00footnotetext:

# Unified mechanism behind the even-parity ground state and neutron halo of
11Be ††thanks: This work was partly supported by the Strategic Priority
Research Program of Chinese Academy of Sciences (Grant No. XDB34000000), the
National Natural Science Foundation of China under Grant No. 12275111 and the
Fundamental Research Funds for the Central Universities (lzujbky-2021–sp41 and
lzujbky-2021–sp36).

Jing Geng1,2) Yi Fei Niu1,2,3) Wen Hui Long1,2,3<EMAIL_ADDRESS>1Frontier
Science Center for Rare isotope, Lanzhou University, Lanzhou 730000, China
2School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000,
China
3Joint Department for Nuclear Physics, Lanzhou University and Institute of
Modern Physics, CAS, Lanzhou 730000, China

###### Abstract

Using the axially deformed relativistic Hartree-Fock-Bogoliubov (D-RHFB)
model, we explore the mechanism behind the parity inversion and halo
occurrence in 11Be, which are well reproduced by the RHF Lagrangian PKA1. It
is illustrated that evidently enhanced deformation effects by the
$\pi$-pseudo-vector and $\rho$-tensor couplings in PKA1 are crucial for
correctly describing both even-parity ground state (GS) and neutron halo of
11Be. Coupling with the deformation, the intrude $1d_{5/2}$ component largely
enhances the couplings between the even-parity orbit $1/2_{2}^{+}$ and the
nuclear core to promise the even-parity GS, whereas the $2s_{1/2}$ component
therein dominates the halo formation in 11Be. Moreover, the deformed halo in
11Be is found to be stabilized by the attractive inherent correlations between
the $1d_{5/2}$ and $2s_{1/2}$ components of the halo orbit $1/2_{2}^{+}$,
instead of pairing correlations, which paves a new way to understand the halo
pictures in deformed unstable nuclei.

###### keywords:

parity inversion, deformation effect, halo structure

###### pacs:

2

1.30.Fe, 21.60.Jz 00footnotetext: $\scriptstyle\copyright$2013 Chinese
Physical Society and the Institute of High Energy Physics of the Chinese
Academy of Sciences and the Institute of Modern Physics of the Chinese Academy
of Sciences and IOP Publishing Ltd

##  1 Introduction

Unstable nuclei far from the stability line in nuclear chart exhibit plenty of
novel phenomena, for instance the emergence of new magic shells and the
disappearance of the conventional ones [1, 2, 3, 4, 5, 6, 7], the island of
inversion [8, 9, 10], and the dilute matter distributions – halo structures
[11, 12, 13], which are attracting the interests of many researchers in the
fields of nuclear physics and related disciplines. While enriching the
knowledge of nuclear physics, the emergence of novelties in unstable nuclei
has also subverted our traditional understanding of atomic nuclei.

Beryllium (Be) isotopes are perfect candidates for exploring the novelty
appearing in unstable nuclei. The novel parity inversion of 11Be is one of the
significant issues. As deduced from the conventional shell model, the ground
state (GS) of 11Be must have odd-parity because the odd neutron should
populate the lower $p_{1/2}$ orbit. However, an early experiment with the
reaction 11B($n,p$)11Be indicated that $J^{\pi}=1/2^{-}$ is not the GS [14],
which was further proofed to be even-parity $1/2^{+}$ by later experiments
[15, 16, 17]. At a rather early stage, it was suggested that the even- and
odd-parity states $1/2^{\pm}$ of 11Be should be very close [18]. After a long
time of silence, there were a number of attempts to interpret the GS parity of
11Be under the particle-core coupling picture [19, 20, 21, 22, 23, 24],
indicating that the quadruple core excitation and pairing blocking can be
essential [20, 21]. Besides, the deformation [25, 26, 27], the continuum [28]
and effects beyond mean field [29, 30] might also be also significant for
correctly giving the even-parity GS of 11Be.

Despite the ingenious GS, special interests were also devoted to the halo
occurrence in 11Be. Experimentally, 11Be has been confirmed to be of one-
neutron halo structure [31, 32, 33]. Theoretically, the energy density
functional (EDF) theory presents dilute matter distribution for 11Be, while it
fails to reproduce the even-parity GS [26, 27, 34]. Based on the variational
shell model, it was indicated that the weak couplings between the core and odd
neutron can be responsible for the one-neutron halo in 11Be [19]. Recently,
the halo effects of 11Be on the one-neutron knockout reactions were studied
using the halo effective filed theory [35, 36]. As a consensus, nuclear halo
structures are contributed by neutrons or protons populating the weakly bound
low-$l$ states, and the neighboring high-$l$ orbits do not contribute to the
halo but stabilize the dilute nucleons via the pairing correlations [37].

However, up to now both novelties in 11Be, i.e., the parity inversion and
neutron halo, still cannot be interpreted in a unified framework, and the halo
occurrence may essentially rely on the even-parity GS. The EDF theory, such as
that adopted by the widely used Skyrme Hartree-Fock models, cannot uniformly
describe the even-parity GS and halo occurrence in 11Be, as aforementioned.
The relativistic mean field (RMF) theory [38, 39, 40, 41, 37, 42], that stands
on the microscopic meson exchange picture of the nuclear force, succeeds in
describing the halo structures of spherical and deformed nuclei [37, 43], but
fails to reproduce the parity inversion of 11Be. Implemented with the Fock
diagrams, the relativistic Hartree-Fock (RHF) theory [44, 45, 46, 47, 48] and
an extension — the relativistic Hartree-Fock-Bogoliubov (RHFB) theory [49, 50]
have achieved many successes in describing various nuclear phenomena [51],
including those of unstable nuclei [52, 53, 54, 55] and superheavy ones [56].

In particular, the degrees of freedom associated with the $\pi$-pseudo-vector
($\pi$-PV) and $\rho$-tensor ($\rho$-T) couplings, important ingredients of
the effective nuclear force, play significant roles in providing self-
consistent descriptions of nuclear shell evolution [57, 58, 59], the pseudo-
spin symmetry restoration [48, 60] and the emergences of new magicity [61,
62]. Recently, considering nuclear deformation that may be significant for
11Be, the axially deformed RHF [63] and RHFB [64] models, respectively called
the D-RHF and D-RHFB models, have been established by utilizing an expansion
of the spherical Dirac Woods-Saxon (DWS) base [65]. It was revealed that the
$\pi$-PV and $\rho$-T couplings, which contribute mainly via the Fock
diagrams, evidently enhance the deformation effects of nuclei [63, 64].

