# On the tower factorization of integers

Jean-Marie De Koninck and William Verreault

###### Abstract

Under the fundamental theorem of arithmetic, any integer $n>1$ can be uniquely
written as a product of prime powers $p^{a}$; factoring each exponent $a$ as a
product of prime powers $q^{b}$, and so on, one will obtain what is called the
tower factorization of $n$. Here, given an integer $n>1$, we study its height
$h(n)$, that is, the number of “floors” in its tower factorization. In
particular, given a fixed integer $k\geq 1$, we provide a formula for the
density of the set of integers $n$ with $h(n)=k$. This allows us to estimate
the number of floors that a positive integer will have on average. We also
show that there exist arbitrarily long sequences of consecutive integers with
arbitrarily large heights.

##  1 Introduction.

According to the fundamental theorem of arithmetic, the factorization of an
integer $n\geq 2$ into primes is unique apart from the order of its prime
factors, a fact which was already known to Euclid (for a modern treatment, see
for instance Theorem 1.16 in the book of Niven, Zuckerman, and Montgomery
[6]). This result is also called the unique factorization theorem.

Hence, given an integer $n$, one can write it in a unique manner (called the
canonical factorization of $n$) as

$n=p_{1}^{a_{1}}p_{2}^{a_{2}}\cdots p_{r}^{a_{r}},$ (1)

where $p_{1}<p_{2}<\cdots<p_{r}$ are primes and $a_{1},a_{2},\ldots,a_{r}$ are
positive integers. What if we were to write each exponent $a_{i}$ which is
larger than 1 as a product of primes, and then do the same with the new
exponents thus generated, and so on ? We would then obtain a tower
factorization. For instance, the number $n_{1}=2\,715\,939\,072$ can be
written

$n_{1}=2^{8}\cdot 3^{9}\cdot 7^{2}\cdot 11=2^{2^{3}}\cdot 3^{3^{2}}\cdot
7^{2}\cdot 11,$

the first of these two representations being the canonical factorization of
$n_{1}$, the second one being the tower factorization of $n_{1}$. This
motivates us to introduce the notion of the height $h(n)$ of an integer $n$,
namely the number of floors in the tower factorization of $n$. Clearly,
$h(n)=1$ if and only if $n$ is square-free, $h(p^{2})=2$ for each prime $p$,
$h(n_{1})=3$ and $\displaystyle{h(65\,536)=h\Big{(}2^{2^{2^{2}}}\Big{)}=4}$.
Interestingly, just as with the canonical factorization, one easily sees that
the tower factorization of any given integer $n\geq 2$ is unique. The concept
of tower factorization is not new. It was first introduced in 2014 by Devlin
and Gnang [2] as they studied the density of the set $M(q)$ of all integers
whose tower factorization contains the prime $q$. They were motivated by
computational considerations on shortest encodings appearing in Gnang,
Radziwiłł, and Sanna [4].

In Section 5, for an integer $k\geq 1$ we obtain a formula for the density of
the set of integers $n$ with $h(n)\leq k$ and for the density of the set of
integers $n$ with $h(n)=k$. We first treat the simpler case $k=2$ (resp.
$k=3$) in Section 3 (resp. Section 4). In Section 6, we prove that a positive
integer has on average approximately $1.4361$ floors, and in Section 7, we
show that there exist arbitrarily long sequences of consecutive integers
having arbitrarily large heights. We end this section with open problems.

##  2 The general setup.

Given a positive integer $k$, let $D_{k}$ (resp. $C_{k}$) stand for the set of
those positive integers $n$ for which $h(n)=k$ (resp. $h(n)\leq k$). Also, let

$d_{k}=\lim_{N\to\infty}\frac{1}{N}\\#\\{n\leq N:h(n)=k\\}\quad\mbox{and}\quad
c_{k}=\lim_{N\to\infty}\frac{1}{N}\\#\\{n\leq N:h(n)\leq k\\}$

be the respective densities (as we will see, which exist) of the sets $D_{k}$
and $C_{k}$. Clearly,

$C_{k}=D_{1}\cup D_{2}\cup\cdots\cup D_{k}\quad\mbox{and}\quad
c_{k}=d_{1}+d_{2}+\cdots+d_{k}.$ (2)

Since the density of square-free integers exists and is equal to $6/\pi^{2}$
(see for instance Theorem 8.25 in [6]), we have that

$c_{1}=d_{1}=\frac{6}{\pi^{2}}\approx 0.6079271019.$

For each positive integer $k$, we introduce the characteristic function
$\lambda_{k}$ of the set $C_{k}$, namely the arithmetic function

$\lambda_{k}(n):=\left\\{\begin{array}[]{lll}1&\mbox{if}&h(n)\leq k,\\\
0&\mbox{if}&h(n)\geq k+1.\end{array}\right.$ (3)

Clearly $\lambda_{1}(n)=\mu^{2}(n)$, where $\mu$ stands for the Möbius
function defined by $\mu(1)=1$, $\mu(n)=0$ if $p^{2}\mid n$ for some prime $p$
and $(-1)^{\omega(n)}$ otherwise. (Here $\omega(n)$ stands for the number of
distinct prime factors of $n$.)

##  3 The set of integers having no more than two floors.

Note that the tower factorization of an integer has at most two floors if each
exponent $a_{i}$ in its canonical factorization is square-free, so that

$\lambda_{2}(n)=\lambda_{2}(p_{1}^{a_{1}}p_{2}^{a_{2}}\cdots
p_{r}^{a_{r}})=\prod_{i=1}^{r}\mu^{2}(a_{i}).$ (4)

Also observe that

$\lambda_{2}(n)=\prod_{i=1}^{r}\lambda_{1}(a_{i}).$ (5)

The function $\lambda_{2}(n)$ is multiplicative, that is,
$\lambda_{2}(mn)=\lambda_{2}(m)\lambda_{2}(n)$ if $m$ and $n$ are coprime.
Indeed, assume that $m=p_{1}^{a_{1}}\cdots p_{r}^{a_{r}}$ for some primes
$p_{1}<\cdots<p_{r}$ and positive integers $a_{1},\ldots,a_{r}$ and that
$n=q_{1}^{b_{1}}\cdots q_{s}^{b_{s}}$ for some primes $q_{1}<\cdots<q_{s}$ and
positive integers $b_{1},\ldots,b_{s}$, where
$\\{p_{1},\ldots,p_{r}\\}\cap\\{q_{1},\ldots,q_{s}\\}=\emptyset$. Then, using
(4),

$\displaystyle\lambda_{2}(mn)$ $\displaystyle=\lambda_{2}(p_{1}^{a_{1}}\cdots
p_{r}^{a_{r}}q_{1}^{b_{1}}\cdots q_{s}^{b_{s}})$
$\displaystyle=\mu^{2}(a_{1})\cdots\mu^{2}(a_{r})\mu^{2}(b_{1})\cdots\mu^{2}(b_{s})$
$\displaystyle=\lambda_{2}(m)\lambda_{2}(n),$

thus proving our claim. Therefore, using the fact that
$\lambda_{2}(p^{r})=\mu^{2}(r)$ for all primes $p$ and integers $r\geq 1$, we
have the Euler product expansion

$\displaystyle\sum_{n=1}^{\infty}\frac{\lambda_{2}(n)}{n^{s}}=\prod_{p}\Big{(}1+\sum_{r=1}^{\infty}\frac{\lambda_{2}(p^{r})}{p^{rs}}\Big{)}=\prod_{p}\Big{(}1+\sum_{r=1}^{\infty}\frac{\mu^{2}(r)}{p^{rs}}\Big{)}\qquad(s>1),$
(6)

where the infinite product runs over all prime numbers $p$. Letting

$\zeta(s):=\sum_{n=1}^{\infty}\frac{1}{n^{s}}=\prod_{p}\left(1-\frac{1}{p^{s}}\right)^{-1}\qquad(s>1)$

be the Riemann zeta function, (6) can be rewritten as

$\sum_{n=1}^{\infty}\frac{\lambda_{2}(n)}{n^{s}}=\zeta(s)\prod_{p}\Big{(}1-\frac{1}{p^{s}}\Big{)}\Big{(}1+\sum_{r=1}^{\infty}\frac{\mu^{2}(r)}{p^{rs}}\Big{)}=\zeta(s)F(s)\qquad(s>1),$
(7)

say.

