$i_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}$ |  `iReg4ptX[`$\hat{d},\\{\hat{\Delta}_{1},\hat{\Delta}_{2},\hat{\Delta}_{3},\hat{\Delta}_{4},\hat{\Delta}_{x}\\}$`]` |  `iRen4ptX[`$d,\\{\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4},\Delta_{x}\\}$`]`

  * •

For example, the amplitude $\hat{i}_{[222]}$ regulated in the half-integer
scheme with $u=1$ and $v_{j}=0$ can be obtained by the command

By default, the regulator is represented by $\epsilon$ while the momentum
magnitudes are $k_{1},k_{2},k_{3},k_{4}$ and $s$ for the Mandelstam variable.

  * •

By default the amplitude is expanded to the highest available order in the
regulator. Since the 3-point amplitudes regulated in the half-integer scheme
are exact, the expression above is not expanded in $\epsilon$ at all. In a
general regularization scheme, the expressions are expanded to order
$\epsilon^{1}$:

  * •

The default settings can be adjusted by passing the following options to
`iReg3pt`, `jReg3pt`, `iReg4ptC` and `iReg4ptX`:

Option | Description
---|---
`Regulator` | Symbol for the regulator
`Momenta` | A list of symbols for the momentum magnitudes
`ExpansionOrder` | The expansion order in the regulator

If no regulator is specified, the half-integer scheme is assumed by default.
Thus, for example, `iReg3pt[3, {2,2,2}]` returns the amplitude in the half-
integer scheme.

Alternatively, say we are interested in the amplitude $\hat{i}_{[222]}$
regulated in the scheme $\hat{d}=3+\delta$ and $\hat{\Delta}_{j}=2$ for
$j=1,2,3$, where $\delta$ is the regulator. Moreover, we want the expansion to
include finite and divergent terms, i.e., terms up to order $\epsilon^{0}$,
and with momentum magnitudes denoted by $p_{1},p_{2}$ and $p_{3}$. This can be
achieved by:

  * •

The explicit expressions for 4-point exchange amplitudes can be quite
involved, particularly in a general regularization scheme. This is due to the
use of the equation (266) and the complicated structure of the scheme changing
terms $i^{\text{div}}(u,v_{j})-i^{\text{div}}(1,0)$. As an example, consider
the amplitude $\hat{i}_{[33,22x2]}$ regulated in the scheme where
$\hat{d}=3-\epsilon$ while conformal dimension $\Delta_{j}$ are not regulated
for all $j=1,2,3,4,x$. The amplitude can be accessed by the command

By passing the option `KeepHeld->True` one can prevent the scheme-changing
terms from expanding. This produces results similar in presentation to those
given in Section 5.5, as illustrated in Figure 7.

Figure 7: Suppressing the expansion of scheme-changing terms.

To fully expand these expressions, use Mathematica’s `ReleaseHold` command.

  * •

Renormalized amplitudes are accessed in a similar fashion, but no regulator is
present. For $i^{\text{ren}}_{[222]}$, for example, we get

  * •

In renormalized amplitudes, by default the renormalization scale is denoted by
$\mu$ while renormalization constants are denoted by $\mathfrak{a}$. These can
be changed by the following options passed to the functions:

Option | Description
---|---
`RenormalizationScale` | Symbol for the renormalization scale
`RenormalizationConstant` | Symbol for the renormalization constants
`Momenta` | A list of symbols for the momentum magnitudes

###  Notebooks

The package is accompanied by a number of Mathematica notebooks which derive
the results presented in this paper.

  * •

`BetaScheme.nb` contains calculations of 2-, 3-, and 4-point functions
regulated in the half-integer scheme (1). The notebook requires the `HypExp`
package Huber:2005yg , which is included in the packet. The raw results of the
calculations are then saved to `Results_BetaScheme.nb`.

  * •

`GeneralScheme.nb` contains calculations of 2-, 3-, and 4-point functions
regulated in a general scheme. It also contains calculations of the 3-point
amplitudes $\hat{i}_{[\Delta_{1}\Delta_{2}\Delta_{3}]}$ and
$\hat{j}_{[\Delta_{1}\Delta_{2}\Delta_{3}]}$ including terms of order
$\epsilon^{1}$ in the regulator. The notebook requires the `HypExp` package as
well as the `TripleK` package Bzowski:2020lip , both of which are included in
the packet. The results of the calculations are saved to
`Results_GeneralScheme.nb`.

  * •

`Checks.nb` provides checks on the results contained in
`Results_BetaScheme.nb` and `Results_GeneralScheme.nb`. Furthermore, it
compares these raw results with the expressions presented in Sections 3 and
5.5 and stored in the package file `HandbooK.wl`.

  * •

`Renormalization.nb` contains the results presented in Section 4. First, we
check that the regulated expressions are renormalized by the counterterms
listed, and that they produce the renormalized amplitudes presented in Section
4.3 and stored in `HandbooK.wl`. Next, we calculate the beta functions of the
symmetric theory from Section 4.5.3, and finally, we check the Callan-Symanzik
equation for all 3- and 4-point functions in the symmetric model.

##  Outlook

This handbook has presented explicit closed-form expressions for all
renormalized holographic $4$-point functions derived from contact and exchange
diagrams of scalar operators with dimensions $\Delta=2,3$ in $d=3$. The
results were obtained using dimensional regularization. We saw that the half-
integer scheme (1) is very convenient for carrying out explicit computations,
and we also discussed how to change scheme and presented our results both in
the half-integer scheme and the general scheme (2). We believe this is the
most extensive set of renormalized correlators of this kind currently
available. Besides the listings given in the text, all our results are
available in the accompanying Mathematica notebooks as explained in the
previous section.

There are many natural extensions of our work:

  * •

Generalizations to cases involving both external and internal operators with
spin.

  * •

Analysis of even-dimensional spacetimes including $d=4$.

  * •

Computations for more general operator dimensions.

In addition, as we hope to report soon toappear , our results have immediate
implications for the computation of wavefunction coefficients and correlators
in de Sitter spacetime.

### Acknowledgments

AB is supported by the NCN POLS grant No. 2020/37/K/ST2/02768 financed from
the Norwegian Financial Mechanism 2014-2021 . PM is supported by the Science
and Technology Facilities Council through an Ernest Rutherford Fellowship
(ST/P004326/2). KS is supported in part by the Science and Technology
Facilities Council (Consolidated Grant ST/T000775/1).

##  Appendix A Conventions and definitions

###  QFT conventions

  * •

We work exclusively in $d=3$ Euclidean spacetime dimensions.

  * •

Scaling dimensions are denoted throughout via square brackets, including where
they appear as superscripts on operators and sources. We assign the dimensions
$[\bm{x}]=-1$ and $[\partial_{\mu}]=1$ for coordinates and their derivatives,
$[\bm{k}]=1$ for momenta, $[\mu]=1$ for the RG scale,
$[\mathcal{O}_{j}]=\Delta$ for operators and $[\phi_{j}]=d-\Delta$ for their
sources.

  * •

Our main focus will be scalar operators $\mathcal{O}^{[2]}$ and
$\mathcal{O}^{[3]}$ of dimensions $\Delta=2,3$, their sources
$\phi^{[1]},\phi^{[0]}$ and the holographically dual bulk fields $\Phi^{[2]}$
and $\Phi^{[3]}$.

  * •

Regarding the relation of the generating functional of connected diagrams $W$
to the action $S$, we follow the conventions of Skenderis:2002wp  and
Bzowski:2016kni  and define $W$ as

$\displaystyle Z=e^{W}=\langle e^{-S}\rangle$ (453)

from which it follows that

$\displaystyle\langle\mathcal{O}_{1}(\bm{x}_{1})\ldots\mathcal{O}_{n}(\bm{x}_{n})\rangle=(-1)^{n}\frac{\delta^{n}W}{\delta\phi_{1}(\bm{x}_{1})\ldots\delta\phi_{n}(\bm{x}_{n})}.$
(454)

  * •

The most general regularization scheme we use is (2). The parameterization is
such that the natural parameters arising in holographic calculations
Bzowski:2015pba ,

$\displaystyle\alpha=\frac{d}{2}-1,$
$\displaystyle\beta_{j}=\Delta_{j}-\frac{d}{2}$ (455)

are regulated according to

$\displaystyle\alpha\longmapsto\hat{\alpha}=\alpha+u\epsilon,$
$\displaystyle\beta_{j}\longmapsto\hat{\beta}_{j}=\beta_{j}+v_{j}\epsilon.$
(456)

Here, $\beta_{j}$ is the index associated with the Bessel function
representing the bulk-boundary propagator for external leg $j$.

Any quantity $f=f(d,\Delta_{j})$ depending on the dimensions $d$ and
$\Delta_{j}$ is regulated according to the selected scheme in (2). We denote
the regulated version of $f$ as $\hat{f}$, i.e.,
$\hat{f}=f(\hat{d},\hat{\Delta}_{j})$.

  * •

Mostly, though, we work in the special ‘half-integer’ scheme (1). This
corresponds to setting $u=1$ and $v_{j}=0$ for all $j=1,2,3,4,x$ in the
general scheme (2). In this scheme the value of the $\beta_{j}$ parameters do
not change:

$\displaystyle\alpha\longmapsto\hat{\alpha}=\alpha+\epsilon,$
$\displaystyle\beta_{j}\longmapsto\hat{\beta}_{j}=\beta_{j}.$ (457)

This is a good renormalization scheme in $d=3$ for operators of dimension
$\Delta=2,3$ in the sense that it regulates all correlation functions of such
operators.

  * •

Divergences and scale-violating terms are parametrized by

$\displaystyle\operatorname{Div}_{a}(k)$
$\displaystyle=\Gamma(a\epsilon)k^{-a\epsilon}$
$\displaystyle=\frac{1}{a\epsilon}-(\log
k+\gamma_{E})+\frac{a\epsilon}{2}\left(\log k^{2}+2\gamma_{E}\log
k+\gamma_{E}^{2}+\frac{\pi^{2}}{6}\right)+O(\epsilon^{2}),$ (458)

where the momentum magnitude $k$ can also be replaced by the RG scale $\mu$.

