# Working Group Report on the Combined Analysis of Muon Density Measurements
from Eight Air Shower Experiments

, for the EAS-MSU1, IceCube2, KASCADE-Grande3, NEVOD-DECOR4, Pierre Auger5,
SUGAR6, Telescope Array7, and Yakutsk EAS Array8 collaborations111for
collaboration lists see PoS(ICRC2019)1177
∗ LIP, Av. Prof. Gama Pinto 2, Lisbon, Portugal E-mail:
1EAS-MSU experiment, Moscow, Russia,
2IceCube Neutrino Observatory, Madison, USA
https://icecube.wisc.edu/collaboration/authors/icrc19_icecube,
3KASCADE-Grande experiment, Karlsruhe, Germany
https://web.ikp.kit.edu/KASCADE,
4NEVOD-DECOR experiment, Moscow, Russia,
5Pierre Auger Observatory, Malargüe, Argentina
Full author list: Pierre Auger Collaboration and additional author(s):
http://www.auger.org/archive/authors_icrc_2019_a.html,
6SUGAR Array, Sidney, Australia,
7Telescope Array Project, Salt Lake City UT, USA
http://www.telescopearray.org/research/collaborators,
8Yakutsk EAS Array, Yakutsk, Russia
https://ikfia.ysn.ru/en/theyakutskarrayteam

###### Abstract:

We present a meta-analysis of recent muon density measurements made by eight
air shower experiments which cover shower energies ranging from PeV to tens of
EeV regarding the muon puzzle in extensive air showers. Some experimental
analyses reported deviations between recorded and simulated muon densities in
extensive air showers, and others reported no discrepancies. Comparisons
between experiments were made using a universal reference scale based on the
relative difference to simulated proton and iron initiated air showers. We
have applied a cross-calibration of energy scales between experiments based on
the isotropic flux of cosmic rays as a reference. Above 10 PeV, most
experimental data show a muon excess with respect to simulated air showers,
including those performed with the recent post-LHC high-energy interaction
models. The discrepancy increases with the shower energy with a slope 8 sigma
away from the predictions by EPOS-LHC and QGSJet-II.04. The effect of
measurements being made at different zenith angles and energy threshold of
muons across different experiments will be addressed.

##  1 Introduction

The determination of the primary mass, arrival direction, and energy of cosmic
rays arriving at Earth with energies exceeding $10^{15}\,{\rm eV}$ is done by
means of detection and analysis of Extensive Air Showers (EAS). Two of the
most used observables to infer the mean logarithmic mass of the primary,
$\langle\ln\\!A\rangle$, are the depth of the shower maximum in the atmosphere
$X_{\rm max}$ and the number of muons in the shower, ${N_{\mu}}$. One of the
dominant sources of systematics in $\langle\ln\\!A\rangle$ stems from the
uncertainty in high energy hadronic interaction models used in air shower
simulations. While $X_{\rm max}$ is linked to the electromagnetic component of
EAS, ${N_{\mu}}$ directly stems from the hadronic cascade, which is more prone
to model uncertainties. By checking the consistency of $X_{\rm max}$ and
${N_{\mu}}$ in terms of mass interpretation, models can be tested, opening a
window for particle physics in phase-space regions beyond the reach of the
LHC. At the same time, models can be improved, allowing a more precise
determination of the mass of the primaries.

In the year 2000, the HiRes/MIA collaboration reported a discrepancy of the
number of muons in simulated and measured air showers between $10^{17}$ to
$10^{18}$ eV [5]. A decade later, NEVOD-DECOR reported [7, 17] an increase of
muon density relative to simulations, and also SUGAR array observed an excess
in data [10]. EAS-MSU [9] reported no muon number discrepancy as well as
KASCADE-Grande [8], which in addition reported differences in the muon number
evolution with zenith angle with respect to the latest hadronic interaction
models tuned to LHC-data. The Pierre Auger Observatory [11, 12] and Telescope
Array [13] also observed a muon deficit in simulations, at $10^{19}$ eV. These
measurements, preliminary data from IceCube [14] and AMIGA [15], and
unpublished data from Yakutsk [16] will be systematically compared. In this
report, we present a meta-analysis of the muon density measurements performed
by these eight EAS experiments.

##  2 Measurement of the muonic density

Figure 1: Air shower experiments have measured the muon density at ground
under various conditions, which are shown here as a function of the shower
energy. Points and lines indicate a measurement in a narrow bin of the
parameter, while boxes indicate integration over a parameter range. _Left:_
Zenith angle of air showers. _Middle:_ Lateral distance of the muon density
measurement. _Right:_ Energy threshold for the muons that are counted in the
experiment.

The muon density at the ground depends on many parameters: cosmic-ray energy
$E$, zenith angle $\theta$, shower age (vertical depth $X$, and zenith angle
of the shower $\theta$), lateral distance $r$ from shower axis, and energy
threshold $E_{\mu,{\rm min}}$ of the detectors for muons. The parameter space
covered by each experiment is shown in Figure 1. A direct comparison of the
muon measurements is not possible, due to the very different conditions and
techniques which were used. Instead, we compare each experiment to air shower
simulations in the same observation conditions by means of a data/MC ratio.

According to the type of shower energy and muon density reconstruction, we can
classify the experiments into three groups:

Shower energy and muon density

The Pierre Auger Observatory (including its muon detector AMIGA), Telescope
Array, and the Yakutsk experiment are all capable of measuring the shower
energy, $E_{\rm cal}$, using Cherenkov or fluorescence light, which can be
converted into the primary energy, $E$, with little model dependence. These
experiments come close to measuring the primary cosmic-ray energy, $E$,
independently of the muon density at the ground.

The IceCube Neutrino Observatory can also be classified under this category,
although it does not observe the air shower optically. It can measure the
shower energy with low model dependence thanks the detector being close to the
average depth of the shower maximum [20].

Muon and electron density

KASCADE-Grande and EAS-MSU measure signals from electrons and muons
separately. These experiments compute the data/MC ratio for showers in the
same electron-density interval, which can contain different primary energies,
compensated by the attenuation produced by differences in the depth of the
shower maximum.

Muon density only

NEVOD-DECOR and SUGAR are pure muon detectors, without a separate energy
estimator. The flux of showers is measured in intervals of muon density, which
is then compared with a simulated flux, computed from an external model.
NEVOD-DECOR uses an average cosmic-ray flux computed from multiple
experiments, while SUGAR uses the flux measured by the Pierre Auger
Observatory.

While most of the shower parameters can be easily matched in simulation and
experiment, the cosmic-ray energy, $E$, is difficult to match. The number of
muons scales as ${N_{\mu}}\propto A^{1-\beta}E^{\beta}$, with $\beta\simeq
0.9$ [18]. This could cause two experiments with an energy-scale offset of 20
% to have a 18 % additional offset in the data/MC ratios. If we take the
cosmic ray flux as a universal reference, and assuming that all deviations in
measured fluxes between different experiments arise from energy-scale offsets,
a relative energy-scale ratio $E_{\rm data}/E_{\rm ref}$ can be found for each
experiment so that the all-particle fluxes overlap. This approach is well-
known and has been applied successfully in other work [21, 22].

Table 1: Table showing the energy-scale adjustment factors obtained from cross-calibration (2nd column), the median $\sec\theta$ of several data sets (3rd column), and the minimum energy of the muons at production, due to energy loss in the atmosphere and detector shielding (see text for explanation). Experiment | $E_{\rm data}/E_{\rm ref}$ | $\sec\theta$ |  $E_{\mu\,{\rm prod}}/$GeV
---|---|---|---
EAS-MSU | - | 1.1 | 11.9
IceCube Neutrino Observatory | 1.19 | 1.0 | 0.7
KASCADE-Grande | - | 1.0 , 1.3 | 1.5 , 2.1
NEVOD-DECOR | 1.08 | 2.3 , 4.8 | 8.4 , 18.6
Pierre Auger Observatory | 0.948 | 1.3 , 2.4 | 1.8 , 4.0
AMIGA | 0.948 | 1.2 | 2.4
SUGAR | 0.948 | 1.0 | 1.9
Telescope Array | 1.052 | 1.3 | 1.4
Yakutsk EAS Array | 1.24 | 1.1 | 2.6

The cross-calibration factors used in this report are given in Table 1, and
are summarized in [6]. We assume that the resulting overall energy-scale has
an uncertainty of at least 10 %.