We will now use Wintner’s theorem, which says that if $f$ and $g$ are two
arithmetic functions satisfying

$\sum_{n=1}^{\infty}\frac{f(n)}{n^{s}}=\zeta(s)\sum_{n=1}^{\infty}\frac{g(n)}{n^{s}}\qquad(s>1)$

and if $\displaystyle{\sum_{n=1}^{\infty}\frac{g(n)}{n^{s}}}$ converges
absolutely at $s=1$, then

$\frac{1}{N}\sum_{n\leq
N}f(n)=\sum_{n=1}^{\infty}\frac{g(n)}{n}+o(1)\qquad(N\to\infty),$

where $o(1)$ stands for a function of $N$ tending to $0$ as $N\to\infty$. For
a proof of Wintner’s theorem, see Theorem 6.13 in the book of De Koninck and
Luca [1].

An infinite product $\prod_{k=1}^{\infty}(1+a_{k})$ converges absolutely if
$\prod_{k=1}^{\infty}(1+|a_{k}|)$ converges. It is known (see for instance [5,
Theorem 2.2.8]) that this happens if and only if $\sum_{k=1}^{\infty}|a_{k}|$
converges. We can show that the infinite product appearing in (7) converges
absolutely at $s=1$. Indeed, first observe that

$F(s)=\prod_{p}\Big{(}1-\frac{1}{p^{s}}\Big{)}\Big{(}1+\frac{1}{p^{s}}+\sum_{r=2}^{\infty}\frac{\mu^{2}(r)}{p^{rs}}\Big{)}=\prod_{p}\Big{(}1+\frac{p^{s}-1}{p^{s}}\sum_{r=2}^{\infty}\frac{\mu^{2}(r)}{p^{rs}}-\frac{1}{p^{2s}}\Big{)}.$

Therefore, by the above remark, we only need to show that if we set

$a_{p}:=\frac{p^{s}-1}{p^{s}}\sum_{r=2}^{\infty}\frac{\mu^{2}(r)}{p^{rs}}-\frac{1}{p^{2s}},$

then $\sum_{p}|a_{p}|$ converges. Since, for any $s\geq 1$,

$\sum_{r=2}^{\infty}\frac{\mu^{2}(r)}{p^{rs}}\leq\sum_{r=2}^{\infty}\frac{1}{p^{rs}}=\frac{1}{p^{2s}}\Big{(}1+\frac{1}{p^{s}}+\frac{1}{p^{2s}}+\cdots\Big{)}=\frac{1}{p^{s}(p^{s}-1)},$

we get

$\sum_{p}|a_{p}|\leq\sum_{p}\Big{(}\frac{p^{s}-1}{p^{s}}\cdot\frac{1}{p^{s}(p^{s}-1)}+\frac{1}{p^{2s}}\Big{)}=\sum_{p}\frac{2}{p^{2s}},$

which converges at $s=1$, thus proving our claim.

It follows that we may apply Wintner’s theorem and conclude that

$\frac{1}{N}\sum_{n\leq N}\lambda_{2}(n)=c_{2}+o(1)\qquad(N\to\infty),$ (8)

where

$\displaystyle c_{2}=F(1)$ $\displaystyle=$
$\displaystyle\prod_{p}\Big{(}1-\frac{1}{p}\Big{)}\Big{(}1+\sum_{r=1}^{\infty}\frac{\mu^{2}(r)}{p^{r}}\Big{)}$
(9) $\displaystyle=$
$\displaystyle\prod_{p}\Big{(}1-\frac{1}{p^{4}}+\frac{1}{p^{5}}-\frac{1}{p^{8}}+\frac{1}{p^{10}}+\cdots\Big{)}\approx
0.9559230262.$

Recall that the constant $c_{2}$ gives the density of the set of those
integers $n$ satisfying $h(n)=1$ or $h(n)=2$. We thus also have

$d_{2}=c_{2}-\frac{6}{\pi^{2}}\approx 0.3479959243.$ (10)

######  Remark.

Estimate (8) is not new. In the study of exponentially square-free numbers,
Subbarao [7] obtained (8) with an error term of $O(N^{-1/2})$, and Wu [9]
improved this error term unconditionally. Tóth [8] further improved it under
the Riemann hypothesis.

##  4 The set of integers having no more than three floors.

Given $n$ in the form (1), it is clear that

$\lambda_{3}(n)=\prod_{i=1}^{r}\lambda_{2}(a_{i}).$

This function is also multiplicative, so the reasoning used in Section 3 still
applies. Namely, using the fact that $\lambda_{3}(p^{r})=\lambda_{2}(r)$, we
have, for $s>1$,

$\displaystyle\sum_{n=1}^{\infty}\frac{\lambda_{3}(n)}{n^{s}}$
$\displaystyle=$
$\displaystyle\prod_{p}\Big{(}1+\sum_{r=1}^{\infty}\frac{\lambda_{3}(p^{r})}{p^{rs}}\Big{)}=\zeta(s)\prod_{p}\Big{(}1-\frac{1}{p^{s}}\Big{)}\Big{(}1+\sum_{r=1}^{\infty}\frac{\lambda_{2}(r)}{p^{rs}}\Big{)}$
$\displaystyle=$
$\displaystyle\zeta(s)\prod_{p}\Big{(}1-\frac{1}{p^{16s}}+\frac{1}{p^{17s}}-\frac{1}{p^{48s}}+\frac{1}{p^{49s}}-\frac{1}{p^{80s}}-\frac{1}{p^{81s}}+\frac{1}{p^{82s}}+\cdots\Big{)}.$

Once again Wintner’s theorem implies that

$\frac{1}{N}\sum_{n\leq N}\lambda_{3}(n)=c_{3}+o(1)\qquad(N\to\infty),$

where

$c_{3}=\prod_{p}\Big{(}1-\frac{1}{p}\Big{)}\Big{(}1+\sum_{r=1}^{\infty}\frac{\lambda_{2}(r)}{p^{r}}\Big{)}\approx
0.9999923551,$ (11)

a number which unsurprisingly is in accordance with the fact that very few
integers have at least 4 floors. Combining (2), (9) and (11), we get

$d_{3}=c_{3}-d_{1}-d_{2}=c_{3}-c_{2}\approx 0.0440693289.$ (12)

Let $T_{k}(x)$ stand for the number of positive integers $n\leq N$ with at
least $k$ floors. Using a computer, one obtains the following table of values
of $T_{3}(x)$ and $T_{4}(x)$ for $x=10^{r}$, $1\leq r\leq 8$, which supports
our previous results.

