# Kempner-like harmonic series

Jean-Paul Allouche111CNRS, IMJ-PRG, Sorbonne, 4 Place Jussieu, F-75252 Paris
Cedex 05, France
<EMAIL_ADDRESS>
Claude Morin
<EMAIL_ADDRESS>
###### Abstract

Inspired by a question asked on the list mathfun, we revisit Kempner-like
series, i.e., harmonic sums $\sum^{\prime}1/n$ where the integers $n$ in the
summation have ``restricted'' digits. First we give a short proof that
$\lim_{k\to\infty}(\sum_{s_{2}(n)=k}1/n)=2\log 2$, where $s_{2}(n)$ is the sum
of the binary digits of the integer $n$. Then we propose two generalizations.
One generalization addresses the case where $s_{2}(n)$ is replaced with
$s_{b}(n)$, the sum of $b$-ary digits in base $b$: we prove that
$\lim_{k\to\infty}\sum_{s_{b}(n)=k}1/n=(2\log b)/(b-1)$. The second
generalization replaces the sum of digits in base $2$ with any block-counting
function in base $2$, e.g., the function $a(n)$ of —possibly overlapping—
$11$'s in the base-$2$ expansion of $n$, for which we obtain
$\lim_{k\to\infty}\sum_{a(n)=k}1/n=4\log 2$.

##  1 Introduction

A nice, now classical, 1914 result of Kempner [23] states that the sum of the
inverses of the integers whose expansion in base $10$ contains no occurrence
of a given digit ($\neq 0$) converges. This fact might seen amazing at first
view, but looking, e.g., at all integers whose decimal expansion has no $9$ in
it, one sees that larger and larger ranges of integers are excluded (think of
all integers between $9\cdot 10^{k}$ and $10^{k+1}-1$). After the 1914 paper
of Kempner [23] and the 1926 paper of Irwin [22], several papers were devoted
to generalizations or extensions of this result, as well as numerical
computations of the corresponding series. The reader can look at, e.g., [1, 7,
8, 9, 12, 16, 17, 18, 19, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35] and
the references therein.

In particular the paper of Farhi [17] proves the somehow unexpected result
that, if $c_{j,10}(n)$ denote the number of occurrences of a fixed digit
$j\in\\{0,1,\dots,9\\}$ in the base-$10$ expansion of $n$, then

$\lim_{k\to\infty}\sum_{\begin{subarray}{c}n\geq 1\\\
c_{j,10}(n)=k\end{subarray}}\frac{1}{n}=10\log 10.$

Replacing base $10$ with base $2$ and letting $c_{1,2}(n)$ denote the number
of $1$'s in the binary expansion of $n$, we could expect that

$\lim_{k\to\infty}\sum_{\begin{subarray}{c}n\geq 1\\\
c_{1,2}(n)=k\end{subarray}}\frac{1}{n}=2\log 2.$

The series on the lefthand side is precisely the one occurring in a recent
question on mathfun, that was forwarded to one of the authors by J. Shallit.
Actually, in the post $c_{1,2}(n)$ is replaced with $s_{2}(n)$ which is of
course the same; the question was to determine the value of the limit when $k$
goes to infinity.

First we give a short proof of the following theorem which answers the mathfun
question.

###### Theorem 1

The following equality holds:

$\lim_{k\to\infty}\sum_{\begin{subarray}{c}n\geq 1\\\
s_{2}(n)=k\end{subarray}}\frac{1}{n}=2\log 2.$ (1)

Then we investigate two natural generalizations of this result. In the first
one we replace the sum of binary digits with the sum of $b$-ary digits.

###### Theorem 2

Let $b\geq 2$ be an integer. Let $s_{b}(n)$ be the sum of digits of $n$ in
base $b$. Then

$\lim_{k\to\infty}\sum_{\begin{subarray}{c}n\geq 1\\\
s_{b}(n)=k\end{subarray}}\frac{1}{n}=\frac{2\log b}{b-1}\cdot$ (2)

In the second generalization we replace the sum of digits in base $2$, i.e.,
the number of $1$'s in base $2$, with $a_{w}(n)$ the number of occurrences of
a word $w$ (a fixed string of consecutive digits) in the binary expansion of
the integer $n$.

###### Theorem 3

Let $w$ be a binary word with $r$ letters. Then

$\lim_{k\to\infty}\sum_{\begin{subarray}{c}n\geq 1\\\
a_{w}(n)=k\end{subarray}}\frac{1}{n}=2^{r}\log 2.$ (3)

###### Remark 4

We have essentially limited the references to ``missing digits'' given before
Theorem 1 to harmonic series and Dirichlet series whose summation indexes
``miss'' digits or combinations of digits. Integers with missing digits in a
given base are called ``ellipsephic''. They occur in several papers (e.g., [5,
6, 15, 10, 11, 13]). Nicholas Yu indicates in [20, Footnote, p. 6]:

> ``This word is a translation of the French ellipséphique, which Mauduit
> coined as a port-manteau of the Greek words ἔλλειψι̌ς (élleipsis,
> ``ellipisis'') and ψηφίο (psifío, ``digit''). We prescribe the English
> pronunciation [””ı.lıp”sEf.ık].''

The word in French was proposed by Christian Mauduit. Its origin is given by
Sylvain Col in [14, p. 12]:

> ``[… les progressions arithmétiques]. C. Mauduit les a baptisés entiers
> ellipséphiques en ré-férence à la superposition de deux mots grecs,
> ελλιπτιϰος (littéralement elliptique) et ψνηον (littéralement petit caillou
> poli par l'eau ; ces cailloux étaient notamment utilisés pour voter et
> réaliser les calculs) et signifie qui a des chiffres manquants. [Parmi
> les…]''

Note that there were some misprints in the Greek words in the original
quotations above. Namely ``ἔλλειψι̌ς'', ``ψηφίο'', `` ελλιπτιϰος'', and
``ψνηον'' should be replaced respectively with ``ἔλλειψις'', ``ψη̃φος'',
``ἐλλειπτιϰός'', and ``ψη̃φος''.

##  2 A short proof of Theorem 1

The fact that the series $A_{k}:=\displaystyle\sum_{\begin{subarray}{c}n\geq
1\\\ s_{2}(n)=k\end{subarray}}\frac{1}{n}$ converges can be easily proved by a
counting argument (adapting, e.g., a proof given in [22]) or by noting that
$s_{2}(n)$ is the number of $1$'s in the binary expansion of $n$ and using the
proof of Lemma 1 in [2]. Let us suppose $k\geq 2$. Splitting the sum into even
and odd indices, and recalling that $s_{2}(2n)=s_{2}(n)$ and
$s_{2}(2n+1)=s_{2}(n)+1$, we obtain:

$A_{k}=\sum_{\begin{subarray}{c}n\geq 1\\\
s_{2}(2n)=k\end{subarray}}\frac{1}{2n}+\sum_{\begin{subarray}{c}n\geq 0\\\
s_{2}(2n+1)=k\end{subarray}}\frac{1}{2n+1}=\sum_{\begin{subarray}{c}n\geq 1\\\
s_{2}(n)=k\end{subarray}}\frac{1}{2n}+\sum_{\begin{subarray}{c}n\geq 0\\\
s_{2}(n)=k-1\end{subarray}}\frac{1}{2n+1}=\frac{1}{2}A_{k}+\sum_{\begin{subarray}{c}n\geq
0\\\ s_{2}(n)=k-1\end{subarray}}\frac{1}{2n+1}$

which we rewrite as

$A_{k}=2\sum_{\begin{subarray}{c}n\geq 0\\\
s_{2}(n)=k-1\end{subarray}}\frac{1}{2n+1}=A_{k-1}+B_{k}$

where $B_{k}:=2\displaystyle\sum_{\begin{subarray}{c}n\geq 1\\\
s_{2}(n)=k-1\end{subarray}}\left(\frac{1}{2n+1}-\frac{1}{2n}\right)$. Thus, we
have, for $k\geq 2$,

$\begin{array}[]{llll}A_{k}&-&A_{k-1}&=\ B_{k}\\\ A_{k-1}&-&A_{k-2}&=\
B_{k-1}\\\ \ldots\\\ A_{2}&-&A_{1}&=\ B_{2}.\end{array}$

Hence, summing these equalities,

$A_{k}-A_{1}=\sum_{2\leq j\leq k}B_{j}=2\sum_{2\leq j\leq
k}\sum_{\begin{subarray}{c}n\geq 1\\\
s_{2}(n)=j-1\end{subarray}}\left(\frac{1}{2n+1}-\frac{1}{2n}\right)$

i.e.,

$A_{k}-A_{1}=2\sum_{\begin{subarray}{c}n\geq 1\\\ s_{2}(n)\leq
k-1\end{subarray}}\left(\frac{1}{2n+1}-\frac{1}{2n}\right).$

The righthand term clearly tends to $2\displaystyle\sum_{n\geq
1}\left(\frac{1}{2n+1}-\frac{1}{2n}\right)$ when $k$ tends to infinity, thus
the lefthand term has a limit and

$\lim_{k\to\infty}A_{k}=A_{1}-2\sum_{n\geq
1}\left(\frac{1}{2n}-\frac{1}{2n+1}\right)$

Now

$A_{1}=\sum_{\begin{subarray}{c}n\geq 1\\\
s_{2}(n)=1\end{subarray}}\frac{1}{n}=\sum_{j\geq 0}\frac{1}{2^{j}}=2.$

Hence

$\lim_{k\to\infty}A_{k}=2\left(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}\dots\right)=2\log
2.\ \ \ \ \ \Box$

Before proving our first generalization (Theorem 2), we will prove a general
result on convergence of sequences and a corollary which will be useful.