###  2.1 Combined measurements

Figure 2: Muon density measurements converted to the $z$-scale for each
hadronic interaction model. When corresponding simulations are missing for an
experiment, no points are shown. Error bars show statistical and systematic
uncertainties added in quadrature (systematic uncertainties are dominant in
nearly all measurements). Figure 3: Data from Figure 2 after applying the
energy-scale cross-calibration. The points for KASCADE-Grande and EAS-MSU were
not cross-calibrated in this report and are included for comparison. Shown for
comparison are $z$-values expected for a mixed composition from optical
measurements (band) based on [1], and from the GSF model (dashed line) [22].

To compare all the measurements, we introduce the $z$-scale

$z=\frac{\ln({N_{\mu}}^{\rm det})-\ln({N_{\mu}}_{p}^{\rm
det})}{\ln({N_{\mu}}_{\rm Fe}^{\rm det})-\ln({N_{\mu}}_{p}^{\rm det})},$ (1)

where ${N_{\mu}}^{\rm det}$ is the muon density estimate as seen in the
detector, while ${N_{\mu}}_{p}^{\rm det}$ and ${N_{\mu}}_{\rm Fe}^{\rm det}$
are the simulated muon density estimates for proton and iron showers,
accounting for detector effects. The $z$-scale ranges from 0 (pure proton
showers) to 1 (pure iron showers), if there is no discrepancy between real and
simulated air showers, in which case one could use
$\langle\ln\\!A\rangle=z\,\ln 56$ to infer the average logarithmic mass.

Shown in Figure 2 are the converted measurements for each hadronic interaction
model. Note that the conversion to $z$ is only possible when ${N_{\mu}}_{\rm
p}^{\rm det}$ and ${N_{\mu}}_{\rm Fe}^{\rm det}$ are available for that model
under the specific experimental conditions. The scatter is drastically reduced
after the cross-calibration, as can be seen in Figure 3. The points move
horizontally by the relative amount $(E_{\rm data}/E_{\rm ref})^{-1}$ and
vertically by $z_{\rm ref}-z_{\rm data}=\frac{\beta\ln(E_{\rm data}/E_{\rm
ref})}{\ln({N_{\mu}}_{\rm Fe}^{\rm det})-\ln({N_{\mu}}_{p}^{\rm det})}$. As
expected, the cross-calibration improves the agreement between data from
different experiments. The points for KASCADE-Grande222Latest KASCADE-Grande
results, presented in this conference [27], were not yet included in this
comparison. and EAS-MSU were not cross-calibrated in this report and are
included for comparison. We emphasize again that the reference energy-scale
after cross-calibration has a remaining uncertainty of at least 10 %. This
means that $z$-values in all plots can be collectively varied by about $\pm
0.25$.

Now, we consider the effect of an energy-dependent mass composition. Shown in
Figure 3 is a band which contains the optical measurements of the depth
$X_{\rm max}$ of shower maximum from several experiments after conversion into
$\langle\ln\\!A\rangle$ based on air shower simulations with EPOS-LHC. We will
use this as an estimate of the mass composition. The band is independent of
the muon measurements and can be used as a reference. Also shown is the
$z_{\rm mass}=\langle\ln\\!A\rangle/\ln 56$ value computed from the GSF model
[22], which is based on optical and muon measurements and averages over
experiments and model interpretations of air shower data.

It can be seen that the measured $z$ values do not follow $z_{\rm mass}$,
implying that the models do not consistently describe the muon density. EPOS-
LHC [2], QGSJet-II.04 [3], and SIBYLL-2.3 [4], and the pre-LHC model QGSJet01
give a reasonable description of data up to a few $10^{16}$ eV. At higher
shower energies, a muon deficit in simulations is observed ($z>z_{\rm mass}$)
in all models. Shown in Figure 4 is the difference $\Delta z=z-z_{\rm mass}$
for EPOS-LHC and QGSJet-II.04, the two latest-generation models with most
cross-calibrated data points in this study. Subtracting $z_{\rm mass}$ is
expected to remove the effect of the changing mass composition. An energy-
dependent trend in $\Delta z$ remains.

###  2.2 Energy-dependent trend

Figure 4: $\Delta z=z-z_{\rm mass}$ for EPOS-LHC and QSGJet-II.04. The
function $\Delta z_{\rm fit}=a+b\log_{10}(E/10^{16}{\rm eV})$ was fitted
assuming a correlation coefficient $\alpha=0.5$ within each data-set,
resulting in $b=0.34\pm 0.04$ for EPOS-LHC and $b=0.30\pm 0.03$ for
QGSJetII.04. The inset shows the average residual per data set with respect to
the fitted line, $\Delta z-\Delta z_{\rm fit}$, as a function of the minimum
energy of muons at the production point high in the atmosphere, $E_{\mu\,{\rm
prod}}$ . (See text for explanations).

To quantify the observed trend in $\Delta z$ as a function of energy, the
function

$\Delta z_{\rm fit}=a+b\,\log_{10}(E/{10^{16}\rm eV}),$ (2)

is fitted to the data shown in Figure 4, with free parameters $a$ and $b$. The
uncertainty of $b$ scales with the uncertainties of the data. The error bars
of most data points are dominated by systematic uncertainties, which are
correlated for data points from a single set. The exact amount of correlation
is not known. We work around this problem by repeatedly fitting the data under
different correlation assumptions. We use the least-squares method assuming a
correlation factor $\alpha$ between points belonging to the same data set and
0 otherwise. The fit was repeated for values of $\alpha$ from 0 to 0.95. To
adjust for over- or underestimated uncertainties in the input, the raw result
was rescaled with the $\chi^{2}$ value and the degrees of freedom $n_{\rm
dof}$ of the fit, $\sigma_{b}=\sigma_{b}^{\rm raw}\sqrt{\chi^{2}/n_{\rm dof}}$
[23]. For EPOS-LHC the slope ranges from $b=0.29\pm 0.04$ up to $b=0.35\pm
0.04$ for $\alpha=0$ and $\alpha=0.95$ respectively. For QGSJet-II.04, it
ranges from $b=0.22\pm 0.03$ up to $b=0.31\pm 0.04$ for $\alpha=0$ and
$\alpha=0.95$ respectively. In both cases, the significance is always larger
than 8-sigma.

As previously mentioned, each experiment has a shielding material $\Delta
X_{\rm det}$ above the actual detector volume, which effectively translates
into an energy threshold for muons $E_{\mu{\rm
min}}(\theta)=\frac{dE_{\mu}}{dX}\Delta X_{\rm det}\sec(\theta)=E_{\mu{\rm
min}}\sec(\theta)$ due to energy loss $\frac{dE_{\mu}}{dX}\simeq 2$ MeV / g
cm-2, above the detector. Energy losses in the atmophere are also taken into
account. Most of the muons are produced approximately at the maximum of the
muon production depth distribution, around333In [24], the observed value is
$\sim 475$ g cm-2 above $2\times 10^{19}$ eV. The elongation rate for proton
and iron simulations is around 25 g cm-2/decade, which then sets a lower bound
of $\sim 300$ g cm-2 at $10^{15}$ eV. It must be also noted that the apparent
muon production depth (MPD) depends on the particular condition of
observations, contrary to the total/true MPD [25]. Therefore, we have chosen
the central value $400$ g cm${}^{-2})$. Assuming an uncertainty of $\sim 100$
g cm-2, the error due to these approximations is around $\pm$0.2 GeV in
$E_{\mu\,{\rm prod}}$. $\sim 400$ g cm-2. Therefore $E_{\mu\rm
atm}(\theta)=\frac{dE_{\mu}}{dX}(X\sec(\theta)-400$ g cm${}^{-2})$, where $X$
is the vertical atmospheric depth of the considered experiment. The minimum
energy required at production for a muon to be detected in each experiment is
then $E_{\mu\,{\rm prod}}=E_{\mu{\rm min}}(\theta)+E_{\mu\rm atm}(\theta)$.
Table 1 shows this energy for the different data sets. When we scan for
different $E_{\mu\,{\rm prod}}$ we are in fact looking for differences in the
energy spectrum of muons at production [26]. Inset of Figure 4 displays the
residuals $\Delta z-\Delta z_{\rm fit}$ as a function of $E_{\mu\,{\rm
prod}}$.

