# Meson-Baryon Couplings Revisited

N. F. Nasrallah

###### Abstract

The theoretical evaluation of the coupling constants $g_{\pi NN}$,
$g_{KN\Lambda}$and $g_{KN\Sigma}$ is undertaken using QCD sum rules. These
quantities were previously calculated with exponential (Borel) kernels used to
suppress the unknown contributions of the hadronic continua. This method
however introduces arbitrariness and instability in the calculation. In order
to avoid these I redo the calculation using polynomial kernels tailored to
vanish at the baryonic resonance masses. The results are $g_{\pi NN}=11.5\pm
1.0$, $g_{KN\Sigma}=5.45\pm.4$, $g_{KN\Sigma}=-(12.7-15.0)$ which are close to
experiment and to the predictions of SU(3) and which , with the corresponding
Goldberger-Treiman Discrepancy satisfy quite well the Dashen-Weinstein
relation.

Faculty of Science, Lebanese University. Tripoli 1300, Lebanon

##  1 Introduction

The $\pi$$N$ coupling constant $g_{\pi NN}$ is one of the most important
parameters in hadron physics. This quantity has been extensively studied since
the works of Reinders, Rubinstein and Yazaki [1] using the method of QCD sum
rules [2] starting with either the correlator of three interpolating fields or
with the pion to vacuum matrix element of two nucleon interpolating currents.
These correlators introduce several different Dirac structures and these were
studied in detail two decades ago [3], [4], [5] and more recently in [6]

All these calculations however involve dispersion integrals which include the
unknown contribution of the continuum with quantum numbers of the nucleon. In
order to minimize this unknown contribution a damping kernel
$e^{-\frac{t}{M^{2}}}$ is introduced. This method however presents problems of
arbitrariness and stability. I present here a different choice for the damping
kernel: a polynomial P(t) which practically vanishes in the nucleon resonance
region. The sum rule method consists of expressing the quantity of interest,
which enters in the residue at the pole of the correlation function $\Pi(t)$
in terms of an integration of $\Pi(t)$ over the contour $c$ shown in Fig.1 in
the complex t-plane .This in turn is the sum of an integral over the cut of
the discontinuity of $\varPi(t)$ and an integral over the circle of large
radius R over which $\Pi(t)$ can be replaced by its QCD expression
$\Pi^{QCD}(t)$ the latter can be brought back to an integral over the cut of
the discontinuity of $\Pi^{QCD}(t).$ The main unknown in the calculation is
the integral over the cut of the hadronic amplitude.

In order to minimize this contribution an integration kernel P(t) is
introduced so that

$\text{Residue x
}P(\text{pole})=\frac{1}{\pi}\int_{th}^{R}dtP(t)Im\Pi(t)+\frac{1}{\pi}\int_{0}^{R}dtP(t)Im\Pi^{QCD}(t)$
(1)

The usual choice for the kernel is $P(t)=e^{-\frac{t}{M^{2}}}$where $M^{2}$
(the Borel mass) is a damping parameter which cannot be too large because the
damping worsens nor can it be too small because the contribution of higher
orders in the Operator Product Expansion (OPE) of $\Pi^{QCD}(t)$ increase
beyond control. An intermediate value of $M^{{}^{2}}$has to be chosen by
stability considerations which is not always possible. The radius R of the
circle is another adjustable parameter.

In order to avoid these problems I shall choose for the kernel a simple
polynomial $P(t)=1-a_{1}t-a_{2}t^{2}$ tailored to practically vanish in the
nucleon resonance region in order to render the first integral on the r.h.s.
of eq. (1) negligible. The value of R is chosen to lie in the stability region
of the second integral on the r.h.s. of eq. (1). This leads to an unambiguous
result.

The method is applied to the calculation of the coupling constants $g_{{}_{\pi
NN}},$$g_{KN\varLambda}$and $g_{KN\Sigma}$.

Fig. 1

##  2 The Pion-Nucleon Coupling Constant gπNN

Consider the correlation function

$\Pi(p,q)=i\int dxe^{iqx}\left\langle
0\left|Tj_{p}(x)j_{n}(0)\right|\pi^{+}(p)\right\rangle$ (2)

where

$j_{p}=\epsilon_{abc}(u_{a}^{T}C\gamma_{\mu}u_{b})\gamma_{5}\gamma_{\mu}d_{c}$
(3)

is the proton interpolating field of Joffe [7]. The interpolating field of the
neutron is obtained by the interchange of $u$ and $d$.

I shall work with the $\gamma_{5}\sigma_{\mu\nu}q_{\mu}q_{\nu}$ structure of
the correlator.

In the soft pion limit the OPE reads [5]:

$\Pi^{QCD}(t)=\sqrt{2}\left\langle
qq\right\rangle\left[\frac{ln(-t)}{12\pi^{2}f_{\pi}}+\frac{4}{3}\frac{f_{\pi}}{t}-\frac{C}{t^{2}}\right]$
(4)

with $t=q^{2}$ ,

$C\simeq\left(\frac{1}{216f_{\pi}}\left\langle
a_{s}GG\right\rangle-\frac{m_{o}^{2}f_{\pi}}{6}\right)\simeq-.0117GeV^{3}$ (5)

at the standard values $\left\langle
a_{s}GG\right\rangle\simeq.012GeV^{{}^{4}},m_{0}^{2}\simeq.8GeV^{2}$

On the hadronic side $\Pi(t)$ has a double nucleon pole, single nucleon poles
and a continuum.

$\Pi(t)=\frac{\sqrt{2}g_{{}_{\pi
NN}}\lambda_{N}^{2}}{(t-m_{N})^{2}}+\text{single poles + continuum}$ (6)

$\lambda_{N}$denotes the coupling of $j_{N}$to the nucleon

$\left\langle 0\left|j_{N}\right|N\right\rangle=\lambda_{N}U_{N}$ (7)

Consider the integral $\frac{1}{2\pi i}$$\int_{c}dt$P(t)$\Pi(t)$ in the
complex t-plane where c is the contour shown in Fig.1.

As argued in the introduction I shall choose $P(t)=1-a_{1}t-a_{2}t^{2}$ with
the coefficients $a_{1,2}$ tailored in order to practically annihilate the
contribution of the continuum. We thus have

$\sqrt{2}g_{\pi
NN}\lambda_{N}^{2}P^{\prime}(m_{N}^{2})+\varDelta=\frac{1}{\pi}\int_{th}^{R}dtP(t)Im\Pi(t)\Pi+\frac{1}{2\pi
i}\oint dtP(t)\Pi^{QCD}(t)$ (8)

$\varDelta$ is the contribution of the single poles. The choice of $P(t)$ is
based on the fact that the contribution of the first integral on the r.h.s. of
eq. (7) arises mostly from the interval [8] $I=2.0GeV^{2}\preceq t\preceq
3.0GeV^{2}$ where the resonances
$N^{+}(1440),N^{-}(1535),N^{-}(1650),N^{+}(1710)$ lie. The parameters
$a_{1}$and $a_{2}$ are chosen so as to minimize the integral
$\int_{2.0GeV^{2}}^{3.0GeV^{2}}dt$ $\left|P(t)\right|^{2}$, numerically
$a_{1}=.807GeV^{-2},a_{2}=-.160GeV^{-4}$.