##  3 A general result on convergence of sequences

###### Theorem 5

Let $P(X)=\sum_{0\leq k\leq d}a_{k}X^{d-k}$ be a polynomial having all its
roots in ${\mathbb{C}}$ of modulus $<1$. For a sequence $(u_{n})_{n\geq 0}$,
define the sequence $(u^{(P)}_{n})_{n\geq d}$ by $u^{(P)}_{n}:=\sum_{0\leq
k\leq d}a_{k}u_{n-k}$. Then the sequence $(u_{n})_{n\geq 0}$ tends to $0$ if
and only if the sequence $(u^{(P)}_{n})_{n\geq d}$ tends to $0$.

Proof. Since one direction is trivial, we only prove that if
$(u^{(P)}_{n})_{n\geq d}$ tends to $0$, then $(u_{n})_{n\geq 0}$ tends to $0$.

First we look at the case $d=1$. Suppose that $z$ is a complex number with
$|z|<1$. We prove that if the sequence $(w_{n})_{n\geq 1}$ tends to $0$, where
$w_{n}:=u_{n}-zu_{n-1}$, then the sequence $(u_{n})_{n\geq 0}$ tends to $0$.
Namely, if $(w_{n})_{n\geq 1}$ tends to $0$, we have

$\forall\varepsilon>0,\ \exists n_{0}\geq 1,\text{\ such that for \ }n\geq
n_{0}\text{\ one has \ }|w_{n}|\leq(1-|z|)\varepsilon.$

But, for all $p\in{\mathbb{N}}$, one has by an easy induction

$u_{n_{0}+p}=\sum_{0\leq k\leq p}z^{k}w_{n_{0}+p-k}+z^{p+1}u_{n_{0}-1}.$

Hence, for $p$ larger than some $p_{0}$, one has
$|u_{n_{0}+p}|\leq\varepsilon+|z|^{p+1}|u_{n_{0}-1}|\leq 2\varepsilon$.

Now we can address the general case where $P(X)=\sum_{0\leq k\leq
d}a_{k}X^{d-k}=\prod_{1\leq j\leq d}(X-z_{j})$, with $|z_{j}|<1$ for all $j$.
Defining $\varphi_{j}((u_{n})_{n}):=((u_{n}-z_{j}u_{n-1})_{n})$, it is easy to
see that
$(u^{(P)}_{n})_{n}=(\varphi_{d}\circ\varphi_{d-1}\circ\dots\circ\varphi_{1})((u_{n})_{n})$.
Thus it suffices to apply $d$ times the case $d=1$ above. $\Box$

###### Corollary 6

Let $b$ be an integer $>1$. Then

$\lim_{n\to\infty}((b-1)u_{n}+(b-2)u_{n-1}+\dots+2u_{n-b+3}+u_{n-b+2})=\ell\text{
\ if and only if \ }\lim_{n\to\infty}u_{n}=\dfrac{2\ell}{b(b-1)}\cdot$

Proof. Of course one can suppose $b>2$. Again one direction is trivial. For
the other direction, up to replacing $(u_{n})_{n\geq 0}$ with
$(u^{\prime}_{n})_{n\geq 0}$, where $u^{\prime}_{n}:=u_{n}-2\ell/(b(b-1))$, we
can suppose that $\ell=0$. Now, in order to apply Theorem 5 with
$P(X)=(b-1)X^{b-2}+(b-2)X^{b-3}+\dots+2X+1$, it suffices to prove that all the
(complex) roots of this polynomial have modulus $<1$. We note that
$(1-X)P(X)=1+X+X^{2}+\dots+X^{b-2}-(b-1)X^{b-1}$. Hence if $P(z)=0$ for some
$z$ with $|z|\geq 1$, then $(b-1)|z|^{b-1}\leq
1+|z|+\dots+|z|^{b-2}\leq(b-1)|z|^{b-2}$, hence $|z|=1$. Furthermore equality
in the triangular inequality implies here that $z$ is real and non-negative,
hence equal to $1$. Since $1$ is not a root of $P$, this gives the desired
contradiction: hence, necessarily $|z|<1$. $\Box$

##  4 A first generalization. Proof of Theorem 2

We begin with a lemma.

###### Lemma 7

We have the following properties.

  * (i)

The sum $\sum_{s_{b}(n)=k}\frac{1}{n}$ is finite.

  * (ii)

For any $j\in\\{0,1,\dots,b-1\\}$,

$\lim_{k\to\infty}\sum_{s_{b}(n)=k}\left(\frac{1}{bn+j}-\frac{1}{bn}\right)=0.$

  * (iii)

For $n\geq 1$, let $H_{n}=1+\frac{1}{2}+\dots+\frac{1}{n}$ be the $n$-th
harmonic number. Then, for $b>1$,

$\lim_{k\to\infty}\sum_{1\leq s_{b}(n)\leq k}\left(\sum_{0\leq j\leq
b-1}\frac{1}{bn+j}-\frac{1}{n}\right)=\log b-H_{b-1}.$

Proof. To prove (i) we note that the number of integer solutions of
$x_{1}+x_{2}+\dots+x_{j}=k$ is equal to $\displaystyle{j+k-1\choose k}$, hence

$\sum_{\begin{subarray}{c}b^{j-1}\leq n\leq b^{j}\\\
s_{b}(n)=k\end{subarray}}\frac{1}{n}\leq\frac{{j+k-1\choose
k}}{b^{j-1}}\sim\frac{j^{k}}{k!b^{j-1}}\cdot$

The convergence of the series $\displaystyle\sum_{j\geq
1}\frac{j^{k}}{k!b^{j-1}}$ implies the existence of
$\displaystyle\sum_{s(n)=k}\frac{1}{n}\cdot$

In order to prove (ii), we note that for any $j\in\\{0,1,\dots,b-1\\}$, we
have

$\sum_{0\leq j\leq b-1}\frac{1}{bn+j}-\frac{1}{n}=\sum_{0\leq j\leq
b-1}\left(\frac{1}{bn+j}-\frac{1}{bn}\right)\text{\ \ is a sum of negative
terms.}$

Hence (ii) holds if and only if
$\displaystyle\lim_{k\to\infty}\sum_{s_{b}(n)=k}\sum_{0\leq j\leq
b-1}\left(\frac{1}{bn+j}-\frac{1}{bn}\right)=0$. But

$\sum_{s_{b}(n)=k}\sum_{0\leq j\leq
b-1}\left(\frac{1}{bn+j}-\frac{1}{bn}\right)=\sum_{s_{b}(n)\leq k}\sum_{0\leq
j\leq b-1}\left(\frac{1}{bn+j}-\frac{1}{bn}\right)\ -\sum_{s_{b}(n)\leq
k-1}\sum_{0\leq j\leq b-1}\left(\frac{1}{bn+j}-\frac{1}{bn}\right).$

Thus, it suffices to prove (iii).

Finally we prove (iii). Define $v_{n}:=\displaystyle\sum_{0\leq j\leq
b-1}\frac{1}{bn+j}-\frac{1}{n}=\sum_{0\leq j\leq
b-1}\left(\frac{1}{bn+j}-\frac{1}{bn}\right).$ We have

$\sum_{1\leq n\leq N}v_{n}=H_{bN+b-1}-H_{b-1}-H_{N}$

which tends to $\log b-H_{b-1}$ when $N$ tends to infinity. Now, $v_{n}\leq
0$, and, if $n\leq b^{k}-1$, then $s_{b}(n)\leq k(b-1)$. Hence

$\sum_{1\leq n\leq b^{k}-1}v_{n}\geq\sum_{1\leq s_{b}(n)\leq k(b-1)}v_{n}\geq\
\sum_{n\geq 1}v_{n}=\log b-H_{b-1}.$

Since $\displaystyle\sum_{1\leq n\leq b^{k}-1}v_{n}$ tends to $\log
b-H_{b-1}$, we obtain the desired result. $\Box$

Proof of Theorem 2

Define $\displaystyle u_{k}:=\sum_{s_{b}(n)=k}\frac{1}{n}$. Then, splitting
the integers according to their value modulo $b$, we have

$u_{k}=\sum_{s_{b}(bn)=k}\frac{1}{bn}+\sum_{1\leq j\leq
b-1}\sum_{\begin{subarray}{c}n\geq 0\\\
s_{b}(bn+j)=k\end{subarray}}\frac{1}{bn+j}$

thus, using that $s_{b}(bn+j)=s_{b}(n)+j$ for $j\in\\{0,1,\dots,b-1\\}$,

$u_{k}=\frac{1}{b}u_{k}+\sum_{1\leq j\leq b-1}\sum_{\begin{subarray}{c}n\geq
0\\\ s_{b}(n)=k-j\end{subarray}}\frac{1}{bn+j}\cdot$

Hence

$\left(1-\frac{1}{b}\right)(u_{1}+\dots+u_{k})=\sum_{j=1}^{b-1}\sum_{\begin{subarray}{c}n\geq
0\\\ s_{b}(n)\leq k-j\end{subarray}}\frac{1}{bn+j}\cdot$

Thus, by subtracting $\left(1-\frac{1}{b}\right)(u_{1}+\dots+u_{k-1})$ with
the definition of the $u_{j}$'s,

$\left(1-\frac{1}{b}\right)u_{k}=\sum_{j=1}^{b-1}\sum_{\begin{subarray}{c}n\geq
0\\\ s_{b}(n)\leq
k-j\end{subarray}}\frac{1}{bn+j}-\left(1-\frac{1}{b}\right)\sum_{1\leq
s_{b}(n)\leq k-1}\frac{1}{n}\cdot$