We have modified equation 2 and added an extra parameter to account for this
possible effect, as $\Delta z_{\rm fit}=a+b\log_{10}(E/{10^{16}\rm
eV})+c\,E_{\mu\,{\rm prod}}$. The results of this modification are not
conclusive because $c$ changes significantly when the data set at
$\sec(78^{\circ})=4.8$ from NEVOD-DECOR is excluded from the fit. At these
extreme zenith angles, the average density of the atmosphere drops very
rapidly in the region where the hadronic cascade develops. Therefore, the
critical energy of the $K/\pi$ mix of the shower will change. A possible
mismatch with respect to models might also produce visible effects, and it is
difficult to disentangle the effects in the current state of analysis. Another
complication is the effective distance of the muon bundles recorded by NEVOD-
DECOR to the shower core and its evolution with primary energy and zenith
angle. This must be carefully accounted for, as it might introduce a further
mismatch with respect to simulations.

##  3 Summary and outlook

A comprehensive collection of muon measurements in air showers with energies
from PeV up to tens of EeV is presented. We developed the $z$-scale as a
comparable measure of the muon density between different experiments and
analyses. The $z$-scale uses air shower simulations as a reference to compare
muon density measurements taken under different conditions. We cross-
calibrated energy-scales of experiments when possible, using the isotropic
all-particle flux of cosmic rays as a reference.

A remarkably consistent picture was obtained. Muon measurements seem to be
consistent with simulations based on the latest generation of hadronic
interaction models, EPOS-LHC, QGSJet-II.04, and SIBYLL-2.3, up to about
$10^{16}$ eV. At higher energies, a growing muon deficit in the simulations is
observed. The slope of this increase in $z$ per decade in energy is 0.22 to
0.35 for EPOS-LHC and QGSJet-II.04, with 8 $\sigma$ significance.

We have looked for other trends in the deviations between simulations and
data. We have calculated the approximated minimum energy of muons at
production for each experiment to be detected. This value ranges from $\sim 1$
GeV up to $\sim 19$ GeV. Current data do not provide conclusive evidence of a
deviation of the muon spectrum at production with respect to models. In this
conference [27], KASCADE-Grande supports previous findings about a problem
with the evolution of the muon content with the zenith angle between 10 PeV
and 1 EeV.

The new generation of ongoing experimental upgrades and more refined analyses
are expected to further reduce the uncertainities of the different
experimental results.

## References

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# Mad families of vector subspaces and the smallest nonmeager set of reals

Iian B. Smythe  Department of Mathematics, Rutgers University – New Brunswick,
110 Frelinghuysen Road, Piscataway, NJ, 08854 www.iiansmythe.com
<EMAIL_ADDRESS>
(Date: September 17, 2019)

###### Abstract.

We show that a parametrized $\diamondsuit$ principle, corresponding to the
uniformity of the meager ideal, implies that the minimum cardinality of an
infinite maximal almost disjoint family of block subspaces of a countable
vector space is $\aleph_{1}$. Consequently, this cardinal invariant is
$\aleph_{1}$ in the Miller model. This verifies a conjecture of the author
from [14].

###### Key words and phrases:

Cardinal invariants, generalized mad families, vector spaces, parametrized
$\diamondsuit$ principles

###### 2010 Mathematics Subject Classification:

Primary 03E17; Secondary 15A03

The author would like to thank Jörg Brendle and Michael Hrušák for helpful
conversations that have contributed to this work. He would also like to thank
the Centro de Ciencias Matemáticas at the Universidad Nacional Autónoma de
México in Morelia and Casa Matemática Oaxaca for providing the time and space
to have these conversations and contemplate the uncountable. Lastly, he must
thank his friends Jeffrey Bergfalk and Sena Aydin for making him feel at home
in Morelia, and the Department of Mathematics at Rutgers University for
funding his travels to Mexico.

##  1\. Introduction

The setting of this note is a countably infinite-dimensional vector space $E$
over a countable field $F$, with a distinguished basis
$(e_{n})_{n\in\mathbb{N}}$. For example, we may take
$E=\bigoplus_{n\in\mathbb{N}}F$ and $e_{n}$ to be the $n$th unit coordinate
vector. If $X$ is a set or sequence of vectors in $E$, then $\langle X\rangle$
will denote its linear span. We also fix a well-ordering of $E$ that will be
used implicitly.

We will focus on _block subspaces_ of $E$, that is, those linear subspaces
which have a basis $(x_{n})_{n\in\mathbb{N}}$, called a _block sequence_ ,
such that for all $n$,

$\max(\mathrm{supp}(x_{n}))<\min(\mathrm{supp}(x_{n+1})),$

where the _support_ of a nonzero vector $v$ is the set $\mathrm{supp}(v)$ of
those $n$ for which the coefficient of $e_{n}$ in the basis expansion of $v$
is nonzero. We will abbreviate $\max(\mathrm{supp}(v))<\min(\mathrm{supp}(w))$
by writing $v<w$. It is easy to see that every infinite-dimensional subspace
contains a block subspace.

Two subspaces $X$ and $Y$ of $E$ are _almost disjoint_ if their intersection
$X\cap Y$ is finite-dimensional. A collection $\mathcal{A}$ of infinite-
dimensional subspaces which are pairwise almost disjoint is called an _almost
disjoint family of subspaces_ , and it is _maximal almost disjoint_ , or _mad_
, if it is not properly contained in any other such family. Note that an
almost disjoint family consisting of block subspaces is maximal if and only if
it is maximal amongst almost disjoint families of block subspaces.

Our focus here is the following cardinal invariant:

$\mathfrak{a}_{\mathrm{vec},F}=\min\\{|\mathcal{A}|:\mathcal{A}\text{ is an
infinite mad family of block subspaces}\\}.$

The basic properties of mad families of block subspaces and
$\mathfrak{a}_{\mathrm{vec},F}$ can be found in [14]. In particular,
$\mathfrak{a}_{\mathrm{vec},F}$ is uncountable (Proposition 2.5 in [14]).

When $|F|=2$, our setting corresponds exactly to the study of combinatorial
subspaces in $\mathrm{FIN}$, the set of finite nonempty subsets of
$\mathbb{N}$. The resulting cardinal invariant, $\mathfrak{a}_{\mathrm{FIN}}$,
was investigated by Brendle and García Ávila [5], who proved the following
lower bound, which the author observed could be extended to
$\mathfrak{a}_{\mathrm{vec},F}$ for general $F$:

######  Theorem 1.1 (Corollary 2.11 in [14]).

$\mathrm{non}(\mathcal{M})\leq\mathfrak{a}_{\mathrm{vec},F}$.

Here, $\mathrm{non}(\mathcal{M})$ is the _uniformity of the meager ideal_
$\mathcal{M}$, that is, the smallest cardinality of a nonmeager subset of the
reals, or equivalently, any perfect Polish space. Consequently, in any model
of set theory in which $\mathrm{non}(\mathcal{M})$ is “large”, i.e., at least
$\aleph_{2}$, so is $\mathfrak{a}_{\mathrm{vec},F}$. Such models include those
satisfying Martin’s Axiom $\mathsf{MA}(\kappa)$ for $\kappa\geq\aleph_{1}$, as
well as the random, Hechler, Laver, and Mathias models (cf. [3]).