$x=10^{r}$ | 10 | $10^{2}$ | $10^{3}$ | $10^{4}$ | $10^{5}$ | $10^{6}$ | $10^{7}$ | $10^{8}$
---|---|---|---|---|---|---|---|---
$T_{3}(x)$ | 0 | 4 | 43 | 440 | 4408 | 44 077 | 440 760 | 4 407 699
$T_{4}(x)$ | 0 | 0 | 0 | 0 | 1 | 15 | 152 | 1527

##  5 The set of integers having no more than $k$ floors.

We now seek a formula for $c_{k}$ and $d_{k}$ for a general $k\geq 2$. Since
it is clear from (2) that

$d_{k}=c_{k}-c_{k-1}\qquad(k\geq 2),$

it suffices to obtain a formula for $c_{k}$. Recalling the definition of
$\lambda_{k}$ given in (3), it is easy to see that (5) can be generalized to

$\lambda_{k}(n)=\prod_{i=1}^{r}\lambda_{k-1}(a_{i})\qquad(k\geq 2).$

Since each $\lambda_{k}(n)$ is multiplicative, it follows as before that

$\displaystyle\sum_{n=1}^{\infty}\frac{\lambda_{k}(n)}{n^{s}}=\zeta(s)\prod_{p}\Big{(}1-\frac{1}{p^{s}}\Big{)}\Big{(}1+\sum_{r=1}^{\infty}\frac{\lambda_{k-1}(r)}{p^{rs}}\Big{)}=\zeta(s)F_{k}(s)\qquad(s>1),$

say, and that $F_{k}(s)$ converges at $s=1$. Wintner’s theorem then gives

$\frac{1}{N}\sum_{n\leq N}\lambda_{k}(n)=c_{k}+o(1)\qquad(N\to\infty),$

where

$c_{k}=\prod_{p}\Big{(}1-\frac{1}{p}\Big{)}\Big{(}1+\sum_{r=1}^{\infty}\frac{\lambda_{k-1}(r)}{p^{r}}\Big{)}\qquad(k\geq
2).$

By definition of $\lambda_{k}(n)$, this is precisely the density of the set of
integers $n$ with ${h(n)\leq k}$.

##  6 Average height.

If we pick a positive integer at random, how many floors should one expect its
tower factorization to have? This average height depends on the various
densities $d_{k}$. Since a positive integer can have any number of floors, one
can prove that

$\lim_{N\to\infty}\frac{1}{N}\sum_{n\leq N}h(n)=\sum_{k=1}^{\infty}kd_{k}.$
(13)

In order to establish that (13) holds, we need to obtain upper bounds for the
densities $d_{k}$ which will ensure that the series
$\displaystyle{\sum_{k=1}^{\infty}kd_{k}}$ does indeed converge. To do so, we
proceed as follows.

For convenience, given an integer $k\geq 2$, we set
$2^{(k)}:=2^{\iddots^{2}}$, the minimal tower with $k$ floors, so that in
particular $2^{(2)}=2^{2}$ and $2^{(3)}=2^{2^{2}}$. We begin by establishing
two important inequalities.

######  Lemma 1.

(i) For any given integer $a\geq 2$, we have that
$\displaystyle{\sum_{p}\frac{1}{p^{a}}<\frac{2}{2^{a}}}$.
(ii) For any given integer $k\geq 2$, we have that
$\displaystyle{d_{k}<\frac{4}{2^{(k)}}}$.

###### Proof.

It is well-known that $\displaystyle{\sum_{p}\frac{1}{p^{2}}=0.452\ldots}$
(see for instance page 95 in the book of Finch [3]), thus proving $(i)$ in the
case of $a=2$. Hence, let $a\geq 3$ and first observe that it is clear that

$\sum_{p}\frac{1}{p^{a}}<\zeta(a)-1.$

In light of this last inequality, the proof of part $(i)$ will be complete if
we can prove that

$\zeta(a)-1<\frac{1}{2^{a-1}}\quad\mbox{for all }a\geq 3,$ (14)

or equivalently that

$\frac{1}{3^{a}}+\frac{1}{4^{a}}+\frac{1}{5^{a}}+\cdots<\frac{1}{2^{a}}\quad\mbox{for
all }a\geq 3.$ (15)

It is immediate that for any integer $a\geq 3$,

$\frac{1}{3^{a}}+\frac{1}{4^{a}}+\frac{1}{5^{a}}+\cdots<\int_{2}^{\infty}\frac{dt}{t^{a}}=\frac{1}{(a-1)2^{a-1}}\leq\frac{1}{2^{a}},$

thus proving (15), from which (14) follows, thereby completing the proof of
part $(i)$.

To prove part $(ii)$, we first note that the case $k=2$ is immediate because
(10) implies that $d_{2}<1$. For the general case $k\geq 3$, observe that any
given $n\in D_{k}$ (that is, an integer $n$ with $h(n)=k$) can be written in a
unique way as $n=p^{a}\cdot m$, where $p$ is the smallest prime $p$ dividing
$n$ for which $p^{a}\mid\mid n$ with $h(a)=k-1$ (implying that $a\geq
2^{(k-1)}$). We then have, using part $(i)$,

$\displaystyle\sum_{n\leq N\atop h(n)=k}1$ $\displaystyle\leq$
$\displaystyle\underset{\begin{subarray}{c}p^{a}m\leq N\\\
h(a)=k-1\end{subarray}}{\sum_{p}\sum_{m\geq 1}}1\leq N\sum_{a\geq
2^{(k-1)}}\sum_{p}\frac{1}{p^{a}}<N\sum_{a\geq
2^{(k-1)}}\frac{2}{2^{a}}=N\frac{4}{2^{(k)}},$

thus completing the proof of part $(ii)$. ∎

Observe that since, trivially, $2^{(k)}\geq 2^{k}$ for all $k\geq 2$, it
follows from part $(ii)$ of Lemma 1 that

$\sum_{k=1}^{\infty}kd_{k}=d_{1}+\sum_{k=2}^{\infty}kd_{k}<1+4\sum_{k=2}^{\infty}\frac{k}{2^{k}}=1+4\cdot\frac{3}{2}=7,$

implying that the series $\displaystyle{\sum_{k=1}^{\infty}kd_{k}}$ does
indeed converge.

So, given a large integer $N$ and letting $K=K_{N}$ be the smallest positive
integer for which $h(n)\leq K$ for all $n\leq N$, we then have

$\frac{1}{N}\sum_{n\leq N}h(n)=\frac{1}{N}\sum_{k=1}^{K_{N}}k\sum_{n\leq
N\atop h(n)=k}1=\sum_{k=1}^{K_{N}}k\frac{1}{N}\sum_{n\leq N\atop h(n)=k}1,$

so that

$\lim_{N\to\infty}\frac{1}{N}\sum_{n\leq N}h(n)=\sum_{k=1}^{\infty}kd_{k},$

which proves (13).

Denoting by $\mathfrak{a}$ the average value of the number of floors of a
positive integer, one can round it to four decimal places by only using the
values of $d_{1}$, $d_{2}$, and $d_{3}$ obtained above (the other $d_{i}$’s
having no incidence on these four decimals). One gets that
$\mathfrak{a}\approx 1.4361$.

##  7 Consecutive integers with arbitrarily large heights and various open
problems.

Tower factorizations raise many interesting questions. For instance, using a
computer one can check that the first occurrence of three consecutive integers
each with height at least 3 is provided by the triplet $n_{0}$, ${n_{0}+1}$,
$n_{0}+2$, where

$n_{0}=248\,750=2\cdot 5^{2^{2}}\cdot 199,\quad n_{0}+1=3^{2^{2}}\cdot 37\cdot
83,\quad n_{0}+2=2^{2^{2}}\cdot 7\cdot 2221.$

What about four consecutive integers with the same property ? What about
greater heights ? Perhaps surprisingly, all are possible. Indeed, one can show
that given arbitrary integers $\ell\geq 2$ and $k\geq 2$, there exist
infinitely many integers $n$ with the property that $h(n+j)\geq k$ for
$j=0,1,\ldots,\ell-1$. Indeed, consider the system of the $\ell$ congruences

$\displaystyle n$ $\displaystyle\equiv$ $\displaystyle
0\pmod{2^{2^{\iddots^{2}}}},$ $\displaystyle n+1$ $\displaystyle\equiv$
$\displaystyle 0\pmod{3^{2^{\iddots^{2}}}},$ $\displaystyle\vdots$
$\displaystyle n+\ell-1$ $\displaystyle\equiv$ $\displaystyle
0\pmod{P_{\ell}^{2^{\iddots^{2}}}},$

where $P_{\ell}$ is the $\ell$-th prime and each of the numbers
$\displaystyle{p_{i}^{2^{\iddots^{2}}}}$ represents a tower of height $k$.
Then, set

$Q_{\ell}:=\displaystyle{2^{2^{\iddots^{2}}}\cdot
3^{2^{\iddots^{2}}}\cdots\;P_{\ell}^{2^{\iddots^{2}}}}.$

By the Chinese remainder theorem one can claim the existence of a number
$n_{0}<Q_{\ell}$ for which $h(n_{0}+j)\geq k$ for ${j=0,1,\ldots,\ell-1}$.
Actually, more is true, namely the existence of infinitely many integers $n$
satisfying $h(n+j)\geq k$ for $j=0,1,\ldots,\ell-1$ by simply considering the
numbers $n=n_{0}+r\cdot Q_{\ell}$ with $r=0,1,2,\ldots$, thus proving our
claim.