In particular, the D-RHFB model has the advantage of a unified treatment of
spin-orbit coupling, tensor force, deformation, pairing correlations and the
continuum [64], the significant mechanisms which are essential for unstable
nuclei. This encourages us to explore the underlying mechanism related to the
even-parity GS and halo occurrence in 11Be. The paper is organized as follows.
In Sec. 2, the decomposition of the canonical single-particle energy and its
physical significance are introduced. Afterwards, the parity inversion and the
halo phenomenon are discussed in Sec. 3. Finally, we give a summary in Sec. 4.

##  2 Canonical s.p. energy and the decomposition

To understand the microscopic picture of 11Be, the language of the basis
expansion is rather helpful. In the D-RHFB model, the spherical DWS base is
used to expand the quasi-particle wave functions $\psi^{V}$ and $\psi^{U}$
[64],

$\psi_{\nu\pi m}^{V}=\sum_{a}C_{\nu\pi m,a}^{V}\psi_{am},\hskip
10.00002pt\psi_{\nu\pi m}^{U}=\sum_{a}C_{\nu\pi m,a}^{U}\psi_{am},$ (1)

where $m$ and $\pi$ denote the angular momentum projection and parity, $\nu$
stands for the index of the orbits in the $\pi m$ block. Imposing axial
symmetry and reflection symmetry, $m$ and $\pi$ remain as good quantum
numbers. For a spherical DWS base, $\psi_{am}$ represents the wave function
and the index $a$ = $(n\kappa)$ carries the principle number $n$ and
$\kappa$-quantity, namely $\kappa=\pm(j+1/2)$ with $j=l\mp 1/2$, where $l$ is
the orbital angular momentum. In the following, the index $i=(\nu\pi m)$ is
used to denote the deformed orbits, while the index $a$ specifies the states
in the spherical DWS base. It is worth noting that the DWS base has the
advantage of providing appropriate asymptotic behaviors for the wave functions
due to the nature of the Woods-Saxon type potential [66]. For unstable nuclei,
e.g. 11Be with a halo structure, this advantage is essential to provide a
reliable description.

For the convenience of analysis, the canonical transformation from the quasi-
particle space is performed in general to get canonical s.p. states, by
diagonalizing the density matrix. In practice, the canonical s.p. space is
taken to be similar to the Hartree-Fock space, for both of which the density
matrices are diagonal. Utilizing the orthogonality of the DWS base, there is a
straightforward way to build the density matrix, which reads as,

$\rho_{aa^{\prime}}^{\pi m}=\sum_{\nu}C_{\nu\pi m,a}^{V}C_{\nu\pi
m,a^{\prime}}^{V}.$ (2)

Notice that for the contributions from the odd-nucleon in the blocked orbit,
the expansion coefficient $C^{V}$ shall be replaced by the $C^{U}$ one in Eq.
(1). The diagonalization of the above density matrix $\rho_{aa^{\prime}}^{\pi
m}$ gives the eigenvalues $v_{i}^{2}$, namely the occupation probability of
orbit $i=m_{\nu}^{\pi}$, and the eigenvectors $\widehat{D}_{\nu}^{\pi m}$, the
set of the expansion coefficients $D_{i,n\kappa}$ of the canonical wave
functions in the spherical DWS base.

Thus, in terms of the spherical DWS base, the canonical s.p. energy
$\varepsilon_{i}$ and wave function $\psi_{i}$ for orbit $i$ can be expressed
as,

$\varepsilon_{i}=\sum_{aa^{\prime}}D_{i,a}h_{aa^{\prime}}^{\pi
m}D_{i,a^{\prime}},\hskip 10.00002pt\psi_{\nu\pi
m}=\sum_{n\kappa}D_{i,n\kappa}\psi_{n\kappa m},$ (3)

where $h_{aa^{\prime}}$ is the s.p. Hamiltonian in the spherical DWS base, the
index $a$ is for spherical DWS basis states. Following the deformation of a
nucleus, orbits which originally had the same spherical $n\kappa$ quantities
cannot degenerate any more, leading to a mixture of spherical
$n\kappa$-components in deformed s.p. orbits. As seen from Eq. (3), the
couplings between various spherical DWS components $a$ and $a^{\prime}$ give
the canonical s.p. energy. Thus, after summing the principle numbers $n$ and
$n^{\prime}$, thsy can be decomposed into the diagonal terms
$E_{d,i}^{\kappa}$ and off-diagonal ones $E_{c,i}^{\kappa\kappa^{\prime}}$ as,

$\varepsilon_{i}=\sum_{\kappa\kappa^{\prime}}E_{i}^{\kappa\kappa^{\prime}}=\sum_{\kappa}p_{i\kappa}^{2}E_{d,i}^{\kappa}+\sum_{\kappa\neq\kappa^{\prime}}p_{i\kappa}p_{i\kappa^{\prime}}E_{c,i}^{\kappa\kappa^{\prime}},$
(4)

where $p_{i\kappa}^{2}$ describes the sum proportion of the $\kappa$-component
of the orbit $m_{\nu}^{\pi}$, and $E_{i}^{\kappa\kappa^{\prime}}$ corresponds
to the coupling effects between the spherical $\kappa$ and $\kappa^{\prime}$
components,

$p_{ik}=\sum_{n}D_{i,n\kappa}^{2},\hskip
10.00002ptE_{i}^{\kappa\kappa^{\prime}}=\sum_{nn^{\prime}}D_{i,n\kappa}h_{aa^{\prime}}^{\pi
m}D_{i,n^{\prime}\kappa^{\prime}}.$ (5)

Using the above conventions, the diagonal term $E_{d,i}^{\kappa}$ and the off-
diagonal one $E_{c,i}^{\kappa\kappa^{\prime}}$ can be defined as,

$E_{d,i}^{\kappa}\equiv E_{i}^{\kappa\kappa}/p_{i\kappa}^{2},\hskip
10.00002ptE_{c,i}^{\kappa\kappa^{\prime}}\equiv
E_{i}^{\kappa\kappa^{\prime}}/\big{(}p_{i\kappa}p_{i\kappa^{\prime}}\big{)}.$
(6)

Further considering the sum over $\kappa$ components, the diagonal
contribution $E_{d,i}$ and the off-diagonal one $E_{c,i}$ to the canonical
s.p. energy $\varepsilon_{i}$ read as,

$E_{d,i}=\sum_{\kappa}E_{i}^{\kappa\kappa},\hskip
10.00002ptE_{c,i}=\sum_{\kappa\neq\kappa^{\prime}}E_{i}^{\kappa\kappa^{\prime}}.$
(7)

As aforementioned, more spreading of the proportion $p_{i\kappa}^{2}$ over
different $\kappa$-components is in general due to enhanced deformation
effects. Thus using the language of the spherical DWS basis expansion, the
$E_{d,i}^{\kappa}$-term can be taken as the centroid energy of the
$\kappa$-component of the deformed orbit $i$, or taken as the single-particle-
like energy of the spherical $\kappa$-fragment of orbit $i$. In contrast, the
$E_{c,i}^{\kappa\kappa^{\prime}}$-term describes the inherent correlation
between the $\kappa$\- and $\kappa^{\prime}$-components, which in fact
reflects the shape evolution of orbit $i$. Following deformation, it is not
hard to deduce that orbit $i$ becomes more bound with the negative $E_{c,i}$
value, and vice versa.

