Applying Fourier Analysis to Judgment Aggregation

The classical Arrow's Theorem answers "how can n voters obtain a collective preference on a set of outcomes, if they have to obey certain constraints?" We give an analogue in the judgment aggregation framework of List and Pettit, answering "how can n judges obtain a collective judgment on a set of logical propositions, if they have to obey certain constraints?" We abstract this notion with the concept of "normal pairs" of functions on the Hamming cube, which we analyze with Fourier analysis and elementary combinatorics. We obtain judgment aggregation results in the special case of "symbol-complete" agendas and compare them with existing theorems in the literature. Amusingly, the non-dictatorial classes of functions that arise are precisely the classical logical functions OR, AND, and XOR.

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1. Introduction

In [9], List and Pettit explore the “doctrinal paradox” put forth in Kornhauser and Sager [8] and prove an impossibility theorem. In their followup work [10], they contrast their theorem with the celebrated Arrow’s Theorem from the social choice literature. List-Pettit’s Theorem is about the difficulty of aggregating judges’ collective judgments on an agenda of logical propositions into a single judgment, and Arrow’s Theorem is about the difficulty of aggregating voters’ preference orderings into a single preference ordering. The “judgment aggregation” framework that comes out of List and Pettit’s work is very powerful, because logical propositions are very expressive and flexible.

[10] concludes that while List-Pettit’s theorem is superficially similar to Arrow’s Theorem, the two theorems are quite different at heart. What, then, is the right analogue to Arrow’s Theorem in the judgment aggregation framework? Arrow’s Theorem is typically phrased in the following way: “if voters wish to obtain a collective preference on a set of outcomes, then it is impossible for the aggregation function to be simultaneously Pareto, Indifferent to Irrelevant Alternatives, and Non-Dictatorial.” In the judgment aggregation literature, Dietrich and List continue this trend in [2] and [3], proving impossibility theorems that cover a large space of situations, including explicitly making an analogy with Arrow’s Theorem in [2].

As one persons’ modus ponens is another’s modus tollens, we explore a different phrasing of Arrow’s Theorem, which is: “if voters wish to obtain a collective preference on a set of outcomes, and the aggregation function must be Pareto and Indifferent to Irrelevant Alternatives, then it must be a dictatorship function.” That is, we think of Arrow’s Theorem as a classification theorem rather than an impossibility theorem.

In this direction, we define an aggregation function of judgments of to be Arrovian

if it satisfies Arrow’s Theorem-like constraints. We classify possible Arrovian aggregation functions in the case when our agenda

of logical propositions is symbol-complete and symbol-connected, a condition special enough to give us a lot of structure but general enough to include many realistic applications. We discover a friendly set of functions: the OR, AND, and XOR functions from the first chapter of any undergraduate book on logic or computer science. Here is a simplified paraphrasing of our main theorem; the details and definitions are made explicit in the paper:

Theorem (Corollary 5.1, paraphrased).

Suppose the agenda is symbol-complete and symbol-connected, containing the compound proposition . If aggregates the judgments of judges on in an Arrovian manner, then either:

• is a “dictatorship” (that is, returns the judgment of a single judge), or

• must be the OR, AND, or XOR function (or negation), and is an “oligarchy” on a distinguished subset of judges, obtained by performing the same function type (or possibly its negation) as on .

We now give two examples.

• if judges are judging on the agenda The defendant lied, the defendant misspoke, the defendant committed the crime, the defendant was guilty and gave false information (i.e. committed the crime and then misspoke or lied), logically representable as , then the only Arrovian method to aggregate the judgments is for the judges to select a single “dictator” judge to represent the entire group.