This allows the neglect of the first integral on the r.h.s. of eq. (8) so
that, using expression (4) for $\Pi^{QCD}(t)$

$\sqrt{2}g_{\pi
NN}\lambda_{N}^{2}P^{\prime}(m_{N}^{2})+\varDelta\cong\sqrt{2}\left\langle
qq\right\rangle\left[\frac{1}{12\pi^{2}f_{\pi}}I_{0}(R)+\frac{4}{3}f_{\pi}-a_{1}C\right]$
(9)

with the definition

$I_{n}(R)=\int_{{}_{0}}^{R}dtt^{n}P(t)$ (10)

The fact that the contrbution of term $a_{1}C$ is small compared to that of
the preceding ones ($\sim 5\%$) justifies the neglect of the next unknown term
proportional to $a_{2}$. This is to be contrasted to the case of exponential
damping where an inifinite number of unknown higher order terms enter. Morever
it can be seen from Fig. 2 that the damping of the continuum is much better in
our case.

Fig. 2

$P(t)$ is our polynomial $1-a_{1}t-a_{2}t^{2}$

$M^{2}\sim 1.0GeV^{2}$as chosen in the litterature

$\lambda_{N}$was obtained using the same approach in [8] starting with the
nucleon correlator

$\Pi_{N}(q)=i\int dxe^{iqx}\left\langle
0\left|T\eta(x)\eta(0)\right|0\right\rangle=\cancel{q}\Pi_{1}(q^{2})+\Pi_{2}(q^{2})$
(11)

The sum rule using $\Pi_{2}(q^{2})$ yields

$(2\pi)^{4}m_{N}\lambda_{N}^{2}P(m_{N}^{2})=-B_{3}I_{1}(R)-B_{7}$ (12)

with the OPE coefficients

$B_{3}=4\pi^{2}(1+\frac{3}{2}a_{s})\left\langle\bar{q}q\right\rangle,\,\,\,\,\,\,\,\,\,\,B_{7}=-\frac{4}{3}\pi^{2}\left\langle
qq\right\rangle\left\langle a_{s}GG\right\rangle$ (13)

Let us now estimate $\varDelta$. The contribution of the $N^{\prime}(1440)$
adds to eq. (6)

$\delta_{N^{\prime}}=\sqrt{2}g_{\pi
NN^{\prime}}\lambda_{N}\lambda_{N^{\prime}}\frac{1}{(t-m_{N}^{2})(t-m_{N^{\prime}}^{2})}$
(14)

which corresponds to

$\varDelta=\sqrt{2}g_{\pi
NN}\lambda_{N}\lambda_{N^{\prime}}\frac{P(m_{N}^{2})-P(m_{N^{\prime}}^{2})}{m_{N}^{2}-m_{N^{\prime}}^{2}}$
(15)

While $P(m_{N^{\prime}}^{2})$ is small by construction $P(m_{N}^{{}^{2}})$ is
not . The contribution of the simple pole is not damped by the polynomial.

$\lambda_{N^{\prime}}$is estimated from a variant of eq. (12) (with no
damping)

$(2\pi)^{4}\lambda_{N^{\prime}}^{2}m_{N^{\prime}}\backsimeq-
B_{3}\int_{(m_{{}_{N}}^{2}+m_{{}_{N^{\prime}}}^{2})/2}^{(m_{N^{\prime}}^{2}+m_{N^{\prime\prime}}^{2})/2}tdt$
(16)

which yields $\frac{\lambda_{N^{\prime}}}{\lambda_{N}}\backsimeq.35$

The coupling constant $g_{\pi NN^{\prime}}$was studied in [9] who give $g_{\pi
NN^{\prime}}=.47\pm.04$ . This yields

$\frac{\varDelta_{N^{\prime}}}{(\text{dominant\,term})}=\frac{g_{\pi
NN^{\prime}}}{g_{\pi
NN}}\frac{\lambda_{N^{\prime}}}{\lambda_{N}}\frac{P(m_{N}^{2})}{P^{\prime}(m_{N}^{2})}\backsimeq.015$
which shows that $\varDelta$ amounts to only a few percent of the dominant
term. Finally

$g_{\pi
NN}\frac{-P^{\prime}(m_{N}^{2})}{P(m_{N}^{2})}=\frac{\left[\frac{1}{12\pi^{2}}I_{0}(R)+\frac{4}{3}f_{\pi}^{2}-a_{1}f_{\pi}C\right]}{\left[\frac{1}{4\pi^{2}}I_{1}(R)(1+\frac{3}{2}a_{s})-\frac{1}{12}\left\langle
a_{s}GG\right\rangle\right]}\frac{m_{N}}{f_{\pi}}$ (17)

As discussed before the radius $R$ of the integration circle is not arbitrary,
it cannot be too small because this would invalidate the OPE nor can it be too
large because the polynomial would start enhancing the unknown continuum
instead of damping it, $R$ should be chosen in the stability region of the
integrals $I_{0,1}(R)$ shown in Fig. 3

The interval of stability is very wide and the results are

$I_{0}(R)=.814GeV^{2}$ similarly $I_{1}(R)=.474GeV^{4}$. With the standard
value $\left\langle a_{s}GG\right\rangle=.012GeV^{4}$ eq. (17) finally gives

$g_{\pi NN}=11.5\pm 1.0$ (18)

The error, estimated about 8% stems mostly from neglected radiative
corrections as well as from $\varDelta.$

Fig. 3

In order to assess the stability of the method other choices of the damping
polynomial can be made, e.g $P(t)=(1-\frac{t}{2.5GeV^{2}})$ or impose
$\int_{2.0GeV^{2}}^{3.0GeV^{2}}dt\,P(t)=0$. In both cases the results are very
close to eq.(18) ($g_{\pi NN}=11.2$ and $g_{\pi NN}=11.0$)

##  3 Kaon-Baryon Couplings

Several attempts were made to evaluate the couplings $g_{KN\Lambda}$and
$g_{KN\Sigma}$using QCD sum rules [10],[11],[12],[13],[14],[15]

which all turned out much smaller than their SU(3) values

$\displaystyle g_{KN\Lambda}$
$\displaystyle=\frac{-1}{\sqrt{3}}(3-2\alpha)g_{\pi NN}\simeq-13.05$ (19)
$\displaystyle g_{KN\Sigma}$ $\displaystyle=(2\alpha-1)g_{\pi NN}\simeq 3.50$

where $\alpha=D/(D+F)\simeq.635$ is the fraction of D-type coupling.

Experimentally, the determination of the coupling constants still involves
uncertainties but the analysis of [16] yields

$g_{KN\varLambda}=-13.5\text{\,\,\,\,\,\,and\,\,\,\,\,\,}g_{KN\Sigma}=4.5$
(20)

Evaluation of the coupling constants using heavy baryon chiral perturbation
theory at the one loop level has also been attempted [17]resulting in values
much smaller than one would expect from eq. (19).