##  3 Unified mechanism for parity inversion and neutron halo in 11Be

Aiming at a unified mechanism behind the even-parity GS and neutron halo of
11Be, we focus on the D-RHFB calculations with PKA1 [48], one of the most
complete RHF Lagrangians which contains the $\pi$-PV and $\rho$-T couplings.
For comparison, the RHF Lagrangians PKO3 [57] containing $\pi$-PV coupling and
PKO2 [57], and the RMF one DD-LZ1 [67] are considered as well. In the pairing
channel, the finite-range Gogny force D1S [68] is utilized as the pairing
force, due to the advantage of a finite range, i.e., natural convergence with
the configuration space. The details of the calculations can be found in Ref.
[64].

To find the global minimum for an odd-$A$ nucleus using the D-RHFB model, the
blocking effects shall be treated carefully [69]. Similar to a popular
treatment in the RMF calculations, the equal filling approximation is adopted
[70, 71]. Within the Bogoliubov scheme, we attempted to block the first quasi-
particle level for various $\pi m$-block to determine the ground state.
According to the mapping relation between the canonical single-particle and
the Bogoliubov quasi-particle states, the blocked quasi-particle orbits are
mapped in general with the s.p. ones closest to the Fermi level. For 11Be, we
tried to block the states $m^{\pi}=$ $1/2^{-}$, $1/2^{+}$, $3/2^{+}$ and
$5/2^{+}$, and only show the results of $m^{\pi}=1/2^{-}$ and $1/2^{+}$, from
which the ground state can be determined by selected models.

###  3.1 Parity inversion of 11Be

We first performed the shape constrained calculations for 11Be using PKA1,
PKO2, PKO3 and DD-LZ1. Figures 3.1 (a) and 3.1 (b) show the binding energy
$E_{B}$ (MeV) of 11Be as a function of the quadruple deformation $\beta$,
which corresponds to the blocking of even- and odd-parity states
$m^{\pi}=1/2^{\pm}$, respectively. It is shown that only PKA1 results in the
same even-parity GS as the experiments [14, 15, 16]. Moreover, a rather large
prolate deformation ($\beta\simeq 1.2$) is obtained by PKA1 for the GS, while
the others give an odd-parity GS with spherical or near spherical shapes.
Compared with the experimental value, which reads as $-$65.478 MeV [72], the
binding energy given by PKA1 is overestimated by roughly $3$ MeV. In fact, the
calculations of nearby nuclei show that the binding energies for light nuclei
are overestimated by PKA1 in general.

(color online) Binding energies $E_{B}$ (MeV) as functions of the quadruple
deformation $\beta$ for 11Be with even [plot (a)] and odd [plot (b)] parity.
The results are calculated by PKA1, PKO2, PKO3 and DD-LZ1. Referring to
$\beta=0$, plots (c) and (d) show the PKA1 contributions of $E_{B}$
respectively with even and odd parity, including the total, the $\pi$-PV and
$\rho$-T ones, and the sum from the other channels.

As an example, Table 3.1 shows the calculated binding energies $E_{B}$ of
10,11,12Be, as well as the one-neutron separation energy $S_{n}$ for 11Be and
the two-neutron one $S_{2n}$ for 12Be, in comparison with the experimental
data [73]. It can be found that the systematics of the binding energies, here
from 10Be to 12Be, are different between PKA1 and the other selected models.
Specifically, all the $E_{B}$ values given by PKA1 are larger than the
experimental ones by $1.4\sim 3.5$ MeV. While the binding energy of 10Be is
underestimated by all other selected Lagrangians, the one of 12Be is
overestimated by different extents. However, for the $S_{n}$ and $S_{2n}$
values, all selected models provide a similar accurate result as the
experimental values. Thus, deviations in the binding energy of a few MeV do
not do not influence the conclusions made in this work substantially.

Binding energies $E_{B}$ (MeV) of 10,11,12Be, along with the one-neutron
separation energy (MeV) $S_{n}$ for 11Be and two-neutron one $S_{2n}$ for 12Be
calculated by PKA1, PKO2, PKO3 and DD-ME2, as compared to the experimental
(Exp.) data [73].

| $E_{B}$ | $S_{n}$ | $S_{2n}$
---|---|---|---
|  10Be |  11Be |  12Be |  11Be |  12Be
Exp. |  $-$64.98 |  $-$65.48 |  $-$68.65 | 0.50 | 3.67
PKA1 |  $-$66.40 |  $-$68.50 |  $-$72.21 | 2.10 | 5.81
PKO3 |  $-$63.63 |  $-$65.82 |  $-$70.13 | 2.19 | 6.50
PKO2 |  $-$63.50 |  $-$65.73 |  $-$69.00 | 2.23 | 5.50
DD-LZ1 |  $-$63.61 |  $-$66.09 |  $-$69.67 | 2.47 | 6.06

In order to clarify the mechanism behind the GS of 11Be, Figs. 3.1 (c) and 3.1
(d) show the contributions of $E_{B}$ referring to $\beta=0$ given by PKA1 for
even- and odd-parity cases, respectively, namely the total, the $\pi$-PV one
that is enclosed in both PKA1 and PKO3, the $\rho$-T one which is uniquely
considered by PKA1, and the sum from other channels, which corresponds to all
degrees of freedom adopted in PKO2 and DD-LZ1. It is interesting to see that
both $\pi$-PV and $\rho$-T contributions are evidently enhanced following the
enlarged deformation. Compared with the odd-parity one [plot (d)], more
remarkable enhancements are presented by the $\pi$-PV and $\rho$-T couplings
for the even-parity case [plot (c)], which are seen to be crucial for
reproducing the even-parity GS of 11Be.