$\operatorname{Div}_{a}(k)$ satisfies the identity

$\displaystyle\operatorname{Div}_{a}(k)-\operatorname{Div}_{a}(\mu)$
$\displaystyle=-\log\left(\frac{k}{\mu}\right)+O(\epsilon).$ (459)

###  Definitions for momenta

  * •

External momenta are denoted $\bm{k}_{j}$, with lengths or magnitudes
$k_{j}=|\bm{k}_{j}|$, where $j=1,2,\ldots$. The Mandelstam variables are

$\displaystyle s=|\bm{k}_{1}+\bm{k}_{2}|,$ $\displaystyle
t=|\bm{k}_{1}+\bm{k}_{3}|,$ $\displaystyle u=|\bm{k}_{2}+\bm{k}_{3}|$ (460)

without squares. For convenience, we also adopt the convention $k_{s}=s$.

  * •

The $3$\- and $4$-point total magnitudes are denoted

$\displaystyle k_{t}=k_{1}+k_{2}+k_{3},$ $\displaystyle
k_{T}=k_{1}+k_{2}+k_{3}+k_{4}.$ (461)

  * •

A contact diagram with $n$ identical particles has symmetry group $S_{n}$
corresponding to permutations of the external momenta:
$\bm{k}_{j}\mapsto\bm{k}_{\sigma(j)}$ for any $\sigma\in S_{4}$.

We use $\sigma_{(m)J}$ to denote the corresponding $m$-th symmetric polynomial
on the set of indices $J$. To be precise, let $J$ be an ordered set of indices
and let $m$ be an integer such that $1\leq m\leq|J|$. Then,

$\displaystyle\sigma_{(m)J}=\sum_{\begin{subarray}{c}L\subseteq J\\\
|L|=m\end{subarray}}k_{L_{1}}\ldots k_{L_{m}},$ (462)

where the sum is taken over all ordered subsets $L\subseteq J$ of cardinality
$m$. In particular

$\displaystyle\sigma_{(1)12}$ $\displaystyle=k_{1}+k_{2},$
$\displaystyle\sigma_{(1)123}$ $\displaystyle=k_{1}+k_{2}+k_{3},$ (463)
$\displaystyle\sigma_{(2)12}$ $\displaystyle=k_{1}k_{2},$
$\displaystyle\sigma_{(2)123}$
$\displaystyle=k_{1}k_{2}+k_{1}k_{3}+k_{2}k_{3},$ (464)
$\displaystyle\sigma_{(3)123}$ $\displaystyle=k_{1}k_{2}k_{3},$ (465)
$\displaystyle\sigma_{(1)1234}$ $\displaystyle=k_{1}+k_{2}+k_{3}+k_{4},$ (466)
$\displaystyle\sigma_{(2)1234}$
$\displaystyle=k_{1}k_{2}+k_{1}k_{3}+k_{1}k_{4}+k_{2}k_{3}+k_{2}k_{4}+k_{3}k_{4},$
(467) $\displaystyle\sigma_{(3)1234}$
$\displaystyle=k_{1}k_{2}k_{3}+k_{1}k_{2}k_{4}+k_{1}k_{3}k_{4}+k_{2}k_{3}k_{4},$
(468) $\displaystyle\sigma_{(4)1234}$ $\displaystyle=k_{1}k_{2}k_{3}k_{4}.$
(469)

We also allow for the indices to take the value $s$, so that, for example,
$\sigma_{(1)12s}=k_{1}+k_{2}+s$ and so on.

  * •

For 4-point exchange diagrams we define the following variables,

$\displaystyle l_{ij-}=k_{i}+k_{j}-|\bm{k}_{i}+\bm{k}_{j}|,$ $\displaystyle
l_{ij+}=k_{i}+k_{j}+|\bm{k}_{i}+\bm{k}_{j}|.$ (470)

In particular all 4-point exchange diagrams will contain the following
combinations,

$\displaystyle l_{12-}=k_{1}+k_{2}-s,$ $\displaystyle l_{12+}=k_{1}+k_{2}+s,$
(471) $\displaystyle l_{34-}=k_{3}+k_{4}-s,$ $\displaystyle
l_{34+}=k_{3}+k_{4}+s,$ (472)

which we will use in addition to the standard momentum variables.

Note that all $l_{ij-}$ and $l_{ij+}$ are non-negative. Furthermore,
$l_{ij+}=0$ corresponds to $\bm{k}_{i}=\bm{k}_{j}=0$, while $l_{ij-}=0$
indicates the collinear limit, $\bm{k}_{i}\parallel\bm{k}_{j}$, when the two
momenta are parallel.

  * •

The highest possible symmetry group of an exchange diagram $12\mapsto 34$,
arising when all external particles are identical, is the dihedral group
$D_{4}$. This contains the eight permutations generated by swapping the
numbers within each pair, $12;34\mapsto 21;34$ and $12;34\mapsto 12;43$, as
well as exchanging the pairs, $12;34\mapsto 34;12$.

The following dilogarithmic quantities then arise in exchange diagrams:

$\displaystyle\mathcal{D}^{(+)}$
$\displaystyle=\operatorname{Li}_{2}\left(\frac{l_{34-}}{k_{T}}\right)+\operatorname{Li}_{2}\left(\frac{l_{12-}}{k_{T}}\right)+\log\left(\frac{l_{12+}}{k_{T}}\right)\log\left(\frac{l_{34+}}{k_{T}}\right)-\frac{\pi^{2}}{6},$
(473) $\displaystyle\mathcal{D}^{(-)}$
$\displaystyle=\operatorname{Li}_{2}\left(\frac{l_{34-}}{k_{T}}\right)-\operatorname{Li}_{2}\left(\frac{l_{12-}}{k_{T}}\right)+\frac{1}{2}\log^{2}\left(\frac{l_{12+}}{k_{T}}\right)-\frac{1}{2}\log^{2}\left(\frac{l_{34+}}{k_{T}}\right).$
(474)

Using the identity

$\displaystyle\operatorname{Li}(-z)+\operatorname{Li}(-z^{-1})=-\frac{\pi^{2}}{6}-\frac{1}{2}\log^{2}z,$
(475)

one can check that $\mathcal{D}^{(+)}$ is invariant under the group $D_{4}$,

$\displaystyle\mathcal{D}^{(+)}(\bm{k}_{1},\bm{k}_{2};\bm{k}_{3},\bm{k}_{4})=\mathcal{D}^{(+)}(\bm{k}_{2},\bm{k}_{1};\bm{k}_{3},\bm{k}_{4})=\mathcal{D}^{(+)}(\bm{k}_{3},\bm{k}_{4};\bm{k}_{1},\bm{k}_{2}),$
(476)

while $\mathcal{D}^{(-)}$ acquires a sign when any two pairs of indices are
exchanged,

$\displaystyle\mathcal{D}^{(-)}(\bm{k}_{1},\bm{k}_{2};\bm{k}_{3},\bm{k}_{4})=\mathcal{D}^{(-)}(\bm{k}_{2},\bm{k}_{1};\bm{k}_{3},\bm{k}_{4})=-\mathcal{D}^{(-)}(\bm{k}_{3},\bm{k}_{4};\bm{k}_{1},\bm{k}_{2}).$
(477)

##  Appendix B Conformal Ward identities

Dilatation and special conformal Ward identities for scalar operators
$\mathcal{O}_{1},\ldots,\mathcal{O}_{n}$ in momentum space can be found in
Bzowski:2013sza  and read

$\displaystyle 0$
$\displaystyle=\left[\sum_{j=1}^{n}\Delta_{j}-(n-1)d-\sum_{j=1}^{n-1}k_{j}^{\alpha}\frac{\partial}{\partial
k_{j}^{\alpha}}\right]\langle\\!\langle\mathcal{O}_{1}(\bm{k}_{1})\ldots\mathcal{O}_{n}(\bm{k}_{n})\rangle\\!\rangle,$
(478) $\displaystyle 0$
$\displaystyle=\left[\sum_{j=1}^{n-1}\left(2(\Delta_{j}-d)\frac{\partial}{\partial
k_{j}^{\kappa}}-2k_{j}^{\alpha}\frac{\partial}{\partial
k_{j}^{\alpha}}\frac{\partial}{\partial
k_{j}^{\kappa}}+(k_{j})_{\kappa}\frac{\partial}{\partial
k_{j}^{\alpha}}\frac{\partial}{\partial
k_{j\alpha}}\right)\right]\langle\\!\langle\mathcal{O}_{1}(\bm{k}_{1})\ldots\mathcal{O}_{n}(\bm{k}_{n})\rangle\\!\rangle.$
(479)

In the absence of renormalization effects, the dilatation Ward identity fixes
the total dimension of the $n$-point function to
$\sum_{j=1}^{n}\Delta_{j}-(n-1)d$.

Defining the operators

$\displaystyle\operatorname{\mathcal{K}}_{\beta}(k)$
$\displaystyle=\frac{\partial^{2}}{\partial
k^{2}}+\frac{1-2\beta}{k}\frac{\partial}{\partial k},$
$\displaystyle\operatorname{L}_{\Delta}(k)$
$\displaystyle=k\frac{\partial}{\partial k}-\Delta,$ (480)

an independent set of special conformal Ward identities for 2-, 3-, and
4-point functions can be written as follows:

  * •

For 2-point functions, we have the single equation

$\displaystyle 0$
$\displaystyle=\operatorname{\mathcal{K}}_{\Delta-\frac{d}{2}}(k)\langle\\!\langle\mathcal{O}_{[\Delta]}(\bm{k})\mathcal{O}_{[\Delta]}(-\bm{k})\rangle\\!\rangle.$
(481)

  * •

For 3-point functions, we have two independent equations

$\displaystyle 0$
$\displaystyle=\left[\operatorname{\mathcal{K}}_{\Delta_{i}-\frac{d}{2}}(k_{i})-\operatorname{\mathcal{K}}_{\Delta_{i}-\frac{d}{2}}(k_{j})\right]\langle\\!\langle\mathcal{O}_{[\Delta_{1}]}(\bm{k}_{1})\mathcal{O}_{[\Delta_{2}]}(\bm{k}_{2})\mathcal{O}_{[\Delta_{3}]}(\bm{k}_{3})\rangle\\!\rangle,$
(482)

with $i,j=1,2,3$.

