# The Unusual AGN Host NGC 1266: Evidence for Shocks in a Molecular Gas Rich
S0 Galaxy with a Low Luminosity Nucleus

Peibin Chen  Department of Astronomy, Xiamen University, Xiamen, Fujian
361005, China<EMAIL_ADDRESS>Yinghe Zhao  Yunnan Observatories, Chinese
Academy of Sciences, Kunming 650011, China<EMAIL_ADDRESS>Junfeng
Wang  Department of Astronomy, Xiamen University, Xiamen, Fujian 361005,
China<EMAIL_ADDRESS>
###### Abstract

NGC 1266 is a lenticular galaxy (S0) hosting an active galactic nucleus (AGN),
and known to contain a large amount of shocked gas. We compare the luminosity
ratio of mid-_J_ CO lines to IR continuum with star-forming galaxies (SFGs),
and then model the CO spectral line energy distribution (SLED). We confirm
that in the mid- and high-_J_ regions ($J_{\rm up}$ = 4–13), the C-type shock
($v_{\rm s}$ = 25 km s-1, $n_{\rm H}$ = $5\times 10^{4}$ cm-3) can reproduce
the CO observations well. The galaxy spectral energy distribution (SED) is
constructed and modeled by the code X-CIGALE and obtains a set of physical
parameters including the star formation rate (SFR, 1.17 $\pm$ 0.47 _M ⊙_yr-1).
Also, our work provides SFR derivation of [C ii] from the neutral hydrogen
regions only (1.38 $\pm$ 0.14 $M_{\odot}$yr-1). Previous studies have illusive
conclusions on the AGN or starburst nature of the NGC 1266 nucleus. Our SED
model shows that the hidden AGN in the system is intrinsically low-luminosity,
consequently the infrared luminosity of the AGN does not reach the expected
level. Archival data from _NuSTAR_ hard X-ray observations in the 3–79 keV
band shows a marginal detection, disfavoring presence of an obscured luminous
AGN and implying that a compact starburst is more likely dominant for the NGC
1266 nucleus.

galaxies: elliptical and lenticular, cD - galaxies: active - galaxies:
individual (NGC 1266) - galaxies: ISM - galaxies: star formation

Accepted by Research in Astronomy and Astrophysics

##  1 Introduction

The morphology of galaxies plays an key role in the galaxy formation and
evolution (e.g., morphological quenching; Martig et al. 2009). The Hubble
tuning-fork diagram divides galaxies into early-type galaxies (ETGs) and late-
type galaxies (LTGs), and in between is their bridge–S0 galaxies. S0 galaxies
appear as a central bulge surrounded by a disk, but they do not show the
spiral patterns (Freeman, 1970; Sandage & Visvanathan, 1978). Its bulge is
almost as large as the disk, so the galaxy appears almost spherical in shape.
ETGs generally contain all types of ellipticals and S0 galaxies (e.g., Xu et
al. 2022). It is believed that they are short of atomic and molecular gas
(Lees et al., 1991), but their disk galaxy progenitors are generally gas-rich
(Kauffmann et al., 2003). The conversion of gas to stars in ETGs is related to
a series of key processes in galaxy evolution, e.g., starburst and AGN
activities.

ETGs are generally redder than LTGs (Sandage & Visvanathan, 1978), and contain
little or no ongoing star formation (SF). Nevertheless, atomic gas can exist
in ETGs, and dust and molecular gas have been observed as well (e.g., Colbert
et al. 2001; Combes et al. 2007). Their related SF activities have been
revealed through the continuum (Leroy et al., 2008) or the emission lines
diagnosis (e.g., Alatalo et al. 2015; Xiao et al. 2016). Some ETGs have higher
gas surface density ($\Sigma_{\rm gas}$) than normal spirals (Davis et al.,
2014), yet their SFR surface density ($\Sigma_{\rm SFR}$) is significantly
lower than expected by the Kennicutt-Schmidt law (Schmidt, 1959; Kennicutt,
1998). Moreover, studies have shown that different environments may have
various effects on the formation and evolution of S0 galaxies (e.g., Deeley et
al. 2021; Xu et al. 2022). Therefore, the studies of S0 galaxies not only help
to understand the quenching or rejuvenation of SF activities in ETGs, but also
can shed light on the nature of the SF law in ETGs (e.g., Xu et al. 2022).