In fact, referring to other selected models that fail to reproduce the GS of
11Be, it is also helpful to understand the role played by the $\pi$-PV and
$\rho$-T couplings. In Figs. 3.1 (a) and 3.1 (b), both PKO2 and DD-LZ1, which
do not contain either $\pi$-PV or $\rho$-T couplings, present the odd-parity
GS with a fairly stable spherical shape and even-parity local minima with a
rather soft prolate shape. For PKO3, which contains the $\pi$-PV coupling, an
odd-parity GS with a rather soft near-spherical shape and an even-parity local
minimum with a solid prolate shape are obtained, and the energy difference
between the even- and odd-parity minima are significantly reduced, as compared
with PKO2 and DD-LZ1. Thus, combined with the PKA1 results, it can be
concluded that both $\pi$-PV and $\rho$-T couplings, which evidently enhance
the deformation effects, play a key role in determining the even-parity GS of
11Be.

(color online) Expansion proportions of neutron orbit $1/2_{2}^{+}$ with
respect to the deformation $\beta$ [plot(a)], and the canonical s.p. energies
$E_{i}$ (MeV) and occupation probabilities (in thick bars) of spherical (Sph.)
11Be and the GS [plot (b)]. The results are extracted from the PKA1
calculations by blocking the state $m^{\pi}=1/2^{+}$, and the
$E_{d,i}^{\kappa}$ values (MeV) and expansion proportions (in thick bars) of
the neutron orbits $1/2_{2}^{\pm}$ of the GS are also shown in plot (b).

To understand why the even-parity GS of 11Be can be correctly reproduced at
large deformation in terms of the spherical DWS base, the expansion
proportions of the even-parity orbit $1/2_{2}^{+}$ occupied by the odd neutron
are shown as functions of the deformation $\beta$ in Fig. 3.1 (a). Figure 3.1
(b) shows the canonical s.p. energies $E_{i}$ (MeV) with occupation
probabilities (in thick bars) for spherical 11Be and the GS, and the
$E_{d,i}^{\kappa}$ values deduced from Eq. (6) with the expansion proportions
(in thick bars) of the dominant $\kappa$ fragments of the orbits
$1/2_{2}^{\pm}$ for the GS. The results are extracted from the PKA1
calculations by blocking the state $m^{\pi}=1/2^{+}$. As shown in Fig. 3.1
(a), the $1d_{5/2}$ component in the orbit $1/2_{2}^{+}$ increases gradually
with a simultaneously reduced $2s_{1/2}$ component when 11Be becomes prolately
deformed. Meanwhile, the inversion of the even- and odd-parity orbits
$1/2_{2}^{\pm}$ from the spherical 11Be to the GS is found in Fig. 3.1 (b) and
is consistent with an even-parity GS.

Moreover, as deduced from the high-lying $\kappa$ fragments of the
$1/2_{2}^{+}$ orbit in Fig. 3.1 (b), the $E_{c,i}$ value given by Eq. (7)
shall be negatively and large, indicating strong attractive inherent
correlations between the $\kappa$ fragments of the $1/2_{2}^{+}$ orbit. In
contrast, the $\kappa$ fragments of the odd-parity orbit $1/2_{2}^{-}$ are
deeply bound, and the deduced large, positively $E_{c,i}$ value reveals strong
repulsive inherent correlations between the $\kappa$ fragments of the
$1/2_{2}^{-}$ orbit. Thus, from the opposite effects carried by the
correlation terms $E_{c,i}$, it is not hard to understand the inversion of the
even- and odd-parity orbits $1/2_{2}^{\pm}$ from spherical 11Be to the
deformed GS, in which the enhancements due to the $\rho$-T and $\pi$-PV
couplings are very important as deduced from the nature of the channels [64].

In fact, the PKA1 results are consistent with the indication by Bohr and
Mottelson that for a $p$-shell nucleus 11Be the even-parity GS can be a
consequence of a strong preference for the orbit $[220,1/2]$ with prolate
deformation [74]. Supplementary, it should be emphasized that the expansion
proportions in the theoretical calculation are different from the
spectroscopic factor, which is a basic quantity characterizing the single-
particle nature of nuclear excitations in both experiments and theories. In
this work, the calculations of 11Be are performed by restrictions on the mean-
field level. Thus, one cannot take the expansion proportions in Fig. 3.1 (a)
as some equivalent concept of the spectroscopic factor, or even compare it
directly with the measured data of the transfer reaction (see Ref. [33] and
references there in).

(color online) Interaction matrix elements (MeV) between a selected neutron
($\nu$) and the core orbits of 11Be. The left and right plots correspond to
the couplings with proton ($\pi$) and neutron ($\nu$) cores, respectively,
including the total (upper plots) and the sum of the $\rho$-T and $\pi$-PV
couplings (lower plots). The results are obtained from spherical RHFB
calculations with PKA1 by blocking neutron $s$ state.

To enhance the understanding of the largely deformed GS and the role played by
the $\pi$-PV and $\rho$-T couplings, Fig. 3.1 shows the interaction matrix
elements (IMEs) between selected neutron states and core orbits of 11Be,
including the total (upper plots) and the sum of the $\rho$-T and $\pi$-PV
couplings (lower plots). The results are given by the spherical RHFB
calculations [49] with PKA1 by blocking the even-parity state $s_{1/2}$. In
fact, the selected even-parity states $\nu 1s_{1/2}$, $\nu 2s_{1/2}$ and $\nu
1d_{5/2}$ are the dominant orbital components of the deformed orbit
$1/2_{2}^{+}$, as seen from Fig. 3.1 (a).

As the general feature of the IMEs in Fig. 3.1, the proton $(\pi)$ core (left
plots) shows stronger couplings with selected neutron orbits than the neutron
$(\nu)$ core (right plots). This can be explained by the effects of the Fock
terms [75]. Specifically, the Fock terms arising from the isoscalar $\sigma$\-
and $\omega$-couplings do not contribute to neutron-proton ($np$) interactions
but present net repulsive neutron-neutron ($nn$) and proton-proton ($pp$)
ones, thus partly canceling their net attractive Hartree terms in the $nn$ and
$pp$ channels. In contrast, as shown in the lower plots of Fig. 3.1, the
isovector $\rho$-T and $\pi$-PV couplings present rather strong attractive
contributions mainly through the Fock terms, particularly for the $np$
interactions (left-lower plot). Compared with other RHF and RMF models, the
$np$ interactions are evidently enhanced due to the strong $\rho$-T coupling
in PKA1 [52, 60].

