The Dayenu Boolean Function Is Almost Always True!

TL;DR

This paper analyzes the Dayenu Boolean function, showing that for n miracles, the number of satisfying truth assignments is 2^n minus (n+1), revealing its near-universal truthfulness.

Contribution

It formally expresses the Dayenu function in conjunctive normal form and derives an exact count of its satisfying truth vectors.

Findings

Number of satisfying truth-vectors is 2^n - (n+1)

The function is almost always true for large n

Provides a formal logical analysis of a cultural song

Abstract

The Boolean function implicit in the famous Dayenu song, sung at the Passover meal, is expressed in full conjunctive normal form, and it is proved that if there are n miracles the number of truth-vectors satisfying it is $2^n -(n+1)$.

Equations

$$G_n(x_1,\dots ,x_n)=x_1\wedge \cdots \wedge x_n=\underset{i=1}{\overset{n}{\bigwedge }}{x}_i,$$

$$D_n(x_1,\dots ,x_n)=\underset{i=1}{\overset{n-1}{\bigvee }}{x}_i\overline{x_{i+1}}.$$

$$\overline{D_n(x_1,\dots ,x_n)}=\underset{i=1}{\overset{n-1}{\bigwedge }}(\overline{x_i}\vee x_{i+1}).$$

$$\overline{D_n(x_1,\dots ,x_n)}=\underset{i=1}{\overset{n+1}{\bigvee }}\left(\underset{j=1}{\overset{n-i+1}{\bigwedge }}\overline{x_j}\underset{j=n-i+2}{\overset{n}{\bigwedge }}{x}_j\right),$$

$$\overline{D_n(x_1,\dots ,x_n)}=\overline{x_1}\overline{x_2}\cdots \overline{x_{n-1}}\overline{x_n}\vee \overline{x_1}\overline{x_2}\cdots \overline{x_{n-1}}{x}_n\vee \dots \vee \overline{x_1}{x}_2\dots x_{n-1}{x}_n\vee x_1{x}_2\cdots x_n.$$

$$\overline{D_n(x_1,\dots ,x_n)}=\overline{D_{n-1}(x_1,\dots ,x_{n-1})}\wedge \left(\overline{x_{n-1}}\vee x_n\right)$$

$$=\overline{D_{n-1}(x_1,\dots ,x_{n-1})}\wedge \overline{x_{n-1}}\vee \overline{D_{n-1}(x_1,\dots ,x_{n-1})}\wedge x_n.$$

$$\overline{D_{n-1}(x_1,\dots ,x_{n-1})}=\overline{x_1}\overline{x_2}\cdots \overline{x_{n-2}}\overline{x_{n-1}}\vee \overline{x_1}\overline{x_2}\cdots \overline{x_{n-2}}{x}_{n-1}\vee \dots \vee \overline{x_1}{x}_2\dots x_{n-2}{x}_{n-1}\vee x_1{x}_2\cdots x_{n-1}.$$

$$\overline{D_{n-1}(x_1,\dots ,x_{n-1})}\wedge \overline{x_{n-1}}$$

$$=\overline{x_1}\overline{x_2}\cdots \overline{x_{n-2}}\overline{x_{n-1}}\overline{x_{n-1}}\vee \overline{x_1}\overline{x_2}\cdots \overline{x_{n-2}}{x}_{n-1}\overline{x_{n-1}}\vee \dots \vee \overline{x_1}{x}_2\dots x_{n-2}{x}_{n-1}\overline{x_{n-1}}\vee x_1{x}_2\cdots x_{n-1}\overline{x_{n-1}}.$$

$$\overline{D_{n-1}(x_1,\dots ,x_{n-1})}\wedge \overline{x_{n-1}}=\overline{x_1}\overline{x_2}\cdots \overline{x_{n-2}}\overline{x_{n-1}}.$$

$$\overline{D_{n-1}(x_1,\dots ,x_{n-1})}{x}_n=$$

$$\overline{x_1}\overline{x_2}\cdots \overline{x_{n-2}}\overline{x_{n-1}}{x}_n\vee \overline{x_1}\overline{x_2}\cdots \overline{x_{n-2}}{x}_{n-1}{x}_n\vee \dots \vee \overline{x_1}{x}_2\dots x_{n-2}{x}_{n-1}{x}_n\vee x_1{x}_2\cdots x_{n-1}{x}_n.$$

$$\overline{D_n(x_1,\dots ,x_n)}=$$

$$\overline{x_1}\overline{x_2}\cdots \overline{x_{n-2}}\overline{x_{n-1}}$$

$$\vee \overline{x_1}\overline{x_2}\cdots \overline{x_{n-2}}\overline{x_{n-1}}{x}_n\vee \overline{x_1}\overline{x_2}\cdots \overline{x_{n-2}}{x}_{n-1}{x}_n\vee \dots \vee \overline{x_1}{x}_2\dots x_{n-2}{x}_{n-1}{x}_n\vee x_1{x}_2\cdots x_{n-1}{x}_n.$$

$$\overline{x_1}\overline{x_2}\cdots \overline{x_{n-2}}\overline{x_{n-1}}=\overline{x_1}\overline{x_2}\cdots \overline{x_{n-2}}\overline{x_{n-1}}\overline{x_n}\vee \overline{x_1}\overline{x_2}\cdots \overline{x_{n-2}}\overline{x_{n-1}}{x}_n,$$

$$\overline{D_n(x_1,\dots ,x_n)}=$$

$$\overline{x_1}\overline{x_2}\cdots \overline{x_{n-2}}\overline{x_{n-1}}\overline{x_n}\vee \overline{x_1}\overline{x_2}\cdots \overline{x_{n-2}}\overline{x_{n-1}}{x}_n\vee$$

