best one(s).” The first phase is recursive, and we would like to push as much as possible of “pick the best” into the recursive computation. In the extreme case of a greedy algorithm, we can thin the intermediate sets all the way down to singletons, but even if the sets are a bit bigger than that we can still reduce the computation cost significantly. A good (but abstract) reference for thinning is the Algebra of Programming book (Bird & de Moor, Reference Bird and de Moor1997, Chapter 8) and more concrete references are the corresponding developments in Agda (Mu et al., Reference Mu, Ko and Jansson2009) and Haskell (Bird & Gibbons, Reference Bird and Gibbons2020). In this subsection, the main focus is on specification and correctness, with Agda-like syntax for the logic part.

The “pick the best” phase is ${\mathit{best}\;\mathit{p}\mathrel{=}\mathit{minimum}\mathbin{\circ}\mathit{map}\;(\lambda \mathit{q}\,\to\,\mathit{evalP}\;\mathit{q}\;\mathit{p})} of type {\mathit{Set}\;(\mathit{Poly}\;\mathit{r})\,\to\,\mathit{r}}$ for some ring of scalars ${\mathit{r}}$ (usually rational numbers). In this context, it is clear that in the generation phase, we can throw away any polynomial which is “uniformly worse” than some other polynomial and this is what we want to use thinning for. We are looking for some “smallest” polynomials, but we only have a preorder, not a total order, which means that we may need to keep a set of incomparable candidates (elements ${\mathit{x}\not = \mathit{y}}$ for which neither ${\mathit{x}\prec\mathit{y}}$ nor ${\mathit{y}\prec\mathit{x}}$ ). We first describe the general setting and move to the specifics of our polynomials later.

We start from a strict preorder ${(\prec)\mathop{:}\mathit{a}\,\to\,\mathit{a}\,\to\,\mathit{Prop}}$ (an irreflexive and transitive relation). You can think of ${\mathit{Prop}}$ as ${\mathbb{B}}$ because we only work with decidable relations and finite sets in this application. As we are looking for minima, we say that ${\mathit{y}}$ dominates ${\mathit{x}}$ if ${\mathit{y}\prec\mathit{x}}$ .

We lift the order relation to sets in two steps. First, ${\mathit{ys}\mathrel{\dot{\prec}}\mathit{x}}$ means that ${\mathit{ys}}$ dominates ${\mathit{x}}$ , meaning that some element in ${\mathit{ys}}$ is smaller than ${\mathit{x}}$ . If this holds, there is no need to add ${\mathit{x}}$ to ${\mathit{ys}}$ because we already have at least one better element in ${\mathit{ys}}$ . Then ${\mathit{ys}\mathrel{\ddot{\prec}}\mathit{xs}}$ means that ${\mathit{ys}}$ dominates all of ${\mathit{xs}}$ .

Finally, we combine subset and domination into the thinning relation:

We will use this relation in the specification of our efficient computation to ensure that the small set of polynomials computed, still “dominates” the big set of all the polynomials generated by ${{\mathit{genAlg}_{\mathit{n}}}\;\mathit{f}}$ .

But first we introduce the helper function ${\mathit{thin}\mathop{:}\mathit{Set}\;\mathit{a}\,\to\,\mathit{Set}\;\mathit{a}}$ which aims at removing some elements, while still keeping the minima in the set. Later, we will use the function ${{\mathit{genAlgT}_{\!\mathit{n}}}\;\mathit{f}}$ specified similarly to ${{\mathit{genAlg}_{\mathit{n}}}\;\mathit{f}}$ but using the helper function ${\mathit{thin}}$ . It has to refine the relation ${\mathit{Thin}}$ which means that if ${\mathit{ys}\mathrel{=}\mathit{thin}\;\mathit{xs}}$ then ${\mathit{ys}}$ must be a subset of ${\mathit{xs}}$ ( ${\mathit{ys}\subseteq\mathit{xs}}$ ) and ${\mathit{ys}}$ must dominate the rest of ${\mathit{xs}}$ ( ${\mathit{ys}\mathrel{\ddot{\prec}}(\mathit{xs} \backslash \backslash \mathit{ys})}$ ). A trivial (but useless) implementation would be ${\mathit{thin}\mathrel{=}\mathit{id}}$ , and any implementation which removes some “dominated” elements could be helpful. The best we can hope for is that ${\mathit{thin}}$ gives us a set of only incomparable elements. If ${\mathit{thin}}$ compares all pairs of elements, it can compute a smallest thinning. In general that may not be needed (and a linear time greedy approximation is good enough), but in some settings almost any algorithmic cost which can reduce the intermediate sets will pay off. We collect the thinning functions in the type class ${\mathit{Thinnable}}$ :

The greedy ${\mathit{thin}}$ starts from an empty set and considers one element ${\mathit{x}}$ at a time. If the set ${\mathit{ys}}$ collected thus far already dominates ${\mathit{x}}$ , it is returned unchanged, otherwise ${\mathit{x}}$ is inserted. The optimal version also removes from ${\mathit{ys}}$ all elements dominated by ${\mathit{x}}$ . It is easy to prove that ${\mathit{thin}}$ implements the specification ${\mathit{Thin}}$ .