Combined with Fig. 3.1, the IMEs in Fig. 3.1 can help us understand the
evident deformation effects given by PKA1 in Fig. 3.1. As seen from Fig. 3.1,
the neutron state $\nu 2s_{1/2}$, namely the $1/2_{2}^{+}$ orbit at $\beta=0$,
shows rather weak couplings with both neutron and proton cores due to the
nodal differences, and thus a loosely bound $2s_{1/2}$ orbit is found in Fig.
3.1 (b), which indicates an unfavorable even-parity configuration for
spherical 11Be. Compared with $\nu 2s_{1/2}$, the $\nu 1d_{5/2}$ state shows
much stronger couplings with the cores. Combined with the gradually enlarged
$1d_{5/2}$ proportion and the simultaneously reduced $2s_{1/2}$ one in Fig.
3.1 (a), the couplings between the $1/2_{2}^{+}$ orbit and the cores are then
notably enhanced following the enlarged deformation, which eventually leads to
a fairly deep bound $1/2_{2}^{+}$ orbit in the GS of 11Be. In addition, as
shown in Fig. 3.1 (a), the $1s_{1/2}$ component is also enfolded slightly into
the orbit $1/2_{2}^{+}$ in the GS, which also plays some role in deepening the
orbit $1/2_{2}^{+}$, due to its rather strong couplings with the core orbits
in Fig. 3.1.

In contrast, for odd-parity 11Be, which is not shown in details, the mixing of
the $1p_{1/2}$\- and $1p_{3/2}$-components also happens in the populated odd-
parity orbits $1/2_{1}^{-}$ and $1/2_{2}^{-}$ following an increase in the
deformation. However, the mixing of high-lying odd-parity components such as
$2p$ and $1f$ are hindered by the $sd$-shell between the $p$\- and
$pf$-shells. This indicates that the sums of the $1p_{3/2}$ and $1p_{1/2}$
proportions in these orbits will not change much, which is confirmed by the
D-RHFB calculations. As a result, the enhancement of the binding energy from
spherical 11Be to the prolate minimum with odd parity is not so remarkable;
see Fig. 3.1 (b). In contrast, the nearby $1d_{5/2}$ component can be easily
enfolded into the even-parity orbit $1/2_{2}^{+}$, following the deformation
from spherical 11Be to the GS. Thus, due to strong couplings of the orbitals
$1d_{5/2}$ and $1s_{1/2}$ with the core orbits, more evident enhancement
following the deformation is found in even-parity 11Be than that in odd-parity
case, which is crucial for correctly determining the even-parity GS of 11Be.

###  3.2 The halo structure of 11Be

(color online) Neutron ($N$) and proton ($P$) densities (fm-3) of 11Be given
by PKA1, including logarithm ones (left plot) along the $z$-axis and contour
maps (right plot) in which the dashed ellipsoid is defined by the deformation
$\beta$ and radius $r$.

On the basis of correctly reproducing the parity of the GS, we further explore
the halo phenomenon in 11Be. Figure 3.2 shows the neutron ($N$) and proton
($P$) density profiles of the GS of 11Be given by PKA1, including the
logarithm ones along the symmetric $z$-axis (left plot) and the contour maps
(right plot), in which the dashed ellipsoid defined by the deformation $\beta$
and radius $r$ is shown as a reference for the size and shape of 11Be. As
shown in Fig. 3.2, the neutrons in 11Be distributed rather extensively along
the $z$-axis. Different from the decoupling picture [43, 76], the neutron
dispersion in 11Be shows a consistent shape with the whole nucleus. Moreover,
it can be seen from the left plot of Fig. 3.2 that the neutron dispersion is
almost fully a results of the even-parity orbit $1/2_{2}^{+}$ (in pattern of
horizontal lines), as referred to in the proton density profiles. It is worth
noting that such results are consistent with the experimental indication of a
one-neutron halo [31, 32, 33, 11].

However, the orbit $1/2_{2}^{+}$ in Fig. 3.1 (b) is somewhat deeply bound for
the GS of 11Be, which seems to conflict with the extensive neutron
distributions in Fig. 3.2. Combined with the results in Fig. 3.1, a picture
becomes apparent. As shown in Fig. 3.1 (b), the neutron orbit $2s_{1/2}$ is
loosely bound for spherical 11Be because of the nodal difference, and for the
same reason, the $s_{1/2}$ fragment (mainly the $2s_{1/2}$ one) remains
loosely bound in the largely deformed GS. Due to the vanishing centrifugal
barrier, this leads to the halo occurrence in the GS of 11Be. In contrast, as
hindered by the centrifugal barrier, the neighboring $d_{5/2}$ fragment does
not contribute to the halo, but plays an essential role in stabilizing the
halo via the attractive inherent correlations with the $s_{1/2}$ fragment, as
indicated by the negative $E_{c,i}$ value of the halo orbit $1/2_{2}^{+}$
aforementioned.

In fact, the enhancements due to the $\rho$-T and $\pi$-PV couplings are not
only crucial for the parity inversion but also significant for the formation
and stabilization of the halo in 11Be. Following the deformation from
spherical 11Be to the GS, the spherical fragment $2s_{1/2}$ remains weakly
bound to form the halo, and the intruding fragment $1d_{5/2}$ in the orbit
$1/2_{2}^{+}$ enhances the couplings with the core orbits of 11Be, resulting
in an even-parity GS, in which the $\rho$-T and $\pi$-PV couplings play an
important role. This also stabilizes the halo via an attractive inherent
correlation with the $2s_{1/2}$ fragment. Moreover, due to the nodal
difference between the $2s_{1/2}$ and $1d_{5/2}$ components, such an inherent
correlation is not strong enough to hinder halo formation. It is worth noting
that this stability mechanism is different from that of halos in spherical
unstable nuclei, which are stabilized in general via enhanced pairing
correlations by neighboring high-$l$ orbits [37].

##  4 Summary

In summary, employing the D-RHFB model, we explored the mechanism behind the
parity inversion and neutron halo of 11Be, which is uniformly interpreted by
the RHF Lagrangian PKA1. Using the language of the spherical DWS basis
expansion, it is illustrated that evidently enhanced deformation effects by
the $\rho$-T and $\pi$-PV couplings are crucial for describing correctly the
even-parity ground state and the neutron halo of 11Be correctly. Moreover, a
new microscopic picture of the halo formation and stability is revealed in
this work, which paves a way to understanding the halo in deformed unstable
nuclei. For light nuclei, the effects of the rotation and vibration
corrections might be important. In the future, further implementations of the
D-RHFB model are expected by considering the angular momentum projection, the
number projection, and other beyond mean field effects.