  * •

For 4-point functions there are three independent equations. In Arkani-
Hamed:2018kmz , these are chosen as

$\displaystyle 0$
$\displaystyle=(D_{m}-D_{n})\langle\\!\langle\mathcal{O}_{[\Delta_{1}]}(\bm{k}_{1})\mathcal{O}_{[\Delta_{2}]}(\bm{k}_{2})\mathcal{O}_{[\Delta_{3}]}(\bm{k}_{3})\mathcal{O}_{[\Delta_{4}]}(\bm{k}_{4})\rangle\\!\rangle,$
(483)

where

$\displaystyle D_{1}$ $\displaystyle=\frac{\partial^{2}}{\partial
k_{1}^{2}}+\frac{1}{s}\frac{\partial}{\partial
s}\left(k_{1}\frac{\partial}{\partial k_{1}}+k_{2}\frac{\partial}{\partial
k_{2}}\right)+\frac{1}{u}\frac{\partial}{\partial
u}\left(k_{1}\frac{\partial}{\partial k_{1}}+k_{4}\frac{\partial}{\partial
k_{4}}\right)-\frac{k_{3}^{2}}{su}\frac{\partial^{2}}{\partial s\partial u}$
$\displaystyle\qquad-2\left(\Delta_{1}-\frac{d}{2}-\frac{1}{2}\right)\frac{1}{k_{1}}\frac{\partial}{\partial
k_{1}}-(\Delta_{1}+\Delta_{2})\frac{1}{s}\frac{\partial}{\partial
s}-(\Delta_{1}+\Delta_{4})\frac{1}{u}\frac{\partial}{\partial u}.$ (484)

The operators $D_{2},D_{3}$ and $D_{4}$ are obtained by cyclic permutation of
the momenta $\bm{k}_{1},\bm{k}_{2},\bm{k}_{3},\bm{k}_{4}$ as well as the
conformal dimensions $\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4}$. Note that
under a single cyclic permutation $s\leftrightarrow u$.

Another useful representation of the special conformal Ward identities is

$\displaystyle 0$
$\displaystyle=D_{ij}\langle\\!\langle\mathcal{O}_{[\Delta_{1}]}(\bm{k}_{1})\mathcal{O}_{[\Delta_{2}]}(\bm{k}_{2})\mathcal{O}_{[\Delta_{3}]}(\bm{k}_{3})\mathcal{O}_{[\Delta_{4}]}(\bm{k}_{4})\rangle\\!\rangle,$
(485)

where

$\displaystyle D_{12}$
$\displaystyle=\operatorname{\mathcal{K}}_{\Delta_{1}-\frac{d}{2}}(k_{1})-\operatorname{\mathcal{K}}_{\Delta_{2}-\frac{d}{2}}(k_{2})+(-k_{3}^{2}+k_{4}^{2})\frac{1}{st}\frac{\partial^{2}}{\partial
s\partial t}$ $\displaystyle\qquad+\frac{1}{t}\frac{\partial}{\partial
t}\left[L_{\Delta_{1}}(k_{1})-L_{\Delta_{2}}(k_{2})-L_{\Delta_{3}}(k_{3})+L_{\Delta_{4}}(k_{4})\right].$
(486)

The other operators $D_{ij}$ are obtained by permuting the momenta and
dimensions. Only three are independent, which can be chosen as
$D_{12},D_{23},D_{34}$ corresponding to cyclic permutations.

When deriving the simplified expressions for special conformal Ward identities
from (479), one uses repeatedly the dilatation Ward identity (478) and its
derivatives. This is the reason why different (equivalent) representations of
the special conformal Ward identities exist in the literature.

##  Appendix C Useful formulae

The Bessel-$I$ function has series expansion

$\displaystyle I_{\beta}(z)$
$\displaystyle=\sum_{n=0}^{\infty}a_{n}(\beta)z^{\beta+2n},\qquad
a_{n}(\beta)=\frac{1}{2^{\beta+2n}n!\Gamma(\beta+n+1)}.$ (487)

Defining the Bessel-$K$ function via

$\displaystyle K_{\beta}(z)$
$\displaystyle=\frac{\pi}{2}\frac{I_{-\beta}(z)-I_{\beta}(z)}{\sin(\beta\pi)},$
(488)

we can concentrate on $\beta\geq 0$ since $K_{\beta}(z)=K_{-\beta}(z)$. Thus,
the series expansion of the bulk-boundary propagator (3) around $z=0$ equals

$\displaystyle\operatorname{\mathcal{K}}_{d,\Delta}(z,k)$
$\displaystyle=\sum_{\sigma=\pm
1}\sum_{n=0}^{\infty}b_{n}(\sigma\beta)z^{\sigma\beta+2n},$ (489)
$\displaystyle b_{n}(\nu)$
$\displaystyle=\frac{(-1)^{n}\Gamma(-\nu-n)}{n!\Gamma(|\nu|)}\left(\frac{p}{2}\right)^{2n+\nu+|\nu|},$
(490)

where $\beta=\Delta-\frac{d}{2}$ as usual.

We already know terms linear in $u$, since we know $\hat{i}(u\epsilon,0)$ to
all orders in $\epsilon$. To find the derivative with respect to $\beta$, we
can move the derivative under the integral. Indeed, for $\epsilon$
sufficiently large the integral converges absolutely and we can move the
derivative inside it. We use

$\displaystyle\left.\frac{\partial
K_{\beta}(z)}{\partial\beta}\right|_{\beta=\frac{1}{2}}$
$\displaystyle=\sqrt{\frac{\pi}{2z}}E_{1}(2z)e^{z},$ (491)
$\displaystyle\left.\frac{\partial
I_{\beta}(z)}{\partial\beta}\right|_{\beta=\frac{1}{2}}$
$\displaystyle=-\frac{1}{\sqrt{2\pi
z}}\left[E_{1}(2z)e^{z}+\operatorname{Ei}(2z)e^{-z}\right],$ (492)

and

$\displaystyle\left.\frac{\partial
K_{\beta}(z)}{\partial\beta}\right|_{\beta=\frac{3}{2}}$
$\displaystyle=\sqrt{\frac{\pi}{2z^{3}}}\left[2e^{-z}+e^{z}(1-z)E_{1}(2z)\right],$
(493) $\displaystyle\left.\frac{\partial
I_{\beta}(z)}{\partial\beta}\right|_{\beta=\frac{3}{2}}$
$\displaystyle=-\frac{1}{\sqrt{2\pi z^{3}}}\left[4\sinh
z+e^{z}(z-1)E_{1}(2z)-e^{-z}(z+1)\operatorname{Ei}(2z)\right],$ (494)

where444See, e.g., functions.wolfram.com.

$\displaystyle E_{1}(z)=\int_{z}^{\infty}\mathrm{d}t\,\frac{e^{-t}}{t},$
$\displaystyle\operatorname{Ei}(z)=-P.V.\int_{-z}^{\infty}\mathrm{d}t\,\frac{e^{-t}}{t}.$
(495)

The two exponential integrals are related, but their continuations to negative
arguments differ. For $x>0$, one has the relations

$\displaystyle\operatorname{Ei}(-x)$ $\displaystyle=-E_{1}(x),$ $\displaystyle
E_{1}(-x\pm\mathrm{i}0)=-\operatorname{Ei}(x)\mp\mathrm{i}\pi.$ (496)

Finally, we have the integrals

$\displaystyle\int_{0}^{\infty}\mathrm{d}z\,z^{\alpha-1}e^{-\mu z}E_{1}(2pz)$
$\displaystyle=\frac{\Gamma(\alpha)}{\alpha(\mu+2p)^{\alpha}}\,{}_{2}F_{1}\left(1,\alpha;\alpha+1;\frac{\mu}{\mu+2p}\right),$
(497) $\displaystyle\int_{0}^{\infty}\mathrm{d}z\,z^{\alpha-1}e^{-\mu
z}\operatorname{Ei}(-2pz)$
$\displaystyle=-\frac{\Gamma(\alpha)}{\alpha(\mu+2p)^{\alpha}}\,{}_{2}F_{1}\left(1,\alpha;\alpha+1;\frac{\mu}{\mu+2p}\right)$
$\displaystyle\qquad\qquad-\frac{\mathrm{i}\pi\Gamma(\alpha)}{\mu^{\alpha}}\theta(-p).$
(498)

Altogether, this allows us to evaluate the derivative
$\partial\hat{i}/\partial\beta_{j}$ exactly to all orders in $u\epsilon$.
Furthermore, the expansions of the hypergeometric function in its argument are
known as well, so one can evaluate terms of any order $\epsilon^{n}$.

##  Appendix D Further shift operators

In this appendix, we collect together some further results on weight-shifting
operators supplementary to our discussion in Section 6. Section D.1 presents
additional shift operators for contact diagrams based on their representation
as a quadruple Bessel integral. Section D.2 shows the weight-shifting shift
operators employed in the cosmological bootstrap paper Arkani-Hamed:2018kmz
are a special case of the $\mathcal{W}^{\sigma_{i}\sigma_{j}}_{ij}$ operator
discussed here. We also give an explicit derivation of an additional weight-
shifting operator for exchange diagrams proposed in Arkani-Hamed:2018kmz .
This operator acts to shifts the exchange dimension provided all external
dimensions are equal to two.