The nearby S0 galaxy NGC 1266 has a luminosity distance of $D_{\rm L}$ = 30.6
Mpc (_z_ = 0.0072, Temi et al. 2009). It contains 1.1$\times$109 M⊙ molecular
gas within a radius of 60 pc (Alatalo et al., 2011), a significant fraction of
which is outflowing. As one of the ETGs that have been mapped
interferometrically111Based on millimeter interferometric IRAM PdBI and
optical integral field unit SAURON observations. ($\sim$ 20 galaxies, Crocker
et al. 2011), the molecular gas in NGC 1266 is remarkable due to its
compactness and the massive outflow. For comparison, previous work found the
existence of a small sample of galaxies (e.g., M82, 3C 293, Mrk 231) with
molecular outflows (e.g., Walter et al. 2002; Emonts et al. 2005; Feruglio et
al. 2010). The molecular outflows were thought to be triggered by mergers or
interactions. Compared to this sample, it is intriguing that the isolated
galaxy NGC 1266 does not show any signs of interaction or merger activity
(e.g., Alatalo et al. 2011).

A series of studies are reported in the literature for NGC 1266, including its
molecular gas properties (e.g., Alatalo et al. 2011; Glenn et al. 2015), SF
processes (e.g., De Looze et al. 2014; Alatalo et al. 2015) and gas excitation
mechanisms (e.g., Davis et al. 2012; Pellegrini et al. 2013). Among these,
Alatalo et al. (2011) confirmed that the existing molecular outflow is driven
by the mechanical energy of the AGN. The derived mass loss rate (110 _M ⊙_
yr-1) was at least 100 times that of the SFR (Alatalo et al., 2015). The
derivation of SFR, however, has many limitations in this system. This is
because of the presence of AGN in NGC 1266, and it also contains a large
amount of shocked gas (Davis et al., 2012). Alatalo et al. (2015) obtained the
SFR through the continuum or the ionized gas emission lines and they suggested
that 0.3 _M ⊙_yr-1 (polycyclic aromatic hydrocarbons, PAHs + UV) is the lower
limit of SFR. Davis et al. (2012) used different methods (line ratio
diagnostics and large-scale velocity structure) to prove that the ionized gas
emission line ratio in the system is consistent with the result of the shock
ionization. Pellegrini et al. (2013) and Glenn et al. (2015) analyzed the CO
transitions with the existing models and obtained evidence that the shock
dominates the gas excitation. Moreover, the dense gas observations (e.g.,
CS(2-1) and HCN(1-0), Alatalo et al. 2015) confirmed that the line-of-sight
gas column density towards the AGN is Compton-thick (Alatalo et al. 2015,
$N_{\rm H}\sim 10^{24}{\rm cm}^{-2}$). These previous studies, however, are
inconclusive on the nature of activity in the center of NGC 1266 (a
radiatively powerful AGN or an ultra-compact starburst, e.g., De Looze et al.
2014; Alatalo et al. 2015).

In this work, we present the analysis of infrared (IR) and sub-millimeter
(sub-mm) observations of NGC 1266, investigating its SFR, the CO spectral line
energy distribution (SLED) and the nature of its nucleus. This paper is
organized as follows. In section 2, we describe the archival data in this
paper, as well as the modeling process. In section 3, we discuss the diagnosis
of the energy sources and the SFR derivation. Consequently, based on the model
fitting of CO SLED, we confirm that shock dominates the mid- to high-_J_ CO
excitation ($J_{\rm up}$ = 4 – 13). We further validate this result by
calculating the luminosity ratio of mid-_J_ CO lines ($J_{\rm up}$ = 4 – 10)
to IR continuum, $R_{\rm mid\ CO}$ value (Lu et al., 2014). Finally, the SED
of the galaxy is constructed and modeled by the code X-CIGALE. In addition,
this work also compares other commonly used methods (e.g., IR continuum, [C
ii]) for SFR derivation. In sections 4 and 5, we discuss the activity within
NGC 1266 and summarize our work. In the Appendix A, we provide the SED fitting
parameter settings.