Finally, it would be nice to obtain explicit expressions for the densities
$c_{k}$ and $d_{k}$, $k\geq 3$, instead of the recursive expressions obtained
here. It is also an open problem to find an explicit expression for the
density of the set $M(q)$ of all integers whose tower factorization contains
the prime $q$, which we alluded to in the Introduction. For instance, when
$q=2$, we certainly have that this density is at least $0.5$, but in fact it
is $\approx 0.577350$. We refer the reader to [2] for more information on this
problem.

  * ACKNOWLEDGMENT.

The authors wish to thank the referees and the editor for their very helpful
comments.

## References

  * 1. De Koninck, J.-M., Luca, F. (2012). Analytic Number Theory: Exploring the Anatomy of Integers, Graduate Studies in Mathematics, Vol. 134. Providence, Rhode Island: American Mathematical Society.
  * 2. Devlin, P., Gnang, E.K. (2014). Primes appearing in the tower factorization of integers. arXiv Preprint, arXiv:1204.5251v2.
  * 3. Finch, S.R. (2003). Mathematical Constants, Encyclopedia of Mathematics and its Applications, Vol. 94. Cambridge: Cambridge University Press.
  * 4. Gnang, E.K., Radziwiłł, M., Sanna, C. (2015). Counting arithmetic formulas. European J. Combin. 47: 40–53.
  * 5. Little, C.H., Teo, K.L., van Brunt, B. (2022). An introduction to infinite products, New York: Springer.
  * 6. Niven, I., Zuckerman, H.S., Montgomery, H.L. (1991). An Introduction to the Theory of Numbers, 5th ed. New York: Wiley.
  * 7. Subbarao, M.V. (1972). On some arithmetic convolutions. In: Gioia, A. A., Goldsmith, D. L., eds. The theory of arithmetic functions: Proceedings of the Conference at Western Michigan University, Berlin: Springer, 247–271.
  * 8. Tóth, L. (2007). On certain arithmetical functions involving exponential divisors II. Ann. Univ. Sci. Budapest. Sect. Comput. 27: 155–166.
  * 9. Wu, J. (1995). Problème de diviseurs exponentiels et entiers exponentiellement sans facteur carré. J. Théor. Nombres Bordeaux. 7(1): 133–141.

  * JEAN-MARIE DE KONINCK

obtained his Ph.D. in mathematics from Temple University in 1972 under the
supervision of Emil Grosswald. After 44 years as a faculty member at
Université Laval in Quebec, Canada, he retired in 2016 and is now Professor
Emeritus. His main research interest is the multiplicative structure of
integers. He is still involved in math outreach and takes pleasure in swimming
and writing books. His latest, co-authored with Nicolas Doyon, is The Life of
Primes in 37 Episodes.

    * Département de mathématiques et de statistique, Université Laval, Québec G1V 0A6, Canada
<EMAIL_ADDRESS>
  * WILLIAM VERREAULT

is a master’s student at Université Laval and a soon-to-be Vanier Scholar at
the University of Toronto. While his main research interests lie in analysis,
he enjoys working on various number theory problems.

    * Département de mathématiques et de statistique, Université Laval, Québec G1V 0A6, Canada
<EMAIL_ADDRESS>


University of Amsterdam, Amsterdam,
NLb.p.allen@uva.nlhttps://orcid.org/0000-0003-0216-3930 Vrije Universiteit
Amsterdam, Amsterdam, NLl.stork@vu.nlhttps://orcid.org/0000-0002-2146-4803
University of Amsterdam, Amsterdam,
NLp.t.groth@uva.nlhttps://orcid.org/0000-0003-0183-6910 Bradley P. Allen, Lise
Stork, Paul Groth 10.1234/0000000.00000000 2023-07-14

# Knowledge Engineering using Large Language Models

Bradley P. Allen  ,  Lise Stork  and  Paul Groth

###### Abstract.

Knowledge engineering is a discipline that focuses on the creation and
maintenance of processes that generate and apply knowledge. Traditionally,
knowledge engineering approaches have focused on knowledge expressed in formal
languages. The emergence of large language models and their capabilities to
effectively work with natural language, in its broadest sense, raises
questions about the foundations and practice of knowledge engineering. Here,
we outline the potential role of LLMs in knowledge engineering, identifying
two central directions: 1) creating hybrid neuro-symbolic knowledge systems;
and 2) enabling knowledge engineering in natural language. Additionally, we
formulate key open research questions to tackle these directions.

###### Key words and phrases:

knowledge engineering, large language models

###### 1991 Mathematics Subject Classification:

[500]Computing methodologies Natural language processing, Computing
methodologies Machine learning, Computing methodologies
Philosophical/theoretical foundations of artificial intelligence, Software and
its engineering Software development methods

##  1\. Introduction

Knowledge engineering (KE) is a discipline concerned with the development and
maintenance of automated processes that generate and apply knowledge [4, 93].
Knowledge engineering rose to prominence in the nineteen-seventies, when
Edward Feigenbaum and others became convinced that automating knowledge
production through the application of research into artificial intelligence
required a domain-specific focus [32]. The period from the mid-nineteen-
seventies into the nineteen-eighties saw the knowledge engineering of rule-
based expert systems for the purposes of the automation of decision making in
business enterprise settings. By the early nineteen-nineties, however, it
became clear that the expert systems approach, given its dependence on manual
knowledge acquisition and rule-based representation of knowledge by highly
skilled knowledge engineers, resulted in systems that were expensive to
maintain and difficult to adapt to changing requirements or application
contexts. Feigenbaum argued that, to be successful, future knowledge-based
systems would need to be scalable, globally distributed, and interoperable
[34].

The establishment of the World Wide Web and the emergence of Web architectural
principles in the mid-nineteen-nineties provided a means to address these
requirements. Tim Berners-Lee argued for a "Web of Data" based on linked data
principles, standard ontologies, and data sharing protocols that established
open standards for knowledge representation and delivery on and across the Web
[11]. The subsequent twenty years witnessed the development of a globally
federated open linked data "cloud" [13], the refinement of techniques for
ontology engineering [51], and methodologies for the development of knowledge-
based systems [86]. During the same period, increasing use of machine learning
and natural language processing techniques led to new means of knowledge
production through the automated extraction of knowledge from natural language
documents and structured data sources [26, 68]. Internet-based businesses in
particular found value in using such technologies to improve access to and
discovery of Web content and data [43]. A consensus emerged around the use of
knowledge graphs as the main approach to knowledge representation in the
practice of knowledge engineering in both commercial and research arenas,
providing easier reuse of knowledge across different tasks and a better
developer experience for knowledge engineers [45].

More recently, the increase in the availability of graphical processing
hardware for fast matrix arithmetic, and the exploitation of such hardware to
drive concurrent innovations in neural network architectures at heretofore
unseen scales [106], has led to a new set of possibilities for the production
of knowledge using large language models (LLMs). LLMs are probabilistic models
of natural language, trained on very large corpora of content, principally
acquired from the Web. Similiar to previous approaches to language modelling ,
given a sequence of tokens LLMs predict a probable next sequence of tokens
based on a learned probability distribution of such sequences. However,
presumably due to the vast amount of content processed in learning and the
large size and architecture of the neural networks involved, LLMs exhibit
remarkable capabilities for natural language processing that far exceed
earlier approaches [60].