An attempt to calculate the coupling constants starting from the Golberger-
Treiman relation (GTR) in $SU(3)\times(SU(3)$ was undertaken in [18]. In this
paper the extrapolation to the nucleon mass shell was done using exponential
kernels to damp the unknown parts of dispersion integrals. I shall here redo
the calculation using polynomial kernels which avoid the arbitrariness and
stability problems.

The GTRs in the strange channel are

$\sqrt{2}f_{K}g_{KN\Lambda}=(m_{N}+m_{Y})g_{A}^{Y}(0)$ (21)

where $Y=\varLambda,\Sigma.$ g${}_{A}^{Y}(0)$ is the axial-vector coupling
constant and $f$K the K-meson decay constant.

The coupling constants could of course be obtained from eq. (21) because the
$g_{{}_{A}}^{Y}$ are measured. eq. (21) involves however an extrapolation in
momentum transfer squared from 0 to $m_{K}^{2}$ which is not a small quantity
on the hadronic scale. In other words explicit chiral symmetry breaking leads
to corrections to the GTR, the GT discrepancies (GTD) which are not small in
the strangeness changing case.

The GTD are defined by

$\varDelta^{Y}=1-(m_{N}+m_{Y})g_{A}^{Y}(0)/(\sqrt{2}f_{K}g_{KNY})$ (22)

The evaluation of the K-nucleon coupling constants will yield the GTD.

Start from the matrix element

$\left\langle
P\left|\partial_{\mu}A_{\mu}^{K^{+}}\right|Y\right\rangle=\Pi(q^{2})\overline{P}i\gamma_{5}Y$
(23)

q denoting the momentum transfer between the baryons. We have

$\Pi(0)=(m_{N}+m_{Y})g_{A}^{Y}(0)$ (24)

The analytic properties of $\Pi(t=q^{2})$in he complex t-plane are known. It
has a pole at $t=m_{K}^{2}$ and a cut along the positive t-axis starting at
$t_{th}=(m_{K}+2m_{\pi})^{2}$

$\displaystyle\Pi(t)$
$\displaystyle=\frac{-\sqrt{2}f_{K}m_{K}^{2}g_{KNY}}{(t-m_{K}^{2})}+\cdots$
(25)

Consider now the usual contour $c$ in the complex t-plane and the integral

$\frac{1}{2\pi
i}\intop_{c}\frac{dt}{t}\Pi(t)=\frac{1}{\pi}\int_{th}^{R}\frac{dt}{t}Im\Pi(t)+\frac{1}{2\pi
i}\oint\frac{dt}{t}\Pi^{QCD}(t)=\Pi(0)-\sqrt{2}f_{K}g_{NKY}$ (26)

The first integral on the r.h.s. running along the cut represents the
contribution of the $0^{-}$strange continuum and provides the main part of the
GTD. In the second integral along the circle $\Pi^{QCD}(t)$ is a good
approximation for $\Pi(t).$In order to overcome the lack of knowledge of
$Im\Pi(t)$ on the cut consider the modified integral

$\displaystyle\frac{1}{2\pi i}\int_{c}dt(\frac{1}{t}-a_{0}-a_{1}t)\Pi(t)$
$\displaystyle=$
$\displaystyle\frac{1}{\pi}\int_{th}^{R}dt(\frac{1}{t}-a_{0}-a_{1}t)Im\Pi(t)+\frac{1}{2\pi
i}\oint dt(\frac{1}{t}-a_{0}-a_{1}t)\Pi^{QCD}(t)$ (27) $\displaystyle=$
$\displaystyle\Pi(0)-\sqrt{2}f_{K}g_{KNY}(1-a_{0}m_{K}^{2}-a_{1}m_{K}^{4})$

The main contribution to the integral over the cut arises from the interval
$I$ : $1.5GeV^{2}\lesssim t\lesssim 3.5GeV^{2}$ which includes the resonances
$K(1460)$ and $K(1830)$. The constants $a_{1}$and $a_{2}$are now chosen so as
to annihilate the kernel $(1-a_{1}t-a_{2}t^{2})$ at $t=1.46^{2}GeV^{2}$and at
$t=1.83^{2}GeV^{2}$ i.e

$a_{0}=.77GeV^{-2}\text{\,\,\,\,\,\,and\,\,\,\,\,\,}a_{1}=-.14GeV^{-4}$ (28)

With this choice the integrand is reduced to only a few percent of its initial
value over the interval $I$ and the integral over the cut becomes negligible.
so

$\Pi(0)-\sqrt{2}f_{K}g_{KNY}(1-a_{1}m_{K}^{2}-a_{2}m_{{}_{K}}^{4})\backsimeq\frac{1}{2\pi
i}\oint dt(\frac{1}{t}-a_{0}-a_{1}t)\Pi^{QCD}(t)$ (29)

It appears from the equation above that chiral symmetry breaking manifests
itself in the presence of the r.h.s. as well as in the deviation of the factor
$(1-a_{1}m_{K}^{2}-a_{2}m_{K}^{4})$ from unity.

In [18] $\Pi^{QCD}(t)$ was obtained from Borel-type sum rules. I shall here
redo the calculation using a polynomial kernel as advocated. Consider the
three-point function

$\varGamma(s=p^{2},t=q^{2})=\int\int dxdye^{-ipx}e^{iqy}\left\langle
0\left|T\psi^{N}(x)\partial_{\mu}A_{\mu}^{K}(y)\psi^{Y}(0)\right|0\right\rangle$
(30)

where $\psi^{N,Y}$are baryonic currents and let

$\varGamma(s,t)=F(s,t)\sigma_{\mu\nu}\gamma_{5}q_{\mu}p^{\prime}_{\nu}+\text{other
tensor structure}$ (31)

Furthermore the residue at the double baryonic pole of the $\varGamma$ is
related to $\Pi(t)$, i.e.

$\varGamma(s,t)=\left(\frac{\lambda_{N}\lambda_{Y}\Pi(t)}{(s-m_{N}^{2})(s-m_{Y}^{2})}+\cdots\right)\sigma_{\mu\nu}\gamma_{5}p_{\mu}^{{}^{\prime}}q_{\nu}+\cdots$
(32)

$\Pi^{QCD}(t)$ is obtained from the above by using $F_{\Lambda}^{QCD}(s,t)$,
extrapolating to the baryons mass shell and identifying terms in (32) and
(33).

$\lambda_{N,Y}$denotes the coupling of the baryonic currents to the
corresponding baryons.

It will be assumed that the contribution of the single poles is small as it
was in the $\pi-N$ case and it will be neglected.

$\Pi^{QCD}(t)$is obtained by using F${}^{QCD}(t)$ and extrapolating to the
baryon mass shell.

$F^{QCD}(s,t)$ is given in [10]

$F_{\Lambda}^{QCD}(s)=\frac{C_{\Lambda}}{st},\,\,\,\,\,\,C_{\Lambda}=[\frac{4}{3}\sqrt{\frac{2}{3}}m_{s}(\left\langle\overline{q}q^{2}\right\rangle+\left\langle\overline{q}q\overline{s}s\right\rangle)]$
(33)

and

$F_{\Sigma}^{QCD}(s,t)\backsimeq 0$ (34)

Consider now the integral

$\frac{1}{2\pi i}\int_{c}dsP_{Y}(s)F(s,t)$ (35)

in the complex s-plane where as before $P_{Y}(s)$ is the damping polynomial
intended to eliminate the contribution of the continuum.