##  2 Archival DATA AND Modeling Process

###  2.1 Archival Data

_Herschel_ Spectral and Photometric Imaging Receiver (SPIRE) and Photodetector
Array Camera and Spectrometer (PACS) observations of NGC 1266 are collected.
We obtain the mid- and high-_J_ CO data from the program _Beyond The Peak_
(BTP; OT1$\\_$jsimth; PI: J. D. Smith), which conducted a survey on 21 nuclear
regions and 2 extra-nuclear regions from the _SINGS_ (Kennicutt et al., 2003)
and _KINGFISH_ (Kennicutt et al., 2011)) surveys. It completed deep,
intermediate-spaced mapping of transitions above CO(3-2) in these nearby
galaxies. Also, as part of the _Herschel_ key program _KINGFISH_ , NGC 1266
has been imaged in the emission lines (e.g., [C ii], [N ii] and [O iii]). The
data analysis is previously presented in Croxall et al. (2012). Our CO data
($J_{\rm up}$ = 4 – 13) come from Pellegrini et al. (2013) (listed in Table
1), and the ionized emission lines data ([C ii] and [N ii] 205 $\rm{\mu m}$)
come from Sutter et al. (2019).

We further obtain archival CO data ($J_{\rm up}$ = 1 – 3) observed with _IRAM_
30 m telescope and Sub-millimeter Array (_SMA_). NGC 1266 is part of the
CO(1-0), (2-1) survey of a flux-limited sample of all ATLAS3D galaxies (Young
et al., 2011), observed with _IRAM_ 30 m telescope. The data consist of a
single pointing to the galaxy center, covering a velocity range of 1300 km s-1
for the CO spectra. The spectral resolutions for CO(1-0) and CO(2-1) are 2.6
km s-1 and 5.2 km s-1, respectively, with 2155 km s-1 as the velocity center
and 4.73 Jy K-1 as the conversion factor (Alatalo et al., 2011). In addition,
the CO(3-2) was observed by _SMA_ at 345 GHz, with an imaging bandwidth of
2600 km s-1 and a spectral resolution of 0.7 km s-1 (centered on the system
velocity). A detailed description of the data can be found in Alatalo et al.
(2011). Our lowest-$J$ (1-0,2-1,3-2) CO transitions listed in Table 1 are
adopted from Alatalo et al. (2011).

Finally, we also utilized unpublished X-ray data from _NuSTAR_ (Lanz et al.,
2016). If the gas column density is indeed as high as $6\times 10^{24}$ cm-2
(Alatalo et al., 2015), the hidden Compton-thick AGN could still be detected
by hard X-ray observations that are sensitive to $>10$ keV. We used the 3–79
keV archival data from _NuSTAR_ hard X-ray observation (obsID 60101048002 with
51.5 ks exposure time, PI: L. Lanz) to constrain this. We run the NuSTAR Data
Analysis Software (NuSTARDAS v2.0.0, Madsen et al. 2020) with the updated
calibration files (CALDB version 20220413) to obtain pipeline processed data.
Our analysis shows that the detected counts on both Focal Plane Module A and B
(FPMA and FPMB) detectors give a net count rate $1.8\pm 0.6\times 10^{-3}$,
indicating a marginal 3$\sigma$ detection. If we accept the large column
density of $6\times 10^{24}$ cm-2 from dense molecular gas confirmed by
Alatalo et al. (2015), using the PIMMS222https://heasarc.gsfc.nasa.gov/cgi-
bin/Tools/w3pimms/w3pimms.pl tool and adopting a power law continuum with a
typical photon index $\Gamma=1.8$, the estimated intrinsic 2–10 keV luminosity
is $4.50\times 10^{40}$ erg s-1. In addition, we adopt the multi-wavelength
photometric data from the _NASA/IPAC Extragalactic Database_
333https://ned.ipac.caltech.edu (NED, listed in Table 2) to construct the SED
of NGC 1266.