These capabilities include the ability to do zero- or few-shot learning across
domains [20], to generalize across tasks, including the ability to perform
domain-independent question answering integrating large amounts of world
knowledge [77], to generate text passages at human levels of fluency and
coherence [28, 96], to deal gracefully with ambiguity and long-range
dependencies in natural language [104], and to reduce or even eliminate the
need for manual feature engineering [98]. LLMs also exhibit the ability to
generate and interpret structured and semi-structured information, including
programming language code [6, 100], tables [46, 53], and RDF metadata [106,
58, 7]. The generalization of language models (termed "foundation models" by
some) to other modalities including images and audio have led to similarly
significant advances in image understanding [23, 117], image generation [38,
79, 83], speech recognition, and text-to-speech generation [78, 105]. Such
capabilities have prompted a significant amount of research and development
activity demonstrating potential applications of LLMs [66, 84, 54]. However,
the means of incorporating LLMs into structured, controllable, and repeatable
approaches to developing and fielding such applications in production use are
only just beginning to be considered in detail [73].

This paper engages with the question of how LLMs can be effectively employed
in the context of knowledge engineering. We start by examining the different
forms that knowledge can take, both as inputs for constructing knowledge
systems and as outputs of such systems. We argue that the distinction between
knowledge expressed in natural language (or other evolved, naturally occurring
modalities such as images or video) and knowledge expressed in formal
languages (for example, as knowledge graphs or rules), sheds light how LLMs
can be brought to bear on the development of knowledge systems.

Based on this perspective, we then describe two potential paths forward. One
approach involves treating LLMs as components within hybrid neuro-symbolic
knowledge systems. The other approach treats LLMs and prompt engineering [57]
as a standalone approach to knowledge engineering111As defined by [57], prompt
engineering is finding the most appropriate prompt or input text to an LLM to
have it solve a given task., using natural language as the primary
representation of knowledge. We then enumerate a set of open research problems
in the exploration of these paths. These problems aim to determine the
feasibility of and potential approaches to using LLMs with existing KE
methodologies, as well as the development of new KE methodologies centered
around LLMs and prompt engineering.

##  2\. Forms of knowledge and their engineering

In the history of the computational investigation of knowledge engineering,
knowledge has been often treated primarily as symbolic expressions. However,
as [39] noted, knowledge is actually encoded in a variety of media and forms,
most notably in natural language (e.g. English) but also in images, video, or
even spreadsheets. This fact becomes even more apparent when looking at
institutional knowledge practices that have developed over centuries, for
example, in the sciences or archives [44]. We now illustrate this point by
describing the many ways in which knowledge manifests itself in the context of
biodiversity informatics.

###  2.1. The multimodal richness of knowledge: an example from biodiversity
sciences

The ultimate goal of biodiversity science is to understand species evolution,
variation, and distribution, but finds applications in a variety of other
fields such as climate science and policy. At its heart is the collection and
observation of organisms, providing evidence for deductions about the natural
world [59]. Such knowledge is inherently multimodal in nature, most commonly
appearing in the form of images, physical objects, tree structures and
sequences, i.e., molecular data.

Historically, organism sightings have been carefully logged in handwritten
field diaries to describe species behaviour and environmental conditions.
Detailed drawings and later photographs were made to capture colour, organs
and other knowledge about an organism’s traits used for identification, which
is best conveyed visually but which is challenging to preserve in natural
specimens. These manuscripts are housed, together with the physical zoological
specimens and herbaria which they describe, in museums and collection
facilities across the world. Both the multimodal nature of these knowledge
sources as well as their distributed nature hamper knowledge integration and
synthesis.

Metadata describes the specimen’s provenance: where specimens were found, who
found them, and provides an attempt at identifying the type of organism (such
as the preserved squid specimen shown in Figure 2). Such knowledge is
paramount, as it allows researchers to understand resources within the context
in which they were produced, enabling researchers to carry out ecological
studies such as distribution modeling over time.

For a systematic comparison of the multitude of resources available, the
biodiversity sciences have had a long-standing tradition of developing
information standards [67]. From Linnaeus’ Systema naturae mid 18th century as
well as his formal introduction of zoological nomenclature, taxonomists have
started categorizing natural specimens according to tree-like hierarchical
structures. The process is challenging, given that biologist up until this day
do not have a full picture of all living organisms on earth, and incomplete,
naturally evolved and fuzzy knowledge is not easily systematized.

Figure 1. A specimen of the Loligo vulgaris Lamarck, 1798 species from the
Naturalis–Zoology and Geology
catalogues.222https://bioportal.naturalis.nl/nl/specimen/RMNH.MOL.5009890
Images free of known restrictions under copyright law (Public Domain Mark
1.0).

The development of digital methods has opened up new pathways for comparison
and analysis. Gene sequencing technology has lead biologist to the genetic
comparison of species, by the calculation of ancestry and construction of
evolutionary tree structures in the study of phylogeny [50]. More importantly,
digital methods allowed the transfer of analog resources, such as specimen
collection scans [14] and metadata, to the digital world. Such techniques have
furthered formalisation and thereby interoperability of collected data through
the use of Web standards, such as globally unique identifiers for species
names [72] as well as shared vocabularies for data integration across
collections [10]. The Global Biodiversity Information Facility (GBIF) and
their data integration toolkit serves as a great example of such integration
efforts [97, 81]. Currently, there is a large emphasis on linking up disparate
digital resources in the creation of an interconnected network of digital
collection objects on the Web, linked up with relevant ecological,
environmental and other related data in support of machine actionability
(i.e., the ability of computational systems to find, access, interoperate, and
reuse data with minimal intervention) for an array of interdisciplinary tasks
such as fact-based decision-making and forecasting [41].Using data standards
for describing and reasoning over collection data can aid researchers counter
unwanted biases via transparency. However, making data comply with data
standards can also lead to oversimplification or reinterpretation [71].

Machine learning and knowledge engineering strategies can help to
(semi-)automatically extract and structure biodiversity knowledge according
[102, 91], for instance using state-of-the-art computer vision or natural
language processing techniques as well as crowd-sourcing platforms for the
annotation of field diaries and other collection objects with formal language
[92, 29]. Nevertheless, a bottleneck in the digitization of collections and
their use for machine actionability is the amount of work and domain expertise
required for the formalisation of such knowledge, and the extraction from
unstructured texts, images and video’s. Historical resources, i.e. handwritten
texts, pose an additional challenge, as they are exceptionally challenging to
interpret within the current scientific paradigm [107].

The variety and usefulness of different forms of knowledge both natural and
formal and the challenges they pose is not limited to the biodiversity domain
as described above. We see the same diversity happening in law [82], medicine
[16, 21] and even self-driving vehicles [9]. To summarize:

  * •

domain knowledge is often best represented in a variety of modalities, i.e.,
images, taxonomies, or free text, each modality with its own data structure
and characteristics which should be preserved, and no easy way of integrating,
interfacing with or reasoning over multimodal knowledge in a federated way
exists;

  * •

provenance of data is paramount in understanding knowledge within the context
in which it was produced;

  * •

fuzzy, incomplete, or complex knowledge is not easily systematized;

  * •

using data standards for describing and reasoning over collection data can aid
researchers counter unwanted biases via transparency;

  * •

making data comply with data standards can lead to oversimplification or
reinterpretation;

  * •

the production of structured domain knowledge, for instance from images or
free text, requires domain expertise, and is therefore labour intensive and
costly;

  * •

knowledge evolves, and knowledge-based systems are required to deal with
updates in their knowledge bases.