Hence

$\left(1-\frac{1}{b}\right)u_{k}=H_{b-1}+\sum_{j=1}^{b-1}\sum_{\begin{subarray}{c}n\geq
1\\\ s_{b}(n)\leq
k-j\end{subarray}}\frac{1}{bn+j}-\left(1-\frac{1}{b}\right)\sum_{1\leq
s_{b}(n)\leq k-1}\frac{1}{n}\cdot$ (4)

Let us define as in Lemma 7(iii) $w_{k-1}$ by

$w_{k-1}:=\sum_{1\leq s_{b}(n)\leq k-1}\left(\sum_{0\leq j\leq
b-1}\frac{1}{bn+j}-\frac{1}{n}\right)$

We have

$\begin{array}[]{lll}w_{k-1}&=&\displaystyle\sum_{1\leq s_{b}(n)\leq k-1}\ \
\sum_{1\leq j\leq b-1}\frac{1}{bn+j}-\left(1-\frac{1}{b}\right)\sum_{1\leq
s_{b}(n)\leq k-1}\frac{1}{n}\\\ &=&\ \ \displaystyle\sum_{1\leq j\leq b-1}\ \
\sum_{1\leq s_{b}(n)\leq
k-1}\frac{1}{bn+j}-\left(1-\frac{1}{b}\right)\sum_{1\leq s_{b}(n)\leq
k-1}\frac{1}{n}\\\ &=&\ \displaystyle\sum_{1\leq j\leq b-1}\ \ \sum_{1\leq
s_{b}(n)\leq k-j}\frac{1}{bn+j}+\displaystyle\sum_{1\leq j\leq b-1}\ \
\sum_{k-j+1\leq s_{b}(n)\leq
k-1}\frac{1}{bn+j}-\left(1-\frac{1}{b}\right)\sum_{1\leq s_{b}(n)\leq
k-1}\frac{1}{n}\\\ &=&\ \displaystyle\sum_{1\leq j\leq b-1}\ \ \sum_{1\leq
s_{b}(n)\leq k-j}\frac{1}{bn+j}+\displaystyle\sum_{2\leq j\leq b-1}\ \
\sum_{k-j+1\leq s_{b}(n)\leq
k-1}\frac{1}{bn+j}-\left(1-\frac{1}{b}\right)\sum_{1\leq s_{b}(n)\leq
k-1}\frac{1}{n}\cdot\\\ \end{array}$

Hence we can write, using Equation (4),

$\left(1-\frac{1}{b}\right)u_{k}=H_{b-1}+w_{k-1}-R_{k}$

with

$R_{k}=\sum_{2\leq j\leq b-1}\ \ \sum_{k-j+1\leq s_{b}(n)\leq
k-1}\frac{1}{bn+j}.$

Then, when $k\to\infty$, we can write, by using Lemma 7(ii),

$\begin{array}[]{lll}R_{k}&=&\displaystyle\sum_{2\leq j\leq b-1}\ \sum_{1\leq
i\leq j-1}\ \sum_{s_{b}(n)=k-i}\frac{1}{bn+j}\\\ &=&\displaystyle\frac{1}{b}\
\sum_{2\leq j\leq b-1}\sum_{1\leq i\leq j-1}u_{k-i}+o(1)\\\
&=&\displaystyle\frac{1}{b}\sum_{1\leq i\leq
b-2}(b-i-1)u_{k-i}+o(1).\end{array}$

This gives

$H_{b-1}+w_{k-1}=\left(1-\frac{1}{b}\right)u_{k}+R_{k}=\left(1-\frac{1}{b}\right)u_{k}+\frac{1}{b}\sum_{1\leq
i\leq b-2}(b-i-1)u_{k-i}+o(1)$

but, from Lemma 7(iii), $H_{b-1}+w_{k-1}$ tends to $\log b$. Hence

$\left(1-\frac{1}{b}\right)u_{k}+\frac{1}{b}\sum_{1\leq i\leq
b-2}(b-i-1)u_{k-i}\to\log b$

or, equivalently,

$\sum_{0\leq i\leq b-2}(b-i-1)u_{k-i}\to b\log b.$

Applying Corollary 6 yields

$u_{k}\to\frac{2\log b}{b-1}\cdot\ \ \ \Box$

##  5 A second generalization. Proof of Theorem 3

Comparing the equalities (see, e.g., [3] and the references therein for the
second equality)

$\lim_{k\to\infty}\sum_{s_{2}(n)=k}\frac{1}{n}=2\log 2\ \ \text{and}\ \
\sum_{n\geq 1}\frac{s_{2}(n)}{n(n+1)}=2\log 2$

it is tempting to prove directly that the two left-hand quantities are equal.
We did not succeed, but we found that a method permitting to prove the second
equality can be used for proving the first one, thus yielding a generalization
to all base-$2$ pattern counting sequences.

Let $w$ be a word of $0$'s and $1$'s. We let $a_{w}(n)$ denote the number of
occurrences of $w$ in the binary expansion of the integer $n$. As usual, if
$w$ begins with $0$ and is not of the form $0^{\ell}$ —the word consisting of
$\ell$ digits equal to $0$— we assume that the binary expansion of $n$ begins
with an arbitrarily long prefix of $0$'s. And if $w=0^{\ell}$, we use the
classical binary expansion of $n$ beginning with $1$. (For example, taking the
respective binary expansions of $5$ and $8$, namely $5=(0...0)101$ and
$8=1000$, one has $a_{01}(5)=a_{01}(0...0101)=2$ and
$a_{000}(8)=a_{000}(1000)=1$.) Also recall that $|w|$ is the length (i.e., the
number of letters) of the word $w$. First we prove the following lemma.

###### Lemma 8

Define $a_{w}(n)$ as above. Let $(f(n))_{n\geq 1}$ be a sequence of positive
reals such that $\sum_{n}f(n)<+\infty$. Let
$c_{k}:=\displaystyle\sum_{\begin{subarray}{c}n\geq 1\\\
a_{w}(n)=k\end{subarray}}f(n)$ and
$d_{k}:=\displaystyle\sum_{\begin{subarray}{c}n\geq 1\\\
a_{w}(n)=k\end{subarray}}\frac{1}{n}$. Then

  * (i)

One has $d_{k}<+\infty$.

  * (ii)

The series $\displaystyle\sum_{k\geq 0}c_{k}$ converges.

  * (iii)

The sequence $(c_{k})_{k\geq 0}$ tends to $0$ when $k$ tends to infinity.

Proof. The first assertion can be found, e.g., in the proof of Lemma 1 in [2].
The third assertion is a consequence of the second one. Finally, to prove that
$\sum c_{k}$ converges, we write (note that all terms are positive):

$\sum_{k\geq 0}c_{k}=\sum_{k\geq 0}\sum_{\begin{subarray}{c}n\geq 1\\\
a_{w}(n)=k\end{subarray}}f(n)=\sum_{n\geq 1}f(n)<+\infty.\ \ \ \ \ \Box$

Proof of Theorem 3

The idea for proving this theorem is to compare the quantity
$\displaystyle\sum_{\begin{subarray}{c}n\geq 1\\\
a_{w}(n)=k\end{subarray}}\frac{1}{n}$ with a series
$\displaystyle\sum_{\begin{subarray}{c}n\geq 1\\\
a_{w}(n)=k\end{subarray}}g_{w}(n)$ whose sum is known and converges to some
limit $A_{w}$ for $k\to\infty$. If furthermore
$g_{w}(n)-1/(2^{|w|}n)={\mathcal{O}}_{w}(1/n^{2})$, then Lemma 8 will imply
the existence and the value of

$\lim_{k\to\infty}\sum_{\begin{subarray}{c}n\geq 1\\\
a_{w}(n)=k\end{subarray}}\frac{1}{n}=2^{|w|}A_{w}.$

The choice of the function $g$ will use [2] where the authors prove that there
exists a rational function $b_{w}$ such that for all $k\geq 0$, one has

$\sum_{\begin{subarray}{c}n\geq 1\\\
a_{w}(n)=k\end{subarray}}\log(b_{w}(n))=-\log 2\ \ \ \text{(independent of
$k$)}.$

The paper [2] explains how to construct $b_{w}$. This construction is given by
a more explicit recursive algorithm in [4]. Taking $g_{w}$ defined by
$g_{w}=-\log(b_{w})$, it will suffice to prove that
$-\log(b_{w}(n))-1/(2^{|w|}n)={\mathcal{O}}_{w}(1/n^{2})$.

Using [4], we have that, if $w=w_{1}w_{2}\dots w_{m}$, then $\log(b_{w}(n))$
is given in [4] by

$\log(b_{w}(n))=Q_{w}(w_{1}w_{2}\dots w_{m-1},w_{m},n),$

where, for $z=z_{1}\dots z_{r}$ and $t$ two binary words, $Q_{w}$ is
recursively defined by:

$Q_{w}(z,t,n):=\begin{cases}\log(2^{|t|}n+\nu(t))-\log(2^{|t|}n+\nu(t)+1)\ \
&\text{if $r=0$},\\\
Q_{w}(\varepsilon,t,n)-Q_{w}(\varepsilon,\overline{z_{r}}t,n)&\text{if $r=1$
and $z$ is a suffix of $w$},\\\ Q_{w}(z_{2}z_{3}\dots
z_{r},t,n)-Q_{w}(\overline{z_{1}}z_{2}\dots z_{r-1},z_{r}t,n)&\text{if $r\geq
2$ and $z$ is a suffix of $w$},\\\ Q_{w}(z_{1}z_{2}\dots
z_{r-1},z_{r}t,n)&\text{if $r\geq 1$ and $z$ is not a suffix of
$w$},\end{cases}$

where, for $x\in\\{0,1\\}$, one defines $\overline{x}:=1-x$, where $\nu(t)$ is
the value of the word $t$ when interpreted as a binary expansion, and
$\varepsilon$ is the empty word (the word with no letter). Also recall that
$|t|$ is the length (i.e., the number of letters) of the word $w$.