###  Contact diagrams of pairwise-equal shifted dimension

In analogy to the triple-$K$ integrals of Bzowski:2013sza , let us define the
quadruple-$K$ integrals as

$\displaystyle
I_{\alpha\\{\beta_{1}\beta_{2}\beta_{3}\beta_{4}\\}}(k_{1},k_{2},k_{3},k_{4})=\int_{0}^{\infty}\mathrm{d}x\,x^{\alpha}\prod_{j=1}^{4}k_{j}^{\beta_{j}}K_{\beta_{j}}(k_{j}x).$
(499)

From (• ‣ 2) we see that the parameter $\alpha=d-1$ where $d$ is the spacetime
dimension. However, by keeping $\alpha$ (or $d$) general we can use additional
identities which relate quadruple-$K$ integrals in different spacetime
dimensions. Our 3-dimensional amplitudes are then

$\displaystyle
i_{[\Delta_{1}\Delta_{2}\Delta_{3}\Delta_{4}]}=\frac{I_{2+2\epsilon\\{\beta_{1}\beta_{2}\beta_{3}\beta_{4}\\}}}{2^{\beta_{t}-4}\Gamma(\beta_{1})\Gamma(\beta_{2})\Gamma(\beta_{3})\Gamma(\beta_{4})}.$
(500)

The identities between quadruple-$K$ integrals can be derived from the
identities satisfied by the Bessel functions following the same reasoning as
in Section 3.2 of Bzowski:2015yxv . By employing the identities

$\displaystyle\frac{\partial}{\partial
k}\left[k^{\nu}K_{\nu}(kx)\right]=-xk^{\nu}K_{\nu-1}(kx),$ $\displaystyle
K_{-\nu}(x)=K_{\nu}(x)$ (501)

we obtain two relations, both raising the value of $\alpha$ by one,

$\displaystyle I_{\alpha+1\\{\beta_{1}-1,\beta_{2}\beta_{3}\beta_{4}\\}}$
$\displaystyle=-\frac{1}{k_{1}}\frac{\partial}{\partial
k_{1}}I_{\alpha\\{\beta_{1}\beta_{2}\beta_{3}\beta_{4}\\}},$ (502)
$\displaystyle I_{\alpha+1\\{\beta_{1}+1,\beta_{2}\beta_{3}\beta_{4}\\}}$
$\displaystyle=\left(2\beta_{1}-k_{1}\frac{\partial}{\partial
k_{1}}\right)I_{\alpha\\{\beta_{1}\beta_{2}\beta_{3}\beta_{4}\\}}.$ (503)

To derive a relation decreasing the value of $\alpha$ consider the following
integral

$\displaystyle\int_{0}^{\infty}\mathrm{d}x\,\frac{\partial}{\partial
x}\left[x^{\alpha}\prod_{j=1}^{4}k_{j}^{\beta_{j}}K_{\beta_{j}}(k_{j}x)\right]=(\alpha-\beta_{t})I_{\alpha-1\\{\beta_{1}\beta_{2}\beta_{3}\beta_{4}\\}}$
$\displaystyle\qquad\qquad-\left[k_{1}^{2}I_{\alpha\\{\beta_{1}-1,\beta_{2}\beta_{3}\beta_{4}\\}}+k_{2}^{2}I_{\alpha\\{\beta_{1},\beta_{2}-1,\beta_{3}\beta_{4}\\}}+k_{3}^{2}I_{\alpha\\{\beta_{1}\beta_{2},\beta_{3}-1,\beta_{4}\\}}+k_{4}^{2}I_{\alpha\\{\beta_{1}\beta_{2}\beta_{3},\beta_{4}-1\\}}\right],$
(504)

where $\beta_{t}=\beta_{1}+\beta_{2}+\beta_{3}+\beta_{4}$. For generic
(regulated) values of $\alpha$ and $\beta_{j}$ parameters the left hand side
vanishes. Thus we obtain,

$\displaystyle
I_{\alpha\\{\beta_{1}\beta_{2}\beta_{3}\beta_{4}\\}}=\frac{1}{\alpha+1-\beta_{t}}\sum_{j=1}^{4}k_{j}^{2}I_{\alpha+1\\{\beta_{i}-\delta_{ij}\\}}$
(505)

We can combine this identity with (502) and (503) in such a way that the value
of the parameter $\alpha$ stays the same. Since in general the amplitudes with
larger conformal dimensions are more complicated, we will focus on increasing
the conformal dimensions. Thus, we will only apply (503). Before we do this,
however, note that for this identity to be useful, the constant
$\alpha+1-\beta_{t}$ should not vanish. If this was the case, its regulated
version, $\hat{\alpha}+1-\hat{\beta}_{t}$, would be of order $\epsilon^{1}$.
This implies that the knowledge of the finite part of order $\epsilon^{0}$ of
the integrals on the right-hand side of (505) would only determine the
divergent piece of the integral on the left hand side. This happens for
$\beta_{t}=d$.

To proceed, let $\mathcal{N}_{\beta_{t}}$ be the constant which multiplies the
sum in (505),

$\displaystyle\mathcal{N}_{\beta_{t}}$
$\displaystyle=\frac{1}{3+2\epsilon-\beta_{t}}=\frac{1}{3-\beta_{t}}-\frac{2\epsilon}{(3-\beta_{t})^{2}}+O(\epsilon^{2}).$
(506)

Furthermore, define the differential operators
$\mathcal{L}_{n},\mathcal{M}_{n}$ as follows,

$\displaystyle\mathcal{L}_{n}$
$\displaystyle=-\frac{1}{k_{n}}\frac{\partial}{\partial k_{n}},$ (507)
$\displaystyle\mathcal{M}_{n}^{(\beta_{n})}$
$\displaystyle=2\beta_{n}-k_{n}\frac{\partial}{\partial k_{n}},$ (508)

These are the operators featuring on the right hand sides of (502) and (503).
They are related by
$k^{2(\beta+1)}\mathcal{L}(k^{-2\beta}f)=\mathcal{M}(\beta)f$, as follows by
conjugating $\mathcal{L}$ with shadow transforms.

A priori the relation (505) contains four different integrals on its right
hand side. This number can be decreased in some special cases. For example, we
can use (503) to express both
$I_{\alpha+1\\{\beta_{1}-1,\beta_{2}\beta_{3}\beta_{4}\\}}$ and
$I_{\alpha+1\\{\beta_{1},\beta_{2}-1,\beta_{3}\beta_{4}\\}}$ in terms of the
same integral, $I_{\alpha\\{\beta_{1}-1,\beta_{2}-1\beta_{3}\beta_{4}\\}}$. In
this way we obtain

$\displaystyle i_{[\Delta_{1}\Delta_{2}\Delta_{3}\Delta_{4}]}$
$\displaystyle=\mathcal{N}_{\beta_{t}}\left[\frac{k_{1}^{2}\mathcal{M}_{2}^{(\beta_{2}-1)}+k_{2}^{2}\mathcal{M}_{1}^{(\beta_{1}-1)}}{4(\beta_{1}-1)(\beta_{2}-1)}i_{[\Delta_{1}-1,\Delta_{2}-1,\Delta_{3}\Delta_{4}]}\right.$
$\displaystyle\qquad\left.+\frac{k_{3}^{2}\mathcal{M}_{4}^{(\beta_{4}-1)}+k_{4}^{2}\mathcal{M}_{3}^{(\beta_{3}-1)}}{4(\beta_{3}-1)(\beta_{4}-1)}i_{[\Delta_{1}\Delta_{2},\Delta_{3}-1,\Delta_{4}-1]}\right].$
(509)

This expression contains only two integrals on its right hand side and
increases the values of the appropriate dimensions. Assume now that the
dimensions are pairwise equal, i.e., $\Delta_{1}=\Delta_{3}$ and
$\Delta_{2}=\Delta_{4}$. In such a case

$\displaystyle
i_{[\Delta_{1}\Delta_{2},\Delta_{3}-1,\Delta_{4}-1]}(k_{1},k_{2},k_{3},k_{4})=i_{[\Delta_{1}-1,\Delta_{2}-1,\Delta_{3}\Delta_{4}]}(k_{3},k_{4},k_{1},k_{2})$
(510)

and the right hand side of (D.1) contains only a single integral.

Additional relations are available in other special cases, for example

$\displaystyle\hat{i}_{[3222]}$
$\displaystyle=\frac{4}{\pi^{2}}I_{\hat{2}\\{\frac{3}{2}\frac{1}{2}\frac{1}{2}\frac{1}{2}\\}}$
$\displaystyle=\frac{4}{\pi^{2}}\mathcal{N}_{3}\left[k_{1}^{2}I_{\hat{3}\\{\frac{1}{2}\frac{1}{2}\frac{1}{2}\frac{1}{2}\\}}+k_{2}^{2}I_{\hat{3}\\{\frac{3}{2},-\frac{1}{2},\frac{1}{2}\frac{1}{2}\\}}+k_{3}^{2}I_{\hat{3}\\{\frac{3}{2}\frac{1}{2},-\frac{1}{2},\frac{1}{2}\\}}+k_{4}^{2}I_{\hat{3}\\{\frac{3}{2}\frac{1}{2}\frac{1}{2},-\frac{1}{2}\\}}\right]$
$\displaystyle=\frac{4}{\pi^{2}}\mathcal{N}_{3}\left[k_{1}^{2}I_{\hat{3}\\{\frac{1}{2}\frac{1}{2}\frac{1}{2}\frac{1}{2}\\}}+(k_{2}+k_{3}+k_{4})I_{\hat{3}\\{\frac{3}{2}\frac{1}{2}\frac{1}{2}\frac{1}{2}\\}}\right]$
$\displaystyle=\frac{4}{\pi^{2}}\mathcal{N}_{3}\left[k_{1}^{2}\mathcal{M}_{1}^{(-\frac{1}{2})}I_{\hat{2}\\{-\frac{1}{2}\frac{1}{2}\frac{1}{2}\frac{1}{2}\\}}+(k_{2}+k_{3}+k_{4})\mathcal{M}_{1}^{(\frac{1}{2})}I_{\hat{2}\\{\frac{1}{2}\frac{1}{2}\frac{1}{2}\frac{1}{2}\\}}\right]$
$\displaystyle=\frac{4}{\pi^{2}}\mathcal{N}_{3}\left[k_{1}^{3}\mathcal{L}_{1}+(k_{2}+k_{3}+k_{4})\mathcal{M}_{1}^{(\frac{1}{2})}\right]I_{\hat{2}\\{\frac{1}{2}\frac{1}{2}\frac{1}{2}\frac{1}{2}\\}}$
$\displaystyle=\mathcal{N}_{3}\left[k_{1}^{3}\mathcal{L}_{1}+(k_{2}+k_{3}+k_{4})\mathcal{M}_{1}^{(\frac{1}{2})}\right]\hat{i}_{[2222]}$
$\displaystyle=\frac{1}{2\epsilon}\left[-k_{1}k_{T}\frac{\partial}{\partial
k_{1}}+(k_{2}+k_{3}+k_{4})\right]\hat{i}_{[2222]},$ (511)

and if we keep going,

$\displaystyle\hat{i}_{[3322]}$
$\displaystyle=\frac{1}{-1+2\epsilon}\left[\left(k_{1}^{2}\mathcal{M}_{2}^{(\frac{1}{2})}+k_{2}^{2}\mathcal{M}_{1}^{(\frac{1}{2})}\right)\hat{i}_{[2222]}+(k_{3}+k_{4})\mathcal{M}_{2}^{(\frac{1}{2})}\hat{i}_{[3222]}\right].$
(512)

###  Relation to ‘cosmological bootstrap’ operators

In this section we discuss briefly the weight-shifting operators for scalar
correlators employed in the cosmological bootstrap paper Arkani-Hamed:2018kmz
. We confirm that, as expected from Baumann:2019oyu , they are a special case
of the more general weight-shifting operators
$\mathcal{W}_{ij}^{\sigma_{i}\sigma_{j}}$. We also provide a detailed
derivation of the operator (549) proposed in Arkani-Hamed:2018kmz  which acts
to shift the dimension of the exchanged operator in the case where all
external dimensions are equal to two.