• if judges are judging on the agenda The defendant committed unspeakable crime 1, the defendant committed unspeakable crime 2, the defendant committed unspeakable crime 3, the defendant committed at least one unspeakable crime, logically representable as , then the Arrovian methods of aggregating the judgments are for a subset of the judges to decide that each proposition is judged true in the aggregate if and only if at least one of think the proposition is true (in other words, an OR function over the judges). Furthermore, if we add to the agenda any other proposition (besides OR’s) that uses the same symbols (such as ), then only the dictatorship option () is possible. If we add logical OR statements to the agenda (such as ), then the OR function over the judges in can still be used.

While our main results offer original consequences and insight (the most interesting, in our opinion, being that the aggregation function is forced to follow the same logical structure as a compound proposition in , applied to possibly a different number of arguments!), they overlap significantly with existing theorems in [3], so we think of our results primarily as a refinement on the existing work when we have at least one symbol-closed compound proposition. We would not be surprised, for example, if our main results would follow from the existing theorems after some work and case analysis. We believe our paper’s main contribution to the literature is actually the act of introducing Fourier analysis techniques, inspired by their success in the closely related social choice framework by works such as [7]. Our work demonstrates that we can prove strong results fairly efficiently with these methods.

In Section 2, we give the preliminaries of Arrow’s Theorem and List-Pettit’s judgment aggregation framework. In Section 3, we abstract the problem and reduce studying aggregation functions to what we call normal pairs, which are pairs of functions on the Hamming cube that satisfy a commutation-like property. We believe normal pairs lie at the heart of many social choice / judgment aggregation problems, and may be very useful in future work. In Section 4, we state and prove our main result about normal pairs, Theorem 4.6, with Fourier analysis. In Section 5, we apply Theorem 4.6 to judgment aggregation and put our work in context with the existing literature, comparing it with existing theorems. Finally, in Section 6, we conclude with philosophical discussions and ideas for future work.

2. Preliminaries and Definitions

2.1. Arrow’s Theorem

In this paper, we call the framework behind Arrow’s Theorem the “social choice framework.” In this setup, there is a set of voters voting on a set of alternatives. We present each voter’s preference as a partial order on , defined as an anti-symmetric, transitive, and reflexive relation. We denote by the set of partial orders on . We then define social preference functions (SPF’s) to be functions . So a SPF aggregates a profile of preferences into a single collective preference.

We now define some properties relevant to a SPF . Suppose , we define to be:

• A SPF is Pareto If for all , , then .

• A SPF has Indifference of Irrelevant Alternatives (IIA) if and are two profiles in the domain of and there exists and such that for all , if and only if , then if and , we must have if and only if .

An enlightening restatement of IIA for our purposes is the following: if we think of a preference as a function from pairs of alternatives to , where we say if and only if , then if , it means only depends on . In other words, for all pairs there exists a function such that . We will use this intuition again in the future.

Suppose is Pareto and IIA. Then we call Arrovian. We rephrase111Usually, Arrow’s Theorem is stated as an “impossibility theorem.” That is, it is impossible for a social preference function to be Pareto, IIA, and not a dictatorship. Because our main result is more of a “classification theorem,” that is, listing the types of functions (including dictatorships) that satisfy Pareto and IIA, we do not use the “impossibility” presentation. the classical Arrow’s theorem in the following form:

Theorem 2.1 (Arrow’s Theorem, [1], paraphrased).

Let . Suppose is an Arrovian SPF, then is a dictatorship (that is, for some ).

2.2. The Judgment Aggregation Framework

In the judgment aggregation framework of List and Pettit [9], judges seek to judge an agenda (set of logical propositions). In this paper, we define an agenda to be a subset of the boolean algebra222In the treatment of [2] and others, the agenda is defined by an arbitrary set related by rules about which subsets of propositions would imply others, without the need of an underlying set of logical symbols. Any such set can actually be embedded in a boolean algebra and we choose to do so in this paper. of a symbol set of logical symbols such that is closed under negation (i.e. for all , we must have ). Given , we call atomic if or belongs to the symbol set and compound otherwise.