The behavior of $Q_{w}(z,t,n)$ when $n$ tends to infinity can be proved by
induction on $|z|\geq 1$. We claim that, for all $t$,

$Q_{w}(z,t,n)=-\frac{1}{2^{|t|+|z|}n}+{\mathcal{O}}_{z,t}\left(\frac{1}{n^{2}}\right)\cdot$

If $|z|=0$, i.e., $z=\varepsilon$, we have

$Q_{w}(z,t,n)=\log(2^{|t|}n+\nu(t))-\log(2^{|t|}n+\nu(t)+1)=-\frac{1}{2^{|t|}n}+{\mathcal{O}}_{t}\left(\frac{1}{n^{2}}\right)=-\frac{1}{2^{|t|+|z|}n}+{\mathcal{O}}_{t}\left(\frac{1}{n^{2}}\right)\cdot$

Suppose that the property holds for $|z|=r-1$ for some $r\geq 1$. Let us prove
that it holds for $|z|=r$. Let $z=z_{1}z_{2}\dots z_{r}$.

If $|z|=r=1$ and $z$ is a suffix of $w$, then, using the case $|z|=0$ above,

$\begin{array}[]{lll}Q_{w}(z,t,n)=Q_{w}(\varepsilon,t,n)-Q_{w}(\varepsilon,\overline{z_{r}}t,n)&=&-\displaystyle\frac{1}{2^{|t|}n}+\frac{1}{2^{|t|+1}n}+{\mathcal{O}}_{z,t}\left(\frac{1}{n^{2}}\right)=-\frac{1}{2^{|t|+1}n}+{\mathcal{O}}_{t}\left(\frac{1}{n^{2}}\right)\\\
&=&-\displaystyle\frac{1}{2^{|t|+|z|}n}+{\mathcal{O}}_{z,t}\left(\frac{1}{n^{2}}\right)\cdot\\\
\end{array}$

If $r\geq 2$ and $z$ is a suffix of $w$, then, using the induction hypothesis

$\begin{array}[]{lll}Q_{w}(z,t,n)&=&\displaystyle Q_{w}(z_{2}z_{3}\dots
z_{r},t,n)-Q_{w}(\overline{z_{1}}z_{2}\dots
z_{r-1},z_{r}t,n)=-\frac{1}{2^{r-1+|t|}n}+\frac{1}{2^{r-1+|t|+1}n}{\mathcal{O}}_{z,t}\left(\frac{1}{n^{2}}\right)\\\
&=&-\displaystyle\frac{1}{2^{r+|t|}n}+{\mathcal{O}}_{z,t}\left(\frac{1}{n^{2}}\right)=-\frac{1}{2^{|z|+|t|}n}+{\mathcal{O}}_{z,t}\left(\frac{1}{n^{2}}\right)\cdot\end{array}$

If $r\geq 1$ and $z$ is not a suffix of $w$, then, using the induction
hypothesis

$Q_{w}(z,t,n)=Q_{w}(z_{1}z_{2}\dots
z_{r-1},z_{r}t,n)=-\frac{1}{2^{r-1+|t|+1}n}+{\mathcal{O}}_{z,t}\left(\frac{1}{n^{2}}\right)=-\frac{1}{2^{|z|+|t|}n}+{\mathcal{O}}_{z,t}\left(\frac{1}{n^{2}}\right)\cdot$

Thus, we obtain

$g_{w}(n)=-\log(b_{w}(n))=-Q_{w}(w_{1}w_{2}\dots
w_{m-1},w_{m},n)=\frac{1}{2^{|w|}n}+{\mathcal{O}}_{w}\left(\frac{1}{n^{2}}\right)\cdot\
\ \ \ \ \Box$

###### Example 9

If $w=11$ (note that the sequence $(-1)^{a_{w}(n)}$ is a classical sequence,
called the Golay-Shapiro or also the Rudin-Shapiro sequence), then

$\lim_{k\to\infty}\sum_{\begin{subarray}{c}n\geq 1\\\
a_{w}(n)=k\end{subarray}}\frac{1}{n}=4\log 2.$

###### Remark 10

A similar study could probably be undertaken for any integer base $b\geq 2$,
combining ideas in [2, 4, 21].

Acknowledgments We warmly thank Jeff Shallit and Manon Stipulanti for
discussions about Kempner series, and B. Morin for sharing her expertise in
ancient Greek.

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# Simultaneous multi-band radio-frequency detection using high-orbital-
angular-momentum states in a Rydberg-atom receiver

Gianluca Allinson  Matthew J. Jamieson
Andrew R. MacKellar  Lucy A. Downes  C. Stuart Adams and Kevin J. Weatherill
Department of Physics, Durham University, South Road, Durham, DH1 3LE
<EMAIL_ADDRESS>
††journal: opticajournal††articletype: Research Article

We demonstrate simultaneous detection of radio-frequency (RF) fields ranging
from the very high-frequency (VHF) band (128 MHz) to terahertz frequencies
(0.61 THz) using a caesium Rydberg-atom receiver. The RF fields are
concurrently applied to a series of atomic transitions involving states of
increasing orbital angular momentum, where the energy separations become
progressively smaller, allowing access to a very wide range of radio
frequencies. We show that the optical response of the system in the presence
of the RF fields can be reproduced theoretically using a simple Lindblad-
master-equation approach. Furthermore, we demonstrate experimentally that a
series of amplitude-modulated tones can be detected simultaneously using
multiple carrier frequencies. This demonstration opens the way for RF
communications across multiple bands simultaneously using a single optical
receiver.

##  1 Introduction

Rydberg atoms, i.e., atoms in which an electron has been promoted to a highly
excited state with large principal quantum number $n$, are increasingly
finding use in technological applications [1]. The polarizability of atoms,
and the transition strength between neighboring electronic states, both
increase with increasing $n$, making Rydberg atoms highly sensitive to both AC
and DC electric fields. This sensitivity has been exploited to demonstrate
atom-based radio-frequency (RF) electric field metrology [2], RF sensing[3],
THz imaging [4, 5] and RF communications[6]. Most Rydberg-atom-based RF
sensors make use of electromagnetically induced transparency (EIT) [7], a
coherent and non-destructive process that couples the properties of the
Rydberg state to an optical transition from the atomic ground state, thereby
allowing the perturbative effect of an incoming RF electric field on the
Rydberg state to be mapped onto an optical probe field, which is detected
using a photodetector [8, 9]. This EIT technique effectively enables extremely
sensitive and SI-traceable measurements of RF electric fields to be extracted
from an optical signal [10]. The last decade has seen an explosion of
development in this research area, with Rydberg-atom receivers demonstrating
high-performance in the measurement of RF field properties, from precision
amplitude [3], frequency [11] and phase [3, 12] measurements to polarization
[13], angle of arrival [14] and simultaneous detection of multiple
frequencies[15, 16]. One of the major strengths of a Rydberg-atom RF receiver,
when compared to other RF technologies, is its potential to detect an enormous
frequency range using a single device[17]. Indeed, Rydberg-atom receivers have
demonstrated RF field detection at frequencies ranging from below 1 kHz [18]
to above 1 THz [19]. In the low frequency, or quasi-DC detection regime[8, 17,
20], Rydberg receivers can achieve continuous frequency coverage but with
limited sensitivity. Whereas, in the higher-frequency, AC detection regime,
state-of-the-art sensitivities can be achieved [2, 3] but only over a small
range of frequencies around the discrete values defined by resonant atomic
transitions. Furthermore, the resonant frequency of Rydberg-Rydberg
transitions decreases with increasing $n$ meaning that states with very high
$n$ (>100) are typically required to access frequencies below 1 GHz in the
sensitive AC regime. Gaining access to low frequencies using higher $n$ states
places demanding requirements on the laser power needed for experiments.
Consequently, recent work has focused on gaining access to lower frequency
bands via the use of higher angular momentum states [21, 22]. This previously
involved the use of a three step laser excitation scheme[23] plus an RF field
to access transitions of the form $n$F$\to n^{\prime}$G. High angular momentum
measurements have extended the range of resonant RF transitions to $\sim$1 GHz
and below at more accessible principal quantum numbers ($n$<70).

In this work, we take the principle of using higher-angular-momentum states
much further, to demonstrate that a Rydberg atom ensemble is capable of
simultaneously receiving multiple RF signals (seven in this case) ranging in
frequency from the very high-frequency (VHF) band to THz frequencies,
ultimately spanning 12 octaves. This is achieved by using two optical photons
and a sequence of cascading RF fields to resonantly couple higher-angular-
momentum states in the atom. Furthermore, we demonstrate that the VHF range
can be accessed at very low principal quantum numbers, $n=17$, where only
modest laser power is required.