##  5 Acknowledgments

_The authors want to thank Prof. S.-G. Zhou, Prof. F. Q. Chen and Dr. X. X.
Sun for fruitful discussions and the Supercomputing Center of Lanzhou
University and the Southern Nuclear Science Computing Center for the computing
resources provided._

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# Data Validation

Mark P.J. van der Loo111Corresponding author<EMAIL_ADDRESS>and Edwin de Jonge
Statistics Netherlands

(This is the submitted version of the following book chapter: “Van der Loo,
M.P.J. and De Jonge, E. (2020) _Data Validation_. In Wiley StatsRef:
Statistics Reference Online” which has been published in final form in
https://onlinelibrary.wiley.com/doi/10.1002/9781118445112.stat08255. )

###### Abstract

Data validation is the activity where one decides whether or not a particular
data set is fit for a given purpose. Formalizing the requirements that drive
this decision process allows for unambiguous communication of the
requirements, automation of the decision process, and opens up ways to
maintain and investigate the decision process itself. The purpose of this
article is to formalize the definition of data validation and to demonstrate
some of the properties that can be derived from this definition. In
particular, it is shown how a formal view of the concept permits a
classification of data quality requirements, allowing them to be ordered in
increasing levels of complexity. Some subtleties arising from combining
possibly many such requirements are pointed out as well.

Informally, data validation is the activity where one decides whether or not a
particular data set is fit for a given purpose. The decision is based on
testing observed data against prior expectations that a plausible dataset is
assumed to satisfy. Examples of prior expectations range widely. They include
natural limits on variables (weight cannot be negative), restrictions on
combinations of multiple variables (a man cannot be pregnant), combinations of
multiple entities (a mother cannot be younger than her child) and combinations
of multiple data sources (import value of country A from country B must equal
the export value of country B to country A). Besides the strict logical
constraints mentioned in the examples, there are often ‘softer’ constraints
based on human experience. For example, one may not expect a certain economic
sector to grow more than 5% in a quarter. Here, the 5% limit does not
represent a physical impossibility but rather a limit based on past
experience. Since one must decide in the end whether a data set is usable for
its intended purpose, we treat such assessments on equal footing.

The purpose of this paper is to formalize the definition of data validation
and to demonstrate some of the properties that can be derived from this
definition. In particular, it is shown how a formal view of the concept
permits a classification of data validation rules (assertions), allowing them
to be ordered in increasing levels of ‘complexity’. Here, the term
‘complexity’ refers to the amount of different types of information necessary
to evaluate a validation rule. A formal definition also permits development of
tools for automated validation and automated reasoning about data validation
(Zio et al., 2015; Van der Loo and De Jonge, 2018, 2019). Finally, some
subtleties arising from combining validation rules are pointed out.

##  1 Formal definition of data validation

Intuitively, a validation activity classifies a dataset as acceptable or not
acceptable. A straightforward formalization is to define it as a function from
the collection of data sets that could have been observed, to
$\\{\textsf{True}{},\textsf{False}{}\\}$. One only needs to be careful in
defining the ‘collection of data sets’, to avoid a ‘set of all sets’ which
recursively holds itself. To avoid such paradoxes, a data set is defined as a
set of key-value pairs, where the keys come from a finite set, and the values
from some domain.

######  Definition 1.

A _data point_ is a pair $(k,x)\in K\times D$, where $k$ is a _key_ , selected
from a finite set $K$ and $x$ is a _value_ from a _domain_ $D$.

In applications the identifier $k$ makes the value interpretable as the
property of a real-world entity or event. The domain $D$ is the set of all
possible values that $x$ can take, and it therefore depends on the
circumstances in which the data is obtained.

As an example, consider an administrative system holding age and job status of
persons. It is assumed that ‘job’ takes values in
$\\{\texttt{"employed"},\texttt{"unemployed"}\\}$ and that ‘age’ is an
integer. However, if the data entry system performs no validation on entered
data, numbers may end up in the job field, and job values may end up in the
age field. In an example where the database contains data on two persons
identified as $1$ and $2$, this gives

$\displaystyle\begin{split}K&=\\{1,2\\}\times\\{\texttt{"age"},\texttt{"job"}\\}\\\
D&=\mathbb{N}\cup\\{\texttt{"employed"},\texttt{"unemployed"}\\}.\end{split}$
(1)

This definition allows for the occurrence of missing values by defining a
special value for them, say ‘NA’ (not available) and adding it to the domain.

Once $K$ and $D$ are fixed it is possible to define the set of all observable
datasets.

######  Definition 2.

A _dataset_ is a subset of $K\times D$ where every key in $K$ occurs exactly
once.

Another way to interpret this is to say that a data set is a total function
$K\to D$. The set of all observable data sets is denoted $D^{K}$. In the
example, one possible dataset is

$\displaystyle\\{((1,\texttt{"age"}),25),((1,\texttt{"job"}),\texttt{"unemployed"}),$
$\displaystyle\phantom{\\{}((2,\texttt{"age"}),\texttt{"employed"}),((2,\texttt{"job"}),42)\\}$

Observe that the key consists of a person identifier and a variable
identifier. Since type checking is a common part of data validation these
definitions deliberately leave open the possibility that variables assume a
value of the wrong type.

Data validation can now be formally defined as follows.

######  Definition 3.

A _data validation function_ is a surjective function

$v:D^{K}\twoheadrightarrow\\{\textsf{False}{},\textsf{True}{}\\}.$

A data set $d\in D^{K}$ for which $v(d)=\textsf{True}{}$ is said to _satisfy_
$v$. A data set for which $v(d)=\textsf{False}{}$ is said to _fail_ $v$.

Recall that surjective means that there is at least one $d\in D^{K}$ for which
$v(d)=\textsf{False}{}$ and at least one $d\in D^{K}$ for which
$v(d)=\textsf{True}{}$. A validation function has to be surjective to have any
meaning as a data validation activity. Suppose
$v^{\prime}:D^{K}\to\\{\textsf{False}{},\textsf{True}{}\\}$ non-surjective
function. If there is no data set $d$ for which
$v^{\prime}(d)=\textsf{False}{}$, then $v^{\prime}$ is always true and it does
not validate anything. If, on the other hand, there is no $d$ for which
$v^{\prime}(d)=\textsf{True}{}$ then no data set can satisfy $v^{\prime}$. In
short, a function that is not surjective on
$\\{\textsf{False}{},\textsf{True}{}\\}$ does not separate valid from invalid
data.