This gives

$\lambda_{N}\lambda_{Y}\Pi^{QCD}(t)\frac{P_{Y}(m_{Y}^{2})-P_{Y}(m_{N}^{2})}{(m_{Y}^{2}-m_{N}^{2})}\backsimeq\frac{1}{2\pi
i}\oint dsP_{Y}(s)F^{QCD}(s)$ (36)

and take
$P_{Y}(s)=(1-\frac{s}{m_{N^{\prime}}^{2}})(1-\frac{s}{m_{Y^{\prime}}^{2}})$
which vanishes at the first nucleon and hyperon excited states (
m${}_{N^{\prime}}^{2}$=1.98GeV2, m${}_{\varLambda^{\prime}}^{2}$= 2.40GeV2,
$\textnormal{m}_{\Sigma^{\prime}}^{2}=2.51$GeV2)

Eqs. (33) and (34) inserted in eq. (35) yield the QCD expressions

$\frac{\lambda_{N}\lambda_{\Lambda}}{(m_{\Lambda}^{2}-m_{N}^{2})}(P_{{}_{\Lambda}}(m_{\Lambda}^{2})-P_{{}_{\Lambda}}(m_{N}^{2}))\Pi^{QCD}(t)\backsimeq\frac{C_{\varLambda}}{t}$
(37)

and

$\frac{\lambda_{N}\lambda_{\Sigma}}{(m_{\Sigma}^{2}-m_{N}^{2})}(P_{\Sigma}(m_{\Sigma}^{2})-P_{\Sigma}(m_{N}^{2}))\Pi_{\Sigma}^{QCD}(t)\backsimeq
0$ (38)

When this is inserted in eq. (29) it gives

$\Pi_{\varLambda}(0)=\sqrt{2}f_{K}g_{KN\Lambda}(1-a_{1}m_{K}^{2}-a_{2}m_{K}^{4})-\frac{a_{1}C_{\Lambda}(m_{{}_{\varLambda}}^{2}-m_{N}^{2})}{[\lambda_{N}\lambda_{\Lambda}(P_{\Lambda}(m_{\Lambda}^{2})-P_{\Lambda}(m_{N}^{2}))]}$
(39)

$\Pi_{\Sigma}(0)=\sqrt{2}f_{K}g_{KN\Sigma}(1-a_{1}m_{K}^{2}-a_{2}m_{K}^{4})$
(40)

$\lambda_{N}$is given by eq. (12)

$(2\pi)^{4}\lambda_{N}^{2}P(m_{N}^{2})=.444GeV^{6}$ (41)

a similar treatment of the $\Lambda$ 2-point function [12] with our polynomial
replacing the damping exponential gives

$(2\pi)^{4}\lambda_{\Lambda}^{2}P_{\Lambda}(m_{\varLambda}^{2})=.481GeV^{6}$
(42)

Finally with the experimental values
$g_{\Lambda}^{A}=-.72,\,\,\,\,\,g_{\Sigma}^{A}=.34$ we obtain

$g_{KN\Sigma}=5.45,\,\,\,\,\,g_{KN\Lambda}=-(11.50\pm 1.17\kappa)$ (43)

$\kappa$ denotes the deviation of the 4-quark condensate from factorization,
it arises from the condensate
$\left\langle\overline{s}s\overline{q}q\right\rangle\backsimeq\kappa\frac{\left\langle\overline{s}s\right\rangle}{\left\langle\overline{q}q\right\rangle}\left\langle(\overline{q}q)^{2}\right\rangle$
which appears in the expression for the constant CΛ.

There is no consensus on the value of $\kappa$. I shall take
$1\preceq\kappa\preceq 3.$

##  4 Results and Conclusions

The $\pi-N$ and K-N coupling constants were calculated using QCD sum-rules
with polynomial kernels tailored to vanish in the baryonic resonance region.
This solves the problems of arbitrariness and stability inherent to the usual
Borel-type QCD sum-rules. The results are

$g_{\pi
NN}=12.4\pm.6,\,\,\,\,\,\,g_{KN\Sigma}=5.45\pm.4,\,\,\,\,\,\,g_{KN\Lambda}=-(12.7-15.0)$
(44)

which are quite close to experiment and to the SU(3) values

The corresponding GTD are large as expected

$\Delta_{GT}^{\Sigma}\simeq.18,\,\,\,\,\,\Delta_{GT}^{\Lambda}\simeq.29$ (45)

It is finally interesting to see how well our results fit the Dashen-Weinstein
[19] relation between the GTD

$g_{\pi
NN}\Delta_{GT}^{N}=\frac{m_{\pi}^{2}}{2m_{K}^{2}}(g_{KN\Sigma}\varDelta_{GT}^{\Sigma}-\sqrt{3}g_{KN\Lambda}\varDelta_{GT}^{\Lambda})$
(46)

With $\varDelta_{GT}^{N}\backsimeq.02$ it is seen that relation eq. (46) is
quite well satisfied.

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# Contestable AI needs Computational Argumentation

Francesco Leofante  Hamed Ayoobi  Adam Dejl  Gabriel Freedman
Deniz Gorur  Junqi Jiang  Guilherme Paulino-Passos  Antonio Rago
Anna Rapberger  Fabrizio Russo  Xiang Yin  Dekai Zhang&Francesca Toni
Computational Logic and Argumentation Group,
Department of Computing, Imperial College London, UK
{f.leofante, h.ayoobi, adam.dejl18, g.freedman22, d.gorur22, junqi.jiang20,
gppassos, a.rago, a.rapberger, fabrizio, x.yin20, dekai.zhang19,
<EMAIL_ADDRESS>
###### Abstract

AI has become pervasive in recent years, but state-of-the-art approaches
predominantly neglect the need for AI systems to be _contestable_. Instead,
contestability is advocated by AI guidelines (e.g. by the OECD) and regulation
of automated decision-making (e.g. GDPR). In this position paper we explore
how contestability can be achieved computationally in and for AI. We argue
that _contestable AI_ requires dynamic (human-machine and/or machine-machine)
explainability and decision-making processes, whereby machines can  1.
interact with humans and/or other machines to progressively explain their
outputs and/or their reasoning as well as assess grounds for contestation
provided by these humans and/or other machines, and  2. revise their decision-
making processes to redress any issues successfully raised during
contestation.  Given that much of the current AI landscape is tailored to
static AIs, the need to accommodate contestability will require a radical
rethinking, that, we argue, computational argumentation is ideally suited to
support.