The method ${\mathit{cmp}}$ is a more informative version of ${(\prec)}: it returns {\mathit{Just}\;\mathit{LT}}$ , ${\mathit{Just}\;\mathit{EQ}}$ , or ${\mathit{Just}\;\mathit{GT}}$ if the first element is smaller, equal, or greater than the second, respectively, or ${\mathit{Nothing}}$ if they are incomparable.

Our use of thinning. Now we have what we need to specify when an efficient ${{\mathit{genAlgT}_{\!\mathit{n}}}\;\mathit{f}}$ computation is correct. Our specification ( ${\mathit{spec}\;\mathit{n}\;\mathit{f}}$ ) states a relation between a (very big) set ${\mathit{xs}\mathrel{=}{\mathit{genAlg}_{\mathit{n}}}\;\mathit{f}}$ and a smaller set ${\mathit{ys}\mathrel{=}{\mathit{genAlgT}_{\!\mathit{n}}}\;\mathit{f}}$ , we get by applying thinning at each recursive step. We want to prove that ${\mathit{ys}\subseteq\mathit{xs}}$ and ${\mathit{ys}\mathrel{\ddot{\prec}}(\mathit{xs} \backslash \backslash \mathit{ys})}$ because then we know we have kept all the candidates for minimality.

We can first take care of the simplest case (for any ${\mathit{n}}$ ). If the function ${\mathit{f}}$ is constant (returning some ${\mathit{b}\mathop{:}\mathbb{B}}$ ), both ${\mathit{xs}}$ and ${\mathit{ys}}$ will be the singleton set containing ${\mathit{res}\;\mathit{b}}$ . Thus, both properties trivially hold.

We then proceed by induction on ${\mathit{n}}$ to prove ${\mathit{S}_{\mathit{n}}\mathrel{=}\forall\,\mathit{f}\mathop{:}\mathit{BoolFun}\;\mathit{n}.\,\,\mathit{spec}\;\mathit{n}\;\mathit{f}}$ . In the base case ${\mathit{n}\mathrel{=}\mathrm{0}}$ the function is necessarily constant, and we have already covered that above. In the inductive step case, assume the induction hypothesis ${\mathit{IH}\mathrel{=}\mathit{S}_{\mathit{n}}}$ and prove ${\mathit{S}_{\mathit{n}\mathbin{+}\mathrm{1}}}$ for a function ${\mathit{f}\mathop{:}\mathit{BoolFun}\;(\mathit{n}\mathbin{+}\mathrm{1})}$ . We have already covered the constant function case, so we focus on the main recursive clause of the definitions of ${{\mathit{genAlg}_{\mathit{n}}}\;\mathit{f}}$ and ${{\mathit{genAlgT}_{\!\mathit{n}}}\;\mathit{f}}$ when the fixpoint definitions have been expanded:

All subfunctions ${\mathit{f}^{\mathit{i}}_{\mathit{b}}\mathop{:}\mathit{BoolFun}\;\mathit{n}}$ used in the recursive calls satisfy the induction hypothesis: ${\mathit{spec}\;\mathit{n}\;\mathit{f}^{\mathit{i}}_{\mathit{b}}}$ . If we name the sets involved in these hypotheses ${\mathit{xs}^{\mathit{i}}_{\mathit{b}}}$ and ${\mathit{ys}^{\mathit{i}}_{\mathit{b}}}$ , we can thus assume ${\mathit{ys}^{\mathit{i}}_{\mathit{b}}\subseteq\mathit{xs}^{\mathit{i}}_{\mathit{b}}}$ and ${\mathit{ys}^{\mathit{i}}_{\mathit{b}}\mathrel{\ddot{\prec}}(\mathit{xs}^{\mathit{i}}_{\mathit{b}} \backslash \backslash \mathit{ys}^{\mathit{i}}_{\mathit{b}})}$ .