Table 1: The measurements of the CO line flux in NGC 1266

Linesa | Trans |  Wave ($\rm{\mu m}$) |  | Flux
---
(10-18 w m-2)
Telescope (Beam size)
${}^{12}{\rm C}{\rm O}$ | 1–0 | 2600.80 |  0.63 ($\pm$ 0.04) |  $IRAM$ 30 m (21.6′′)
${}^{12}{\rm C}{\rm O}$ | 2–1 | 1300.40 |  3.90 ($\pm$ 0.15) |  $IRAM$ 30 m (12′′)
${}^{12}{\rm C}{\rm O}$ | 3–2 | 867.00 |  10.20 ($\pm$ 1.10) |  $SMA$ (30′′)
${}^{12}{\rm C}{\rm O}$ | 4–3 | 650.70 |  26.90 ($\pm$ 11.40) |  $Herschel$ (42.8′′)
${}^{12}{\rm C}{\rm O}$ | 5–4 | 520.60 |  38.40 ($\pm$ 6.90) |  $Herschel$ (35.2′′)
${}^{12}{\rm C}{\rm O}$ | 6–5 | 433.90 |  42.80 ($\pm$ 3.50) |  $Herschel$ (31.2′′)
${}^{12}{\rm C}{\rm O}$ | 7–6 | 371.90 |  47.60 ($\pm$ 2.80) |  $Herschel$ (35.9′′)
${}^{12}{\rm C}{\rm O}$ | 8–7 | 325.50 |  47.10 ($\pm$ 3.60) |  $Herschel$ (40.1′′)
${}^{12}{\rm C}{\rm O}$ | 9–8 | 289.30 |  46.40 ($\pm$ 6.50) |  $Herschel$ (19.0′′)
${}^{12}{\rm C}{\rm O}$ | 10–9 | 260.40 |  39.40 ($\pm$ 6.00) |  $Herschel$ (17.4′′)
${}^{12}{\rm C}{\rm O}$ | 11–10 | 236.80 |  40.30 ($\pm$ 6.10) |  $Herschel$ (17.3′′)
${}^{12}{\rm C}{\rm O}$ | 12–10 | 217.10 |  28.50 ($\pm$ 3.00) |  $Herschel$ (16.9′′)
${}^{12}{\rm C}{\rm O}$ | 13–12 | 200.40 |  26.50 ($\pm$ 5.80) |  $Herschel$ (16.6′′)

  * a

The ground CO lines (Jup = 1–3) are from Alatalo et al. (2011), and the rest
of CO (Jup = 4–13) is from Pellegrini et al. (2013) by _Herschel_
observations.