##  2 Method

Figure 1: (a) Experimental layout of the Rydberg receiver with 852 nm probe,
519 nm coupling, and 0.607 THz fields overlapping within the Cs vapor cell. DC
- dichroic mirror, OAP - off-axis parabolic mirror, (P)BS, (polarizing) beam-
splitter, FG - frequency generator, PD - photodiode, $\lambda$/2 - half-
waveplate. A pyramidal RF gain horn and five whip antennas provide the RF
fields. (b) An energy-level diagram showing the electronic states used. A
zoomed inset shows the 17$\ell$ states with $\ell\geq 6$ for clarity.
Transitions are color coded in (a) to their counterpart in (b) and subsequent
figures.

Simultaneous multi-band RF detection is implemented using laser spectroscopy
of an atomic vapor. A schematic diagram of the experimental arrangement is
shown in Fig 1 (a). A linearly polarized 852 nm probe beam (solid red line),
resonant with the $6$S${}_{1/2},F=4\rightarrow 6$P${}_{3/2},F^{\prime}=5$
transition, was passed through a 1 cm long cuboidal cuvette containing caesium
(Cs) vapor at room temperature (23 ∘C) and then detected using a photodiode
(PD). A counter-propagating and linearly polarized 519 nm coupling laser
(solid green line) with detuning $\Delta_{\rm c}$ from the
$6$P${}_{3/2}~{}\rightarrow~{}19$D5/2 transition was overlapped with the probe
field using a dichroic mirror (DC). The probe and coupling beams have powers
of 20 $\upmu$W and 21 mW and 1/$e^{2}$ beam waists of 1.20(7) mm and 0.96(7)
mm respectively. The probe beam is derived from a Toptica DLpro laser and
stabilized using polarization spectroscopy [24]. The coupling beam is derived
from a home-built second-harmonic generation system of a 1030 nm ECDL based on
the design of Legaie et al.[25]. Cavity power build-up is stabilized by a
modified Hänsch-Couillaud lock [26, 27]. The coupling laser is frequency
stabilized by an excited-state lock using nonlinear polarization spectroscopy
[28]. A linearly polarized terahertz (THz) beam (solid purple line) resonant
with the $19$D${}_{5/2}\rightarrow 17$F7/2 transition at 0.607 THz is launched
from a diagonal horn antenna, collimated using a polytetrafluoroethylene
(PTFE) lens (not shown) and focused in the Cs cell using an off-axis parabolic
(OAP) mirror. The OAP contains a 2 mm through-hole to allow the laser beams to
copropagate with the THz beam. The THz beam is derived from a Virginia Diodes
amplifier multiplier chain (AMC) and is estimated to have a power of 10
$\upmu$W and beam waist of approximately 1.5 mm within the cell. The
transmission of the probe beam is recorded as the frequency of the coupling
beam ($\Delta_{\rm c}$) is varied. The EIT spectrum obtained in the presence
of the THz field is presented in Fig 2 (a). The RF fields are generated using
frequency generators and emitted via commercial Wi-Fi, cellular, and radio
$\lambda/4$ monopole whip antennas that are placed approximately 10-20 cm from
the cell where there is clear line-of-sight. The inclusion of each additional
field changes the transmission profile of the probe beam as shown in Fig 2 (b)
- (g).

Atomic transition | Final $\ell$ | Transition Frequency (GHz) | DME ($ea_{0}$)
---|---|---|---
$19$D5/2 $\to$ $17$F7/2 | 3 | 607 | 354
$17$F7/2 $\to$ $17$G9/2 | 4 | 34.9 | 423
$17$G9/2 $\to$ $17$H11/2 | 5 | 6.01 | 414
$17$H11/2 $\to$ $17$I13/2 | 6 | 1.85 | 406
$17$I13/2 $\to$ $17$K15/2 | 7 | 0.700 | 395
$17$K15/2 $\to$ $17$L17/2 | 8 | 0.310 | 383
$17$L17/2 $\to$ $17$M19/2 | 9 | 0.128 | 368
Table 1: A list of the electronic states that are coupled by the series of
cascading RF fields. DME - (Radial) dipole matrix element. Whilst the fine
structure splitting is not resolved beyond F, we label the strongest coupled
$j$ state for completeness. Note that by convention ‘J’ is omitted as the
symbol for the $\ell$ = 7 azimuthal quantum number.

The transitions used are shown schematically in Fig 1 (b) with their
frequencies and numerically calculated dipole matrix elements (DMEs) presented
in Table 1 [29]. We exploit the fact that beyond $\ell\geq 3$, transitions of
the form $n\ell\to n(\ell+1)$ allow for lower frequency bands to be addressed.
The rapidly decreasing overlap of the electron wavefunction with the atomic
core results in extremely small quantum defects, and hence small energy
separations, of high-$\ell$ states [30]. Each RF field is applied such that it
is resonant with the transition to the next highest angular momentum atomic
state. A higher $\ell$ state can only be coupled if the lower $\ell$ state is
coupled by the relevant RF field. Consequently, the 17L$\to$17M coupling at
128 MHz is dependent on all previous states to be coupled by their resonant RF
and optical fields.

When the high $\ell$ states are all simultaneously coupled by their respective
resonant RF fields, we can amplitude modulate the fields and detect the
baseband tones on the optical signal. For all data, readout is taken from the
photodiode via USB-DAQ of an oscilloscope and no lock-in
detection/amplification is used.

##  3 Results

###  3.1 Multi-level EIT Cascade Scheme

The experimental spectra in Fig 2 can be understood in terms of a $N$-level
EIT cascade scheme [31]. With the probe laser locked, a transparency window in
the probe absorption profile is expected around $\Delta_{\rm c}$ = 0,
resulting in a coherent transmission feature as the coupling laser is detuned
across this resonance. This phenomenon is known as electromagnetically induced
transparency (EIT)[32]. Applying the THz field results in a reduction in
transmission of the probe field at $\Delta_{\rm c}$ = 0. For large THz field
strengths, this results in the Autler-Townes (AT) splitting seen prominently
in Fig 2 (a). With the addition of another field, there is again transparency
in the line center. This is apparent in the data in the left-hand column of
Fig 2, showing the distinct increase in the probe transmission at zero
detuning of the coupling field for an odd number of levels. For an even number
of levels, there is absorption about the line center, shown in the right hand
column of Fig 2. When the subsequent RF fields are applied, the previous
resultant AT splitting is further shifted from the line center and decreases
in amplitude. These features become hard to distinguish in the lineshape,
nevertheless modulation of these fields causes a change to the transmission of
the probe beam.

Figure 2: Absolute probe transmission traces from the experiment (green) with
each subsequent addition of a field in the ladder system overlaid with an
n-level master equation model (red). The frequency scale is calibrated via the
6P3/2 $F$ = 4, $F$ = 5 separation. (a) The probe and coupling field on with
the THz field resonant to the 19D${}_{5/2}\to$17F7/2 transition. The
subsequent microwave fields are then resonant with the (b) 17F${}_{7/2}\to$17G
(c) 17G$\to$17H (d) 17H$\to$17I (e) 17I$\to$17K (f) 17K$\to$17L (g) and
17L$\to$17M transitions.

The atomic ensemble is modelled using a Lindblad master equation approach,
which gives a time evolution of the density matrix, $\hat{\rho}$:

$\frac{d\hat{\rho}}{dt}=-\frac{i}{\hbar}[\hat{H},\hat{\rho}]+\hat{L}_{\textrm{atom}}+\hat{L}_{\textrm{dephasing}},$
(1)

where $\hat{H}$ is the Hamiltonian of the atom-light system,
$\hat{L}_{\textrm{atom}}$ is the Linblad superoperator describing spontaneous
decay, and $\hat{L}_{\textrm{dephasing}}$ is dephasing due to the laser
linewidths or collisional effects. The series of differential equations formed
are then solved in the steady state to give the susceptibility of the atomic
ensemble [33]. The system is repeatedly solved for varying velocity classes
and integrated over a bounded velocity distribution to obtain the Doppler
broadened profile. Outside the weak probe regime, numerical solutions are
needed. Such a computation takes a significant amount of time on modern
computer systems, leaving computer minimization for such a problem
impractical. Model inputs, such as the Rabi frequencies for each field, were
varied around the experimental values to give the best fit by eye.

The resulting model spectra are shown by the red lines in Fig 2, along with
those from the experiment. Only one $m_{j}$ state is used for each level;
including all sub-levels would result in a model consisting of 110 levels.
Fine structure beyond the F state is ignored but note that the
$j=\ell+\frac{1}{2}$ states are coupled the strongest and are used in the
calculation of dipole matrix elements for G, H and beyond. We find good
agreement with the model in the generality of adding additional levels. The
parameters used for the model were $\Omega_{\rm p}=2\pi\times 1.8$ MHz,
$\Omega_{\rm c}=2\pi\times 3.7$ MHz and $\Omega_{\rm THz}=2\pi\times 46$ MHz.
The remaining Rabi frequencies of the RF fields varied by a few MHz but were
approximately $2\pi\times 20$ MHz.

Figure 3: Spectral map of the I $\to$ K transition as the RF field is swept
and coupling laser is detuned across resonance. (a) A color map made up of the
resultant probe transmission traces in response to scanning the RF field over
resonance for the 17I$\to$17K transition at $\sim$700 MHz. (b) A plot of the
probe transmission against the coupling detuning for an RF field of 0.703 GHz
- a grey bar shows the EIT trace which makes up that respective part of the
color map.

Fig 3 (a) shows behaviour of the lineshape in response to detuning the RF
field from resonance. Here, the probe laser power ($\sim$100 $\upmu$W) and the
power of the RF fields are increased in order to improve signal to noise and
visibility of the distinct AT splittings arising from each of the RF field
couplings. The contrast is seen in Fig 3 (b) when compared to Fig 2 (e)
whereby each of the distinct AT peaks are separated and can be distinguished.