##  1 Introduction

AI has become pervasive in recent years, with applications ranging from
autonomous driving (Muhammad et al., 2020) to finance (Cao, 2022) and
healthcare (Shaheen, 2021). Current state-of-the-art AI systems focus on
algorithmic solutions, often built from data, generating outputs in the forms
of predictions, recommendations and/or decisions and, in some cases,
explanations thereof. These existing solutions, though, mostly neglect the
need for these AI systems to be _contestable_. Instead, the need to
accommodate contestability is a crucially important problem if AI systems are
to be deployed in and benefit society. Indeed, contestability is prominently
advocated in some frameworks for AI ethics and in regulations. For example,
the Organisation for Economic Co-operation and Development (OECD) states that
information should be provided “to enable those adversely affected by an AI
system to challenge its outcome based on plain and easy-to-understand
information on the factors, and the logic that served as the basis for the
prediction, recommendation or decision” (Principle
1.3111https://oecd.ai/en/dashboards/ai-principles/P7). Moreover, the ACM
Global Technology Policy Council for Responsible Algorithmic Systems advocates
‘Contestability and Auditability’ as one of nine
principles.222https://www.acm.org/media-center/2022/october/tpc-statement-
responsible-algorithmic-systems Contestability is required by law in some
jurisdictions, e.g. GDPR, article 22(3),333https://gdpr-
text.com/read/article-22/ states that the data subject’s rights to be
safeguarded shall include “at least the right […] to contest the decision”.
Finally, contestability and redress are also identified as key principles
underpinning the UK AI regulation
framework444https://www.gov.uk/government/publications/ai-regulation-a-pro-
innovation-approach/white-paper, whereby “Guidance should clarify existing
‘formal’ routes of redress offered by regulators in certain scenarios”. This
is a major departure point from “classical” requirements for explainable
systems, which typically do not require correcting the AI when faulty
behaviours are exposed.

Contestability is seen by some (Hicks, 2022) as a means to facilitate
accountability, e.g. to prevent the use of Uber’s algorithm for banning
drivers because it is not taking into account bias of customers when giving
bad reviews. A handful of recent works (overviewed in Section 2) bring
contestability to the attention of AI researchers, practitioners and users,
while providing insights into its possible interpretations in practice.
Moreover, there is evidence, in the form of user studies, that contestability
can affect users’ perception of AI fairness (Yurrita et al., 2023). However,
formal/computational forms of contestability are mostly lacking in the
literature, with some exceptions, in the form of  1. indications that, in the
case of machine learning, methods generating counterfactual explanations in
terms of “actionable changes that an individual can make to flip the
prediction of the classifier” may be interpreted as offering (limited)
contestability (Venkatasubramanian and Alfano, 2020) and  2. (Russo and Toni,
2023), offering contestability in the very specific setting of causal
discovery.  Finally, existing works mostly see contestability as a post-hoc
process, after predictions/recommendations/decisions have been computed.

In this position paper (specifically in Section 3) we consider what it means
for (any) _algorithmic decision systems (ADS)_ (Henin and Métayer, 2022) to be
contestable and contested (by humans or other ADSs). Our take-away message is
that formalising/realising computationally contestable AI requires:
_explanations_ (by the ADSs which humans/other ADSs may want to contest);
_grounds_ for contestation (by the contesters); ability (by the contested) to
_redress_ any issues successfully raised during contestation; and ability to
_interact_ (by both contested and contester). We then advocate computational
argumentation (CA) as being ideally suited to support contestability,
providing evidence from the literature on CA (Section 4). We finally conclude
(Section 5) with some pointers for future directions in what we believe is a
very promising direction of research for the KR community.

##  2 AI Contestability in the literature

Here, we summarise recent works focusing on advocating or proposing forms of
contestability in AI.555We have identified these works by searching DBLP with
‘Contestable’ and ‘Contestability’, restricting attention to papers on AI from
2014 onwards published in peer-reviewed venues, while also considering
additional references therein. We focus on the main contestability dimensions
that emerge in this literature, as well as providing pointers to other (less)
related research.

Contested entities. Prior work has focused on contesting various aspects of an
ADS, including its general design (Almada, 2019; Alfrink et al., 2022;
Yurrita, Balayn, and Gadiraju, 2023), training data (Kluttz, Kohli, and
Mulligan, 2022), training procedure (Almada, 2019), its inputs (Alfrink et
al., 2022) and outputs (Hirsch et al., 2017; Ploug and Holm, 2020; Tubella et
al., 2020; Lyons, Velloso, and Miller, 2021; Henin and Métayer, 2022; Hicks,
2022) as well as the ADS as a whole (Lyons, Velloso, and Miller, 2021; Vaccaro
et al., 2021; Henin and Métayer, 2022; Hicks, 2022; Alfrink et al., 2023;
Russo and Toni, 2023). Note that the possibility of humans being contested by
an ADS (e.g., if their assumptions about the subject domain are flawed) is not
typically considered.

Contesting entities. All approaches we identified focus on humans as the
contesting entities. However, there are considerable differences between the
categories and assumed skill sets of people engaged in contestation. Several
argue that decision subjects should be empowered to directly contest ADS
decisions affecting them, potentially without possessing detailed knowledge of
the subject domain (Almada, 2019; Ploug and Holm, 2020; Lyons, Velloso, and
Miller, 2021; Vaccaro et al., 2021; Henin and Métayer, 2022; Hicks, 2022).
Some works suggest that individual decision subjects may not always be able to
effectively contest on their own, and propose contestation via third-party
representatives (Alfrink et al., 2022) or as part of a group in a “class
action” (Lyons, Velloso, and Miller, 2021). Others also consider contestation
by professionals, subject matter experts and regulators (Tubella et al., 2020;
Henin and Métayer, 2022; Kluttz, Kohli, and Mulligan, 2022; Alfrink et al.,
2023; Russo and Toni, 2023). Note that the possibility of ADSs being the
contesting entities is not typically considered.

Contestation methods. Several methods have been proposed for facilitating
contestation. In the realm of non-technical solutions, participatory design
has been advocated as a way to involve various stakeholders in ADS
development, enabling advance mitigation of possible issues and risks
associated with the use of the ADS (Almada, 2019; Vaccaro et al., 2021;
Alfrink et al., 2023). The framework proposed in (Alfrink et al., 2022)
suggests following a set of development practices over the ADS lifecycle,
including incorporating ex-ante safeguards, gathering end-user feedback,
implementing quality assurance, mitigating possible risks and allowing for
third-party oversight. More technical approaches envision usage of operation
logs (Hicks, 2022) or model explanations (Almada, 2019; Ploug and Holm, 2020;
Kluttz, Kohli, and Mulligan, 2022; Russo and Toni, 2023; Yurrita, Balayn, and
Gadiraju, 2023). We discuss these and their use of explanations in greater
detail next.