We usually describe the agenda by a basis , which contains exactly one of each of and for every ; note we can recover the agenda from the basis

by adding negations, much in the same way that we can recover a vector space from the basis by allowing multiplication by scalars. For example, a possible symbol set could be

, with and . Each judge’s judgment can then be defined as an assignment of (True) or (False) to each proposition in the agenda; formally, a function .

If the judges are to make judgments on a proposition, it seems natural for them (at least some of the time) to also be making judgments on its symbols. For example, if the judges were to judge on “the defendant committed crime A or crime B,” we would assume they would also argue about whether the defendant committed the individual crimes separately. We say that a compound proposition is symbol-closed if all of its symbols are in . For example, in , is symbol-closed. However, in , is not symbol-closed.

We say that a set of propositions is symbol-complete if the symbol set is in . Note that when is symbol-closed, and its symbols form a symbol-complete subset of . We denote this subset by . For example, .

In this paper, we typically assume our agenda is symbol-complete. This is a very strong assumption. The existing results in [2] and [3] do not make this assumption and still hold for very general agendas. Compared to these nice results, our restriction gives results that are less generally applicable but stronger when applied. We discuss this trade-off more in Sections 5 and 6.

In the social choice framework, the rationality (i.e. if , , then ) of a preference is embedded in the requirement that the preference is a partial order. In the judgment aggregation framework, we analogously define , the fully rational judgments on , to be the set of judgments on such that there exists some assignment of truth values to the symbol set such that each proposition in the judgment (as a function of the symbol set) is consistent with the assignment; for example, a fully rational judgment on could be , corresponding to . Note that in a fully rational judgment, different truth values must be assigned to and for all . From this, we can see that while we care about the agenda , in it suffices to provide judgment on just the basis ; in other words, is well-defined. Thus, we usually look at just the judgment restricted to instead of to avoid redundancy, losing no information. We call the set of functions judgment aggregation rules (JAR’s) on to aggregate fully rational judgments, as an analogue of social preference functions being used to aggregate non-cyclic preferences. In [2], JAR’s are simply called “aggregation rules.” We add “judgment” to disambiguate the many uses of “aggregation” in this paper.

In a symbol-complete with atomic propositions, there is a bijection between and the assignments of to the atomic propositions, as every assignment of the atomic propositions uniquely extends to a fully rational assignment to the propositions in .

2.3. Logic

We define two propositions (or sets) in to be symbol-disjoint if the logical symbols the two propositions (or sets) use are disjoint. As an example, is symbol-disjoint from , but not from . Consider a graph where the vertices are the propositions and we draw an edge between two propositions if and only if they share a symbol. We say that is symbol-connected if is a connected graph. As an example, if , the set is symbol-disjoint from the set of the other propositions, so the graph splits into two connected components and is therefore not symbol-connected.

Given a set , let be the set of -valued functions on . One of our main ideas (which lets us use Fourier analysis) is that we interchangeably think of a proposition as an element of the boolean algebra and as a -valued function on , where are the symbols in the proposition. We now make some relevant notations and definitions:

• For any function where , we say that is (-) pivotal at if , , is defined to be with the -indexed element flipped, and . Note this requires both and to be in the domain . Also note that is -pivotal at if and only if it is -pivotal at as well.

• In the setup above, we say that is irrelevant for the function if is not -pivotal anywhere. We say is relevant otherwise.

Note that we can assume that any symbol appearing in a compound proposition must be relevant for (for example, is irrelevant to , but we can simplify to in this case). This removes some pathological cases.

We say that form a set of consistent values for a set of propositions if there exist assignments of to the symbol set such the induced logical values of each equals . We denote the set of sets of consistent values for by . Equivalently, a set is in if it is the restriction of some element in to . For a proposition and a set of propositions , we say that is determined by if can be written as a function from to . For example, and determine . A recurring theme is that a compound proposition is determined by its symbols.