A data validation function is reminiscent of a predicate as defined in first-
order logic. Informally, a predicate is a statement about variables that can
be True or False. The variables can take values in a predefined set (referred
to as the _domain_ of the predicate). Since validation functions map elements
of $D^{K}$ to $\\{\textsf{False}{},\textsf{True}{}\\}$, it is tempting to
equate a validation function as a predicate over $D^{K}$. However, the
elements of $D^{K}$ are instances of possible data sets, and validation is
based on statements involving variables of a single observed data set. It is
therefore more convenient to adopt the following definition.

######  Definition 4.

A _validation rule_ is a predicate over an instance $d\in D^{K}$ that is
neither a tautology nor a contradiction.

The elements of $D^{K}$ are sufficiently similar so that any validation rule
over a particular data set $d\in D^{K}$ is also a validation rule over another
data set in $d^{\prime}\in D^{K}$, where the truth value of a validation rule
depends on the chosen data set. This also allows us to interpret a tautology
as a predicate that is True for every element of $D^{K}$ and a contradiction
as a predicate that is False for every element of $D^{K}$.

The equivalence between an assertion about a data set and a function
classifying possible data sets as valid or invalid instances is a rather
obvious conclusion. The actual value of the above exercise is the strict
definition of a data point as a key-value pair. As will be shown below,
inclusion of the key permits a useful classification of data validation rules.

To demonstrate that many types of validation rules can be expressed in this
framework a few examples based on the example of Equation (1) will be
considered. The following rule states that all ages must be integer.

$\forall((u,\texttt{"age"}),x)\in d:x\in\mathbb{N}.$

Here, $((u,\texttt{"age"}),x)$ runs over the person-value pairs where the
value is supposed to represent an age. Similarly, it is possible express a
nonnegativity check on the age variable.

$\forall((u,\texttt{"age"}),x)\in d:x\geq 0.$

In these examples a data set fails a validation rule when not all elements
satisfy a predicate. Such rules are not very informative when it comes to
pinpointing what elements of the data set cause the violation. It is customary
to perform data validation on a finer level, for example by checking
nonnegativity element by element. Based on the definitions introduced here
this is done by fixing the key completely.

$\displaystyle\forall((1,\texttt{"age"}),x)\in d:x\geq 0$
$\displaystyle\forall((2,\texttt{"age"}),x)\in d:x\geq 0.$

Now consider the cross-variable check that states that employed persons must
be $15$ or older.

$\displaystyle\forall((u,\texttt{"age"}),x),(u,\texttt{"job"}),y)\in
d:y=\texttt{"employed"}\Rightarrow x\geq 15,$

where $\Rightarrow$ denotes logical implication (if $y=\texttt{"employed"}$
then $x\geq 15$). Finally, consider the cross-person check that states that we
expect the average age in the data set to be larger then or equal to 5.

$\frac{\sum_{(u,\texttt{"age"},x)\in d}x}{\sum_{(u,X,x)\in
d}\delta(X,\texttt{"age"})}\geq 5,$

where $\delta(X,Y)=1$ when $X=Y$ and otherwise 0.

In practical applications validation rules are often expressed more directly
in terms of variable names, such as $age\geq 0$. Such expressions are
specializations where one silently assumes extra properties such as that the
rule will be evaluated for the entry for age in every record.

### Remark 1.

In Definition 3, (and also 4) a validation function is defined as a surjection
$D^{K}\twoheadrightarrow\\{\textsf{False}{},\textsf{True}{}\\}$. In practical
applications it is often useful to also allow the value NA (not available) for
cases where one or more of the data points necessary for evaluating the
validation rule are missing. In that case the domain $D$ in the Equation (1)
must be extended to $D\cup\\{\texttt{NA}\\}$. See Van der Loo and De Jonge
(2019) for an implementation.

### Remark 2.

The current formalization of data validation excludes checking for
completeness of the key set: it is assumed by definition that each data set in
$D^{K}$ is a complete set of key-value pairs where the keys cover all of $K$.
The above definitions therefore create a clean distinction between what is
metadata (the keys and their interpretation) and data (key-value pairs). It is
possible to check for uniqueness of variables.

### Remark 3.

The formal definition of data validation rules also allows a formalization of
data validation software tools or domain specific languages, such as in Van
der Loo and De Jonge (2019).

### Remark 4.

In official (government) statistics, validation rules are referred to as _edit
rules_ rather than data validation rules. See De Waal et al. (2011) and
references therein.

##  2 Classification of validation rules

Validation rules defined by domain experts may be easy for humans to
interpret, but can be complex to implement. Take as an example the assertion
‘the average price of a certain type of goods this year, should not differ
more then ten percent from last year’s average’. To evaluate this assertion
one needs to combine prices of multiple goods collected over two time periods.
A practical question is therefore if the ‘complexity of evaluation’, in terms
of the amount of information necessary, can somehow be measured. In this
section a classification of data validation rules is derived that naturally
leads to an ordering of validation rules in terms of the variety of
information that is necessary for their evaluation.

One way to measure the amount of information is to look at the predicate
defining a data validation rule and to determine how many different $(k,x)$
pairs are needed to evaluate it. This is not very useful for comparing
complexity across different data sets that are not from the same $D^{K}$ since
the numbers will depend on the size of the key set $K$. One measure that does
generalize to different $D^{K}$ is the measure ‘does a rule need one, or
several $(k,x)$ to be evaluated?’.

This measure is not very precise, but it can be refined when a key consists of
multiple meaningful parts such as in the running example where the key
consists of the id of a person and the name of a variable. One can then
classify a rule according to two questions: ‘are multiple person id’s
necessary for evaluation?’, and: ‘are multiple variables necessary for
evaluation?’. This gives a four-way classification: one where both questions
are answered with ‘no’, two where one of the questions is answered with ‘yes’
and one where both questions are answered with ‘yes’. Although this refinement
improves the accuracy of the classification, it only allows for comparing
validation rules over data sets with the exact same key structure. It would
therefore be useful to have a generic key structure that can be reused in many
situations. One such structure can be found by considering in great generality
how a data point is obtained in practice.

Figure 1:  A unit $u$ of type $U$ exists from $t_{u}$ to $t^{\prime}_{u}$.
From $t_{u}$ onward it has an attribute $X$ with its value $x$ possibly
changing over time. At some time $\tau$ this value is observed. The observed
value is thus fully characterized by the quartet ($U$, $\tau$,$u$, $X$).