When, then, is $\mathfrak{a}_{\mathrm{vec},F}$ “small”, i.e., equal to
$\aleph_{1}$? It is shown in [14] that $\mathfrak{a}_{\mathrm{vec},F}$ is
small in the Cohen and (iterated or side-by-side) Sacks models. It was
conjectured in [14] that this is also the case in the Miller model (described
at the end of §2 below), and moreover, that a certain _parametrized
$\diamondsuit$ principle_, in the sense of Moore, Hrušák, and Džamonja [12],
suffices. It is this conjecture that we verify in Theorem 3.1 below.111The
conjecture in [14] states “$\diamondsuit(\mathbb{N}^{\mathbb{N}},=^{\infty})$
implies $\mathfrak{a}_{\mathrm{vec},F}=\aleph_{1}$”; this is, possibly,
slightly stronger than what we prove here, cf. Lemma 2.4. As a consequence of
the main result from [12] (Theorem 2.2 below), in essentiall all “canonical”
models of set theory, i.e., those obtained by $\omega_{2}$-length countable
support iterations of definable proper forcings,
$\mathfrak{a}_{\mathrm{vec},F}=\mathrm{non}(\mathcal{M})$.

There are a few important precursors to this work that deserve mention. The
first is Hrušák’s proof [8] that the principle $\diamondsuit_{\mathfrak{d}}$
implies that $\mathfrak{a}$, the minimum cardinality of an infinite mad family
of subsets of $\mathbb{N}$, is $\aleph_{1}$, and later [12] that
$\diamondsuit(\mathbb{N}^{\mathbb{N}},\not>^{*})$, also known as
$\diamondsuit(\mathfrak{b})$, suffices. Since $\mathfrak{b}\leq\mathfrak{a}$,
this determines the value of $\mathfrak{a}$ in all of the aforementioned
canonical models.

In the sphere of generalized mad families, to which this work belongs, the
most relevant precursors are results on the cardinal invariants
$\mathfrak{a}_{e}$, $\mathfrak{a}_{p}$, and $\mathfrak{a}_{g}$, the minimum
cardinalities of a maximal eventually different family of functions on
$\mathbb{N}$, a maximal almost disjoint family of permutations of
$\mathbb{N}$, and a maximal cofinitary group of permutations of $\mathbb{N}$,
respectively. A characterization of $\mathrm{non}(\mathcal{M})$ (Theorem 2.1
below) implies that $\mathrm{non}(\mathcal{M})\leq\mathfrak{a}_{e}$, and
moreover, $\diamondsuit(\mathbb{N}^{\mathbb{N}},=^{\infty})$ implies
$\mathfrak{a}_{e}=\aleph_{1}$ [12]. Likewise, Brendle, Spinas and Zhang [6]
showed that $\mathrm{non}(\mathcal{M})\leq\mathfrak{a}_{p},\mathfrak{a}_{g}$,
and Kastermans and Zhang [9] showed that
$\diamondsuit(\mathbb{N}^{\mathbb{N}},=^{\infty})$ implies
$\mathfrak{a}_{p}=\mathfrak{a}_{g}=\aleph_{1}$.

This note is arranged as follows: §2 reviews the theory of cardinal invariants
and parametrized $\diamondsuit$ principles, and isolates the principle
$\diamondsuit(\mathrm{bb}^{\infty}(E^{2}),=^{\infty})$ which we will use. The
statement and proof of our main result, Theorem 3.1, is given in §3.

##  2\. Cardinal invariants and parametrized $\diamondsuit$ principles

Following the terminology in [12], an _invariant_ is a triple $(A,B,R)$ where
$A$ and $B$ are sets and $R\subseteq A\times B$ is a relation which satisfies:

  1. (i)

for every $a\in A$, there is a $b\in B$ such that $aRb$, and

  2. (ii)

for every $b\in B$, there is an $a\in A$ such that $a\\!\not\\!\\!Rb$.

$(A,B,R)$ is _Borel_ if $A$ and $B$ are Borel subsets of Polish spaces $X$ and
$Y$, and $R$ is Borel in $X\times Y$.

The standard reference for the theory of such invariants is [3], though we
also point to the recent survey [7], which treats them in the context of
computability theory, where they are called _Weihrauch problems_.

The _evaluation_ of an invariant $(A,B,R)$ is

$\langle A,B,R\rangle=\min\\{|X|:X\subseteq B\text{ and }\forall a\in A\exists
b\in B(aRb)\\}.$

When $A=B$ we will just write $(A,R)$ and $\langle A,R\rangle$, respectively.

For example, $\langle\mathbb{N}^{\mathbb{N}},\not>^{*}\rangle=\mathfrak{b}$,
where $f>^{*}g$ means $\forall^{\infty}n(f(n)>g(n))$.222$\forall^{\infty}$ and
$\exists^{\infty}$ are abbreviations for “for all but finitely many” and
“there exists infinitely many”, respectively. If by $f=^{*}g$ we mean
$\exists^{\infty}n(f(n)=g(n))$, then we have the following characterization of
$\langle\mathbb{N}^{\mathbb{N}},=^{\infty}\rangle$, due to Bartoszyński:

######  Theorem 2.1 (cf. Theorem 2.4 in [1]).

$\langle\mathbb{N}^{\mathbb{N}},=^{\infty}\rangle=\mathrm{non}(\mathcal{M})$.

A streamlined proof of Theorem 2.1 can be found in §4 of [7].

Given a Borel invariant $(A,B,R)$, the corresponding _parameterized
$\diamondsuit$ principle_ [12] is the statement:

> $\diamondsuit(A,B,R)$: For every Borel $F:2^{<\omega_{1}}\to A$, there is a
> $g:\omega_{1}\to B$ such that for every $f:\omega_{1}\to 2$, the set
> $\\{\alpha<\omega_{1}:F(f\upharpoonright\alpha)\,R\,g(\alpha)\\}$ is
> stationary.

Here, $F:2^{<\omega_{1}}\to A$ is _Borel_ if its restriction to $2^{\delta}$
is Borel for every $\delta<\omega_{1}$. If $g$ is as above, we say that $g$ is
a _$\diamondsuit(A,B,R)$ -sequence for $F$_, and if
$F(f\upharpoonright\alpha)\,R\,g(\alpha)$, then $g$ _guesses $f$ at $\alpha$_.

The principle $\diamondsuit(A,B,R)$ is a weakening of the classical
$\diamondsuit$ principle, and has a similar relationship to $\langle
A,B,R\rangle$ as $\diamondsuit$ has to $\mathfrak{c}$. In particular,
$\diamondsuit(A,B,R)$ implies $\langle A,B,R\rangle\leq\aleph_{1}$
(Proposition 2.5 in [12]). The key result about $\diamondsuit(A,B,R)$ is as
follows:

######  Theorem 2.2 (Theorem 6.6 in [12]).

Let $(A,B,R)$ be a Borel invariant. Suppose that
$(\mathbb{P}_{\alpha}:\alpha<\omega_{2})$ is a sequence of Borel partial
orders333In an iteration, a Borel partial order is really a sequence of codes,
interpreted in successive stages of the iteration. such that for each
$\alpha<\omega_{2}$, $\mathbb{P}_{\alpha}$ is forcing equivalent to
$\mathcal{P}(\\{0,1\\})^{+}\times\mathbb{P}_{\alpha}$.444The non-triviality
assumption that $\mathbb{P}_{\alpha}$ is equivalent to
$\mathcal{P}(\\{0,1\\})^{+}\times\mathbb{Q}_{\alpha}$ is very mild; in nearly
all of classical forcing notions, this can be accomplished by coding a
condition in $\mathcal{P}(\\{0,1\\})^{+}$ into the “first coordinate” of the
generic object. Let $\mathbb{P}_{\omega_{2}}$ be the countable support
iteration of this sequence. If $\mathbb{P}_{\omega_{2}}$ is proper, then
$\mathbb{P}_{\omega_{2}}$ forces $\diamondsuit(A,B,R)$ if and only if
$\mathbb{P}_{\omega_{2}}$ forces $\langle A,B,R\rangle\leq\aleph_{1}$.

At the risk of overstatement, we will call models obtained by forcing
iterations satisfying the hypotheses of Theorem 2.2, _canonical models_.