  * •

Weight-shifting operators acting on three-dimensional CFT correlators are
discussed in Section 5 and Appendix E of Arkani-Hamed:2018kmz . The relation
of these operators to the $\mathcal{W}_{ij}^{\sigma_{i}\sigma_{j}}$ in (375) -
(378) is not immediately apparent, however, since the operators in Arkani-
Hamed:2018kmz  are restricted to act on 4-point functions of the form
$f(u,v,s),$ where

$u=\frac{s}{k_{12}},\qquad v=\frac{s}{k_{34}},\qquad k_{ij}=k_{i}+k_{j}.$
(513)

As a starting point, this ansatz is clearly valid for $s$-channel exchange and
contact diagrams with all $\Delta=2$, since propagators with $\beta=1/2$
reduce to plane waves.

Using (392) and the identities

$\frac{1}{k_{1}}+\frac{1}{k_{2}}=\frac{s}{uk_{1}k_{2}},\qquad
k_{1}^{2}+k_{2}^{2}=\frac{s^{2}}{u^{2}}-2k_{1}k_{2},\qquad\partial_{k_{1}}f=\partial_{k_{2}}f=-\frac{u^{2}}{s}\partial_{u}f,$
(514)

one can show that the action of $\mathcal{W}_{12}^{--}$ on this ansatz is

$\displaystyle\mathcal{W}_{12}^{--}f(u,v,s)=\frac{1}{2k_{1}k_{2}}\Big{(}(1-u^{2})\partial_{u}(u^{2}\partial_{u}f)-(d-1)u\partial_{u}f\Big{)}.$
(515)

The raising operator $\mathcal{W}_{12}^{++}$ then follows from (381). If we
explicitly set $\Delta_{1}=\Delta_{2}=2$ and $d=3$, the result can be written
in the form

$\displaystyle\mathcal{W}_{12}^{++}f(u,v,s)=\frac{s^{2}}{2}\Big{(}1+\frac{k_{1}k_{2}}{s^{2}}u^{3}\partial_{u}\Big{)}\Big{[}\Big{(}\frac{1-u^{2}}{u^{2}}\Big{)}\partial_{u}(uf)\Big{]}=s^{2}U_{12}f.$
(516)

where $U_{12}$ is defined in (5.10) of Arkani-Hamed:2018kmz . For this latter
definition, note that the operator $\mathcal{O}_{12}$ given in (5.3) of
Arkani-Hamed:2018kmz  can be rewritten as

$\mathcal{O}_{12}f(u,v,s)=\Big{(}1-\frac{k_{1}k_{2}}{k_{12}}\partial_{k_{12}}\Big{)}f\Big{(}\frac{s}{k_{12}},\frac{s}{k_{34}},s\Big{)}=\Big{(}1+\frac{k_{1}k_{2}}{s^{2}}u^{3}\partial_{u}\Big{)}f(u,v,s)$
(517)

so indeed

$U_{12}f(u,v,s)=\frac{1}{2}\mathcal{O}_{12}\Big{[}\Big{(}\frac{1-u^{2}}{u^{2}}\Big{)}\partial_{u}(uf)\Big{]}$
(518)

as per (5.10) of Arkani-Hamed:2018kmz . As the dilatation Ward identity fixes
the 4-point function to be $f(u,v,s)=s^{\Delta_{t}-9}\hat{f}(u,v)$, one then
has

$\hat{f}(u,v)\Big{|}_{\Delta_{1}=\Delta_{2}=3}=U_{12}\hat{f}(u,v)\Big{|}_{\Delta_{1}=\Delta_{2}=2}.$
(519)

This is also consistent with (5.11) of Arkani-Hamed:2018kmz , since on the
right-hand side the 4-point function with all $\Delta_{j}=2$ for
$j=1,\ldots,4$ is written as $f=s^{-1}\hat{f}(u,v)$ and so

$\mathcal{W}_{12}^{++}\mathcal{W}_{34}^{++}f=s^{4}U_{12}U_{34}f=s^{3}U_{12}U_{34}\hat{f}(u,v).$
(520)

  * •

Acting on the ansatz $f(u,v,s)$, the Casimir operator (414) with $d=3$ and
$\Delta_{1}=\Delta_{2}=2$ reduces to

$\displaystyle\mathcal{C}_{12}\,f(u,v,s)=(2-\Delta_{u})f(u,v,s),$ (521)

where

$\Delta_{u}=u^{2}(1-u^{2})\partial_{u}^{2}-2u^{3}\partial_{u}=u^{2}\partial_{u}\Big{(}(1-u^{2})\partial_{u}\Big{)}$
(522)

as defined in (2.30) of Arkani-Hamed:2018kmz . For general $\Delta_{x}$ and
$\Delta_{3}=\Delta_{4}=2$, we then have

$\displaystyle E_{12,\Delta_{x}}$
$\displaystyle\equiv\mathcal{C}_{12}+\Delta_{x}(\Delta_{x}-3)=-\Delta_{u}+(\Delta_{x}-1)(\Delta_{x}-2),$
(523) $\displaystyle E_{34,\Delta_{x}}$
$\displaystyle\equiv\mathcal{C}_{34}+\Delta_{x}(\Delta_{x}-3)=-\Delta_{v}+(\Delta_{x}-1)(\Delta_{x}-2).$
(524)

compatible with, e.g., (4.57). Let us now define the shift operator

$\displaystyle
S_{u,\Delta_{x}}\equiv(1-u^{2})\partial_{u}+\frac{1-\Delta_{x}}{u}$ (525)

which obeys the intertwining relations

$\displaystyle E_{12,\Delta_{x}+1}S_{u,\Delta_{x}}$
$\displaystyle=S_{u,\Delta_{x}+2}E_{12,\Delta_{x}},$ (526) $\displaystyle
E_{34,\Delta_{x}+1}S_{v,\Delta_{x}}$
$\displaystyle=S_{v,\Delta_{x}+2}E_{34,\Delta_{x}}.$ (527)

Given a homogeneous solution $h_{\Delta_{x}}=h_{\Delta_{x}}(u,v,s)$ of the
Casimir equations

$E_{12,\Delta_{x}}h_{\Delta_{x}}=E_{34,\Delta_{x}}h_{\Delta_{x}}=0,$ (528)

we find

$h_{\Delta_{x}+1}\equiv S_{u,\Delta_{x}}S_{v,\Delta_{x}}h_{\Delta_{x}}$ (529)

is a shifted homogeneous solution satisfying

$E_{12,\Delta_{x}+1}h_{\Delta_{x}+1}=E_{34,\Delta_{x}+1}h_{\Delta_{x}+1}=0.$
(530)

  * •

The shift operator $S_{u,\Delta_{x}}$ can itself be derived from the solution
of the homogeneous Casimir equation $E_{12,\Delta_{x}}=0$. Making use of the
standard quadratic transformations of the hypergeometric function,555 See,
e.g., 15.8.13 in the online Digital Library of Mathematical Functions. this
can be expressed as

$h(u)=c_{1}z^{a}{}_{2}F_{1}(a,a,2a;z)+c_{2}z^{1-a}{}_{2}F_{1}(1-a,1-a,2(1-a);z)$
(531)

where

$z=\frac{2u}{1+u},\qquad u=\frac{z}{2-z},\qquad a=\Delta_{x}-1.$ (532)

Here, the two solutions in (531) are related by the shadow transformation
$\Delta\rightarrow 3-\Delta$ which sends $a\rightarrow 1-a$. In $z$ variables,
the shift operator reads

$S_{u,\Delta_{x}}=2(1-z)\partial_{z}+\frac{a(z-2)}{z}$ (533)

and one can verify that it acts as expected, namely

$\displaystyle
S_{u,\Delta_{x}}\Big{(}z^{a}{}_{2}F_{1}(a,a,2a;z)\Big{)}=-\frac{a^{2}}{2(1+2a)}z^{a+1}{}_{2}F_{1}(a+1,a+1,2(a+1);z),$
(534) $\displaystyle
S_{u,\Delta_{x}}\Big{(}z^{1-a}{}_{2}F_{1}(1-a,1-a,2(1-a);z)\Big{)}=2(1-2a)z^{-a}{}_{2}F_{1}(-a,-a,1-2a;z).$
(535)

To construct $S_{u,\Delta_{x}}$ in the first place, we concatenate the
standard ${}_{2}F_{1}$ shift relations

$\displaystyle\partial_{z}\,{}_{2}F_{1}(a,b,c;z)$
$\displaystyle=\frac{ab}{c}{}_{2}F_{1}(a+1,b+1,c+1;z),$ (536)
$\displaystyle\big{(}(z-1)\partial_{z}+a+b-c\big{)}{}_{2}F_{1}(a,b,c;z)$
$\displaystyle=\Big{(}a+b-c-\frac{ab}{c}\Big{)}{}_{2}F_{1}(a,b,c+1;z)$ (537)

then eliminate the second derivative using the hypergeometric equation. This
gives the first-order shift operator

$\frac{1}{z}\big{(}c(1-z)\partial_{z}-ab){}_{2}F_{1}(a,b,c;z)=-\frac{ab(c-a)(c-b)}{c(c+1)}{}_{2}F_{1}(a+1,b+1,c+2;z)$
(538)

which, on specializing to $(a,b,c)=(a,a,2a)$, reads

$-\frac{2(2a+1)}{a^{2}z}\Big{(}2(1-z)\partial_{z}-a\Big{)}{}_{2}F_{1}(a,a,2a;z)={}_{2}F_{1}(a+1,a+1,2(a+1);z).$
(539)

Multiplying by $z^{1+a}$ and pushing this factor inside the derivative, we
recover

$-\frac{2(2a+1)}{a^{2}}\Big{(}2(1-z)\partial_{z}+\frac{a(z-2)}{z}\Big{)}z^{a}{}_{2}F_{1}(a,a,2a;z)=z^{a+1}{}_{2}F_{1}(a+1,a+1,2(a+1);z)$
(540)

which is precisely (534).