###  2.2. KE as the transformation of knowledge expressed in natural language
into knowledge expressed in a formal language

This sort of rich and complex array of modalities for the representation of
knowledge has traditionally posed a challenge to knowledge engineers [33].
Much of the literature on knowledge engineering methodology has focused on the
ways in which knowledge in these naturally-occurring forms can be recast into
a structured symbolic representation, e.g., using methods of knowledge
elicitation from subject matter experts [88], for instance by the formulation
of competency questions for analysing application ontologies [12]. One way to
think about this is as the process of expressing knowledge presented in a
natural, humanly evolved language in a formally-defined language. This notion
of the transformation of natural language into a formal language as a means of
enabling effective reasoning has a deep history rooted in methodologies
developed by analytical philosophers of the early twentieth century [24, 69],
but dating even further back to Liebniz’s lingua rationalis [35] and the
thought of Ramón Lull [37]. Catarina Dutilh Novaes [69] has argued that formal
languages enable reasoning that is less skewed by bias and held beliefs, an
effect achieved through de-semantification, i.e., the process of replacing
terms in a natural language with symbols that can be manipulated without
interpretation using a system of rules of transformation. Coupled with
sensorimotor manipulation of symbols in a notational system, people can reason
in a manner that outstrips their abilities unaided by such a technology.

While Dutilh Novaes’ analysis focuses on this idea of formal languages as a
cognitive tool used by humans directly, e.g. through the manipulation of a
system of notation using paper and pencil, she notes that this manipulation of
symbols is the route to the mechanization of reasoning through computation.
When externally manifested as a function executed by a machine through either
interpretation by an inference engine, or through compilation into a machine-
level language, this approach of formalization yields the benefits of
reliability, greater speed and efficiency in reasoning.

This idea captures precisely the essence of the practice of knowledge
engineering: Starting from sources of knowledge expressed in natural language
and other modalities of human expression, through the process of formalization
[51, 95], knowledge engineers create computational artifacts embodying this
knowledge. These computational artifacts then enable us to reason using this
knowledge in a predictable, efficient, and repeatable fashion. This is done
either by proxy through the action of autonomous agents, or in the context of
human-mediated decision-making processes.

###  2.3. LLMs as a general-purpose technology for transforming natural
language into formal language

Until recently, there have been two ways in which this sort of formalization
could be performed: through the manual authoring of symbolic/logical
representations, e.g., as in the traditional notion of expert systems [34], or
through the use of machine learning and natural language processing to extract
such representations automatically from natural language text [61]. But what
has become evident with the emergence of LLMs, with their capabilities for
language learning and processing, is that they provide a new and powerful type
of general purpose tool for mapping between natural language333Again, we note
that natural language should be read to include all modalities. Hence, the
term “foundation model”[15] to refer to LLMs. and formal language, as well as
other modalities. LLMs have shown state-of-the-art performance on challenging
NLP tasks such as relation extraction [5] or text abstraction/summarization
[114], and have been used to translate between other modalities, such as
images and text (called vision-language models [119, 77]) in computer vision
tasks, or from natural language to code [113, 47], in which a pretrained task-
agnostic language model can be zero-shot and few-shot transferred to perform a
certain task [20, 52]. If one accepts the position that KE can be generally
described as the process of transforming knowledge in natural language into
knowledge in formal language, then it becomes clear that LLMs provide an
advance in our ability to perform knowledge engineering tasks.

##  3\. The use of LLMs in the practice of knowledge engineering: two
scenarios

Given the above discussion, the natural question that arises is: what might be
the utility and impact of the use of LLMs for the transformation of natural
language into formal language, when applied in the context of the practice of
knowledge engineering?

When LLMs emerged as a new technology in the mid-2010s, two views of the
relationship between LLMs and knowledge bases (KBs) were put forward. One was
the LLM can be a useful component for various processes that are part of a
larger knowledge engineering workflow (i.e. "LMs for KBs" [3]); the other was
that that the LLM is a cognitive artifact that can be treated as a knowledge
base in and of itself (i.e., "LMs as KBs" [75]). We exploit this dichotomy to
formulate a pair of possible future scenarios for the use of LLMs in the
practice of KE. One is to use LLMs as a technology for or tool in support of
implementing knowledge tasks that have traditionally been build using older
technologies such as rule bases and natural language processing (NLP). Another
is to use LLMs to remove the need for knowledge engineers to be fluent in a
formal language, i.e., by allowing knowledge for a given knowledge task to be
expressed in natural language, and then using prompt engineering as the
primary paradigm for the implementation of reasoning and learning. We now
explore each of these scenarios in turn, and consider the open research
problems that they raise.

###  3.1. LLMs as components or tools used in knowledge engineering

We illustrate the first scenario through reference to CommonKADS [86], a
structured methodology that has been used by knowledge engineers since the
early 2000’s. CommonKADS is the refinement of an approach to providing a
disciplined approach to the development of knowledge systems. This approach
saw initial development in the nineteen-eighties as a reaction to both the ad-
hoc nature of early expert systems development [111] and to the frequency of
failures in the deployment of expert systems in an organizational context
[34]. Stemming from early work on making expert systems development
understandable and repeatable [42], CommonKADS is distinguished from
methodologies more focused on ontology development (e.g., NeON [94], Kendall
and McGuinness’s "Ontology 101" framework [51], and Presutti’s ontology design
patterns [76]) in that it provides practical guidance for specification and
implementation of knowledge systems components in a broader sense. It attempts
to provide a synoptic guide to the full scope of activities involved in the
practice of KE, and show how it relates to the activities of the organization
in which that engineering is taking place. As such, in the context of this
paper we can use it as a framework to explore for what tasks and in what ways
LLMs can be used for KE.

Some tasks identified by CommonKADS as part of the KE process may remain
largely unchanged by the use of LLMs. These include knowledge task
identification and project organizational design. But others can involve the
use of LLMs. LLMs can assist knowledge engineers and/or knowledge providers in
the performance of knowledge engineering tasks. They can also be a means for
the implementation of modules performing knowledge-intensive tasks. Examples
of these uses include the following:

Knowledge acquisition and elicitation:

LLMs can be used to support knowledge acquisition and elicitation in a given
domain of interest. Engineers can create prompts that target specific aspects
of the domain, using the responses as a starting point for building the
knowledge base. Dialogs between LLMs trained using such prompts and knowledge
providers, the subject matter experts, can support the review, validation, and
refinement of the acquired knowledge [8].

Knowledge organization:

LLMs can be used to organize the acquired knowledge into a coherent structure
using natural language, making it easy to understand and update. Prompt
engineering can be used to develop a set of prompts that extract formal
language using the LLM, e.g., for text to graph generation [40] or vice versa
[18, 2]. Moreover, LLMs are used for program synthesis [113, 47], the
generation of metadata [56] or for fusing knowledge graphs [118].

Data augmentation:

LLMs can be used to generate synthetic training data to aid in testing the
knowledge system by evaluating its performance on instances of the specific
task [116].

Testing and refinement:

Feedback from subject matter experts and users can be used to prompt an LLM to
refine the natural language knowledge base and improve the system’s accuracy
and efficiency through self-correction of prompts and tuning of the LLM model
settings as needed to optimize the system’s performance [110].

Maintenance:

LLMs can be used to monitor new information and trends, and to then propose
new prompts integrating those updates into the knowledge base.

Consider the CommonKADS knowledge task hierarchy shown in Figure 2. Synthetic
knowledge-intensive tasks, e.g. design or configuration, are amenable to
generative approaches [109]; analytic knowledge-intensive tasks can involve
LLM components within a hybrid neuro-symbolic knowledge system.

Figure 2. Hierarchy of knowledge-intensive task types from CommonKADS ([86],
p.125)

A shortcoming of using CommonKADS for our purposes, however, is that it
predates the widespread use of machine learning and statistical natural
language processing in KE. A number of architectural approaches have since
been developed that extend the CommonKADS concepts of a knowledge-intensive
task type hierarchy and knowledge module templates. These include modeling the
fine-grained data flows and workflows associated with knowledge systems that
combine components that ingest, clean, transform, aggregate and generate data,
as well as generate and apply models built using machine learning [103, 19,
27, 31, 101]. These architectures are put forward as providing a general
framework for composing heterogeneous tools for knowledge representation and
inference into a single integrated hybrid neuro-symbolic system. The design
pattern notations put forward in recent work [103, 101, 31] treat data,
models, and symbolic representations as the inputs and outputs of components
composed into a variety of knowledge system design patterns. Generalizing
these into natural language and formal language inputs and outputs can provide
a simple way to extend these design notations to accommodate both LLMs as well
as a richer set of knowledge representations.