Returning to our vector space setting, let $\mathrm{bb}^{\infty}(E^{2})$ be
the space of all _$2$ -block sequences_ in $E$, that is, sequences
$(x_{n}^{0},x_{n}^{1})_{n\in\mathbb{N}}$ of pairs of nonzero vectors in $E$
such that for all $n$,

$x_{n}^{0}<x_{n}^{1}<x_{n+1}^{0}<x_{n+1}^{1}.$

Note that both $\mathrm{bb}^{\infty}(E)$ and $\mathrm{bb}^{\infty}(E^{2})$ are
homeomorphic to $\mathbb{N}^{\mathbb{N}}$ (see Theorem I.7.7 in [10]).

We will use the following variation on
$\diamondsuit(\mathbb{N}^{\mathbb{N}},=^{\infty})$:

> $\diamondsuit(\mathrm{bb}^{\infty}(E^{2}),=^{\infty})$: For every Borel
> $F:2^{<\omega_{1}}\to\mathrm{bb}^{\infty}(E^{2})$, there is a
> $g:\omega_{1}\to\mathrm{bb}^{\infty}(E^{2})$ such that for every
> $f:\omega_{1}\to 2$, the set
> $\\{\delta<\omega_{1}:F(f\upharpoonright\delta)=^{\infty}g(\delta)\\}$ is
> stationary.

Given (Borel) invariants $(A_{0},B_{0},R_{0})$ and $(A_{1},B_{1},R_{1})$, a
_(Borel) Tukey reduction_ from $(A_{0},B_{0},R_{0})$ and $(A_{1},B_{1},R_{1})$
is a pair of (Borel) maps $F:A_{0}\to A_{1}$ and $G:B_{1}\to B_{0}$ such that
for all $a\in A_{0}$, $b\in B_{1}$,

$F(a)R_{1}b\quad\text{implies}\quad aR_{0}G(b).$

These are called _morphisms_ in [3] and [7]. We write
$(A_{0},B_{0},R_{0})\leq_{B}(A_{1},B_{1},R_{1})$ if there is a Borel Tukey
reduction from $(A_{0},B_{0},R_{0})$ to $(A_{1},B_{1},R_{1})$. The following
lemma is immediate from the definitions.

######  Lemma 2.3.

If $(A_{0},B_{0},R_{0})\leq_{B}(A_{1},B_{1},R_{1})$, then

  1. (i)

$\langle A_{0},B_{0},R_{0}\rangle\leq\langle A_{1},B_{1},R_{1}\rangle$,

  2. (ii)

$\diamondsuit(A_{1},B_{1},R_{1})$ implies $\diamondsuit(A_{0},B_{0},R_{0})$.∎

The natural invariant corresponding to $\mathrm{non}(\mathcal{M})$ is
$(\mathcal{M},\mathbb{N}^{\mathbb{N}},\not\ni)$,555$(\mathcal{M},\mathbb{N}^{\mathbb{N}},\not\ni)$
is not, technically, Borel, but it has an equivalent Borel presentation. and
so $\diamondsuit(\mathcal{M},\mathbb{N}^{\mathbb{N}},\not\ni)$ is usually
written as $\diamondsuit(\mathrm{non}(\mathcal{M}))$.

######  Lemma 2.4.

  1. (a)

$(\mathbb{N}^{\mathbb{N}},=^{\infty})\leq_{B}(\mathrm{bb}^{\infty}(E^{2}),=^{\infty})$.

  2. (b)

$(\mathrm{bb}^{\infty}(E^{2}),=^{\infty})\leq_{B}(\mathcal{M},\mathbb{N}^{\mathbb{N}},\not\ni)$.

  3. (c)

$\langle\mathrm{bb}^{\infty}(E^{2}),=^{\infty}\rangle=\mathrm{non}(\mathcal{M})$.

###### Proof.

(a) It makes no difference if we replace $\mathbb{N}^{\mathbb{N}}$ with
$(\mathbb{N}\setminus\\{0\\})^{\mathbb{N}}$, so we do so. Consider the map
$F:(\mathbb{N}\setminus\\{0\\})^{\mathbb{N}}\to\mathrm{bb}^{\infty}(E^{2})$
defined by

$F(f)(k)=(v,w),$

where $(v,w)$ is the least pair of nonzero vectors $v<w$ in $E$ with supports
above the vectors in $F(f)(k-1)$ and such that $|\mathrm{supp}(v)|=f(k)$, for
all $k$. Let
$G:\mathrm{bb}^{\infty}(E)\to(\mathbb{N}\setminus\\{0\\})^{<\mathbb{N}}$ be
given by

$G((x_{n}^{0},x_{n}^{1})_{n\in\mathbb{N}})(k)=|\mathrm{supp}(x_{k}^{0})|$

for all $k$. Then, $F,G$ forms a Borel Tukey reduction as desired.

(b) As mentioned above, $\mathrm{bb}^{\infty}(E^{2})$ is homeomorphic to
$\mathbb{N}^{\mathbb{N}}$, so we will instead show that
$(\mathrm{bb}^{\infty}(E^{2}),=^{\infty})\leq_{B}(\mathcal{M},\mathrm{bb}^{\infty}(E^{2}),\not\ni)$,
mimicing the usual proof that
$(\mathbb{N}^{\mathbb{N}},=^{\infty})\leq_{B}(\mathcal{M},\mathbb{N}^{\mathbb{N}},\not\ni)$.

Let $F:\mathrm{bb}^{\infty}(E^{2})\to\mathcal{M}$ be defined by

$F(X)=\\{Y=(y_{n}^{0},y_{n}^{1})_{n\in\mathbb{N}}\in\mathrm{bb}^{\infty}(E^{2}):\forall^{\infty}n((y_{n}^{0},y_{n}^{1})\neq(x_{n}^{0},x_{n}^{1}))\\},$

where $X=(x_{n}^{0},x_{n}^{1})_{n\in\mathbb{N}}$. It is easy to check that
$F(X)$ is a meager $F_{\sigma}$ set in $\mathrm{bb}^{\infty}(E^{2})$. Let
$G:\mathrm{bb}^{\infty}(E^{2})\to\mathrm{bb}^{\infty}(E^{2})$ be the identity
map. If $F(X)\not\ni Y$, then $X=^{\infty}Y$, showing that $F,G$ give a Tukey
reduction from $(\mathrm{bb}^{\infty}(E^{2}),=^{\infty})$ to
$(\mathcal{M},\mathrm{bb}^{\infty}(E^{2}),\not\ni)$. The maps $F,G$ are
clearly Borel.

(c) follows from (a), (b), and Theorem 2.1, by Lemma 2.3. ∎

In particular, $\diamondsuit(\mathcal{M},\mathbb{N}^{\mathbb{N}},\not\ni)$
implies $\diamondsuit(\mathrm{bb}^{\infty}(E^{2}),=^{\infty})$, which in turn
implies $\diamondsuit(\mathbb{N}^{\mathbb{N}},=^{\infty})$. In fact,
$(\mathcal{M},\mathbb{N},\not\ni)\leq_{B}(\mathbb{N}^{\mathbb{N}},=^{\infty})\ast(\mathbb{N}^{\mathbb{N}},=^{\infty})$
(Proposition 4.14 in [7]),666Curiously, a result of Zapletal [15] implies that
$(\mathcal{M},\mathbb{N},\not\ni)\not\leq_{B}(\mathbb{N}^{\mathbb{N}},=^{\infty})$.
where the latter is the (Borel) sequential composition of the invariant
$(\mathbb{N}^{\mathbb{N}},=^{\infty})$ with itself (cf. Definition 4.10 in [3]
or §4.1 of [7]). As a result, in any canonical model, the principles
$\diamondsuit(\mathbb{N}^{\mathbb{N}},=^{\infty})$,
$\diamondsuit(\mathrm{bb}^{\infty}(E^{2}),=^{\infty})$, and
$\diamondsuit(\mathrm{non}(\mathcal{M}))$, are equivalent.777The question of
whether apparently different $\diamondsuit$ principles which have the same
evaluation are _actually different_ remains, stubbornly, open.