  * •

Having solved the homogeneous problem, let us now turn to the inhomogeneous
one. From (419), exchange diagrams with external dimensions $\Delta_{j}=2$
satisfy

$E_{12,\Delta_{x}}f_{\Delta_{x}}=E_{34,\Delta_{x}}f_{\Delta_{x}}=\hat{i}_{[2222]}=\frac{1}{k_{T}}=\frac{uv}{s(u+v)}\equiv
C,$ (541)

where the same contact term appears on the right-hand side for all
$\Delta_{x}$.

To solve this inhomogeneous equation, let us make the ansatz

$f_{\Delta_{x}+1}=\gamma_{\Delta_{x}}(S_{u,\Delta_{x}}S_{v,\Delta_{x}}f_{\Delta_{x}}+g),$
(542)

where $\gamma_{\Delta_{x}}$ is a constant depending on $\Delta_{x}$ and
$g=g(u,v)$ is a function that we will assume (in order to find a simple
solution as below) is independent of $\Delta_{x}$. This gives

$\displaystyle C=E_{12,\Delta_{x}+1}f_{\Delta_{x}+1}$
$\displaystyle=\gamma_{\Delta_{x}}E_{12,\Delta_{x}+1}(S_{u,\Delta_{x}}S_{v,\Delta_{x}}f_{\Delta_{x}}+g)$
$\displaystyle=\gamma_{\Delta_{x}}(S_{u,\Delta_{x}+2}S_{v,\Delta_{x}}E_{12,\Delta_{x}}f_{\Delta_{x}}+E_{12,\Delta_{x}+1}g)$
$\displaystyle=\gamma_{\Delta_{x}}(S_{u,\Delta_{x}+2}S_{v,\Delta_{x}}C+E_{12,\Delta_{x}+1}g).$
(543)

Thus, $g$ must solve the inhomogeneous Casimir problem

$E_{12,\Delta_{x}+1}g=(\gamma_{\Delta_{x}}^{-1}-S_{u,\Delta_{x}+2}S_{v,\Delta_{x}})C\equiv
G(u,v).$ (544)

Evaluating the right-hand side, the only terms with dependence on $\Delta_{x}$
are

$G=-\frac{\Delta_{x}(\Delta_{x}-1)}{s(u+v)}+\frac{(\gamma_{\Delta_{x}}^{-1}-\Delta_{x})uv}{s(u+v)}+\ldots$
(545)

These terms must match those on the left-hand side, which, since $g$ is
independent of $\Delta_{x}$ by assumption, are

$\Delta_{x}(\Delta_{x}-1)g+\ldots$ (546)

This gives

$\gamma_{\Delta_{x}}^{-1}=c_{1}\,\Delta_{x}(\Delta_{x}-1)+\Delta_{x}+c_{2},\qquad
g=\frac{c_{1}\,uv-1}{s(u+v)}$ (547)

where $c_{1}$ and $c_{2}$ are $\Delta_{x}$-independent constants. Here $c_{2}$
encodes the $\Delta_{x}$-independent part of $\gamma_{\Delta_{x}}^{-1}$ which
is unconstrained by the above. Plugging this trial solution back into (544),
we find $c_{1}=c_{2}=-1$ and hence the solution

$\gamma_{\Delta_{x}}^{-1}=-(\Delta_{x}-1)^{2},\qquad
g=-\frac{(uv+1)}{s(u+v)}.$ (548)

We thus have the shift relation

$\displaystyle\hat{i}_{[22,22\,x\,\Delta_{x}+1]}=-\frac{1}{(\Delta_{x}-1)^{2}}\Big{(}S_{u,\Delta_{x}}S_{v,\Delta_{x}}\hat{i}_{[22,22\,x\,\Delta_{x}]}-\frac{(uv+1)}{s(u+v)}\Big{)},$
(549)

with $S_{u,\Delta_{x}}$ as given in (525) and $u,v,$ variables as defined in
(513). This result is equivalent to (4.58) in Arkani-Hamed:2018kmz 666Note the
sign of the final inhomogeneous term in (4.58) of Arkani-Hamed:2018kmz  is
misprinted., noting the $\Delta_{\sigma}$ there is
$\Delta_{\sigma}=\Delta_{x}+1$, i.e., the final dimension of the exchanged
operator after applying the shift relation.777This can be seen from (5.6) in
Baumann:2020dch , correcting the typo in the sign of the first term.

Note that if we were to drop the simplifying assumption that $g$ is
independent of $\Delta_{x}$, we could instead attempt to solve the
inhomogeneous Casimir problem (544) through the method of variation of
parameters. Using the two homogeneous solutions $h_{1}(u)$ and $h_{2}(u)$ from
(531), this gives

$g=-h_{1}(u)\int^{u}\mathrm{d}\tilde{u}\,(1-\tilde{u}^{2})G(\tilde{u},v)h_{2}(\tilde{u})+h_{2}(u)\int^{u}\mathrm{d}\tilde{u}\,(1-\tilde{u}^{2})G(\tilde{u},v)h_{1}(\tilde{u})$
(550)

where the Wronskian is $W(\tilde{u})=(1-\tilde{u}^{2})^{-1}$. Evaluating these
integrals is difficult in general however, and for the special cases that can
be evaluated (e.g., $\Delta_{x}=2$) we obtain complicated solutions involving
the dilogarithm. Thus, the assumption that $g$ is independent of $\Delta_{x}$
is a useful one in that it leads to a simple shift relation that can be
obtained without having to evaluate any integrals.

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##  1 Results and outlook

We use the notation ${\bf{h}}=[h_{1},h_{2},....,h_{L}]$ to represent a network
with $L$ hidden layers, where we denote the number of neurons in a fully
connected layer, or the number of kernels in a convolutional layer, as $h_{i}$
in layer $i$. In recent works that apply DNNs to decode ECCs, the training set
explodes rapidly as the source word length grows. For example, with a rate 0.5
$(n=1024,k=512)$ ECC, one epoch consists of $2^{512}$ possibilities of
codewords of length 1024, which results in very large complexity and makes it
difficult to train and implement DNN-based decoding in practical systems
[deeplearningECC1, deeplearningECC2, deeplearningECC3, deeplearningECC4].
However, we note that in CS decoding, this problem does not exist since CS
source words are typically considerably shorter, possibly up to a few dozen
symbols [Textbook, Franaszek, 8B10B_Ref, DC-freeRLL_Immink, FixedLength2,
ImminkVLCodes, Motwani, BalancedModulation, FixedLength3, Pearson1,
SystematicPearsonCoding, MyMinimalSet, MyJSAC, myPearson, My_MinimalSet,
DNAStorageImmink]. This property fits deep learning based-decoding well.

###  BER performance

####  Frame-by-frame decoding

First we consider frame-by-frame transmission, where the 4B6B codewords are
transmitted and decoded one-by-one, i.e., $|{\bf{v}}|=6$. We will later
consider processing multiple frames simultaneously to improve the system
throughput. Note that in the VLC standard, two 4B6B look-up tables can be used
simultaneously [VLC1].

We compare performance of deep learning-based decoding with traditional table
look-up decoding that generates hard-decision bits. That is, the traditional
detector estimates the hard decision $\hat{\bf{v}}$, and the CS decoder
attempts to map $\hat{\bf{v}}$ to a valid source word to generate
$\hat{\bf{x}}$. If the decoder is not able to locate $\hat{\bf{v}}$ in the
code table due to erroneous estimation at the detector, the decoder determines
the codeword that is closest to $\hat{\bf{v}}$ in terms of Hamming distance,
and then outputs the corresponding source word. We also implement the maximum
likelihood (ML) decoding of CS codes, where the codeword with the closest
Euclidean distance to the received noisy version of codeword is selected and
the corresponding source word is decoded. We assume equiprobable 0s and 1s in
source sequences, thus ML decoding is equivalent to MAP decoding of CS codes
since each codeword has an equal occurrence probability.

Table LABEL:parameters shows the parameters of the MLP networks and the CNNs
for a variety of tasks. The DNNs are trained at an SNR of 1 dB, using _Adam_
for stochastic gradient descent optimization [Adam]. With $|{\bf{v}}|=6$, the
MLP network we trained for frame-by-frame decoding has three hidden layers
[32,16,8] and 924 trainable parameters. Its BER performance is shown in Fig.
1, which shows that DNN-based decoding achieves a BER that is very close to
MAP decoding of CS codes, and outperforms conventional table look-up decoding
by $\sim$2.2 dB.

[width=]fig1.eps

Figure : comparison of BER performance with frame-by-frame transmission, i.e.,
$|{\bf{v}}|=6$

We then investigate employing CNNs for this task. With ECC decoding, CNNs and
MLP networks have roughly the same complexity, i.e., a similar number of
weights in order to achieve similar performance [deeplearningECC3]. In what
follows we outline our findings that are unique to CS decoding.