The role of explanations. That explanations are needed to support
contestability is acknowledged by some (e.g. by Almada (2019); Lyons, Velloso,
and Miller (2021); Alfrink et al. (2022); Wachter, Mittelstadt, and Russell
(2017)), but the problem of using explanations to support contestability in
practice has received little attention in explainable AI (XAI) (or AI for that
matter) to date. A notable exception is (Russo and Toni, 2023), where causal
discovery with neural networks guides human feedback for contesting the
discovered causal relations, which can be seen as a form of global
explanation. Also, some current XAI methods can be seen as offering ground for
some limited contestability when they suggest actionable recourse, as is the
case for counterfactual explanations in terms of “actionable changes that an
individual can make to flip the prediction of the classifier”
(Venkatasubramanian and Alfano, 2020). However, these works disregard that
algorithmic decisions may actually be incorrect, e.g. because they are based
on incorrect or incomplete data. Moreover, these methods are one-shot,
providing no opportunities for follow-up inquiry, and shallow, revealing no
information about the steps or logic that led to the explained output, and
thus offer little ground for contestability. Some other XAI methods provide
information on the AIs’ deliberation and insecurities (Wang and Vasconcelos,
2019); however, they do not support any form of contestability. Overall,
explanations, as understood in state-of-the-art XAI, are seen as inadequate to
support contestability (Henin and Métayer, 2022). Specifically, Yurrita,
Balayn, and Gadiraju (2023) advocate a generic notion of explanations that
capture the rationales behind the development and deployment of the ADS
(referred to therein as “process-centric explanations”).

Post-hoc and ex-ante contestability. Some take the view that contestability
needs to be supported by regulatory frameworks, involving in particular policy
modelling and normative reasoning (Tubella et al., 2020). These approaches see
contestability as post-hoc processes, detached from explainability, “to review
algorithmic decisions” (Lyons, Velloso, and Miller, 2021). Ex-ante
contestability is envisaged by some as a design principle (Hirsch et al.,
2017; Alfrink et al., 2022), whereby systems are designed to enable users to
interactively contest these systems and interaction between humans and systems
is needed for “critique and correction” (Lyons, Velloso, and Miller, 2021),
but no technical solutions exist on how to support it.

Contestability as an interaction process. Kluttz, Kohli, and Mulligan (2022)
see contestability as an interaction process with humans, at development and
deployment times, to allow humans, rather than just data, to train systems.
Several other works (e.g. (Hirsch et al., 2017; Lyons, Velloso, and Miller,
2021)) envisage post-hoc contestability as an interaction process, again
between humans and ADSs alone.

Other related work. Works on _corrigibility_ , such as (Soares et al., 2015;
Carey and Everitt, 2023; Russell, Dewey, and Tegmark, 2015) focus on human
overseers providing feedback/instructions to ADSs so that they align with the
intentions/values of their user(s) (Gabriel, 2020; Hadfield-Menell et al.,
2016). While corrigibility can be viewed as enabling human users to contest
the decisions of ADSs, it is narrowly focused on ADSs which optimise for
reward when they have incentives to manipulate users (Ward, Toni, and
Belardinelli, 2022), ignore instructions (Hadfield-Menell et al., 2016), and
disempower humans/other ADSs (Turner et al., 2021).

##  3 An abstract view of AI contestability

In a nutshell, towards supporting contestability, we see ADSs and humans as in
Figure 1, assuming that the contested entity is an ADS and the contester is
either a human or another ADS.666For simplicity, we assume that the contested
ADS is equipped with a single $M$ and accompanying $E$ and $R$, but in general
the same ADS may be equipped with several. Similarly, we assume that the
contester is equipped with a single $G$, but in general the same contester may
be equipped with several (one per contested model). Further, the same ADS
could be both contested and contester.

Figure 1: An abstract view of AI contestability: the contested ADS (left) is
equipped with a model ($M$), an _explanation_ method ($E$), and a _redress_
method ($R$); the contester (right) is a human or an ADS equipped with a
_ground generator_ for contestations ($G$); both contested ADS and contester
are able to interact ($I$).

Let $M:I\rightarrow O$ be a _model_ computing outputs in $O$ for inputs in
$I$. $M$ could be. e.g., a machine learning model, an expert system, or a
combination of the two. $M$ is part of the (contested) ADS, and the outputs of
$M$ are the ADS’s decisions. For this ADS to be contestable (by a human or
another ADS), it needs to be able to process _contestations_ , which may be in
reference to one of the following settings:777We ignore here the trivial
setting where contestations are directed at specific inputs, as these do not
require altering $M$.

(A) output $y=M(x)$ for some specific input $x$, or

(B) how $M$ determines $y=M(x)$ from input $x$, or

(C) the full model $M$.
The first setting (A) amounts to contesting the model behaviour without
referring to its internal “reasoning” process. The second setting (B) amounts
to contesting (part of) the reasoning process of the model for the particular
input. The third setting (C) amounts to contesting the model in general, e.g.
its “philosophy”, such as the underpinning input and output distributions
and/or that it is biased.

######  Example 1.

As a simple illustration, consider $M$ given by a (transparent) set of rules
for binary classification amounting to awarding a loan (class loan), or not
(class $\neg\texttt{loan}$), with binary features $employed$ (emp in short)
and $career\\_breaks$ (breaks in short):888These rules may correspond to the
behaviour of a machine learning classifier, where continuous_emp is a “latent
feature” of the model. We use them informally here (e.g. they may be
implications in classical logic or logic programming rules, with $\neg$ as
negation as failure).

$\displaystyle\forall
X[\texttt{emp}(X)\wedge\neg\texttt{breaks}(X)\rightarrow\texttt{continuous\\_emp}(X)]$
$\displaystyle\forall
X[\texttt{continuous\\_emp}(X)\rightarrow\texttt{loan}(X)]$
$\displaystyle\forall
X[\neg\texttt{continuous\\_emp}(X)\rightarrow\neg\texttt{loan}(X)]$

For an input (loan applicant) $x$ characterised by features emp and breaks,
classified as $\neg\texttt{loan}$ by $M$, in the first setting a contestation
may amount to objecting to the classification $\neg\texttt{loan}$, whereas in
the second setting it may amount to objecting to the instances for $X=x$ of
the top and bottom rules (i.e.
$\texttt{emp}(x)\wedge\neg\texttt{breaks}(x)\rightarrow\texttt{continuous\\_emp}(x)$
and $\neg\texttt{continuous\\_emp}(x)\rightarrow\neg\texttt{loan}(x)$). In the
third setting, the middle rule could be the object of contestation, without
reference to any specific input.

Explanations for contestability. In line with some of the literature, we
envisage that contestability needs to be supported by explanations for $M$
and/or its outputs for specific inputs. We thus assume that the ADS, in
addition to $M$, is equipped with an _explanation method_ $E$. Differently
from Yurrita, Balayn, and Gadiraju (2023), we believe that technical notions
of explanations from the literature in XAI may already prove useful. These
explanations may be _local_ (i.e. relating to a specific input, output pair)
or _global_ (i.e. relating to all input, output pairs). Different forms of
contestability may require and/or benefit from different forms of explanations
and thus explanation methods, including as follows:  1. a _feature-attribution
explanation_ method, e.g. as provided by model-agnostic LIME (Ribeiro, Singh,
and Guestrin, 2016) or SHAP (Lundberg and Lee, 2017) or by model-specific
(tailored to neural classifiers) LRP (Montavon et al., 2019) or DeepLIFT
(Shrikumar, Greenside, and Kundaje, 2017);  2. an _abductive explanation_
method, e.g. as in (Ignatiev, Narodytska, and Marques-Silva, 2019);  3. a
_counterfactual explanation_ method e.g. as first envisaged in (Wachter,
Mittelstadt, and Russell, 2017; Tolomei et al., 2017);  4. a _rule-based
explanation_ method e.g as in (Ribeiro, Singh, and Guestrin, 2018; Guidotti et
al., 2019; Ferreira et al., 2022) 5. a _mechanistic (local) explanation_
method, e.g. as provided in DAX (Albini et al., 2020), SpArX (Ayoobi, Potyka,
and Toni, 2023), and (Wang et al., 2023).  6. a _mechanistic (global)
explanation_ e.g as in (Ayoobi, Potyka, and Toni, 2023);  7. a _surrogate
model_ for interpreting $M$, e.g. (Kenny and Keane, 2019; Tan and Kotthaus,
2022; Potyka, Yin, and Toni, 2023);  8. $M$itself, if already an
_interpretable model_ (e.g. a decision tree, or a set of rules).