Lastly, we briefly recall the description of the Miller model: Let
$\mathbb{Q}$ denote _Miller forcing_ [11], the set of all trees
$q\subseteq\mathbb{N}^{<\mathbb{N}}$ such that for each $t\in q$, the
corresponding branching set $\\{n:t^{\smallfrown}(n)\in q\\}$ is infinite
(such trees are called _rational_ or _superperfect_). We order $\mathbb{Q}$ by
containment. By the _Miller model_ we mean the result of an
$\omega_{2}$-length countable support iteration of Miller forcing over a model
$\mathsf{CH}$. The relevant fact about the Miller model is:

######  Theorem 2.5 (cf. Theorem 7.3.46 in [2]).

$\mathrm{non}(\mathcal{M})=\aleph_{1}$ in the Miller model.

Since the Miller model is canonical, Theorems 2.5 and 2.2 imply:

######  Corollary 2.6.

$\diamondsuit(\mathrm{bb}^{\infty}(E^{2}),=^{\infty})$ holds in the Miller
model.∎

##  3\. The result

We can now state and prove our main result.

######  Theorem 3.1.

$\diamondsuit(\mathrm{bb}^{\infty}(E^{2}),=^{\infty})$ implies
$\mathfrak{a}_{\mathrm{vec},F}=\aleph_{1}$.

We will need the following lemma, which in the $n=0$ case is just a
restatement of Lemma 2.3 in [14]. The general case can be obtained by applying
this fact repeatedly.

######  Lemma 3.2.

Let $A_{0},\ldots,A_{n}$ be block subspaces of $E$. For any $m\in\mathbb{N}$,
there is an $M\in\mathbb{N}$ such that for any finite-dimensional subspace $V$
of $E$ with $\mathrm{supp}(x)\subseteq[0,m]$ for all $x\in V$, if $y>M$, then
for all $i\leq n$,

$(V+\langle y\rangle)\cap A_{i}=V\cap A_{i}\quad\text{if and only if}\quad
y\notin A_{i}.\qed$

###### Proof of Theorem 3.1..

To define $F:2^{<\omega_{1}}\to\mathrm{bb}^{\infty}(E^{2})$, by a standard
coding, we will instead define $F$ on pairs of the form
$((A_{\xi})_{\xi<\delta},B)$ where $\\{A_{\xi}:\xi<\delta\\}$ is an almost
disjoint family of (possibly finite-dimensional) block subspaces of $E$,
$\omega\leq\delta<\omega_{1}$, and $B$ an infinite-dimensional subspace of $E$
which is almost disjoint from every $A_{\xi}$. On inputs not of this type, we
will take $F$ to be some arbitrary constant value in
$\mathrm{bb}^{\infty}(E^{2})$. $F$ is clearly Borel.

Fix bijections $e_{\delta}:\omega\to\delta$ for each
$\omega\leq\delta<\omega_{1}$. Given $((A_{\xi})_{\xi<\delta},B)$ as above,
begin by letting $F((A_{\xi})_{\xi<\delta},B)(0)=(x_{0}^{0},x_{0}^{1})$ to be
the least pair of nonzero vectors in $B$ with $x_{0}^{0}<x_{0}^{1}$.
Continuing recursively, suppose we have defined
$F((A_{\xi})_{\xi<\delta},B)(i)$ for $i<n$. Define
$F((A_{\xi})_{\xi<\delta},B)(n)=(x_{n}^{0},x_{n}^{1})$ as follows: Let
$x_{n}^{0}$ be the least $x\in B$ above $x_{n-1}^{1}$. By Lemma 3.2, choose an
$M\geq\max(\mathrm{supp}(x_{n}^{0}))$ such that for any subspace $V$ of $E$
with supports contained in $[0,\max(\mathrm{supp}(x_{n}^{0}))]$, if
$y\notin\bigcup_{i<n}A_{e_{\delta}(i)}$ and $y>M$, then

$(V+\langle y\rangle)\cap A_{e_{\delta}(i)}=V\cap A_{e_{\delta}(i)}.$

Let $x_{n}^{1}$ be the least $y\in B$ above $M$ and such that
$y\notin\bigcup_{i<n}A_{e_{\delta}(i)}$. This can be arranged since $B$ is
almost disjoint from each $A_{e_{\delta}(i)}$, for $i<n$.

######  Remark.

One might expect that we would define
$F((A_{\xi})_{\xi<\delta},B)\in\mathrm{bb}^{\infty}(E)$, choosing the $n$th
coordinate in $B$ so that it does not effect the intersection of the subspace
we’re building with $A_{e_{\delta}(i)}$ for $i<n$, and that any block sequence
infinitely often equal it will suffice for the rest of the argument. However,
vectors in such a sequence may fail to be far enough “above” previous vectors.
It is for this reason that we choose _two_ vectors at each stage. The first,
$x_{n}^{0}$, acts as a firewall, preventing earlier vectors in any element of
$\mathrm{bb}^{\infty}(E^{2})$ whose $n$th entry coincides with the pair
$(x_{n}^{0},x_{n}^{1})$ from getting too close to the second vector,
$x_{n}^{1}$, which we will actually use.

Let $g$ be a $\diamondsuit(\mathrm{bb}^{\infty}(E^{2}),=^{\infty})$-sequence
for $F$. For each $n\in\mathbb{N}$, denote $g(\delta)(n)$ by
$(g(\delta)_{n}^{0},g(\delta)_{n}^{1})$. We define
$\mathcal{A}=\\{A_{\delta}:\delta<\omega_{1}\\}$ recursively. First, let
$(A_{n})_{n<\omega}$ be any sequence of disjoint, infinite-dimensional block
subspaces. Having defined $(A_{\xi})_{\xi<\delta}$ for
$\omega\leq\delta<\omega_{1}$, we define
$A_{\delta}=\langle(w_{n})_{n\in\mathbb{N}}\rangle$ where
$(w_{n})_{n\in\mathbb{N}}$ is a block sequence chosen as follows: First, set
$w_{0}=g(\delta)_{0}^{1}$. Having defined
$w_{0},\ldots,w_{n-1}=g(\delta)_{k_{n-1}}^{1}$, we scan across the sequence
$g(\delta)$, starting at index $k_{n-1}+1$, “attempting” to set
$w_{n}=g(\delta)_{k}^{1}$. We do so when we find the least $k\geq k_{n-1}+1$
such that for all $i<n$,

$\langle w_{0},\ldots,w_{n-1},g(\delta)_{k}^{1}\rangle\cap
A_{e_{\delta}(i)}=\langle w_{0},\ldots,w_{n-1}\rangle\cap A_{e_{\delta}(i)},$

if such a $k$ exists. If such a $k$ does not exist, then $w_{m}$ remains
undefined for all $m\geq n$, i.e., $A_{\delta}$ is a finite-dimensional space.
In any case, $A_{\xi}\cap A_{\delta}$ is finite-dimensional for all
$\xi<\delta$.

Consider the set $S=\\{\delta<\omega_{1}:A_{\delta}\text{ is infinite}\\}$. If
$B$ is such that $((A_{\xi})_{\xi<\delta},B)$ is as described in the first
paragraph above, and $g$ guesses $F((A_{\xi})_{\xi<\delta},B)$, then
$\delta\in S$. Since countable almost disjoint families of block subspaces
fail to be maximal, it follows that $S$ is stationary.