First, the subset property: we want to prove that ${{\mathit{genAlgT}_{\!\mathit{n}\mathbin{+}\mathrm{1}}}\;\mathit{f}\subseteq{\mathit{genAlg}_{\mathit{n}\mathbin{+}\mathrm{1}}}\;\mathit{f}}$ , or equivalently, ${\forall\,\mathit{y}.\,\,(\mathit{y}\in {\mathit{genAlgT}_{\!\mathit{n}\mathbin{+}\mathrm{1}}}\;\mathit{f})\Rightarrow (\mathit{y}\in {\mathit{genAlg}_{\mathit{n}\mathbin{+}\mathrm{1}}}\;\mathit{f})}$ . Let ${\mathit{y}\in {\mathit{genAlgT}_{\!\mathit{n}\mathbin{+}\mathrm{1}}}\;\mathit{f}}$ . We know from the specification of ${\mathit{thin}}$ and the definition of ${{\mathit{genAlgT}_{\!\mathit{n}\mathbin{+}\mathrm{1}}}\;\mathit{f}}$ that ${\mathit{y}\mathrel{=}\mathit{pic}\;\mathit{i}\;\mathit{y}_{0}\;\mathit{y}_{1}}$ for some ${\mathit{y}_{0}\in \mathit{ys}^{\mathit{i}}_{\mathrm{0}}}$ and ${\mathit{y}_{1}\in \mathit{ys}^{\mathit{i}}_{\mathrm{1}}}$ . The subset part of the induction hypothesis gives us that ${\mathit{y}_{0}\in \mathit{xs}^{\mathit{i}}_{\mathrm{0}}}$ and ${\mathit{y}_{1}\in \mathit{xs}^{\mathit{i}}_{\mathrm{1}}}$ . Thus, we can see from the definition of ${{\mathit{genAlg}_{\mathit{n}\mathbin{+}\mathrm{1}}}\;\mathit{f}}$ that ${\mathit{y}\in {\mathit{genAlg}_{\mathit{n}\mathbin{+}\mathrm{1}}}\;\mathit{f}}$ .

Now for the “domination” property we need to show that ${\forall\,\mathit{x}\in \mathit{xs} \backslash \backslash \mathit{ys}.\,\,\mathit{ys}\mathrel{\dot{\prec}}\mathit{x}}$ where ${\mathit{xs}\mathrel{=}{\mathit{genAlg}_{\mathit{n}\mathbin{+}\mathrm{1}}}\;\mathit{f}}$ and ${\mathit{ys}\mathrel{=}{\mathit{genAlgT}_{\!\mathit{n}\mathbin{+}\mathrm{1}}}\;\mathit{f}}$ . Let ${\mathit{x}\in \mathit{xs} \backslash \backslash \mathit{ys}}$ . Given the definition of ${\mathit{xs}}$ it must be of the form ${\mathit{x}\mathrel{=}\mathit{pic}\;\mathit{i}\;\mathit{x}_{0}\;\mathit{x}_{1}}$ where ${\mathit{x}_{0}\in \mathit{xs}^{\mathit{i}}_{{\mathbf{0}}{}}}$ and ${\mathit{x}_{1}\in \mathit{xs}^{\mathit{i}}_{{\mathbf{1}}{}}}$ . The (second part of the) induction hypothesis provides the existence of ${\mathit{y}_{\mathit{b}}\in \mathit{ys}^{\mathit{i}}_{\mathit{b}}}$ such that ${\mathit{y}_{\mathit{b}}\prec\mathit{x}_{\mathit{b}}}$ . From these ${\mathit{y}_{\mathit{b}}}$ we can build ${\mathit{y'}\mathrel{=}\mathit{pic}\;\mathit{i}\;\mathit{y}_{0}\;\mathit{y}_{1}}$ as a candidate element to “dominate” ${\mathit{xs}}$ .

We can now show that ${\mathit{y'}\prec\mathit{x}}$ by polynomial algebra:

We are not quite done, because ${\mathit{y'}}$ may not be in ${\mathit{ys}}$ . It is clear from the definition of ${{\mathit{genAlgT}_{\!\mathit{n}\mathbin{+}\mathrm{1}}}\;\mathit{f}}$ that ${\mathit{y'}}$ is in the set ${\mathit{ys'}}$ sent to ${\mathit{thin}}$ , but it may be “thinned away.” But, either ${\mathit{y'}\in \mathit{ys}\mathrel{=}\mathit{thin}\;\mathit{ys'}}$ in which case we take the final ${\mathit{y}\mathrel{=}\mathit{y'}}$ or there exists another ${\mathit{y}\in \mathit{ys}}$ such that ${\mathit{y}\prec\mathit{y'}}$ and then we get get ${\mathit{y}\prec\mathit{x}}$ by transitivity.