A data point usually represents the value of an attribute of an object or
event in the real world: a person, a web site, an e-mail, a radio broadcast, a
country, a sensor, a company, or anything else that has observable attributes.
In what follows, an object or event is referred to as a ‘unit’ for brevity. A
data point is created by observing an attribute $X$ from a unit $u$ of a
certain type $U$ at time $\tau$, as in Figure 1. Using the same reasoning as
above, this yields a $2^{4}=16$-way classification of validation rules: for
each element $U$, $\tau$, $u$, $X$ a validation rule requires a single or
multiple instances to be evaluated. However, there are some restrictions. Any
unit $u$ can only be of one type. So evaluating a validation rule will never
require multiple types and a single unit. Second, the type of a unit fixes its
properties. So a validation rule will never need a single variable for
multiple types. With these restrictions considered, the number of possible
classes of validation rules reduces from sixteen to ten.

To distinguish the classes the following notation is introduced. For each
element $U$, $\tau$, $u$, $X$ assign an $s$ when a validation rule pertains to
a single value of that element and assign an $m$ when a validation rule
pertains to multiple values of that element. For example, a validation rule of
class $sssm$ needs a single type, a single measurement, a single object and
multiple variables to be evaluated.

The ten possible classes can themselves be grouped into _validation levels_ ,
according to whether a class is labeled with no, one, two, three, or four
$m$’s. A higher validation level indicates that we need a larger variety of
information in order to evaluate the rule. The classification, and their
grouping into validation levels is summarized in Table 1.

Table 1: The ten possible classes of validation rules, grouped into
‘validation levels’. A higher level indicates that a wider variety of
information is necessary to evaluate a validation rule. Validation level
---
0 | 1 | 2 | 3 | 4
ssss | sssm | ssmm | smmm | mmmm
| ssms | smsm | msmm |
| smss | smms |  |

Going from level 0 to 4 corresponds to a workflow that is common in practice.
One starts with simple tests such as range checks. These are of level zero
since a range check can be performed on a single data point. That is, one only
needs a value that corresponds to a single type, measurement, unit, and
variable. Next, more involved checks are performed, for instance, involving
multiple variables ($sssm$, e.g. the ratio between two properties of the same
unit must be in a given range), multiple units ($ssms$, e.g. the mean of a
variable over multiple units must be within a specified range), or multiple
measurements ($smss$, e.g. the current value of the property of a unit cannot
differ too much from a past value of the same property of the same unit).
Going up in level, even more complex rules are found until rule evaluation
involves multiple variables of multiple units of multiple types measured at
multiple instances ($mmmm$).

This classification also has an immediate interpretation for data stored in a
data base that is normalized in the sense of Codd (1970). In such a data base,
records represent units, columns represent variables, and tables represent
types. The ‘time of measurement’ is represented as a variable as well. The
classification indicates whether a rule implementation needs to access
multiple records, columns or tables.

##  3 Properties of validation rule sets

Definition 4 implies that a validation rule is a predicate over a data set
that is not a tautology nor a contradiction. This means that combining two
validation rules with $\land$ or $\lor$ does not automatically yield a new
validation rule. Consider for example the rules $x\geq 0$ and $x\leq 1$ (using
shorter notation for brevity). The rule $x\geq 0\lor x\leq 1$ is a tautology.
The rule $x\geq 0\land x\leq-1$ is a contradiction. In fact, the only
operation that is guaranteed to transform a validation rule into another
validation rule is negation.

The fact that validation rules are not closed under conjugation ($\land$) or
disjunction ($\lor$) has practical consequences. After all, defining a set of
validation rules amounts to conjugating them together into a single rule since
a data set is valid only when all validation rules are satisfied. A set of
rules may be such that their conjugation is a contradiction. Such a rule set
is called _infeasible_. More subtle problems involve unintended consequences,
including _partial infeasibility_ (Bruni and Bianchi, 2012), and introduction
on fixed values or range restrictions. Other problems involve the introduction
of several types of redundancies (Dillig et al., 2010; Paulraj and Sumathi,
2010), which make rule sets both harder to maintain and hamper solving
problems such as error localization (Bruni, 2005; De Jonge and Van der Loo,
2019a). In the following, some examples of unintended effects and redundancies
are discussed. The examples shown here are selected from a more extensive
discussion in Van der Loo and De Jonge (2018, Chapter 8). For simplicity of
presentation the rules are expressed as simple clauses, neglecting the key-
value pair representation.

Partial inconsistency is (often) an unintended consequence implied by a pair
of rules. For example the rule set

$\displaystyle gender$ $\displaystyle=\texttt{"male"}\Rightarrow income>2000$
$\displaystyle gender$ $\displaystyle=\texttt{"male"}\Rightarrow income<1000,$

is feasible, but it can only be satisfied when $gender\not=\texttt{"male"}$.
Thus, the combination of rules (unintentionally) excludes an otherwise valid
gender category.

A simple redundancy is introduced when one rule defines a subset of valid
values with respect to another rule. For example if $x\geq 0$ and $x\geq 1$,
then $x\geq 0$ is redundant with respect to $x\geq 1$. More complex cases
arise in sets with multiple rules. A more subtle redundancy, called
‘nonrelaxing clause’ occurs in the following situation.

$\displaystyle x\geq 0\Rightarrow y\geq 0$ $\displaystyle x\geq 0.$

Here, the second rule implies that the condition in the first rule must always
be true. Hence the rule set can be simplified to

$\displaystyle y\geq 0$ $\displaystyle x\geq 0.$

Another subtle redundancy, called a ‘nonconstraining clause’ occurs in the
following situation.

$\displaystyle x>0$ $\displaystyle\Rightarrow y>0$ $\displaystyle x<1$
$\displaystyle\Rightarrow y>1$

Now, letting $x$ vary from $-\infty$ to $\infty$, the rule set first implies
that $y>1$. As $x$ becomes positive, the rule set implies that $y>0$ until $x$
reaches $\infty$. In other words, the rule set implies that $y$ must be
positive regardless of $x$ and can therefore be replaced with

$\displaystyle y>0$ $\displaystyle x<1$ $\displaystyle\Rightarrow y>1$

Methods for algorithmically removing such issues and simplifying rule sets
have recently been discussed by Daalmans (2018), and have been implemented by
De Jonge and Van der Loo (2019b). In short, the methods are based on
formulating Mixed Integer Programming (MIP) problems and detecting their
(non)-convergence after certain manipulations of the rule sets. General theory
and methods for rule manipulation have also been discussed by Chandru and
Hooker (1999) and Hooker (2000) in the context of optimization.

##  4 Conclusion

Data validation can be formalized equivalently in terms of certain predicates
over a data set or as a surjective Boolean function over a well-defined set of
observable data sets. It is possible to define a very general classification
of validation rules, based on a generic decomposition of the metadata.
Combining validation rules into a set can lead to subtle and unintended
consequences that can be solved in some cases with algorithmic methods.

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