###  3.2. Knowledge engineering as prompt engineering

Given that LLMs enable knowledge modeling in natural language, it is
conceivable that the programming of knowledge modules could take place
entirely in natural language. Consider that prompt programming is "finding the
most appropriate prompt to allow an LLM to solve a task" [57]. One can through
this lens view knowledge engineering as the crafting of dialogues in which a
subject matter expert (SME) arrives at a conclusion by considering the
preceding context and argumentation [80, 109, 89, 60]. This framing of
knowledge engineering as prompt engineering is the second scenario we wish to
explore.

From the perspective of the CommonKADS knowledge-intensive task type
hierarchy, this would involve a redefinition of the types and hierarchy to use
LLMs and prompt programming design patterns, e.g. as described in [57].
Several aspects of this redefinition could include:

Natural language inference:

LLMs can be used to build natural language inference engines that use the
organized knowledge to perform the specific task by taking input queries and
generate output using prompt engineering to guide the LLM towards generating
accurate inferences, e.g. using zero- or few-shot chain-of-thought design
patterns. The benefit here is that the gap between the knowledge engineer,
knowledge provider (the subject matter expert) and the user is smaller since a
translation to a formal language (the language of the engineer) is no longer
required.

Knowledge-intensive task execution through human/machine dialog:

LLMs can be used to a conversational interface that allows users to interact
with the knowledge system and receive task-specific support.

Testing and refinement through human/machine dialog:

Feedback from subject matter experts and users can be used to prompt an LLM to
refine the natural language knowledge base and improve the system’s accuracy
and efficiency through self-correction of prompts and tuning of the LLM model
settings as needed to optimize the system’s performance.

One possible benefit of this approach would be that the barrier to adoption of
knowledge engineering as a practice could be lowered significantly. Knowledge
elicitation could be conducted entirely within natural language, meaning that
subject matter experts without training in formal knowledge representations
could perform these tasks directly. However, this approach assumes that
predictable inference [101] using natural language is satisfactory. The
propensity of current LLMs to "hallucinate", i.e., to confabulate facts, is an
obstacle to the realization of this idea [48]. Multiple efforts have been
devoted to the creation of prompt programming patterns that address this
issue, ranging from chain-of-thought approaches [108] to retrieval-assisted
generation, i.e. the augmentation of LLMs with authoritative document indexes
and stores [84, 65]. Recent work [73] has described ways in which knowledge
graphs as a formal language can be integrated with natural language and LLM-
based language processing and reasoning to provide knowledge systems
architectures that directly address this issue. [115] surveys work in this
direction.

##  4\. Open research questions

Using the scenarios outlined above, we can identify a number of open research
questions to be addressed to realize either or both of these two possible
approaches to the use of LLMs in knowledge engineering. These questions touch
on three general areas: the impact of LLMs on the methodologies used to build
knowledge systems, on the architectural design of knowledge systems
incorporating and/or based on LLMs, and on the evaluation of such systems. For
each of these open questions, we provide a link back to the biodiversity
scenario discussed in Section 2.1 denoted by a .

###  4.1. Methodology

####  4.1.1. How can knowledge engineering methodologies best be adapted to
use LLMs?

How can we harmoniously meld the considerable body of work on knowledge
engineering methodologies [51, 36, 76, 94, 87, 85, 90] with the new
capabilities presented by LLMs?

Schreiber’s conceptualization of knowledge engineering as the construction of
different aspect models of human knowledge [86], as discussed above, offers a
framework for further elaboration. The distinctive characteristics of LLMs,
coupled with prompt engineering, present unique challenges and opportunities
for building agents within a knowledge system, one that is consistent with the
CommonKADS approach.

While the role definitions within KE methodologies might mostly remain the
same, the skills required for knowledge engineers will need morphing to adapt
to the LLM environment. This evolution of roles calls for an extensive
investigation into what these new skills might look like, and how they can be
cultivated. Additionally, the adaptability of the various knowledge-intensive
task type hierarchies described by CommonKADS and its descendants in the
literature on hybrid neuro-symbolic systems (e.g., as described in [19]) to
accommodate LLMs is another fertile area for exploration.

LLM-based applications, likened to synthetic tasks within these knowledge
engineering frameworks, raise compelling research questions regarding accuracy
and the prevention of hallucinations. LLM-based applications have a lower bar
to reach with respect to notions of accuracy and avoidance of hallucinations,
but still must provide useful and reliable guidance to users and
practitioners.

Connecting back to the biodiversity domain, answering these questions would
provide guidance on the appropriate methodology to adopt when developing a new
specimen curation and collection knowledge management system that needs to
deal with multimodal assets like handwritten text or images.

####  4.1.2. How do principles of content and data management apply to prompt
engineering?

Applying content and/or data management principles to collections of prompts
and prompt templates, integral to work with LLMs, is an area ripe for
exploration. Properly managing these resources could improve efficiency and
guide the development of improved methodologies in knowledge engineering. This
calls for a rigorous investigation of current data management practices, their
applicability to LLMs, and potential areas of refinement. Ensuring the
reproducibility of LLM engineering from a FAIR data standpoint [112] is a
crucial yet complex challenge. Developing and validating practices and
protocols that facilitate easy tracing and reproduction of LLM-based processes
and outputs is central to this endeavour.

Addressing this challenge will aid researchers in applying LLM engineering in
a FAIR way. Doing so is critical for biodiversity research and science in
general where precision, reproducibility and provenance are key for knowledge
discovery and research integrity.

####  4.1.3. What are the cognitive norms that govern the conduct of KE?

A crucial area of inquiry involves the identification and understanding of
cognitive norms, as described by Menary [62], that govern the practice of
knowledge engineering. Cognitive norms are established within a human
community of practice as a way of governing the acceptable use of "external
representational vehicles to complete a cognitive task" [63]. As the consumer
adoption of LLM technology has progressed, we see a great deal of controversy
about when and how it is appropriate to use, e.g. in the context of education
or the authoring of research publications. Understanding how these norms shape
the use of LLMs in this context is an under-explored field of study. By
unravelling the interplay between these cognitive norms and LLM usage, we can
gain valuable insights into the dynamics of knowledge engineering practices
and possibly foster more effective and responsible uses of LLMs.

In the biodiversity sciences, this means understanding the cognitive norms
specific to the domain, to understand how LLMs can be used in a way that
respects the domain’s practices and standards.

####  4.1.4. How do LLMs impact the labor economics of KE?

A related but distinct question pertains to the impact of LLMs on the economic
costs associated with knowledge engineering. The introduction and application
of LLMs in this field may significantly alter the economic landscape, either
by driving costs down through automation and efficiency or by introducing new
costs tied to system development, maintenance, and oversight. Thoroughly
exploring these economic implications can shed light on the broader effects of
integrating LLMs into knowledge engineering.

The realm of labor economics as it pertains to hybrid or centaur systems [1],
is another area ripe for investigation. Understanding how the deployment of
these systems influences labor distribution, skill requirements, and job roles
could provide valuable input into the planning and implementation of such
technologies. Additionally, it could reveal the potential societal and
economic impacts of this technological evolution.

Developments for LLM-based KE can help mitigate labour of knowledge experts in
the biodiversity sciences, for instance by the development of more efficient
KE workflows for the digitization of museum specimens or manuscripts.