$$\overline{x_1}\overline{x_2}\cdots \overline{x_{n-2}}\overline{x_{n-1}}{x}_n\vee \overline{x_1}\overline{x_2}\cdots \overline{x_{n-2}}{x}_{n-1}{x}_n\vee \dots \vee \overline{x_1}{x}_2\dots x_{n-2}{x}_{n-1}{x}_n\vee x_1{x}_2\cdots x_{n-1}{x}_n.$$

$$\overline{D_n(x_1,\dots ,x_n)}=$$

$$\overline{x_1}\overline{x_2}\cdots \overline{x_{n-2}}\overline{x_{n-1}}\overline{x_n}\vee \overline{x_1}\overline{x_2}\cdots \overline{x_{n-2}}\overline{x_{n-1}}{x}_n\vee \overline{x_1}\overline{x_2}\cdots \overline{x_{n-2}}{x}_{n-1}{x}_n\vee \dots \vee \overline{x_1}{x}_2\cdot x_n\vee x_1{x}_2\cdots x_{n-1}{x}_n.$$

Full text

The Dayenu Boolean Function Is Almost Always True!

Doron ZEILBERGER

Dedicated to Hemi and Yael Nae

[Recall, from Logic 101, that for any two statements, $X,Y$, $X\vee Y$ means that “either $X$ or $Y$ or both happened”, $X\wedge Y$ means that both $X$ and $Y$ happened, while $\bar{X}$ means that $X$ did not happen. For typographical clarity, $X\wedge Y$ is often written $XY$.]

Two nights ago, my wife Jane and I were fortunate to be guests in a wonderful Passover seder at the house of Hemi††1

Hemi and I were dorm-mates at the Weizmann Institute, way back in the early seventies. His son is also called Doron, so we call him ‘little Doron’, while I am ‘big Doron’. ‘Little Doron’ is no longer so little (he is $25$-years-old), but Hemi commented that he knows me (‘big Doron’) much longer than he knows ‘little Doron’. and Yael Nae. Soon enough we came to the number-one hit song, Dayenu, praising God for doing $15$ amazing miracles, let’s call them $x_{1},\dots,x_{15}$, where $x_{1}$ stands for “Took us out of Egypt”, and $x_{15}$ stands for “Built us our Temple”. The exact nature of the other $13$ miracles (and for that matter the above two ) are irrelevant to this mathematical paper, but can be easily looked-up in any Haggadah (and of course, nowadays, on the internet).

It so happened that what God actually did can be described by the Boolean function (with $n=15$)

[TABLE]

whose only truth vector is the all-true vector $T^{n}$, i.e., assuming, that the probability of any one miracle occurring is $p$ (and that they are independent events), has the tiny probability of $p^{n}$.

But the author of Dayenu asserts that God was an over-achiever, and we, the children of Israel, should have been content if the following Boolean function would have been satisfied.

[TABLE]

The following question immediately came to my mind: How many (and which) truth-vectors are satisfied by the Dayenu function, and what is the probability that God, deciding randomly which miracles to perform and which not to perform, would have satisfied the minimum requirement demanded by the anonymous author of Dayenu?

Thanks to De Morgan we have

[TABLE]

We are now ready for

The Dayenu Theorem. The full disjunctive normal form of the negation of the Dayenu Boolean function is given by

[TABLE]

or, more concretely:

[TABLE]

Proof: By induction on $n$. It is true when $n=2$ (check!). Assume that it is true when $n$ is replaced by $n-1$. Note that

[TABLE]

[TABLE]

By the inductive hypothesis:

[TABLE]

Regarding $\overline{D_{n-1}(x_{1},\dots,x_{n-1})}\,\wedge\,\overline{x_{n-1}}$ we have

[TABLE]

[TABLE]

Since $x_{n-1}\,\overline{x_{n-1}}$ is FALSE, all the above terms, except the first, vanish, and since $\overline{x_{n-1}}\,\overline{x_{n-1}}=\overline{x_{n-1}}$, we have

[TABLE]

Regarding $\overline{D_{n-1}(x_{1},\dots,x_{n-1})}\,\wedge\,x_{n}$ we have

[TABLE]

[TABLE]

Combining, we have

[TABLE]

[TABLE]

[TABLE]

Since

[TABLE]

we get

[TABLE]

[TABLE]

[TABLE]

Since the second and third term above are the same (and $X\vee X=X$), we finally get

[TABLE]

[TABLE]

Corollary: Assuming that the probability of each miracle is ${1\over 2}$, and that they are done independently, the probability of not meeting the Dayenu requirement is ${n+1\over 2^{n}}$ and hence of meeting it is $1-{n+1\over 2^{n}}$, that happens to be, for $n=15$, ${2047\over 2048}=0.9995117\dots$  .

Comments:

1. Surprisingly, what God actually did, performing all the miracles, is not part of the truth-set of the Dayenu Boolean function, since there is always at least one miracle that is not performed.

2. A faster proof, without Boolean logic, for getting the set of true-false vectors not satisfying the Dayenu function $D_{n}$ (and hence proving the above Dayenu theorem), can be gotten by finding the set of all members of $\{T,F\}^{n}$ for which an $F$ never (immediately) follows a $T$. Of course, this set is $\{F^{n},F^{n-1}T,\dots,T^{n}\}$.

16 Nisan 5777. Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger , http://www.math.rutgers.edu/~zeilberg/pj.html, and arxiv.org .

Doron Zeilberger, Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA. Email: DoronZeil at gmail dot com  .