Table 2: The NGC 1266 Global Flux Densities in Janskys

Filters |  | $\lambda_{\rm eff}{\textsuperscript{a}}$
---
($\rm{\mu m}$)
| Flux (observed)
---
(Jy)
|  Obs$\\_$uncertaintyb
---
(Jy)
Aλ/AVc |  | Flux (corrected)
---
(Jy)
|  Corrected$\\_$uncertaintyb
---
(Jy)
GALEX$\\_$FUV | 0.152 |  4.90 $\times$ 10-5 |  7.00 $\times$ 10-6 | 0.78 |  0.10 $\times$ 10-3 |  0.85 $\times$ 10-6
GALEX$\\_$NUV | 0.231 |  2.90 $\times$ 10-4 |  4.00 $\times$ 10-5 | 0.90 |  0.63 $\times$ 10-3 |  0.49 $\times$ 10-5
B | 0.460 |  2.00 $\times$ 10-2 |  4.00 $\times$ 10-4 | 0.39 |  3.00 $\times$ 10-2 |  0.49 $\times$ 10-4
V | 0.538 |  3.60 $\times$ 10-2 |  4.00 $\times$ 10-4 | 0.305 |  5.00 $\times$ 10-2 |  0.49 $\times$ 10-4
R | 0.652 |  3.70 $\times$ 10-2 |  4.00 $\times$ 10-4 |  2.36 $\times$ 10-1 |  5.00 $\times$ 10-2 |  0.49 $\times$ 10-4
I | 0.866 |  3.50 $\times$ 10-2 |  4.00 $\times$ 10-4 |  1.73 $\times$ 10-1 |  4.00 $\times$ 10-2 |  0.49 $\times$ 10-4
2MASS$\\_$J | 1.20 |  1.20 $\times$ 10-1 |  4.40 $\times$ 10-4 |  0.89 $\times$ 10-1 |  1.30 $\times$ 10-1 |  0.53 $\times$ 10-4
2MASS$\\_$H | 1.67 |  1.30 $\times$ 10-1 |  6.14 $\times$ 10-4 |  0.56 $\times$ 10-1 |  1.40 $\times$ 10-1 |  0.75 $\times$ 10-4
2MASS$\\_$Ks | 2.20 |  1.20 $\times$ 10-1 |  5.86 $\times$ 10-4 |  0.35 $\times$ 10-1 |  1.20 $\times$ 10-1 |  1.26 $\times$ 10-4
IRAC1 | 3.56 |  5.50 $\times$ 10-2 |  8.00 $\times$ 10-3 |  1.35 $\times$ 10-2 |  0.60 $\times$ 10-1 |  0.97 $\times$ 10-3
IRAC2 | 4.51 |  4.20 $\times$ 10-2 |  6.00 $\times$ 10-3 |  0.86 $\times$ 10-2 |  0.40 $\times$ 10-1 |  0.73 $\times$ 10-3
IRAC3 | 5.76 |  5.70 $\times$ 10-2 |  8.00 $\times$ 10-3 |  0.58 $\times$ 10-2 |  0.60 $\times$ 10-1 |  0.97 $\times$ 10-3
IRAC4 | 7.69 |  9.00 $\times$ 10-2 |  1.20 $\times$ 10-2 |  0.89 $\times$ 10-2 |  0.90 $\times$ 10-1 |  0.15 $\times$ 10-2
WISE$\\_$W3 | 12.8 |  9.31 $\times$ 10-2 |  5.15 $\times$ 10-4 |  1.07 $\times$ 10-2 |  0.90 $\times$ 10-1 |  0.63 $\times$ 10-4
MIPS$\\_$24 | 23.80 |  8.80 $\times$ 10-1 |  4.00 $\times$ 10-2 |  0.58 $\times$ 10-2 |  8.80 $\times$ 10-1 |  0.49 $\times$ 10-2
MIPS$\\_$70d | 70 | 12.70 | 0.95 |  0.69 $\times$ 10-3 | 12.7 | 0.12
PACS$\\_$70d | 72 | 14.50 | 0.70 |  0.69 $\times$ 10-3 | 14.50 | 0.09
PACS$\\_$100 | 103 | 15.90 | 0.80 |  0.31 $\times$ 10-3 | 15.90 | 0.10
MIPS$\\_$160 | 159 | 10.30 | 0.13 |  0.12 $\times$ 10-3 | 10.30 | 0.02
PACS$\\_$160 | 160 | 11.30 | 0.60 |  0.12 $\times$ 10-3 | 11.30 | 0.07
SPIRE$\\_$250 | 250 | 3.61 | 0.54 |  0.60 $\times$ 10-4 | 3.61 | 0.07
SPIRE$\\_$350 | 350 | 1.48 | 0.23 |  0.21 $\times$ 10-4 | 1.48 | 0.03
SPIRE$\\_$500 | 500 |  4.15 $\times$ 10-1 |  7.03 $\times$ 10-2 |  0.12 $\times$ 10-4 |  4.20 $\times$ 10-1 |  0.85 $\times$ 10-2

  * a

The filter central wavelengths are computed via Equation (1) and (2) from Dale
et al. (2017), and they are arranged in ascending order.

  * b

The uncertainties include statistical and systematic effects.

  * c

Assuming an extinction to reddening ratio AV/E(B-V) = 3.1, E(B-V) = 0.098 mag
from Schlegel et al. (1998).

  * d

Shifting this effective wavelength to the modified value given by Brown et al.
(2014a).