The transmission window in the lineshape persists over a large detuning range,
$\Delta_{\rm RF}=\pm 20$ MHz. As this is an even level system, the induced
transparency from the presence of the previous field slowly shifts about the
line center and splits until symmetric at $\Delta_{\rm RF}=0$ MHz. Because of
the high field strengths used, the resonance of the feature is seen at
$\sim$705 MHz compared to the previously observed 700 MHz used in Fig 2 (e).
This can be attributed to the large AC Stark shifts that culminate for each RF
field that is on resonance.

###  3.2 Amplitude Modulation of Resonant RF Fields

The RF fields were modulated to demonstrate simultaneous detection across the
various frequencies that are resonant with the atomic transitions. A change in
each RF field amplitude will independently change the transmission of the
probe beam. When amplitude modulated, the baseband tones on each RF field can
then be extracted via performing a fast Fourier transform (FFT) of the probe
transmission signal. In a Rydberg atom detector, demodulation of the RF
frequencies are performed by the atom in and of itself, so no additional
demodulation of the signal is necessary. Each electronic state detailed above
were simultaneously coupled by the probe, coupling, THz and the 6 RF fields
with powers similar to those used in the spectra of Fig. 3. A baseband tone in
the kHz frequency range was modulated on each carrier RF field and detailed in
Fig. 4. The modulation index of the AM used on all of the carrier frequencies
was 0.5. The resulting modulus of the fast Fourier transform of the probe
signal is shown in Fig 4 (a).

The data show that the baseband tone from each of the independent carriers can
be recovered from the probe signal. The decreasing strength of the FFT signal
as the carrier frequency decreases can somewhat be attributed to the
decreasing DME of the transitions. In order to remove servo noise that we
observed in the FFT, the data presented is taken by disengaging the coupling
laser frequency lock and subsequently saving a trace. Due to technical
limitations, the THz source and 34.9 GHz MW generator could not be amplitude
modulated. However, there is nothing to suggest that amplitude modulation of
these transitions could not be detected.

Figure 4: Simultaneous detection of five independent AM tones using five of
the seven RF carriers. (a) The modulus of an FFT of the probe transmission at
$\Delta_{\rm c}=0$. The amplitude-modulated tones used were 7.2 kHz, 6.05 kHz,
4.15 kHz, 3.1 kHz, and 2.66 kHz on carrier frequencies of 6.01 GHz, 1.85 GHz,
700 MHz, 310 MHz, and 128 MHz, respectively. A dashed line is shown on each of
the peaks for visual aid. (b) An inset showing a typical scan of an EIT
feature during modulation of the RF fields. A vertical dashed line shows the
point at which the coupling laser is locked. (c) A typical trace before it is
discrete Fourier transformed.

##  4 Discussion

In principle, the number of fields that could be applied is limited by the
possible angular momentum states, ($n-1$). At higher principal quantum number,
the sensitivity to the fields would increase, and the splitting between
neighboring $\ell$ states would decrease. For example, neighboring angular
momentum transitions $\ell\geq 8$ at higher $n$ (40 -70) would correspond to
resonant fields in the medium frequency (300 kHz - 3 MHz) or high frequency (3
MHz - 30 MHz) radio bands.

We note that at low $n$, many perturbing effects such as DC Stark induced
state mixing and Rydberg-Rydberg collisions are significantly reduced [30].
Despite the lower sensitivity to external RF fields, electrometry at low
principal quantum number may be more suitable for calibration of an RF field
or metrology applications. A sensitivity assessment of the method to each of
the applied RF fields is beyond the scope of this paper as the sensitivity for
each transition depends upon the driving strength of all previous transitions
and the parameter space is very large. We note that the experimental
requirements used here are relatively modest, as we use low coupling-laser
power and do not rely on lock-in detection. The use of other well-
characterized antennas for the lower frequency fields would allow a better
estimation of the electric field strengths involved. The $\lambda/4$ monopole
antennas used are not well-characterized, are omnidirectional, and have poor
efficiency, resulting in significant power loss before the cell.

The method presented above could be used to investigate higher-angular-
momentum states which are not as well studied as the S, P, D, and F states.
There is currently no published work on quantum defects in Cs higher than G7/2
known to the authors [34, 35], whilst some exists for rubidium [36, 37].
Transitions were found in this work by sweeping the frequency of the RF fields
to observe the largest change in transmission of the probe beam at low RF
power. Initial estimations of the energy levels were given by `arc`, which
uses an approximation to estimate $\delta_{\ell}$ for $\ell\geq 4$ based on
known quantum defects [38]. The observed resonances were found to be in
general agreement and were additionally compared to the scarce available
literature [39]. Discrepancies on the order of several MHz are found for
$\ell\geq 7$, which could be attributed to the culminating AC Stark shift of
the many resonant fields. There is no control of stray magnetic/electric
fields which may cause further splitting of the numerous $m_{J}$ states.

##  5 Conclusion

Simultaneous detection of RF fields across 12 octaves by an optically coherent
method is demonstrated. The method opens up a new way of conducting radio
frequency electrometry using a cascade of RF fields across adjacent angular
momentum coupled states. Further work should be done to generalize and
optimize the technique both for higher principal quantum number and 3-photon
electrometry methods.

Funding We acknowledge the UK Engineering and Physical Sciences Research
Council under grants EP/R002061, EP/V030280, EP/W009404 and EP/W033054.

Acknowledgments The authors would like to thank Robert Potvliege and Popoola
Waisu for useful discussions, Ifan Hughes for careful reading of the
manuscript, Jonathan Pritchard for SHG laser designs and Mike Tarbutt for the
loan of equipment.

Disclosures The authors declare no conflicts of interest.

Data Availability All data are available at doi:10.15128/r12f75r8059.

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# Extracting Qualitative Causal Structure with Transformer-Based NLP

Scott E. Friedman1111Contact Author&Ian H. Magnusson1,2&Sonja M. Schmer-
Galunder1 1SIFT, Minneapolis, MN, USA
2Northeastern University, MA, USA
{sfriedman, imagnusson<EMAIL_ADDRESS>
###### Abstract

Qualitative causal relationships compactly express the direction, dependency,
temporal constraints, and monotonicity constraints of discrete or continuous
interactions in the world. In everyday or academic language, we may express
interactions between quantities (e.g., sleep decreases stress), between
discrete events or entities (e.g., a protein inhibits another protein’s
transcription), or between intentional or functional factors (e.g., hospital
patients pray to relieve their pain). This paper presents a transformer-based
NLP architecture that jointly identifies and extracts (1) variables or factors
described in language, (2) qualitative causal relationships over these
variables, and (3) qualifiers and magnitudes that constrain these causal
relationships. We demonstrate this approach and include promising results from
in two use cases, processing textual inputs from academic publications, news
articles, and social media.

##  1 Introduction

We express qualitative causal relationships in our everyday language and our
scientific texts to capture the relationship between quantities or entities or
events, compactly communicating how one event or purpose or quantity might be
affected by another. These causal relations are not complete mechanism
descriptions in themselves, but we use them frequently in everyday language
and formal instruction to express causality, allowing us to avoid unnecessary
detail or to hedge when details are uncertain.

Identifying these causal relationships from natural language—and also properly
identifying the factors that they relate—remains a challenge for NLP systems.
This difficulty is due in part to the expressiveness of our language, e.g.,
the multitude of ways we may describe how an experimental group scored higher
on an outcome than a control group, and also due to the complexity of the
systems we describe.

This paper describes an approach to automatically extracting (1) entities that
are the subject of causal relationships, (2) qualitative causal relationships
describing mechanisms, purposes, monotonicity, and temporal priority, and (3)
multi-label attributes to further characterize the causal structure. Our
primary claim is that context-sensitive language models can detect and
characterize the qualitative causal structure of everyday and scientific
language. As evidence, we present our SpEAR transformer-based NLP model based
on BERT Devlin et al. (2019) and SpERT Eberts and Ulges (2020) that extracts
causal structure from text as knowledge graphs, and we present promising
initial results on (1) characterizing scientific claims and (2) representing
and traversing descriptive mental models from ethnographic texts.

The causal, semantic graphs produced by SpEAR do not conform to a strict
ontology and therefore do not presently support the formal reasoning afforded
by ontologies with strict selectional restrictions and relational constraints;
however, we demonstrate that these outputs allow traversal across concepts to
characterize meaningful causal influences.

We continue with a review of related work in qualitative causal
representations (Section 2.1) and transformer-based NLP (Section 2.2). We then
describe our approach (Section 3) and preliminary results in two domains
(Section 4). We conclude with a discussion of future work in this area.

Figure 1: SpEAR knowledge graph output for the text “This sex predisposition
might be associated with the much higher smoking rate in men than in women in
China.” Includes a correlation, a comparison with a qualitative increase,
magnitudes, and a location qualifier.

Figure 2: SpEAR knowledge graph output for “Movement restriction greatly
reduced the number of infections from 5 February onwards.” Includes a causal
association, a qualitative decrease, a magnitude, and a temporal qualifier.

##  2 Background and Related Work

We review related work in representing causal relations, which informs the
present approach. We then review previous work in transformer-based NLP,
including the SpERT system Eberts and Ulges (2020) which is a subsystem of our
architecture.