In Table LABEL:cnnparameters we outline the structure of the CNNs we apply for
CS decoding. ReLU is used as the activation function for each convolutional
layer. We note that CS codes always have inherent constraints on their
codewords such that they match the characteristics of the transmission or
storage channel. For example, the 4B6B code in Table LABEL:4B6B always has an
equal number of logic ones and logic zeros in each codeword, and that the
runlength is limited to four in the coded sequence for flicker reduction.
These _low-level features_ can be extracted to enable CNNs to efficiently
learn the weights of the kernels, which results in significantly reduced
complexity compared to MLP networks. For example, although similar BER
performance is achieved by the [32, 16, 8] MLP network and the [6, 10, 6] CNN,
the number of weights in the CNN is only $82\%$ of that in the MLP network.
With larger networks the complexity reduction is more significant, as we will
show in the next subsection.

Another finding we observe during training of a CNN is that pooling layers
which are essential component structures in CNNs for visual tasks, may not be
required in our task. The reason is that in visual tasks, pooling is often
used to extract _high-level_ features of images such as shapes, edges or
corners. However, CS codes often possess low-level features only, and we find
that adding pooling layers may not assist CS decoding. Therefore, no pooling
layer is used in our CNNs, as indicated in Table LABEL:cnnparameters. Fig. 1
shows that use of a CNN achieves similar performance to the use of an MLP
network, and that it also approaches the performance of MAP decoding.

####  Improving the throughput

We now consider processing multiple frames in one time slot in order to
improve the system throughput. The system throughput can be enhanced by
increasing the optical clock rate, which has its own physical limitations, or
by processing multiple 4B6B codewords in parallel. The VLC standard allows two
4B6B codes to be processed simultaneously [VLC1]. Now we show that DNNs can
handle larger input size where $|{\bf{v}}|$ is a multiple of 6, thus system
throughput can be enhanced by using one of those DNNs or even using multiple
DNNs in parallel.

[width=]fig2.eps

Figure : BER performance of MLP networks with multiple frames processed
simultaneously, i.e., $|{\bf{v}}|=12,18,24,30$

[width=]fig3.eps

Figure : BER performance of CNNs with multiple frames processed
simultaneously, i.e., $|{\bf{v}}|=12,18,24,30$

Figs. 1 and 1 present the BER performance of MLP networks and CNNs
respectively, where the parameters of those networks are shown in Table
LABEL:parameters. These figures demonstrate that both MLP networks and CNNs
are able to achieve BERs very close to MAP decoding, while the CNNs have
significantly lower complexity than the MLP networks for the reason outlined
above. With larger $|{\bf{v}}|$, it becomes easier for the CNN to extract the
low-level features from the longer input seqeuences and learn the weights, and
thus the complexity reduction with CNN is more significant. For example, when
processing five frames simultaneously, the CNN has less than $1/5$ of the
parameters that need to be trained for the MLP network. Note that we consider
the networks shown in Table LABEL:parameters to be small (ResNet [resnet] has
a few million parameters to train). Therefore, we anticipate that it will be
practical to achieve further improvement on system throughput with larger
networks.

###  Paving the way to capacity-achieving CS codes

As we outlined in Section I, it has not been an easy task to design fixed-
length capacity-achieving CS codes. As determined by equations eq1-eq3, the
capacity of a constraint is most likely irrational, which can be approached
with fixed-length codes of rate $R=k/n$ only with very large $k$ and $n$
values. This hinders implementation of table look-up encoding and decoding for
capacity-achieving CS codes. For example, as we show in Section II-B, the 4B6B
code achieves $84.12\%$ of the capacity of a DC-free code with 5 different RDS
values, which has $C=0.7925$. In Table 1 we list, for increasing values of
$k$, values of $n$ such that code rate $R$ approaches capacity. This table
shows that with $k=11,15,19$, it could be possible to construct fixed-length
codes with efficiencies that exceed $99\%$. With $k$ as large as $79$ and
$n=100$, the code would have rate 0.79 and efficiency $99.68\%$. However, a
table look-up codebook with $2^{k}$ source word-to-codeword mappings becomes
impractical to implement as $k$ grows large. Other examples are the
$k$-constrained codes recently developed for DNA-based storage systems in
[DNAStorageImmink]. Those fixed-length 4-ary $k$-constrained codes have rates
very close to the capacity, however they require very large codebooks. For
example, with method B in [DNAStorageImmink], for the $k$-constrained code
with $k=1,2,3,4$, the codebooks have $4^{10},4^{38},4^{147},4^{580}$ source
word-to-codeword mappings respectively.

Table : The highest code rate and efficiency with fixed-length CS codes for the DC-free constraint with five different RDS values, $C=0.7925$ $k$ | $n$ | $R$ | $\eta$ | $k$ | $n$ | $R$ | $\eta$
---|---|---|---|---|---|---|---
1 | 2 | 0.5000 | 63.09% | 11 | 14 | 0.7857 | 99.14%
2 | 3 | 0.6667 | 84.12% | 12 | 16 | 0.7500 | 94.64%
3 | 4 | 0.7500 | 94.64% | 13 | 17 | 0.7647 | 96.49%
4 | 6 | 0.6667 | 84.12% | 14 | 18 | 0.7778 | 98.14%
5 | 7 | 0.7143 | 90.13% | 15 | 19 | 0.7895 | 99.62%
6 | 8 | 0.7500 | 94.64% | 16 | 21 | 0.7619 | 96.14%
7 | 9 | 0.7778 | 98.14% | 17 | 22 | 0.7727 | 97.51%
8 | 11 | 0.7273 | 91.77% | 18 | 23 | 0.7826 | 98.75%
9 | 12 | 0.7500 | 94.64% | 19 | 24 | 0.7917 | 99.89%
10 | 13 | 0.7692 | 97.06% | 20 | 26 | 0.7692 | 97.06%

[width=]fig5.eps

Figure : BER performance of DNN-based decoding for a constrained sequence code
with $2^{20}$ possibilities of mappings

With DNNs, however, it becomes practical to handle a large set of source word-
to-codeword mappings which has long been considered impractical with table
look-up decoding. This paves the way for practical design and implementation
of fixed-length capacity-achieving CS codes. Appropriate design of such codes
is a practice of using standard algorithms from the rich theory of CS coding,
such as Franaszek’s recursive elimination algorithm [Franaszek], or the
sliding-block algorithm [ACH1, ACH2] with large $k$ and $n$ values to
determine the codebooks. We propose implementing both the encoder and the
decoder with DNNs. Although here we focus on decoding, similar to DNN-based
decoders, CS encoders map noiseless source words to codewords, and can also be
implemented with DNNs.

We now demonstrate that the proposed DNN-based decoders are able to map long
received words from a CS code to their corresponding source words. We
concatenate five 4B6B codebooks, where each 4B6B codebook is randomly shuffled
in terms of its source word to codeword mappings, to generate a large codebook
with $2^{20}$ entries of mappings. We train and test a CNN with
${\bf{h}}=[16,32,12]$ that has 9536 weights to perform decoding. From Fig. 1,
we can see that this CNN is capable of decoding the received noisy version of
the large set of received words. The BER is close to MAP decoding, and
outperforms the table look-up decoding approach that we implemented as a
benchmark. Therefore, the design and implementation of DNN-based CS decoding
is practical with long CS codewords because of their low-level features that
we can take advantage of to simply decoding.

We also note that DNNs that are proposed in communication systems could have
many more parameters than the 9536 we use in this network. For example, [SCMA]
proposes an MLP network with 4 hidden layers where each layer has 512 neurons,
which results in at least $512\times 512\times 3=786432$ parameters. Thus a
larger CNN can be trained to decode fixed-length capacity-approaching CS codes
with longer codewords.


# Beyond Traditional Neural Networks: Toward adding Reasoning and Learning
Capabilities through
Computational Logic Techniques

Andrea Rafanelli Department of Computer Science
University of Pisa, Pisa, ItalyDepartment of Information Engineering, Computer
Science and Mathematics
University of L’Aquila, Italy<EMAIL_ADDRESS>
###### Abstract

Deep Learning (DL) models have become popular for solving complex problems,
but they have limitations such as the need for high-quality training data,
lack of transparency, and robustness issues. Neuro-Symbolic AI has emerged as
a promising approach combining the strengths of neural networks and symbolic
reasoning. Symbolic Knowledge Injection (SKI) techniques are a popular method
to incorporate symbolic knowledge into sub-symbolic systems. This work
proposes solutions to improve the knowledge injection process and integrate
elements of ML and logic into multi-agent systems (MAS).

##  1 Introduction and Problem Description

Deep Learning (DL) models have gained popularity for addressing complex
problems in various domains. However, they face challenges. Insufficient or
biased training data can result in poor model performance and limited
generalization. DL models heavily rely on statistical models which can
restrict their ability to reason about concepts not explicitly represented in
the data. This limitation hampers their capacity to understand the world and
draw accurate conclusions from available information. To address these issues,
Neuro-Symbolic AI has emerged, which seeks to combine the strengths of neural
networks and symbolic reasoning to overcome the limitations of purely
statistical models. There are various techniques for integrating neural and
symbolic systems (for a better understanding c.f., e.g., [7, 13, 22]). This
work focuses mainly on methods that aim to combine these two systems by
injecting symbolic knowledge into neural networks, known as Symbolic Knowledge
Injection (SKI) techniques. Additionally, we propose the creation of
collaborative networks that combine the strengths of logic-based agents and
neural agents within a Neuro-Symbolic architecture, creating a more
comprehensive and efficient system. This approach offers a unique synergy,
paving the way for more advanced and intelligent systems.

##  2 Background and Existing Literature

In this section, we will provide a quick overview of the concepts and methods
considered in our work.

### Symbolic Knowledge Injection

SKI is a strategy used to improve the performance of sub-symbolic predictors,
such as neural networks, by integrating useful symbolic knowledge into them.
It has several benefits, such as alleviating the problem of insufficient
training data, reducing the required time and computational resources for
learning, increasing the predictive accuracy of sub-symbolic predictors, and
providing human-interpretable frameworks. There are several ways the knowledge
can be injected into predictors, mainly:  _(i)_ loss constraining[4, 14, 29],
i.e., the inclusion of penalty terms in the loss function. Penalty terms are
indicative of the limitations imposed by prior knowledge;  _(ii)_ structure
constraining[5, 20, 30], i.e., the alteration of the structure of the sub-
symbolic model to reflect symbolic knowledge expressed as constraints, e.g. by
changing the number and size of hidden layers or by setting the connections
between neurons;  _(iii)_ knowledge embedding[6, 8, 28], i.e., incorporating
additional domain-specific knowledge by generating numeric data from the
symbolic ones.