To sum up, we have now proved that we can push a powerful ${\mathit{thin}}$ step into the recursive enumeration of all cost polynomials in such a way that any minimum is guaranteed to reside in the much smaller set of polynomials thus computed.

The specific properties we need from ${(\prec)}$ (in addition to the general requirements for thinning) are that ${(\mathit{pos}\mathbin{+})}$ and ${(\mathit{pos}\times)}$ are monotonic (for polynomials ${\mathrm{0}\prec\mathit{pos}}$ ) and that ${\mathit{q}_{0}\prec\mathit{q}_{1}}$ implies ${\mathit{evalP}\;\mathit{q}_{0}\;\mathit{p}{\leqslant} \mathit{evalP}\;\mathit{q}_{1}\;\mathit{p}}$ for all ${\mathrm{0}{\leqslant} \mathit{p}{\leqslant} \mathrm{1}}$ .

3.3 Memoization

The call graph of ${{\mathit{genAlgT}_{\!\mathit{n}}}\;\mathit{f}}$ is the same as the call graph of ${{\mathit{genAlg}_{\mathit{n}}}\;\mathit{f}}$ and, as mentioned above, it can be exponentially big. Thus, even though thinning helps in making the intermediate sets exponentially smaller, we still have one source of exponential computational complexity to tackle. Fortunately, the same subfunctions often appear in many different nodes and this means we can save a significant amount of computation time using memoization.

The classical example of memoization is the Fibonacci function. Naively computing ${\mathit{fib}\;(\mathit{n}\mathbin{+}\mathrm{2})\mathrel{=}\mathit{fib}\;(\mathit{n}\mathbin{+}\mathrm{1})\mathbin{+}\mathit{fib}\;\mathit{n}}$ leads to exponential growth in the number of function calls. But if we fill in a table indexed by ${\mathit{n}}$ with already computed results we can compute ${\mathit{fib}\;\mathit{n}}$ in linear time.

Similarly, here we “just” need to tabulate the result of the calls to ${{\mathit{genAlg}_{\mathit{n}}}\;\mathit{f}}$ so as to avoid recomputation. The challenge is that the input we need to tabulate is now a Boolean function, which is not as nicely structured as a natural number index. Fortunately, thanks to Hinze (Reference Hinze2000), Elliott, and others we have generic Trie-based memo functions only a hackage library away ^{Footnote 5} . The ${\mathit{MemoTrie}}$ library provides the ${\mathit{Memoizable}}$ class and suitable instances and helper functions for most types. We only need to provide a ${\mathit{Memoizable}}$ instance for ${\mathit{BDD}}$ s, and we do this using ${\mathit{inSig}}$ and ${\mathit{outSig}}$ from the ${\mathit{BDD}}$ package (decision-diagrams). They expose the top-level structure of a ${\mathit{BDD}}: {\mathit{Sig}\;\mathit{bf}}$ is isomorphic to ${\mathit{Either}\;\mathbb{B}\;(\mathit{Index},\mathit{bf},\mathit{bf})}$ where ${\mathit{bf}\mathrel{=}\mathit{BDDFun}}$ . We define our top-level function ${\mathit{genAlgThinMemo}}$ by applying memoization to ${{\mathit{genAlgT}_{\!\mathit{n}}}}$ (or, more specifically, to ${\mathit{genAlgStepThin}}$ ).

3.4 Comparing polynomials

As argued in Section 3.2, the key to an efficient computation of the best cost polynomials is to compare polynomials as soon as possible and throw away those which are “uniformly worse.” The specification of ${\mathit{p}\prec\mathit{q}}$ is ${\mathit{p}\;\mathit{x}{\leqslant} \mathit{q}\;\mathit{x}}$ for all ${\mathrm{0}{\leqslant} \mathit{x}{\leqslant} \mathrm{1}}$ and ${\mathit{p}\;\mathit{x}\mathbin{<}\mathit{q}\;\mathit{x}}$ for some ${\mathrm{0}\mathbin{<}\mathit{x}\mathbin{<}\mathrm{1}}$ . Note that ${(\prec)}$ is a strict preorder — if the polynomials cross, neither is “uniformly worse” and we keep both. A simple example of two incomparable polynomials is ${\mathit{xP}}$ and ${\mathrm{1}\mathbin{-}\mathit{xP}}$ which cross at ${\mathit{p}\mathrel{=}{1}/{2}}$ .