We stress that satisfying some desirable properties will make the explanation
methods more useful to the contestation. Notably, faithfulness requires that
the explanations reflect the true reasoning of the explained models (Lakkaraju
et al., 2019). This will guide the interactions in contestations towards the
correct direction, and allow for more effective redress. Another important
property is robustness (Jiang et al., 2024b). Non-robust explanation methods
could result in drastically different explanations for two users with similar
inputs (Leofante and Potyka, 2024), which may jeopardise their explanatory
function. In contrast, robust methods (Jiang et al., 2023; Leofante and
Lomuscio, 2023; Leofante, Botoeva, and Rajani, 2023; Jiang et al., 2024a)
improve the consistency and trustworthiness of explanations and may be better
suited to support contestability.

Note that, in setting (A), an explanation is not strictly necessary, e.g. a
loan applicant could contest not having received a loan no matter the reason.
However, if an explanation is present, the contestation can be richer, e.g.,
if a loan applicant finds out that the reason for the loan refusal is their
career-breaks, then they can contest the decision with evidence that they were
encouraged to take career-breaks by the employer for training purposes.

Note also that, in settings (B-C), explanations are essential “windows” over
the model, without which contestation cannot take place. Specifically, in the
second setting, explanations are crucial and, arguably, they need to reveal
the reasoning by the model for obtaining the output, e.g. as given by
(faithful) mechanistic explanations. Indeed, input-output explanations alone
(such as those computed by feature-attribution or counterfactual explanation
methods) cannot always provide grounds for contestation. For instance,
consider the model $M$ in Example 1 and applicant $x^{\prime}$ characterised
again by features emp and breaks, with the latter due to parental leaves: an
explanation including the (instances for $X\\!=\\!x^{\prime}$ of the) first
and third rule in the model empowers the loan applicant to contest the
decision by objecting to its bias against people who take parental leave; this
is more powerful than contesting based on the feature $\texttt{breaks}(x)$
alone in a feature-attribution explanation.

While in the first two settings (A-B) local explanations suffice, in the third
setting global explanations are needed. For illustration, a bank manager or
regulator, inspecting (interpretable) model $M$ in Example 1, may realise of a
possible bias underpinning the strict definition of $continuous\\_emp$ by the
first rule, disregarding the possibility of career breaks by applicants.

For further illustration, consider the case of a black-box text classifier
which, taken an input text, returns a classification, e.g. the sentiment of
the text, or that the text is about a certain topic. Feature-attribution
explanations for the classifications (such as those provided by LIME) can be
used to pinpoint words in the input text deemed responsible for the
classification, but not how the model determines the classification based on
those input words – being thus unsuitable for the second and third settings.

Interaction for contestability. The process of contestation needs to be
supported by suitable forms of interaction between the contested ADS and the
(human or ADS) contester. Interaction may be in the form of conversations in
natural language, e.g. using explanations generated by Large Language Models
(Bills et al., 2023), especially if the contester is a human. Alternatively,
they may be in structured format, using a formal agent communication language,
e.g. in the spirit of FIPA (Poslad, 2007).999http://www.fipa.org/ In Figure 1,
we indicate with $I$ the method used by contested ADS and contester to engage
in the interactions necessary for contestability.

The viewpoint of the contester. So far, we have taken the viewpoint of the ADS
being contested. The contester may be a human, as in all the existing works on
AI contestability, or, alternatively another ADS. In either case, for
contestations to be acceptable, they need to be accompanied by some _grounds_
for contestation, as in all earlier illustrations. Thus, we assume that the
contester is equipped with a _ground generator method_ $G$. For instance, in
setting (A) for Example 1, if a loan applicant finds out that the reason for
the loan refusal is their career-breaks and they decide to contest the
decision with evidence that they were encouraged to take career-breaks by the
employer for training purposes, this evidence forms the grounds for the
contestation.

Redress. While explanations empower contestation, its complete realisation
needs the ADS to have the ability to redress any issues (successfully) brought
about in the contestation. Thus, we assume that the ADS is also equipped with
a _redress method_ $R$. For illustration, in the loan application Example 1,
$R$ may amount to revising the first rule by allowing for exceptions to be
made when the breaks are due to training or parental leaves. As an additional
illustration for text classifiers, redress may result from the a post-hoc
reasoning process with the classifier’s outputs and external knowledge encoded
in argument schemes, as in (Carstens and Toni, 2017), or with the classifier’s
outputs and explanations therefor, e.g. as in (Freedman et al., 2024).

Note that we see contestation, and thus redress, as a post-training process,
rather than model debugging during training, such as via data augmentation
(e.g. as in (Teso and Kersting, 2019)) or regularisation (e.g. as in (Ross,
Hughes, and Doshi-Velez, 2017; Rieger et al., 2020; Shao et al., 2021; Zhang,
Williams, and Toni, 2024)). Thus, our view is that contestation is, in
general, different from explanation-based model debugging (e.g. as in (Ghai et
al., 2021) for tabular data, (Lertvittayakumjorn and Toni, 2021) for text
classification, or (Popordanoska, Kumar, and Teso, 2020) for image
classification). We, however, envisage that, in the case of models trained
from data, redress may at times involve fine-tuning or retraining steps in
which case the above methods, as well as methods for repairing AI models, e.g.
(Henriksen, Leofante, and Lomuscio, 2022; Almog and Kalech, 2023), could be
drawn upon. In the illustrative case of a black-box text classifier, an
example of contestation of the full model (the third setting) is offered by
FIND (Lertvittayakumjorn, Specia, and Toni, 2020): here, LRP (Montavon et al.,
2019) is used to associate output neurons in the feature extractor of a text
classifier with word clouds. Users can then contest the use of individual
neurons by disabling them and fine-tuning the model to no longer rely on them.
This allows, specifically, to decrease model bias and reliance on artifacts
(Lertvittayakumjorn, Specia, and Toni, 2020). Dreyer et al. (2024), similarly,
aim to expose visual concepts learned by an image classifier using concept
activation vectors (Kim et al., 2018) and fine-tune the model to mitigate the
use of specific concepts using gradient regularisation.

##  4 The Role of Computational Argumentation

In this position paper we argue that computational argumentation (CA) is
ideally suited to support AI contestability computationally. CA, broadly
understood as in (Atkinson et al., 2017; Baroni et al., 2018), is a branch of
Knowledge Representation & Reasoning which represents information in terms of
arguments and dialectical relations (of attack and, possibly, support) between
them. CA is equipped with semantics to reach some form of consensus regarding
conclusions to be drawn. As such, CA is ideally and uniquely suited to cover
all aspects of the abstract view in Figure 1 organically. We support this view
with reference to several lines of work in the CA literature, as follows.