2.4. Fourier Analysis on the Hamming Cube

We present some elementary Fourier analysis tools in style of [7] to use in Section 4. See e.g. [13] for a survey, including what we use here and beyond.

Given a function , we can decompose the function as

 f(s1,…,sn)=∑R⊂[n]^f(R)sR,

where we define . These Fourier coefficients satisfy .

Given the desired values of , we can solve for

via interpolation because the function

returns on a single list and for all other lists. Doing so for a few key functions gives:

• the constant function is the function , so its only nonzero Fourier coefficient is .

• the XOR function is the function , so its only nonzero Fourier coefficient is .

• the NXOR function is the negation of the XOR function, or . It also only has a single nonzero Fourier coefficient .

• the AND function is the function , so for all , the Fourier coefficient is . For the empty set we must subtract the in the beginning to get .

• the OR function is the function , so for all , the Fourier coefficient is . For the empty set we must add the in the beginning to get .

Given a function , for and , we say that forces with at index if whenever with , we have . In this case, we say argument is forceable for . Otherwise, we say index is free for . We say such a function is forceful if is forceable at every index.

Lemma 2.2.

Suppose where is a non-constant forceful function. Then it can be written as

 f(r1,…,rn)=−c0+c012n−1(1+c1r1)(1+c2r2)⋯(1+cnrn)

for arbitrary choices of . In this case, for all , and .

Proof.

If is forceful, then forces some with at index for all . We must have all the ’s equal to a constant , otherwise we can force two contradictory results since (if forces and forces , then we get a contradiction when looking at any set of arguments including and in the -th and -th spots, respectively). We now know that takes the same value at every argument but one (namely, when we pick for each index . Linear interpolation of at the then gives the above form. ∎

In particular, note that OR and AND are forceful functions. For example, the OR function forces (True) if any argument equals , and can be written .

2.5. The Main Question

Since our goal is an analogue of Arrow’s Theorem, we now give some analogous definitions in the judgment aggregation framework to the concepts used in Arrow’s Theorem. We follow the nomenclature of [3].

• A JAR is Unanimity-Preserving (UP) if for every proposition , if for all , (respectively ), then (respectively ).

• A JAR is Propositionally Independent (PI) if for each proposition there is a function such that

 f(p1,…,pn)(x)=fx(p1(x),p2(x),…,pn(x))

for all .

These are completely analogous to Pareto and IIA, respectively. As in Section 2, we say that a judgment aggregation rule is Arrovian if it is both UP and PI. We mention a couple of similar notions from [9]:

• A JAR has Anonymity if is invariant under permutations on the judges. In other words, rearranging the arguments of does not change the judgment.

• A JAR has Systematicity if there exists a function such that, for any proposition , .

These were originally used in List-Pettit’s impossibility theorem in [9]. In Section 5, we look at some consequences of our main result when an Arrovian JAR is also forced to have these properties.

The main question, then, is:

For a symbol-complete basis , what Arrovian JAR’s of exist?

3. Setting up the Main Problem

We give a few definitions and Lemmas that help us reduce the problem into smaller parts in prepration for Section 4. If is an JAR with judges on a basis , then for any subset , restricting each judgment from to gives a function , which we call the restriction of to . We make a few observations:

• For any , if is an Arrovian JAR on , the restriction of to is also Arrovian.

• If a set of propositions is not symbol-connected, then decomposes into a disjoint union of symbol-connected components, and the set of Arrovian JAR’s for is in bijection with the cartesian product of the sets of Arrovian JAR’s on .

• In the above decomposition, if is symbol-complete, then each of the is also symbol-complete.

The proofs of these observations are obvious and we omit them. The first observation allows us to freely consider subsets of our agenda without worrying about . In particular, we mostly apply this when we analyze for a symbol-closed compound proposition . The second and third observations say that instead of doing classifying symbol-complete directly, we can (and should) limit our attention to symbol-connected subsets of , who themselves must be symbol-complete.