###  4.2. Architecture

####  4.2.1. How can hybrid neuro-symbolic architectural models incorporate
LLMs?

Design patterns for hybrid neuro-symbolic systems, as described in [103],
offer a structured approach to comprehend the flow of data within a knowledge
system. Adapting this model to account for the differences between natural and
formal language could significantly enhance our ability to trace and manage
data within knowledge systems. A salient research question emerging from this
scenario pertains to the actual process of integrating LLMs into knowledge
engineering data processing flows [27]. Understanding the nuances of this
process will involve a deep examination of the shifts in methodologies,
practices, and the potential re-evaluations of existing knowledge engineering
paradigms. The perspective of KE enabled by LLMs as focused on the
transformation of natural language into formal language provides insights that
can be used to improve the motivation for hybrid neuro-symbolic systems; e.g.,
[19] references [17] in using dual process theories of reasoning (i.e. the
"System 1/System 2" model described in [49]) as a motivation for hybridization
in knowledge systems, but more recent analyses [69, 64] cast doubt on the
validity of such models, and point to more nuanced perspectives that provide a
better grounding for the benefits of hybridization.

Addressing these questions would shed light on tasks for which hybridization
using LLMs would prove favourable, e.g., image classification of species.

####  4.2.2. How can prompt engineering patterns support reasoning in natural
language?

One fundamental question that arises is how prompt engineering patterns can be
utilized to facilitate reasoning in natural language. Exploring this topic
involves understanding the mechanics of these patterns and their implications
on natural language processing capabilities of LLMs. This line of research
could open new possibilities for enhancing the functionality and efficiency of
these models.

A related inquiry concerns the structure, controllability, and repeatability
of reasoning facilitated by LLMs. Examining ways to create structured,
manageable, and reproducible reasoning processes within these models could
significantly advance our capacity to handle complex knowledge engineering
tasks and improve the reliability of LLMs.

The interaction of LLMs and approaches to reasoning based on probabilistic
formalisms is also an underexplored area of research. A particularly evocative
effort in this area is that described in [113], which describes the use of
LLMs to transform natural language into programs in a probabilistic
programming language, which can then be executed to support reasoning in a
particular problem domain. We note that this work provides an excellent
example of the knowledge engineering as the transformation of natural language
into formal language perspective and of the impact of LLMs in advancing that
perspective. Investigating how to automatically generate and assess other
nuanced forms of knowledge within LLMs could lead to a more refined
understanding of these models and their capabilities.

Given that biodiversity knowledge is often best represented in a variety of
modalities each with their own data structures and characteristics, research
may explore how LLMs can act as natural language interfaces to such multimodal
knowledge bases.

####  4.2.3. How can we manage bias, trust and control in LLMs using knowledge
graphs?

Trust, control, and bias in LLMs, especially when these models leverage
knowledge graphs, are critical areas to explore. Understanding how to detect,
measure, and mitigate bias, as well as establish trust and exert control in
these models, is an essential aspect of ensuring ethical and responsible use
of LLMs. Furthermore, investigating methods to update facts in LLMs serving as
knowledge graphs is a crucial area of research. Developing strategies for
efficient and reliable fact updating could enhance the accuracy and usefulness
of these models.

Another key question involves understanding how we can add provenance to
statements produced by LLMs. This line of research could prove vital in
tracking the origin of information within these models, thus enhancing their
reliability and usability. It opens the door to more robust auditing and
validation practices in the use of LLMs.

Addressing this challenge can help biodiversity researchers detect and
mitigate biases, as use of LLMs might further exacerbate knowledge gaps, e.g.,
groups of individuals omitted from historical narratives in archival
collections. Moreover, novel update mechanisms can aid researchers to reliably
update facts or changing knowledge structures learned by LLMs, for instance
when domain knowledge evolves.

####  4.2.4. Is extrinsic explanation sufficient?

A significant area of interest pertains to how we can effectively address the
explainability of answers generated using LLMs [30]. This exploration requires
a deep dive into the functioning of LLMs and the mechanisms that govern their
responses to prompts. Developing a thorough understanding of these processes
can aid in creating transparency and trust in LLMs, as well as fostering their
effective use.

The need for explanation in LLMs also leads to the question of whether
extrinsic explanation is sufficient for the purposes of justifying a knowledge
system’s reasoning, as argued in general for the intelligibility of knowledge
systems by Cappelen and Devers [22], or if intrinsic explainability is a
necessary requirement [55]. This question calls for a thoughtful exploration
of the value and limitations of both extrinsic and intrinsic explanation
methodologies, and their implications for the understanding and usage of LLMs.
An exciting research avenue arises from the work of Tiddi [99], concerning
explainability with formal languages. The exploration of this topic could
reveal significant insights into how we can leverage formal languages to
enhance the explainability of LLMs. This could pave the way for new methods to
increase transparency and intelligibility in these models.

In the sciences in general, answering these questions would aid explainability
of LLM-generated answers via curated facts, increasing transparency and trust.

####  4.2.5. How can LLMs support the engineering of hybrid human/machine
knowledge systems?

Another topic of interest involves exploring the potential of hybrid systems
that combine human cognition with machine capabilities within a dialogical
framework [64, 70]. As an exciting example of the possibilities for new
approaches to human/machine collaboration in this vein, we point to the recent
results reported by [74] on the creation of conversational agents that
simulate goal-directed human conversation and collaboration on tasks. One can
imagine coupling LLM-based agents with human interlocutors working
collaboratively in this manner on specific knowledge-intensive tasks.
Understanding how to develop these types of systems, and what their
implications might be for the practice of knowledge engineering presents a
fertile research line. It requires the careful analysis of human-machine
interaction, the study of system design principles, and the investigation of
their potential impact.

Research in this avenue can help mitigate the workload of the knowledge
expert, for instance in the elicitation of domain knowledge, or crowdsourcing
of annotations from unstructured sources such as herbaria or manuscripts.

###  4.3. Evaluation

####  4.3.1. How do we evaluate knowledge systems with LLM components?

The first point of interest involves the evaluation of knowledge-based
systems, with a focus beyond just logic. This area calls for innovative
methodologies to assess the system’s capacity to manage and utilize knowledge
efficiently, going beyond traditional logical evaluations. This topic of
evaluation naturally extends to the question of how we evaluate ontologies and
design patterns within knowledge engineering. Evaluating these aspects would
require a deep dive into the structures and mechanisms underpinning these
elements, potentially leading to the development of refined evaluation metrics
and methodologies.

Interestingly, the long-standing paradigm of machine learning evaluation,
relying on benchmarking against a standard train/test dataset, seems to falter
in the era of LLMs [25]. This presents an intriguing challenge for researchers
and engineers alike. It is quite possible that traditional methods may need to
be significantly buttressed by methodologies and supporting tools for the
direct human evaluation of knowledge system performance. This has implications
concerning the cost and speed of evaluation processes, encouraging the rethink
of current approaches to perhaps develop new strategies that balance accuracy,
cost-effectiveness, and timeliness. Reimagining evaluation methodologies in
this new context could provide transformative insights into how we can gain
confidence in the reliability engineering of knowledge systems that use LLMs.

Developments in this direction may aid biodiversity researchers to get a
better understanding of the real-world efficacy of employing knowledge-based
systems with LLM components in their institutions. One can think of improving
access to collections, knowledge discovery, or accuracy in describing
institutional knowledge.

####  4.3.2. What is the relationship between evaluation and explainability?

Lastly, there is an inherent dependency of evaluation on effective solutions
for explainability within knowledge systems. Understanding this relationship
could help in the creation of more comprehensive evaluation models that take
into account not only the performance of a system but also its explainability.

##  5\. Summary

In this paper, we have advocated for a reconsideration of the practice and
methodology of knowledge engineering in light of the emergence of LLMs. We
argued that LLMs allow naturally-occurring and humanly-evolved means of
conveying knowledge to be brought to bear in the automation of knowledge
tasks. We described how this can enhance the engineering of hybrid neuro-
symbolic knowledge systems, and how this can make knowledge engineering
possible by people who do not necessarily have the experience of recasting
natural language into formal, structured representation languages. Both of
these possibilities will involve addressing a broad range of open questions,
which we have attempted to outline above. Given the rapid pace of the
development of this area of research, it is our earnest hope that the coming
months and years will yield results shedding light on these questions.

##### Acknowledgements

This work was partially supported by the EU’s Horizon Europe research and
innovation programme within the ENEXA project (grant Agreement no. 101070305).

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