We claim that $\mathcal{A}=\\{A_{\delta}:\delta\in S\\}$ is a mad family of
subspaces. Towards a contradiction, suppose that $B$ is an infinite-
dimensional subspace almost disjoint from each $A_{\delta}$, for $\delta\in
S$. Then, $g$ guesses $F((A_{\xi})_{\xi<\delta},B)$ at some
$\delta\geq\omega$, and by the preceding paragraph, $\delta\in S$. Let
$k\in\mathbb{N}$ be such that
$g(\delta)(k)=F((A_{\xi})_{\xi<\delta},B)(k)=(x_{k}^{0},x_{k}^{1})$ and
suppose that $n$ is the first value for which we “attempt” to set
$w_{n}=g(\delta)_{k}^{1}$. In particular, $n\leq k$. By definition of
$\mathrm{bb}^{\infty}(E^{2})$, we know that $w_{0},\ldots,w_{n-1}$ have
supports contained in $[0,\max(\mathrm{supp}(x_{k}^{0}))]$, and thus
$x_{k}^{1}$ has the property that for all $i<k$, and in particular, for $i<n$,

$\langle w_{0},\ldots,w_{n-1},x_{k}^{1}\rangle\cap A_{e_{\delta}(i)}=\langle
w_{0},\ldots,w_{n-1}\rangle\cap A_{e_{\delta}(i)},$

and so we must have set $w_{n}=x_{k}^{1}$. Since this happens for all values
of $k$ for which $g(\delta)(k)=F((A_{\xi})_{\xi<\delta},B)(k)$, it follows
that $A_{\delta}$ has infinite-dimensional intersection with $B$, contrary to
our supposition. ∎

Combining this with Corollary 2.6, we have:

######  Corollary 3.3.

$\mathfrak{a}_{\mathrm{vec},F}=\aleph_{1}$ in the Miller model. ∎

As mentioned in the introduction, our Theorem 3.1, together with results from
[12], [6], and [9], respectively, implies that
$\mathfrak{a}_{\mathrm{vec},F}$, $\mathfrak{a}_{e}$, $\mathfrak{a}_{g}$, and
$\mathfrak{a}_{p}$ are all equal to $\mathrm{non}(\mathcal{M})$ in canonical
models, and in particular, to each other. While these cardinals can be
separated from $\mathrm{non}(\mathcal{M})$ using Shelah’s technique [13] of
template iterations, see [4], it would be very interesting to find a model in
which _some_ of $\mathfrak{a}_{e}$, $\mathfrak{a}_{p}$, $\mathfrak{a}_{g}$,
and $\mathfrak{a}_{\mathrm{vec},F}$ (for some $F$) are unequal. This question
was asked for $\mathfrak{a}_{p}$ and $\mathfrak{a}_{g}$ in [9], and we believe
that it deserve renewed attention.

## References

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  * [7] N. Greenberg, R. Kuyper, and D. Turetsky.  Cardinal invariants, non-lowness classes, and Weihrauch reducibility.  Submitted.
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# First Place Solution of 2023 Global Artificial Intelligence Technology
Innovation Competition Track 1

Xiangyu Wu1 Hailiang Zhang1 Yang Yang1 Jianfeng Lu1 1 Nanjing University of
Science and Technology
<EMAIL_ADDRESS>
###### Abstract

In this paper, we present our champion solution to the Global Artificial
Intelligence Technology Innovation Competition Track 1: Medical Imaging
Diagnosis Report Generation. We select CPT-BASE as our base model for the text
generation task. During the pre-training stage, we delete the mask language
modeling task of CPT-BASE and instead reconstruct the vocabulary, adopting a
span mask strategy and gradually increasing the number of masking ratios to
perform the denoising auto-encoder pre-training task. In the fine-tuning
stage, we design iterative retrieval augmentation and noise-aware similarity
bucket prompt strategies. The retrieval augmentation constructs a mini-
knowledge base, enriching the input information of the model, while the
similarity bucket further perceives the noise information within the mini-
knowledge base, guiding the model to generate higher-quality diagnostic
reports based on the similarity prompts. Surprisingly, our single model has
achieved a score of 2.321 on leaderboard A, and the multiple model fusion
scores are 2.362 and 2.320 on the A and B leaderboards respectively, securing
first place in the rankings.

##  1 Introduction

Medical imaging diagnosis report generation [5, 13, 21, 3] is an important
research in the field of medical artificial intelligence. With the development
of natural language processing [17, 16, 25, 19] technology, it has become
feasible to automatically generate medical imaging diagnostic reports. The
traditional diagnostic process relies on the professional knowledge and
experience of radiologists to write detailed diagnostic reports, a process
that is time-consuming and easily influenced by subjective factors.

Figure 1: The sample of the dataset. Clinical and Description are denoted as
the inputs, while Diagnosis is the output. Note that in the competition, all
words have undergone desensitization processing, which means that the text is
desensitized at the character level, separated by spaces (e.g., 88 29 17 55
72).

In recent years, with the advancement of artificial intelligence technology
[24, 29, 4, 23], researchers have begun to explore how to use deep learning
models [30, 18, 12] to automatically generate accurate diagnostic reports.
These models typically generate natural language descriptions based on
sequence generation models such as Recurrent Neural Networks (RNN) [31, 33,
28] or Transformers [22, 14, 27, 26] and require a large amount of annotated
data, including descriptions of medical imaging diagnoses and corresponding
expert-written diagnostic reports.

Due to data desensitization, it is not feasible to directly fine-tune open-
source pre-trained models. Therefore, we select Chinese CPT-BASE [20] as our
base model and append the desensitized numbers to the vocabulary of the pre-
trained model. To reduce the gap between pre-train and downstream tasks, we
remove the Masked Language Modeling task from CPT-BASE and adopt a span mask
[11] strategy to perform the Denoising Auto-Encoding pre-training task, with
an increasingly larger mask ratio to enlarge the difficulty of the pre-
training tasks. This is beneficial for the pre-training tasks to capture
deeper levels of textual context information.

In the fine-tuning stage, we note the rapid development of retrieval
augmentation [6, 9, 7] strategies in the field of natural language processing.
Therefore, we introduce and improve retrieval augmentation technology to adapt
to the competition. For each input sample, we use the embedding of Description
as the query and key to retrieve similar Description Diagnosis pairs, adding
them to the input as a mini-knowledge base for the sample, enriching the input
information of the model. Additionally, we originally design a noise-aware
similarity bucketing prompt strategy, distributing the training data into
different buckets according to their noise levels. Different buckets represent
different quantities and qualities. This training method can force the model
to generate higher-quality diagnostic reports during the inference stage. We
achieve first place with a score of 2.320 on the final leaderboard through
additional general tricks (i.e., FGM, R-Dropout, EMA, and Model Ensemble).

##  2 Related Works

###  2.1 Text Generation in NLP

Text generation [34, 10, 8, 32, 1] is a widely researched direction in the
field of natural language processing. It involves using computer systems to
generate text that resembles human language, and it has extensive applications
in areas such as machine translation, intelligent customer service, literary
creation, and more. In recent years, significant progress has been made in
text generation techniques, which mainly include text summarization and text
generation. In the field of text summarization, algorithms based on TF-IDF and
sequence-to-sequence (Seq2Seq) models from deep learning are commonly used
methods. TF-IDF measures the importance of words in documents, while Seq2Seq
models consist of an encoder and a decoder for generating summaries.
Additionally, the large language model based on the Transformer architecture
in natural language generation has also gained increasing attention. It
implements text generation through an attention mechanism and encoder-decoder
structure.

###  2.2 Retrieval Augmentation in NLP

Retrieval Augmented Generation (RAG) technology [6, 9, 7, 15, 2] in the field
of natural language processing represents an innovative breakthrough.
Traditional NLP techniques primarily rely on large language models, but their
accuracy and depth may be limited when dealing with complex queries that
require extensive background knowledge. To overcome this limitation, RAG
combines conventional information retrieval methods with modern generative
language models, aiming to enhance the model’s text generation capabilities by
incorporating external knowledge sources. The core principle is to integrate
retrieval and generation techniques, allowing the model to access and utilize
a vast amount of external information before generating text. RAG excels in
addressing knowledge-intensive NLP tasks such as question answering, fact
verification, and more. In recent years, RAG systems have evolved from a
primary stage to an advanced stage, and then to a modular stage, to improve
performance, cost-effectiveness, and efficiency.

##  3 Methods

###  3.1 Pre-training Stage

Figure 2: The sample of the dataset. Clinical and Description are denoted as
the inputs, while Diagnosis is the output. Note that in the competition, all
words have undergone desensitization processing, which means that the text is
desensitized at the character level, separated by spaces (e.g., 88 29 17 55
72).  Figure 3: The strategy of Retrieval Augmentation.