### Abductive Logic Programming

ALP is a programming paradigm [17] that combines logic programming with
abductive reasoning [19] to generate hypotheses (or abducibles) that explain
certain observations. Abduction is formally described [21] as a framework
where, given a knowledge base $KB$ and observations $O$, one seeks to find a
hypothesis $H$, s.t. $KB\cup H\models O.$

ALP is a version of traditional logic programming in which the reasoner
generates abductive hypotheses by assuming the truth of abducibles (a set of
open predicates) given some integrity constraints. Integrity constraints are
logical formulas of the form :-$Body.$, where Body is a conjunction of logical
literals. Intuitively, an integrity constraint specifies a set of conditions
that must not hold for any valid solution to the problem. According to [18],
an abductive framework is known as a triple $<P,A,I_{c}>$, where $P$ is a
collection of clauses, $A$ is a set of predicate symbols, called abducibles,
and $I_{c}$ is a set of closed formulae. Clauses in $P$ are of the form
$H\leftarrow L_{1},L_{2},..,L_{k},\enspace k\geq 0$, where $H$ is an atom with
a predicate symbol not present in $A$ and $L_{i}$ is a single literal.
Therefore, $P$ is a general logic program with the limitation that predicates
present in $A$ do not have definitions in $P$. Here, the idea is to link to an
abductive framework a collection of models–commonly called generalized stable
models, or GSMs [15]– that are assumed to reflect the framework itself and may
be used to characterise abduction within this framework. Hence, any
observation has an abductive explanations if it is consistent with at least
one of these GSM models. Let $<P,A,I_{c}>$ represent an abductive framework,
whereas $O$ represents an observation. Then $O$ is explained abductively given
a set of hypotheses $\Delta$ iff exists a GSM, $M(\Delta)$, s.t.
$M(\Delta)\models O.$ In general, the main aim of abduction, given an
observation $O$, is to find a set of abductive explanation $\Delta\subseteq
A$, s.t.:  _(i)_ $M\models O.$, and  _(ii)_ $M\models I_{c}.$

### DALI

DALI [9, 10] is a programming language developed to construct logical agents.
It is an extension of the Prolog
language111https://en.wikipedia.org/wiki/Prolog, and any Prolog program can be
considered a DALI agent. The agent orientation is provided through events as
first-class objects in the language, as well as two types of rules: reactive
rules and proactive rules. Reactive rules permit agents to interact with and
respond to their surroundings. These rules enable the agents to react to
external events. External events are triggered by the environment of the agent
and are denoted by the postfix E. An agent may choose to react to an external
event by employing a reactive rule containing the event in its head. Proactive
rules, on the other hand, are used by DALI agents to take initiative and start
activities when they deem it suitable. Using internal events, that are denoted
by the postfix I, agents can act independently of their environment and other
agents. DALI agents use actions to influence their surroundings, potentially
in response to external or internal events. Actions may have pre-conditions,
which are specified in the action rule. Otherwise, they are just an action
atom denoted with the postfix A. Agents can communicate with and affect their
environment through actions.

##  3 Goal of the Research

This work aims to address challenges in the field of DL models by exploring
the integration of neural models and computational logic. It will focus on two
main tools: SKI and MAS. The main goals are:  _(i)_ RG 1: use abduction as a
way of integrating targeted knowledge into the injection process.  _(ii)_ RG
2: develop metrics to assess the goodness-of-fit of injection mechanisms from
several perspectives, both qualitative and efficiency-related.  _(iii)_ RG3:
formalise an integrated MAS in which one or more agents have purely perceptual
tasks and other purely logical tasks.

RG 1 aims to use abduction to improve the SKI process. In supervised learning,
the observation comprises input-output pairs encoded as a model
$f(\mathcal{X})\to\mathcal{Y}$. When using SKI, we want to modify $f(\cdot)$
to align it with symbolic knowledge. However, selecting the right rules for
each output becomes challenging with increasing rule count, causing possible
scalability issues. One solution is to leverage abduction, which helps
identify plausible rules for a given output. Here, abduction can be seen as a
rule $Y:-A.$, where $Y$ is the observation (ground-truth) and $A$ are
abducibles generated by the reasoner. The injection procedure can then be
modified to incorporate abducibles, making the predictions of $f(\cdot)$
consistent with the provided abducibles.

RG 2 aims to investigate different metrics to gain a deeper understanding of
the potential benefits and limitations of SKI from multiple perspectives. In
the field of SKI, the effectiveness of the injection procedure is often
measured by comparing the performance of the SKI predictor with its
counterpart (non-injected predictor), using metrics such as accuracy, F1 score
or MSE. However, these metrics may not fully capture all aspects of knowledge
injection, such as whether such injection resulted in a more sustainable
predictor in terms of allocated resources or a more robust predictor, and so
on. Therefore, RG 2 seeks to study the injection process through a broader
lens, including factors such as the sustainability and robustness of the
injection procedure.

Finally, the aim of RG 3 is to create a MAS, in which ML and logic-based
elements can collaborate and interact with each other. In practice, given a
neuro-symbolic architecture, the main objective is to allocate the different
modules of the architecture to different collaborating agents. The integration
of ML and logic-based elements into MAS could potentially lead to more
efficient and effective decision-making processes.

### Results

We provided a rough idea of RG 1 in a conference position paper [25], in which
we proposed the incorporation of abductive reasoning through two possible
frameworks. The idea is applied to image segmentation to address issues of
data scarcity and low robustness. In this proposal, we suggest the use of
either ABL [12] or a combination of ABL and knowledge injection. The article
aims to indicate possible uses of abduction within the realm of Neuro-Symbolic
integration.

We presented a preliminary idea of RG 2 in [3], where we provide the first -
to the best of our knowledge - set of Quality-of-Service (QoS) metrics for
SKI, with a focus on quantifying robustness and efficiency advantages owing to
injection. In this paper, we offer an initial formulation for the following
metrics:  _(i)_ memory footprint efficiency, i.e., the gain in model
complexity;  _(ii)_ energy efficiency, i.e., the gain in total energy required
to train and deploy a sub-symbolic model;  _(iii)_ latency efficiency, i.e.,
improvements in terms of the time required for inference;  _(iv)_ data
efficiency, i.e., the improvement in terms of the amount of data required to
optimise a sub-symbolic model;  _(v)_ robustness, i.e., the capability of the
injection mechanism to adapt to variations of input data and knowledge;
_(vi)_ comprehensibility, i.e., the capability of the injection mechanism to
produce more intelligible models.  In addition, this work discusses the
potential impact of these metrics within the field of MAS, considering that a
probable inefficiency of the sub-symbolic models incorporated in the agents
affects their functioning in multiple ways (e.g. energy inefficiency, data
inefficiency, computational inefficiency, and so on).

In a subsequent paper, [2], we extended and enhanced RG 2 by providing a more
rigorous mathematical definition and implementation of the following metrics:
memory footprint, energy efficiency, latency efficiency, and data efficiency.
The software tools necessary for the practical use of these metrics were also
provided, and the metrics were evaluated on three different datasets using
three different injection models222The repository containing the experiments
can be found at the following link: https://github.com/pikalab-unibo/ski-qos-
jaamas-experiments-2022. The results provide valuable insights into the
performance of different injection predictors on various datasets,
highlighting the importance of adopting specific metrics when evaluating them.
The paper emphasises that the use of these metrics in the context of MAS could
be a crucial tool for comparing different predictors and selecting the most
suitable one for a given task.

In [23], an initial approach to symbolic and sub-symbolic integration within a
MAS was proposed. This paper was later extended in [24]. These two works are a
first step towards RG 3. In [23], we present the potential capabilities of an
integrated system consisting of logical agents and a neural network
specialized in monitoring flood events for civil protection purposes. The
paper describes a framework composed of a group of intelligent agents
performing various tasks and communicating with each other to efficiently
generate alerts during flood crisis events. In [24], the work is extended, and
an initial implementation of the framework is provided. The paper presents a
preliminary prototype of a MAS that autonomously collects weather warnings,
categorises related images using a DL module, filters the results, and alerts
human operators only if there is a reasonable certainty that a risk situation
is occurring. To this end, we implemented a system333The repository containing
the experiments can be found at the following link: https://github.com/AAAI-
DISIM-UnivAQ/MAS_Py_FLOOD using a combination of logical agents and a DL
component. The neural network is trained on eight classes of topographic
entities to segment the images. Once the images have been segmented, a Logical
Image Descriptor (LID) is used to generate a logical description of the
segmented mask predicates. This description is then submitted to a logical
agent that performs the reasoning. In our system, the reasoning is performed
using a perception-fusion approach, where the MAS agents use their perceptions
to reason about the environment and make decisions collectively. The paper
emphasises the adoption of agents based on computational logic with a logical
basis to provide verifiability, explainability, and reliability.

##  4 Future Works

In the future, potential research directions include:  _(i)_ for RG 1: explore
abductive reasoning technique to enhance knowledge injection efficiency and
effectiveness. Evaluate the framework’s performance through extensive
experiments in various domains;  _(ii)_ for RG 2: refine and expand evaluation
metrics, including comprehensibility metrics, for better assessment of
injected models. Validate metrics across different scenarios and datasets;
_(iii)_ for RG 3: explore collaborative frameworks and hybrid models combining
logical and neural reasoning. Possibly evaluate their performance in real-
world applications.

Overall, these future works aim to improve knowledge injection, evaluation
methodologies, and the integration of logical and neural agents, contributing
to more efficient and interpretable machine learning models.

## Acknowledgements

This PhD work is conducted under the supervision of professors Stefania
Costantini (University of L’Aquila, Italy), Fosca Giannotti (Scuola Normale
Superiore, Pisa, Italy), and Andrea Omicini (Alma Mater Studiorum–University
of Bologna, Italy).

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