Lemma 3.1.

If is symbol-complete and symbol-connected, then every compound proposition is determined by .

Proof.

As is compound, it is a function of its symbols and thus determined by its symbols. As is symbol-complete, contains all the symbols of and thus determines . ∎

Corollary 3.2.

If is symbol-closed in , then is determined by .

We warn the reader that the adjectives “symbol-complete” and “compound” in Lemma 3.1 are necessary, suggesting that the problem is much harder without them. If were not symbol-complete, does not have to determine ; e.g. . Even in the symbol-complete , the atomic proposition is not determined by the other propositions (though e.g. if and , we can deduce , but if and we know nothing about ).

We now give an important and rather surprising lemma that does not depend on symbol-completeness. Given a function , define to be the function with the same domain and codomain such that for all . Note that , so is an involution on the space of such functions.

Lemma 3.3.

Let depend on . Furthermore, assume that is determined by . If is an Arrovian JAR on , we must have or .

Proof.

Since is dependent on , must be -pivotal somewhere. In other words, there exists consistent judgments and such that , , and , but for all other propositions , .

Now, fix . For each , if or respectively, let or respectively. Let be the aggregated judgment. For example, the following illustrates the aggregation for :

 π(a1)π(a1)π(a1)π(a1)π(a1)→π(a1)π(a2)π(a2)π(a2)π(a2)π(a2)→π(a2)⋮⋮⋮⋮⋮→⋮TFTTF→fx(T,F,T,T,F)⋮⋮⋮⋮⋮→⋮π(am)π(am)π(am)π(am)π(am)→π(am)↓↓↓↓↓↓w=π(y)¬w=¬π′(y)ww¬w→fy(w,¬w,w,w,¬w)

By construction, all propositions not equal to or are identically judged by all the judgments . By UP, this means for all propositions . By the constraint that judgments must be logically consistent, if we must have (because and match on all the non- propositions and is determined by the other propositions) and if we must have . By construction, the truth values of and in will either be always equal (in which case their values in the aggregation are also equal) or always opposite (in which case their values in the aggregation are also opposite). As we can do this for all choices of , we see that for any input in , the function is always consistently equal to (when ) or the negation of applied to the negation of its input (when ), so our result is proved. ∎

Corollary 3.4.

Let be symbol-complete and symbol-connected and be an Arrovian JAR on . Then for every two propositions , we must have or .

Proof.

Consider every instance of a symbol (which must be in since it is symbol-complete) that appears in a compound proposition . By Lemma 3.3 and Lemma 3.1, we have or . Because is symbol-connected, there is a path between every two propositions using only edges between symbols and compounds (no two symbols share a symbol, and if compounds share a symbol, they must each have an edge to the symbol). Thus, because is an involution, any two propositions must actually be connected by or operation, which is what we desired. ∎

Corollary 3.5.

Let be symbol-complete and symbol-connected and there exists an Arrovian JAR on . Then there exists representing the same agenda as and for all .

Proof.

First, note that replacing a proposition in by preserves , but . For any , define to be all the elements (including ) with and define to be all the elements with . Now, we may replace all the propositions by . The resulting set of propositions has the same agenda as , but now has for all . ∎

Corollary 3.4 is very strong; it tells us that we can restrict our attention to where all the ’s guaranteed by the PI condition on an Arrovian JAR are actually the same function, which means we almost have Systematicity333A similar phenomenon happens in proofs of Arrow’s Theorem (see [6] for a short example), where potentially different aggregation functions on each “component” are forced to be identical. In the judgment aggregation framework, equivalent ideas have also been found in e.g. [2] and [3]. (the difference between our situation and Systematicity is that on , the functions are all flips of the functions in ), despite assuming only PI. If this happens, we say that the is in normal form for and denote, with abuse of notation, the unified function (that is, for all and ).