###  2.1 Qualitative Causal Relations

The knowledge representations described in this paper are motivated by
previous work in qualitative reasoning and simulation Forbus (2019). For
example, _qualitative proportionalities_ describe how one quantity impacts
another, in a directional, monotonic fashion. In this work, we designate
$\langle a,\textbf{q+},b\rangle$ (and respectively, $\langle
a,\textbf{q–},b\rangle$) as qualitative proportionalities from $a$ to $b$,
such that increasing $a$ would increase (and respectively, decrease) $b$. This
is motivated by formal quantity-to-quantity $\alpha_{Q+/-}$ relations Forbus
(1984) and $M^{+/-}$ relations in qualitative simulation Kuipers (1986), but
our semantics are less constrained than either of these, due to tendencies in
language to express an increase from an event to a quantity (e.g., “smoking
that cigarette will increase your risk of cancer”) or from entities to
activities (e.g., “the prime increased participants’ retrieval of the cue”),
and so on.

Previous work in philosophy Dennett (1989) and cognitive psychology Lombrozo
and Carey (2006) has acknowledged intentional (i.e., psychological, goal-
based) and teleological (i.e., functional, design-based) relationships as
types of causal relations. Previous work has represented these as lexical
qualia or affordances Pustejovsky (1991). In this work, we represent
purposeful, intentional actions as a qualitative relationship $\langle
a,\textbf{forPurpose},b\rangle$, such that the actor of action $a$ may have
intended the purpose or goal $b$, e.g., “they prayed for a safe pregnancy.” We
represent teleological (i.e., functional or design-based) causal relations as
$\langle a,\textbf{hasFunction},b\rangle$ to indicate that the action or
artifact $a$ is designed or otherwise has a function to achieve $b$, e.g.,
“the artifacts provide protection for pregnant women.”

###  2.2 Causal and Transformer-Based NLP

Transformer-based methods for NLP utilize neural networks to encode a sequence
of textual tokens (i.e., words or sub-words) into large vector-based
representations for each token, sensitive to the context of the surrounding
tokens Devlin et al. (2019). This is widely regarded as a state-of-the-art
methodology for NLP, and has been used to process text to extract knowledge
graphs, e.g., of people and relations Eberts and Ulges (2020). The
architecture we present in this paper has been applied to datasets of
scientific claims Magnusson and Friedman (2021) and hate speech Friedman et
al. (2021). Many existing transformer models—similar to the architecture
presented in this paper—require hundreds (sometimes thousands) of labeled
training examples to reach high proficiency.

Previous symbolic semantic parsers extract scientific claims and assertions
from text with explicit relational knowledge representations Allen et al.
(2015), but many of these approaches rely on rule-based engines with hand
tuning, which requires NLP experts to maintain and adapt to new domains. By
contrast, our approach is designed to extract causal relationships at a level
of expressiveness comparable to these symbolic systems while using the
advances in transformer-based models such as SpERT Eberts and Ulges (2020) to
learn graph-based representation from examples alone.

Other NLP approaches use machine learning to extract features from scientific
texts, e.g., to identify factors and directions of influence Mueller and
Abdullaev (2019); however these approaches do not explicitly infer the
relations between elements in the claim, as shown with the present approach.

##  3 Approach

We describe our graph schema for representing the entities, attributes, and
qualitative relationships extracted from text. We discuss the general problem
definition and then we explain the specific graph schemas in two domains: (1)
scientific claims and (2) ethnographic mental models.

###  3.1 Knowledge Graphs

The SpEAR knowledge graph format include the following three types of
elements: entities, attributes, and relations. We describe each of these
before defining the problem and describing the architecture.

Figure 3: SpEAR knowledge graph for “Therefore, the women prayed to prevent
any complications,” including forPurpose and q- relations.

Figure 4: SpEAR knowledge graph for “I prayed that God should free me from the
severe pain I was going through.”

#### Entities.

Entities are labeled spans within a textual example. These are the nodes in
the knowledge graph. The same exact span cannot correspond to more than one
entity type, but two entity spans can overlap. Entities comprise the nodes of
Figures 2-5 upon which attributes and relations are asserted. Unlike most
ontologically-grounded symbolic parsers (e.g., Das et al. (2010); Allen et al.
(2015)), these entity nodes are not ontologically grounded in a class
hierarchy; rather, these entity nodes are associated with a token sequence
(e.g., “smoking rate” in Figure 2) and a corresponding entity class (e.g.,
Factor). These entities also have high-dimensional vectors from the
transformer model, which approximates the distributed semantics.

#### Attributes.

Attributes are Boolean labels, and each entity (i.e., graph node) may have
zero or more associated attributes. Attribute inference is therefore a multi-
label classification problem. The previous SpERT transformer model was not
capable of expressing these; this is a novel contribution of SpEAR, as
described in Section 3.5. In Figures 2-5, attributes are rendered as
parenthetical labels inside the nodes, e.g., Correlation and Increases in the
Figure 2 nodes for “associated with” and “higher,” respectively. The multi-
label nature allows the Figure 2 “higher” node to be categorized
simultaneously as Increases and Comparison.

#### Relations.

Relations are directed edges between labeled entities, representing semantic
relationships. These are critical for expressing what-goes-with-what over the
set of entities. For example in the sentence in Figure 2, the relations (i.e.,
edges) indicate that the “higher” association asserts the antecedent (arg0)
“men” against (comp_to) “women” for the consequent (arg1) “smoking rate.”
Without these relations, the semantic structure of this scientific hypothesis
would be ambiguous. Note that in Figures 2-5 the unlabeled arrows are all
modifier relations, left blank to avoid clutter.

###  3.2 Problem Definition

We define the multi-attribute knowledge graph extraction task as follows: for
a text passage $\mathcal{S}$ of $n$ tokens $s_{1},...,s_{n}$, and a graph
schema of entity types $\mathcal{T}_{e}$, attribute types $\mathcal{T}_{a}$,
and relation types $\mathcal{T}_{r}$, predict:

  1. 1.

The set of entities $\langle
s_{j},s_{k},t\in\mathcal{T}_{e}\rangle\in\mathcal{E}$ ranging from tokens
$s_{j}$ to $s_{k}$, where $0\leq j\leq k\leq n$,

  2. 2.

The set of relations over entities $\langle
e_{head}\in\mathcal{E},e_{tail}\in\mathcal{E},t\in\mathcal{T}_{r}\rangle\in\mathcal{R}$
where $e_{head}\neq e_{tail}$,

  3. 3.

The set of attributes over entities $\langle
e\in\mathcal{E},t\in\mathcal{T}_{a}\rangle\in\mathcal{A}$.

This defines a directed multi-graph without self-cycles, where each node has
zero to $|\mathcal{T}_{a}|$ attributes. SpEAR does not presently populate
attributes on relations.

Figure 5: SpEAR knowledge graph output for “Obese patients have a higher mean
IOP and lower fow velocity than non-obese patients.” The two qualitative
comparisons “higher” and “lower”, support qualitative increase and decrease
relations, respectively.

###  3.3 Knowledge Graph Schemata

We briefly describe a subset of the graph schemata for our two use-cases:
scientific claims and ethnographic mental models. These two schemata share
some qualitative causal representations but vary in other domain-specific
descriptions. In follow-on work, these schemata may be integrated into a
single schema.

#### Scientific Claims.

Our scientific claim schema is designed to capture associations between
factors (e.g., causation, comparison, prediction, proportionality),
monotonicity constraints across factors, epistemic status, and high-level
qualifiers. This model is used for qualitative reasoning to help characterize
the replicability and reproducibility of scientific claims Alipourfard et al.
(2021); Gelman et al. (2021). We describe the entities, attributes, and
relations of the schema, referencing the graphed examples rendered by our
system in Figures 2, 2, and 5.

This schema includes six entity types: Factors are variables that are tested
or asserted within a claim (e.g., “smoking rate” in Figure 2); Associations
are explicit phrases associating one or more factors in a causal, comparative,
predictive, or proportional assertion (e.g., “associated with” and “reduced”
in Figures 2 and 2, respectively); Magnitudes are modifiers of an association
indicating its likelihood, strength, or direction (e.g., “might” and “much” in
Figure 2); Evidence is an explicit mention of a study, theory, or methodology
supporting an association; Epistemics express the belief status of an
association, often indicating whether something is hypothesized, assumed, or
observed; Qualifiers constrain the applicability or scope of an assertion
(e.g., “in China” in Figure 2 and “from 5 February onwards” in Figure 2).

This schema includes the following attributes, all of which apply solely to
the Association entities: Causation expresses cause-and-effect over its
constituent factors (e.g., “reduced” span in Figure 2); Comparison expresses
an association with a frame of reference, as in the “higher” statement of
Figure 2 and the “higher” and “lower” statements of Figure 5; Increases
expresses high or increased factor value; Decreases expresses low or decreased
factor value; Indicates expresses a predictive relationship; and Test
indicates a statistical test employed to test a hypothesis.

We encode six relations: arg0 relates an association to its cause, antecedent,
subject, or independent variable; arg1 relates an association to its result or
dependent variable; comp_to is a frame of reference in a comparative
association; modifier relates entities to descriptive elements, e.g., all
leftward arrows in Figures 2 through 5 (unlabeled for simplicity); q+ and q-
indicate positive and negative qualitative proportionality, respectively,
where increasing the head factor increases or decreases (the amount or
likelihood of) the tail factor, respectively.

#### Ethnographic Mental Models.

In our preliminary ethnographic mental modeling domain, we utilize a slightly
different schema to capture intentional and functional causality in addition
to culturally-specific attributes such as gender and spirituality.

This schema includes attributes for Spirituality (e.g., “God” and “prayed” in
Figure 4), Action/event (e.g., “prayed” and “free” in Figure 4), Influence for
causally-potent elements (e.g., “prevent” in Figure 4), and others.