###  2.2 Modeling Process

####  2.2.1 CO SLED Fitting

To diagnose the emission mechanism of the dominant spectral line in NGC 1266,
we combined with previous studies (Glenn et al., 2015; Israel et al., 2015)
and firstly used the non-local thermodynamic equilibrium (non–LTE) code RADEX
(van der Tak et al., 2007) to model the CO observations. There are many
parameters in this model, here we only focus on the non-default ones: kinetic
temperature ($T_{\rm kin}$), gas density ($n_{\rm H_{2}}$) and CO column
density ($N_{\rm CO}$). The statistical equilibrium calculation performed by
RADEX includes collisional and radiative processes. For CO molecular data
(Schöier et al. 2005444home.strw.leidenuniv.nl/moldata) and the collisional
rate coefficient with H2, we follow Yang et al. (2010) and assume that H2 is
the only collision counterpart. In our run, these parameters ranged from:
20–40 K ($T_{\rm kin}$), 103–105 cm-3 ($n_{\rm H_{2}}$), and 1014–1017 cm-2
($N_{\rm CO}$). The excitation temperature in the model should cover a wide
range, but we again performed verification work on the basis of previous work
(Glenn et al., 2015; Israel et al., 2015), so we narrowed this range. For a
wide range of input parameters, the code is able to generate a grid of model
CO SLED (e.g., Glenn et al. 2015).

Figure 1: The observed CO SLED (blue circles with error bars) with the model
fit. The best-fitting result is given by the black circles (reduced
$\chi^{2}\sim 2.0$). The parameters for the best fit are: $v_{\rm s}$ = 25
($\pm$ 5) km s-1, $n_{\rm H}$ = 5$\times$104 cm-3 (shock model); $T_{\rm kin}$
= 30 ($\pm$ 5) K, $n_{\rm H_{2}}$ = 1$\times$104 cm-3 (RADEX model). The
shadow areas show the optimal solutions of the C-type shock models (magenta,
the upper limit means $v_{\rm s}$ = 30 km s-1, and the lower limit means
$v_{\rm s}$ = 20 km s-1, and $n_{\rm H}$ = 5$\times$104 cm-3) and RADEX models
(yellow, the upper limit means $T_{\rm kin}$ = 35 K, and the lower limit means
$T_{\rm kin}$ = 25 K, and $n_{\rm H_{2}}$ = 1$\times$104 cm-3) for
observations. The observed CO lines flux is listed in Table 1.

Secondly, due to the existence of shocked gas in NGC 1266, we combined with
previous work (van der Werf et al., 2010; Spinoglio et al., 2012) and
considered using the shock code (Flower & Pineau des Forêts, 2015). The code
has a number of input files, including files that specify initial values for
dynamical and chemical variables, chemical reaction networks, data for
calculating atomic and molecular intensities, and variables that control the
precision of the numerical integration process (Flower & Pineau des Forêts,
2015). Combining with previous work (Draine et al., 1983; Lesaffre et al.,
2013; Meijerink et al., 2013; Flower & Pineau des Forêts, 2015),
parameters555The magnetic field intensity is related to the hydrogen number
density, which is defined as: $B(\mu G)=b[n_{\rm H}(\rm cm^{-3})]^{1/2}$. The
$n_{\rm H}$ is defined as $n_{\rm H}=n(\rm{H})+2n(\rm{H_{2}})+n(\rm{H^{+}})$.
of this model are set in our run as: C-type shock; the number of the fluid is
3 (for general C-type shock); 1.0–5.0 (magnetic field intensity $B$ = 101.5 –
5 $\times$ 107/2 $\mu G$); $10^{3}$–$10^{7}$ cm-3 (density of hydrogen nuclei,
$n_{\rm H}$); 10–50 km s-1 (shock velocity, $v_{\rm s}$). The large velocity
gradient (LVG) calculations are part of the model, which yield the density and
emission per unit volume in various molecular lines (Flower & Pineau des
Forêts, 2015).

The two models will produce a series of meshes in these parameter spaces.
Using a chi-square ($\chi^{2}$) fit between the CO observations and these
simulated values, we obtain the best fit of the observations (the black
circles in Figure 1). The best-fitting parameters are: $v_{\rm s}$ = 25
($\pm$5) km s-1, $n_{\rm H}$ = 5$\times$104 cm-3 (shock model); $T_{\rm kin}$
= 30 ($\pm$ 5) K, $n_{\rm H_{2}}$ = 1$\times$104 cm-3 (RADEX model).