We now have a very clean and symmetric way to think about how an Arrovian JAR interacts with a symbol-closed compound proposition in . is a symbol-closed and symbol-connected subset of . Arrange the judgments on the propositions in a matrix , where is the judgment of judge on proposition . We can think of as a function that takes the values of the as input, applied to each column. Because is in normal form, we have a function applied to each row. In this setup, we are asking the following two operations to give the same answer:

• For each row , we compute to obtain aggregate propositional judgments , then compute .

• For each column , we compute to obtain aggregate compound judgments , then compute .

We make one more reduction. Recall that we assumed all the symbols in are relevant to . Similarly, we can assume all judges are relevant for by removing irrelevant judges. To obtain solutions for the general case, all we have to do is to add an arbitrary number of irrelevant judges back. In fact, a dictatorship is just the identity function444This observation allows us to think about Arrow’s Theorem in another way, which is “if we have at least alternatives, then the only Arrovian SPF where all the voters are relevant is the identity function .” with any number of irrelevant judges! Thus, we say that a pair of boolean functions and form a normal pair if:

• For all sets of , we have the equation

 f(g(M11,…,Mm1),…,g(M1n,…,Mmn))=g(f(M11,…,M1n),…,f(Mm1,…,Mmn)).
• Every index is relevant for .

• Every index is relevant for .

• Neither nor is the constant function (this restriction does not rule out anything interesting for our purposes. We want to be UP, so it cannot be constant. We want to be a compound proposition, which also cannot be constant if all its symbols are relevant).

Thus, studying compound propositions really come down to studying normal pairs. To be explicit, when we have an Arrovian JAR on a (not necessarily symbol-complete or symbol-connected!) basis , each symbol-closed compound proposition gives a , on which we can assume is in normal form. We then know is a normal pair where is UP. Our main question is, then, the following slight generalization (removing ’s UP requirement):

What normal pairs exist?

4. The Main Theorem

We now classify normal pairs with the aid of some elementary Fourier analysis555We have also obtained the relevant results, Propositions 4.3 and 4.5, with purely combinatorial proofs, but we decided to showcase the Fourier analysis proofs because our emphasis is on the techniques being new. on Boolean functions. In this section, we think of as and as , so we consider and as functions and respectively.

As before, we encode the normal pair as a matrix where each cell in the last row corresponds to applied to the cells above it and each cell in the last column corresponds to applied to the cells to the left of it. Being a normal pair, the last row and the last column are consistent with and respectively.

 M11M12⋯M1n→x1M21M22⋯M2n→x2⋮⋮⋱⋮→⋮Mm1Mm2⋯Mmn→xm↓↓↓↓y1y2⋯yn→z.
Lemma 4.1.

Suppose form a normal pair. Then consider any subset . For each , let , and for each let . Now, define

 RU={r∈[m]|for some y,(r,y)∈U},SU={s∈[n]|% for some x,(x,s)∈U}.

Then we have

 (∑S′⊃SU^f(S′)^g(∅)|S′−SU|)(∏s∈SU^g(Xs))=(∑R′⊃RU^g(R′)^f(∅)|R′−RU|)(∏r∈RU^f(Yr)).
Proof.

For a subset , we fill the upper-left submatrix of with ’s and ’s such that we place a at if and only if . For a normal pair, we need:

 f(g(M11,…,Mm1),…,g(M1n,…,Mmn))=g(f(M11,…,M1n),…,f(Mm1,…,Mmn)).

Expanding the left-hand side, we get

 f(∑R⊂[m]^g(R)∏r∈RMr1,…,∑R⊂[m]^g(R)∏r∈RMrn) = ∑S⊂[n]^f(S)∏s∈S(∑R^g(R)∏r∈RMrs).