We include additional relations “agent/poss” to describe the actor or
possessor of an element, temporal precedence t+ relations to indicate one
event preceding another, and intentional forPurpose and functional hasFunction
relations to indicate the goal (i.e., intention or function, respectively) of
an action or artifact.

These relatively simple statements in Figure 4 and Figure 4 originate from an
ethnographic article Aziato et al. (2016) that includes interview snippets.
Despite their simplicity, the SpEAR knowledge graphs illustrate rich multi-
step causality: Figure 4 indicates that prayer has the purpose of reducing the
incidence (or severity of) complications, and Figure 4 plots a similar
structure for praying for the purpose of preventing pain of the speaker.

###  3.4 Scientific Claims Dataset

Our preliminary dataset for the scientific claims domain consists of 515
examples from Social and Behavior Science (SBS) literature and abstracts from
PubMed and the CORD-19 dataset Wang et al. (2020). Each example consists of a
single sentence labeled by a trained NLP expert with one or more spans
(possibly nested) identified as entities, zero or more attributes on each
entity, and zero or more relations over entities pairs (label counts are
listed in Table 1 support). Most datasets for transformer-based information
extraction are an order of magnitude larger—and a larger version of our
dataset will be validated in the near future—but despite the sparse dataset,
our model achieves favorable performance.

Figure 6: Our transformer-based model extends the SpERT components (a, b, and
c) with attribute classification (d) that performs multi-label inference on
identified entity spans.

###  3.5 Model Architecture

Our SpEAR model architecture extends SpERT with an attribute classifier. The
original architecture provides components (Figure 6 a–c) for joint entity and
relation extraction on potentially-overlapping text spans. The parameters of
the entity, attribute, and relation classifiers, as well as the parameters of
the BERT language model (initialized with its pre-trained values) are all
trained end-to-end on our dataset.

The tokens $s_{1},...,s_{n}$ of the text passage $\mathcal{S}$ are each
embedded by BERT Devlin et al. (2019) as a sequence
$\textbf{e}_{1},...,\textbf{e}_{n}$ of high-dimensional vectors representing
the token and its context. BERT also provides an additional “[CLS]” vector
output, $\textbf{e}_{0}$, designed to represent information from the complete
text input. For all possible spans, $span_{j,k}=s_{j},...,s_{k}$, up to a
given length, the word vectors associated with a span,
$\textbf{e}_{j},...,\textbf{e}_{k}$, are combined by maxpooling to produce a
single vector, $\textbf{e}(span_{j,k})$, where each element contains the
maximum value across the token vectors for that dimension. The final span
representation, $\textbf{x}(span_{j,k})$ is made by concatenating together
$\textbf{e}(span_{j,k})$ and $\textbf{e}_{0}$ along with a width embedding,
$\textbf{w}_{l}$, that encodes the number of words, $l$, in $span_{j,k}$. Each
valid span length $l$ looks up a different vector of learned parameters,
$\textbf{w}_{l}$.

The span representation $\textbf{x}(span_{j,k})$ is classified into mutually-
exclusive entity types by a linear classifier (Figure 6a). Only spans
identified as entities move on to further analysis (Figure 6b). All pairings
of the remaining entities are classified for relations by a multi-label linear
classifier (Figure 6c), where pairs are represented by the concatenated
vectors of the two spans with the “[CLS]” context vector replaced by the
maxpool of the token vectors between the entities.

We implemented the component in (Figure 6d) to infer multi-label attributes on
the identified entities using $\textbf{x}(span_{j,k})$ as input to another
multi-label linear classifier. We take only identified entity spans as input
to the attribute classifier, as this approach provided best performance and
aligns with the finding by Eberts and Ulges (2020) that training on downstream
tasks is best done on strong negative samples consisting of ground truth
entities (i.e., teacher forcing).

Figure 7: A graph traversal from the concept “pray” to the concept “pregnant” after parsing an ethnography about spirituality in pregnancy in Ghana. | Dimension | P | R | F1 | Support
---|---|---|---|---|---
Entities | factor | 90.13 | 86.71 | 88.39 | 1,604
evidence | 72.73 | 80.00 | 76.19 | 139
epistemic | 93.33 | 100.00 | 96.55 | 178
association | 95.89 | 93.33 | 94.59 | 837
magnitude | 94.44 | 94.44 | 94.44 | 415
qualifier | 86.96 | 68.97 | 76.92 | 216
| Micro-Averaged | 91.29 | 87.89 | 89.56 |
Attributes | causation | 88.24 | 93.75 | 90.91 | 204
comparison | 79.17 | 90.48 | 84.44 | 234
indicates | 80.00 | 66.67 | 72.73 | 44
increases | 75.86 | 95.65 | 84.62 | 262
decreases | 100.00 | 100.00 | 100.00 | 134
correlation | 94.74 | 94.74 | 94.74 | 199
| test | 100.00 | 66.67 | 80.00 | 24
| Micro-Averaged | 84.62 | 91.67 | 88.00 |
Relations | arg0 | 82.93 | 76.40 | 79.53 | 865
arg1 | 76.71 | 71.79 | 74.17 | 883
comp_to | 81.82 | 69.23 | 75.00 | 137
modifier | 84.78 | 74.29 | 79.19 | 1,080
q+ | 77.78 | 56.00 | 65.12 | 295
q- | 60.00 | 85.71 | 70.59 | 138
subtype | 85.71 | 75.00 | 80.00 | 106
| Micro-Averaged | 81.00 | 72.97 | 76.78 |
Table 1: Precision, recall, F1 and support (i.e., occurrences in dataset) for
each label on 10% held-out dataset using SpEAR with rectifier and filtering
model.

##  4 Results

We describe two different results of using SpEAR with our qualitative causal
schemata: (1) precision, recall, and F1 measure in the scientific claims
domain and (2) traversal through an ethnographic qualitative causal model.
This provides empirical evidence of the effectiveness of our approach and the
expressiveness of the qualitative causal schema, respectively.

###  4.1 Information Extraction for Scientific Claims

For our scientific claims dataset, we use the fine-tuned SciBERT transformer
variant Beltagy et al. (2019) as the input layer of our architecture.

We partitioned our dataset into a randomized 90% train/test split of 464 and
51 examples, respectively. We trained our SpEAR model for 20 epochs and then
run our evaluation. The per-class evaluations are listed in Table 1, divided
across the various entities, attributes, and relations. Table 1 reports the
micro-averaged results for entities, attributes, and relationships, as well as
support scores to show how many examples of each element are in the full
515-example dataset. Despite the small size of our preliminary dataset, the
model achieves promising results on most classes.

Importantly, the relations and attributes cannot be correct if the entities
they are defined over are incorrect. This means that we expect relations and
attributes to have lower precision and recall, all else being equal. This is
especially the case for relations, which require both of their constituent
entities (i.e., head and tail nodes) to be properly characterized in order to
be scored as correct. The relations q+ and q- achieved the lowest performance,
due in part to the lower support in the training data, and also due to the
greater distance between these spans in the text, all else being equal.

These results support our claim that qualitative causal structure can be
characterized by context-sensitive NLP models.

###  4.2 Traversing Ethnographic Causal Models

In the ethnographic domain, we trained SpEAR on labeled examples from
Anthropology papers describing religious beliefs surrounding pregnancy in
Ghana Aziato et al. (2016). We then ran SpEAR to extract information from
these and other sentences from the same literature, resulting in a global
causal graph comprising the disconnected knowledge graphs from each sentence
from the literature. These preliminary results include the use of human-
labeled training data, so we consider this a proof-of-concept study of the
practicality of the causal structure.

We then built a traversal system to walk to and/or from any concept in this
global causal graph, along the nodes and edges extracted by SpEAR. The source
and destination concepts are given by the user, and then the system identifies
all source and destination nodes by vector- or lemma-distance to the user’s
inputs. Given these source and destination nodes, it identifies global paths
from one concept to the other.

The traversal produces a graph such as the one in Figure 7, which shows paths
from the source “pray” (including “prayed,” “prayer,” “prayers,” etc.) to the
destination “pregnant” (also including “pregnancy.”) These traversals describe
prayer for the purpose of preventing unfortunate consequences, ensuring safe
pregnancy, and qualitatively increasing faith, hope, and confidence in
delivery. These graph structures only contain complete paths, so the
extraneous structure has been pruned for readability at the expense of
completeness.

These results support our claim that the causal models extracted by our
transformer-based NLP can support coarse-level reasoning and traversal across
concepts.

##  5 Conclusion

This paper describes a transformer-based NLP model for extracting entities,
attributes, and relationships that describe qualitative causal structure. We
demonstrated the approach in the two different domains of scientific claims
and descriptions of mental models from ethnography. Our datasets are still
under development, but despite their relative sparsity they support
encouraging results with respect to F1-measure and traversal.

One limitation of this work is that the nodes are not grounded in a formal
hierarchical ontology. This means that many of the assumptions about the
arguments to q+ and q- may not hold in SpEAR’s output: q+ may be expressed
over quantities, over events, over adjectives, or any heterogeneous mix of
these, and a downstream reasoner has no formal _a priori_ indicator of which
these are. One remedy to this is to use our node-based attributes to express
these different types of elements, but whether the transformer-based NLP model
can accurately classify these abstract categories is an empirical question.

Our near-term future work is to expand our datasets and utilizing SpEAR’s
results in downstream systems, e.g., for estimating the reproducibility of a
scientific claim, automatically organizing and combining insights from
academic literature, and globally traversing descriptive mental models to
identify culturally-specific causally-potent concepts and purposes.

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