####  2.2.2 SED Fitting

We use the X-CIGALE code (Yang et al., 2020) to construct its SED, which is
modified on the basis of the python code CIGALE (Boquien et al., 2019). In
fact, the model is constructed from many components: SFH, single-age stellar
populations (SSPs), ionized gas templates including lines and continuum,
flexible dust decay curves, dust emission templates, synchrotron emission, and
finally the intergalactic medium influences (Boquien et al., 2019). Each
component contains different modules and is independently computed by these
modules. For example, dust attenuation can be modeled alternatively with the
Charlot & Fall (2000) or the Calzetti et al. (2000) models. These photometric
measurements were unified in the work of Dale et al. (2017), so we refer to
their work (see their Table 2). For Galactic extinction correction, we used
the color excess E(B-V) = 0.098 mag from Schlegel et al. (1998). Adopting the
extinction to reddening ratio $A_{\rm V}$/E(B-V) = 3.1, the V-band extinction
is $A_{\rm V}$ = 0.305 mag. Table 2 lists the flux after the Galactic
extinction correction and the corresponding $A_{\lambda}$/$A_{\rm V}$ value of
each band. In the Appendix A, we provide the SED fitting parameter settings.

##  3 ANALYSIS

The shape of the CO ladder constructed by multiple CO transitions in NGC 1266
(Figure 2) is similar to that of the shock-dominant galaxy NGC 6240 (Meijerink
et al. 2013) and the AGN-dominant galaxy Mrk 231 (van der Werf et al. 2010),
and the ratios of the CO luminosity ($J_{\rm up}$ = 1–13) to IR continuum
($L_{\rm CO}$/$L_{\rm IR}$666$L_{\rm CO}$ is the sum of all $J$ ladders from
$J_{\rm up}$ = 1 to $J_{\rm up}$ = 13, $L_{\rm IR}$ is calculated by using the
IRAS four-band fluxes, and the equation is $L_{\rm IR}=4\pi D_{\rm
L}^{2}f_{\rm IR}$ from Sanders & Mirabel (1996), and the $f_{\rm IR}=1.8\times
10^{-14}(13.48f_{12}+5.16f_{25}+2.58f_{60}+f_{100}$ (w$\cdot$m-2).) are also
very close (${L_{\rm CO}/L_{\rm IR}}_{\rm NGC1266}$ = 0.54$\times$10-3,
${L_{\rm CO}/L_{\rm IR}}_{\rm NGC6240}$ = 0.67$\times$10-3). Both are an order
of magnitude larger than that of the AGN-dominant galaxy Mrk 231
(0.70$\times$10-4). This suggests that it seems reasonable to assume that
shocks dominate the gas excitation in NGC 1266. This comparative approach has
been demonstrated in the study of NGC 6240 by Meijerink et al. (2013). Indeed,
Pellegrini et al. (2013) and Glenn et al. (2015) have confirmed the evidence
of shock dominance under the comparisons of existing models (photo-
dissociation regions(PDRs), RADEX and shock).

Figure 2: Comparison of the CO ladder of NGC 1266 (red crosses) to those
obtained for NGC 6240 (blue circles, previous studied by Meijerink et al.
2013) and Mrk 231 (green triangles, previous studied by van der Werf et al.
2010). The observed CO lines flux (listed in Table 1) is normalized by the
corresponding $L_{\rm IR}$.

In this work, the heating mechanism of the galaxy is first analyzed using the
${\rm R_{mid\ CO}}$ value. Then the CO SLED is fitted with the help of
existing models (RADEX+shock). The two methods yield consistent results,
confirming previous work (e.g., Pellegrini et al. 2013; Glenn et al. 2015).
Due to the difficulty of SFR derivation in NGC 1266, we used different methods
to calculate its SFR (section 3.2).