Consider the coefficient of the term in this expression. In the outermost sum, we must select such that , otherwise it would be impossible to get all the terms. After a is selected, for each in the product, we must pick for and for . Each contributes and each contributes . As the same for each choice of , they can be factored out, and we obtain the left-hand side of our expression. Looking for the coefficient of in the right-hand side of the original equation gives a similar expression. As the form a basis of functions , the two coefficients must equal for all , which gives our result. ∎

Picking an and and letting be the rectangle in Lemma 4.1, we immediately obtain:

Corollary 4.2.

Suppose form a normal pair. Then for all nonempty and ,

 (∑S′⊃S^f(S′)^g(∅)|S′−S|)^g(R)|S|=(∑R′⊃R^g(R′)^f(∅)|R′−R|)^f(S)|R|.

Note that if either or equals , the problem is almost trivial: one function (the function with a single argument) must be the identity or the negative identity function (to make its only index relevant, the function cannot be constant), and the other function can be any function where every index is relevant. Thus, it suffices to consider only the case where and . We call these nontrivial normal pairs.

Proposition 4.3.

Suppose form a nontrivial normal pair and is the XOR or NXOR function. Then is also the XOR or NXOR function.

Proof.

Recall that has and all other coefficients equal to . Take a maximal (under set inclusion) such that . This means summing in Corollary 4.2 only picks up . Corollary 4.2 applied to and then gives . Since , . Thus, must be the only nonzero Fourier coefficient in . Because every index is relevant in , we must have , else there would be a symbol not appearing in the Fourier expansion of . This shows that is indeed the XOR function (or its negation). ∎

We are now ready to justify why we singled out XOR with Proposition 4.3:

Proposition 4.4.

Suppose form a nontrivial normal pair. Then one of these must hold:

• Both and are the XOR or NXOR functions.

• Both and are forceful functions.

Proof.

To start, we assume neither or are the XOR or NXOR functions (which, as we know from Proposition 4.3, would force the other function to also be of this form). We now show a contradiction arises if either of the functions (without loss of generality, ) is not forceful. This means we can assume that some index (without loss of generality, ), is free for .

We claim that for some and , is not -pivotal at . Otherwise, is -pivotal at every point for every index; it is easy to see by induction that only XOR and NXOR have this property, which we ruled out. Without loss of generality, we can assume . Because all the indices are relevant for , there must also be at least one such that is -pivotal at . Thus, we can conclude that is not -pivotal at some but is -pivotal at some . We now define some more values:

1. Since every index is relevant for , there is some such that is -pivotal at .

2. For each , because index is free for , there exist such that .

3. Let be arbitrary and .

We construct the following evaluation of :

 x1,1x1,2⋯x1,n−1s1→s′1x2,1x2,2⋯x2,n−1s2→s′2⋮⋮⋱⋮⋮→⋮xm−2,1xm−2,2⋯xm−2,n−1sm−2→s′m−2xm−1,1xm−1,2⋯xm−1,n−1sm−1→s′m−1z1z2⋯zn−1sm→s′m↓↓↓↓↓↓j1j2⋯jn−1jn→j.

Here, the rows are correct by our choices of and and we evaluate the by evaluating .

Now, change to in the -entry of the matrix. Because is not -pivotal at , does not change. As the first columns were unaffected, the other do not change either, so remains constant. However, because is -pivotal at , the -entry changes from to . Because is -pivotal at , the last column forces to change to . This gives a contradiction. ∎

Proposition 4.5.

Suppose form a nontrivial normal pair where and are forceful functions. Then can only be the AND function (resp. the OR function); must also be the AND function (resp. the OR function).

Proof.

By Lemma 2.2, we can find and , all in , such that

 g(r1,…,rm) =−c0+c012m−1(1+c1r1)(1+c2r2)⋯(1+cmrm), f(r1,…,rn) =−d0+d012n−1(1+d1r1)(1+d2r2)⋯(1+dnrn).

The Fourier coefficients of are (where ) for all and ; in particular, we know for all . We obtain similar conditions for . Apply Corollary 4.2 to and nonempty to obtain:

 (∑S′⊃S^f(S′)^g