Task definition: We define Clinical information as $\mathcal{C}$, Description
information as $\mathcal{D}$, and the output Diagnostic report as
$\mathcal{O}$. Therefore, this competition can be regarded as a Diagnostic
report generation task conditional on Clinical and Description information,
i.e., $\mathbf{F}(\mathcal{C},\mathcal{D})\longrightarrow\mathcal{O}$, where
$\mathbf{F}$ is the encoder-decoder language model.

As shown in Figure 2, we select Chinese CPT-Base as the base model for text
generation, which consists of 12 layers of transformers as the encoder and 2
layers of transformers as the decoder. The size of the original vocabulary of
the CPT model is 51271. We sequentially append the anonymized numbers of this
competition to the end of the vocabulary and remove the numbers that already
exist in the original vocabulary, resulting in a new vocabulary size of
$\mathrm{51271+347}$.

In the pre-training stage, we remove the MLM (Masked Language Modeling) pre-
training task from the CPT model and retain only the DAE (Denoising Auto-
encoder) task for pre-training. This is done to maintain consistency between
the pre-training task and the downstream task, reducing the gap between them.
We concatenate $\mathcal{C}$ (Clinical), $\mathcal{D}$ (Description), and
$\mathcal{O}$ (Diagnostic report), and use the $[\mathbf{SEP}]$ token to
separate them, resulting in the input $[\mathcal{C},\mathcal{D},\mathcal{O}]$
for the pre-training stage.

Regarding the MASK strategy, we notice that the text content in $\mathcal{C}$,
$\mathcal{D}$, and $\mathcal{O}$ usually appears in chunked form, so we chose
the span mask strategy as our final masking approach. We use a Poisson
distribution to generate the mask length, biasing it towards smaller values to
match the characteristics of text length. Furthermore, we find that the pre-
training in the first competition stage could train for 150 epochs, while in
the second competition stage, the model training saturated at 40 epochs. This
inspired us to increase the difficulty of the pre-training task, gradually
increasing the proportion of masking as the number of epochs increased.
Specifically, we set an initial mask proportion of 0.3, and after every 10
epochs of pre-training, we perform fine-tuning of the downstream task. If the
performance of the fine-tuning is lower than the previous one, we increase the
mask proportion by 0.05 and continue with pre-training. Ultimately, we
increase the number of pre-training epochs to 140, which significantly
improves the text generation performance of the downstream task.

Figure 4: The strategy of Noise-aware Similarity bucketing Prompt.

###  3.2 Fine-tuning Stage

####  3.2.1 Retrieval Augmentation.

Figure 3 illustrates our iterative retrieval augmentation strategy, which
consists of three main parts: the construction of the retrieval knowledge
base, nearest neighbor retrieval, and retrieval iterations.

Retrieval knowledge base. We divide the entire training set into training and
validation sets in a $9:1$ ratio, where the training set is used to construct
the retrieval knowledge base, and the validation set is used for testing the
performance of the model. For each sample in the knowledge base, we extract
the embedding of $\mathcal{D}$(Description) as the key and the original
$\mathcal{O}$(Diagnostic report) corresponding to the
$\mathcal{D}$(Description) as the value. Therefore, each sample in the
training set can form a key-value pair.

Nearest Neighbor Retrieval. For the construction of the training set with
retrieval knowledge, we use $\mathcal{D}$(Description) as the query and
calculate the similarity with the key of each key-value pair in the knowledge
base (e.g., vector inner product, L2 distance, or cosine similarity). If the
similarity is larger than the threshold k, we call it an effective retrieval.
We retrieve this key-value pair and concatenate the value to the end of the
query as the new training sample corresponding to the query. For the val set
and test set, we use the same retrieval method to construct the val set and
test set with retrieval knowledge.

Retrieval Iterations. For the first retrieval augmentation, the embeddings of
key-value pairs are computed using a model trained on a training set without a
knowledge base. However, as retrieval augmentation progresses, the performance
of the model also gradually improves, which suggests that we can use the
augmented model to recalculate the embeddings of key-value pairs. This not
only results in more accurate representations of the embeddings but also
improves the accuracy of retrieval. Therefore, we design an iterative
retrieval augmentation strategy that uses the augmented model to continue the
retrieval augmentation process, iteratively training an even better model.

####  3.2.2 Similarity Bucketing.

It is worth noting that with each iteration of retrieval augmentation, a new
retrieved Diagnostic report will be added to the end of the sample. After n
iterations, at least 0 pseudo Diagnostic reports will be added, but up to n
pseudo Diagnostic reports could be appended. While more Diagnostic reports can
bring a greater diversity of information, it also means that the sample will
contain more noisy information. So, how can we add as many pseudo Diagnostic
reports as possible while also ensuring that the model is influenced as little
as possible by the noise?

As shown in Figure 4, we innovatively design a noise-aware similarity
bucketing prompt strategy. For each sample in the training set, we calculate
the similarity between the input and output, where the input refers to
$\mathcal{C}$ (Clinical), $\mathcal{D}$ (Description), and retrieved
$\mathcal{O}$ (Diagnostic reports), while the output is the corresponding
$\mathcal{O}$ (Diagnostic report) label for the sample. Based on this
similarity, we divide the training set into n buckets, with the first bin
representing samples with high similarity and the last bin representing
samples with low similarity. We believe that higher similarity indicates that
the retrieved diagnostic reports are more similar to the labeled diagnostic
report, meaning it is a high-quality sample. To integrate the bucket signal
into the model’s input, we use [’best match’, ’good match’, ’not good match’,
’noisy match’] to represent different buckets, and add these signals to the
front of the $\mathcal{C}$ (Clinical) information.

The distribution of dataset similarity shows a normal distribution, where
higher similarity indicates that the sample $\mathcal{D}$ (Description) and
$\mathcal{O}$ (Diagnostic reports) are highly relevant. Lower similarity
suggests that the $\mathcal{D}$ (Description) and $\mathcal{O}$ (Diagnostic
reports) are less relevant, and it is likely to be a noisy sample. Each sample
belongs to a certain bin, and through training, each sample is associated with
the prompt of its bin. During inference, we fix the prompt to ’best match’,
forcing the model to generate the most similar, best-matched, and highest-
quality $\mathcal{O}$ (Diagnostic reports).

###  3.3 Model Tricks

FGM. Introducing noise to the embeddings during training and regularizing the
model parameters can enhance the robustness and generalization ability of the
model.

R-Dropout. R-Dropout applies regularization constraints to the output
predictions of different combinations of neurons, thereby improving the
model’s robustness and generalization.

EMA. Averaging the weights of the model at different times makes the weight
updates smoother, enhancing the model’s generalization and stability.

Model Ensemble. From the predictions of n models, one is selected as the
candidate answer, and the rest are used as references. The CIDEr score between
the candidate answer and all references is calculated, and the total score is
used as the score for that candidate answer. The candidate answer with the
highest score is selected as the final integrated answer.

##  4 Experiments

###  4.1 Dataset.

The training set for the first stage of the competition consists of 20,000
samples, while the Test Set A/B each has 3,000 samples. In the second stage,
the training set comprises 80,000 samples, and the Test Set A/B each has 7,500
samples. Clinical information data is only provided in the second stage.

###  4.2 Leadboards.

Method | Cider | Bleu | Score
---|---|---|---
Baseline | 3.0793 | 0.4043 | 2.1876
Span Mask | 3.1446 | 0.4058 | 2.2317
Retrieval-1 | 3.2130 | 0.4241 | 2.2834
Retrieval-2 | 3.2374 | 0.4291 | 2.3013
Bucketing | 3.2553 | 0.4288 | 2.3132
Tricks | 3.2735 | 0.4342 | 2.3271
Ensemble | 3.3242 | 0.4384 | 2.3622
Table 1: Results of each component.

Table 1 shows the improvement in model performance by each of our components.
It can be seen that the SPAN mask strategy with an increasing mask ratio and
the first retrieval strategy significantly improved text generation
performance. The score of the single model also reached 2.3271, surpassing the
scores of most teams’ model ensembles. In the end, we ensemble 10 CPT-Base
models, achieving scores of 2.362 and 2.320 on Leaderboards A and B,
respectively, securing first place in the final competition.

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