If we have two polynomials ${\mathit{p}}$ and ${\mathit{q}}$ , we want to know if ${\mathit{p}{\leqslant} \mathit{q}}$ for all inputs in the interval ${[\mskip1.5mu \mathrm{0},\mathrm{1}\mskip1.5mu]}$ . Equivalently, we need to check if ${\mathrm{0}{\leqslant} \mathit{q}\mathbin{-}\mathit{p}}$ in that interval.

As the difference is also a polynomial, we can focus our attention to locating polynomial roots in the unit interval.

If there are no roots (Figure 9a) in the unit interval, the polynomial stays on “one side of zero,” and we just need to check the sign of the polynomial at any point. If there is at least one single root (Figure 9b), the original polynomials cross and we return ${\mathit{Nothing}}$ . Similarly for triple roots or roots of any odd order. Finally, if the polynomial only has roots of even order (some double roots, or quadruple roots, etc. as in Figure 9c) the polynomial stays on one side of zero, and we can check a few points to see what side that is. (If the number of distinct roots is ${\mathit{r}}$ we check up to ${\mathit{r}\mathbin{+}\mathrm{1}}$ points to make sure at least one will be nonzero and thus tell us on which side of zero the polynomial lies.)

Fig. 9. To compare two polynomials ${\mathit{p}}$ and ${\mathit{q}}$ we use root counting for ${\mathit{q}\mathbin{-}\mathit{p}}$ and these are the three main cases to consider.

To compare polynomials, we thus need to implement the root-counting functions ${\mathit{numRoots}}$ and ${\mathit{numRoots'}}$ :

We will not provide all the code here, because that would take us too far from the main topic of the paper, but we will illustrate the main algorithms and concepts for root counting in Section 3.5. The second function computes real root multiplicities: ${\mathit{numRoots'}\;\mathit{p}\mathrel{=}[\mskip1.5mu \mathrm{1},\mathrm{3}\mskip1.5mu]} means {\mathit{p}}$ has one single and one triple root in the open interval ${(\mathrm{0},\mathrm{1})}$ . From this we get that ${\mathit{p}}$ has ${\mathrm{2}\mathrel{=}\mathit{length}\;[\mskip1.5mu \mathrm{1},\mathrm{3}\mskip1.5mu]}$ distinct real roots and ${\mathrm{4}\mathrel{=}\mathit{sum}\;[\mskip1.5mu \mathrm{1},\mathrm{3}\mskip1.5mu]}$ real roots if we count multiplicities.

Using the root-counting functions, the top level of the polynomial partial order implementation is as follows:

3.5 Isolating real roots and Descartes rule of signs

This section explains how to do root counting by combining Yun’s algorithm and Descartes rule of signs. As explained in Section 3.4, the root counting is the key to implementing comparison, which is needed for thinning. First out is Yun’s algorithm (Yun, Reference Yun1976) for square-free factorization: given a polynomial ${\mathit{p}}$ it computes a list of polynomial factors $p_i$ , each of which only has single roots, and such that $p = C \prod_{i} {p_i}^i$ . Note the exponent ${\mathit{i}}$ : the factor ${\mathit{p}_{2}}$ , for example, appears squared in ${\mathit{p}}$ . If ${\mathit{p}}$ only has single roots, the list from Yun’s algorithm has just one element, ${\mathit{p}_{1}}$ , but in any case we get a finite list of polynomials, each of which is “square-free.”^{Footnote 6}

Second in line is Descartes rule of signs that can be used to determine the number of real zeros of a polynomial function. It tells us that the number of positive real zeros in a polynomial function ${\mathit{f}}$ is the same, or less than by an even number, as the number of changes in the sign of the coefficients. Together with some polynomial transformations, this is used to count the zeros in the interval [0,1).

If the rule gives zero or one, we are done: we have isolated an interval [0,1) with either no root or exactly one root. For our use case, we do not need to know the actual root, just if it exists in the interval or not.

If the rule gives more than one, we do not quite know the exact number of roots yet (only an upper bound). In that case, we subdivide the interval into the lower $[0,1/2)$ and upper $[1/2, 1)$ halves. Fortunately, the polynomial coefficients can be transformed to make the domain the unit interval again so that we can call ourselves recursively. After a finite number of steps, this bisection terminates, and we get a list of disjoint isolating intervals where we know there is exactly one root in each. (The number of steps is on the order of the two-logarithm of the minimum distance between two distinct roots.)

Combining Yun and Descartes, we implement our “root counter,” and thus our partial order on polynomials.