CA for explanation. CA has been widely used for XAI (see (Vassiliades,
Bassiliades, and Patkos, 2021; Cyras et al., 2021) for recent overviews). It
can provide abstractions of several existing, widely used models, e.g. as in
(Potyka, Yin, and Toni, 2023; Ayoobi, Potyka, and Toni, 2023; Prakken and
others, 2020; Fan, 2018; Cyras et al., 2019b), and can itself directly serve
as the basis of models (Rago, Cocarascu, and Toni, 2018; Rago et al., 2020;
Cocarascu et al., 2020). Natural forms of explanations can be obtained from CA
abstractions, e.g. dispute trees (Fan and Toni, 2015b; Čyras et al., 2019a),
defence sets (Arioua, Tamani, and Croitoru, 2015) and attribution scores, e.g.
gradient-based argument attributions (Yin, Potyka, and Toni, 2023), Shapley-
based relation attributions (Amgoud, Ben-Naim, and Vesic, 2017; Yin, Potyka,
and Toni, 2024), amongst several.

CA for redress. CA provides methods for revising (Snaith and Reed, 2017;
Baumann and Brewka, 2015; Falappa, Kern-Isberner, and Simari, 2009) and
repairing knowledge bases (Ulbricht and Baumann, 2019). Much research has been
investigated in the context of forgetting (Berthold, Rapberger, and Ulbricht,
2023; Baumann, Gabbay, and Rodrigues, 2020) and enforcement (Rapberger and
Ulbricht, 2023; Baumann et al., 2021); researchers investigated the effect of
expansions (Prakken, 2023; Oikarinen and Woltran, 2011; Cayrol, de Saint-Cyr,
and Lagasquie-Schiex, 2010) and changes in the knowledge base (Doutre and
Mailly, 2018; Booth et al., 2013; Moguillansky et al., 2013; Niskanen, 2020).
In addition, incomplete argumentation frameworks (Baumeister et al., 2021;
Alfano et al., 2023) incorporate uncertainty which enables redress on a
conceptual level.

These generic lines of work in CA are useful starting points for supporting
redress of models which can be abstracted argumentatively. In addition,
reasoning with argumentation frameworks drawn from models is the basis for
forms of redress of some (machine learning) models, e.g. as in (Carstens and
Toni, 2017) and (Freedman et al., 2024) for natural language processing, and
of some (scheduling) models, as in (Cyras et al., 2019b).

CA for interaction. Interactions between agents, often modelled as dialogues,
have been shown to be effectively supported by various forms of CA, e.g.
argument schemes (Panisson, McBurney, and Bordini, 2021) or abstract
argumentation frameworks (de Tarlé, Bonzon, and Maudet, 2022), in a number of
settings. CA’s formal nature allows for principled desiderata of argumentative
agent protocols in these settings, as defined in (McBurney, Parsons, and
Wooldridge, 2002). These settings include: computational persuasion (Fan and
Toni, 2012; Hunter, 2018; Calegari, Riveret, and Sartor, 2021; Donadello et
al., 2022), framed as selecting the most effective arguments for changing the
mind of the other agents; information-seeking and inquiry (Black and Hunter,
2007; Fan and Toni, 2015a), where agents share information comprising
arguments which are private or public; and other areas such as the handling of
maliciousness amongst agents (Kontarinis and Toni, 2015). Approaches to multi-
agent argumentation such as these have been been shown to be useful in various
real-world applications, e.g. regulatory compliance (Raymond et al., 2022),
recommender systems (Briguez et al., 2014; Teze, Godo, and Simari, 2018; Rago
et al., 2020, 2021) and, more recently, interactive XAI (Madumal et al., 2019;
Calegari et al., 2022; Paulino-Passos and Toni, 2022). One such approach which
looks to have particular promise is that of Rago, Li, and Toni (2023), where
_argumentative exchanges_ frame interactive explanation between agents as a
conflict resolution problem, while accounting for humans’ cognitive biases.
These CA-based approaches to modelling interactions look to have promise for
human interaction with Large Language Models, as in (Freedman et al., 2024).

CA and the viewpoint of the contester. CA-based dialogue and interaction
methods, e.g. as in the aforementioned (Rago, Li, and Toni, 2023), already
have the potential to accommodate the contester’s viewpoint, leveraging on the
use of CA to provide abstractions for underpinning $G$. Indeed, e.g. in (Rago,
Li, and Toni, 2023), contested and contester are interchangeably seen as
argumentation frameworks.

##  5 Discussion and Future Work

In this position paper, we have proposed an abstract view of contestable AI
and advocated computational argumentation (CA) as ideally positioned to
support this view.

Our analysis is restricted, for simplicity and brevity, but could be naturally
extended to cover broader scenarios. For example, we have focused on two-party
scenarios where a single ADS is contested and another ADS or a human is doing
the contesting; however multi-party scenarios are also possible, e.g. when, in
addition to a bank customer, a regulator contests a financial institution.
Also, in line with the existing literature on contestability, we have assumed
that only ADSs can be contested, but it may be useful to consider the
possibility that humans may be contested by ADSs, e.g. in the spirit of
(Miller, 2023). In addition to broadening our view to these and other more
complex scenarios, future work will be needed to provide evidence of our
claim, by building forms of contestable AI that naturally use CA. This
involves connecting what we see are the main requirements of contestability
(explanations, grounds for contestation, redress, and interaction) in concrete
applications and scenarios. CA has already been used to solve several
individual challenges, e.g. see (Borg and Bex, 2020; Lawrence, Visser, and
Reed, 2023; Cyras et al., 2019b). However, considerable engineering work is
needed to combine solutions towards fully-fledged end-to-end contestable
systems. For instance, in order to support contestability with modern AI
systems, we envisage that considerable effort will be required to build large-
scale argumentation solvers.

Finally, we have advocated CA for contestability, but other KR techniques
could provide useful support for some contestability aspects. Specifically,
formal verification of AI models could be used to ensure the functional
correctness of ADSs, e.g. as in (Kouvaros et al., 2023). Similarly, rigorous
logic-based explainability techniques (Narodytska et al., 2019; Darwiche,
2020) could be used to ensure the faithfulness and trustworthiness of
explanations.

## Acknowledgments

This research was partially supported by ERC under the EU’s Horizon 2020
research and innovation programme (grant agreement No. 101020934), by J.P.
Morgan and the Royal Academy of Engineering under the Research Chairs and
Senior Research Fellowships scheme, by Imperial College through an Imperial
College Research Fellowship and by UKRI through the CDT in Safe and Trusted
Artificial Intelligence (Grant No. EP/S023356/1) and the CDT in AI for
Healthcare (Grant No. EP/S023283/1). Any views or opinions expressed herein
are solely those of the authors. We thank Francis Rhys Ward for pointing the
authors towards the literature on corrigibility, and Adam Gould, Avinash Kori
and Andria Stylianou for helpful discussions.

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