###  3.1 Powering Source of the Molecular Line Emission

The molecular gas can be usually traced by its brightest CO lines (Young &
Scoville, 1991; Solomon & Vanden Bout, 2005; Bolatto et al., 2013). Lu et al.
(2014) found that the ${\rm R_{mid\ CO}}$ is largely independent of the FIR
color (C(60/100)) in 65 luminous IR galaxies (LIRGs), and the defined ${\rm
R_{mid\ CO}}$ can be used to distinguish the energy source of galaxies.
According to the definition of ${\rm R_{mid\ CO}}$, we find that in NGC 1266,
log ${\rm R_{mid\ CO}}=-$3.69 $\pm$ 0.10, while Lu et al. (2014) found that
log ${\rm R_{mid\ CO}}$ in SFGs is $-$4.13 $\pm$ 0.10 (an average value). This
could be attributed to the fact that shocks can heat the gas more efficiently
through a mechanical process, making a larger contribution to the mid- and
high-_J_ CO emissions. Therefore, the ratio in NGC 1266 is greater than that
in SFGs. By comparing the ${\rm R_{mid\ CO}}$ values, we obtain qualitative
evidence that shocks may dominate the gas excitation in NGC 1266.

In addition, the CO SLED constructed with multiple CO lines is also one of the
powerful tools for analyzing ISM (e.g., van der Werf et al. 2010; Meijerink et
al. 2013; Pellegrini et al. 2013). Subsequently, we modeled the CO
observations by using existing models (RADEX+shock), and the results are shown
in Figure 1. In this figure, the blue circles are the observations of CO, and
the black circles are the best-fitting (reduced $\chi^{2}$ = 2.0) combination
of the existing models. The three shaded areas represent: the error in the
data (green), the optimal solution for the C-type shock (magenta) and RADEX
(yellow). The best-fitting C-type shock has a hydrogen number density of
$n_{\rm H}$ = $5\times 10^{4}$ cm-3 and velocity of $v_{\rm s}$ = $25\pm 5$ km
s-1, and RADEX has a hydrogen number density of $n_{\rm H_{2}}$ = $1\times
10^{4}$ cm-3 and Kinetic temperature of $T_{\rm kin}$ = $30\pm 5$ K.

The results show that the C-type shock can reproduce the CO emission lines of
mid- and high-$J$ well ($J_{\rm up}$ = 4–10), while the low-$J$ CO emissions
can be simulated by RADEX model. Through the simulation of CO SLED, we
confirmed the results of previous work that shock dominates the excitation of
the gas in NGC 1266 (Pellegrini et al., 2013). This result is consistent with
the above-mentioned diagnostic result using $R_{\rm mid\ CO}$. The shock
velocity in our best-fitting parameters is consistent with previous work,
while nH is more than twice that of the previous work (Pellegrini et al.
2013). This is to be expected since the uncertainty in the shock velocity is a
few km s-1, and nH only changes the final $\chi^{2}$ value (Meijerink et al.,
2013). For RADEX, the result is consistent with the result of Glenn et al.
(2015).

###  3.2 Various methods to calculate SFR

There are a number of reasons that the calculation of the SFR in NGC 1266 is
challenging. The nuclear region of NGC 1266 is highly obscured ($N_{\rm
H}\thicksim 10^{24}$ cm-2), and contains large amounts of shocked gas. Davis
et al. (2012) indicated that the shock will contaminate the ionized gas
emission lines, which hinders SFR calculation based on these lines from the H
ii regions. Furthermore, the hidden AGN (Davis et al., 2012) has non-
negligible contribution to MIR (10 $\rm{\mu m}$ in Figure 3, the AGN
contribution is up to 10$\%$), and the ionized photons from the AGN also
enhance the UV emission (Alatalo et al., 2015). The mid-_J_ CO emissions can
also be used for the SFR calculation (e.g., Liu et al. 2015; Lu et al. 2015),
but the mid- and high-_J_ CO emissions are seriously affected by the shock
(Fig 1), and the result must be severely overestimated (Table 3).

Figure 3: Panchromatic SED for NGC 1266 based on the photometry measurements
listed in Table 2 overlaid with the best-fitting SED model inferred from the
SED fitting tool X-CIGALE (black curve). The remaining informations are given
in the legend of the figure. The panel at the bottom shows the comparison
between the residuals of the model and the observed fluxes in each waveband.

We construct the FUV-to-submm multi-wavelength SED, and perform fitting with
the code X-CIGALE. In Figure 3, the best-fitting SED from X-CIGALE is overlaid
on the observed data. In order to describe the bump clearly present in the
optical band in the fitting process, the age of the main stellar population
(SP) in the model is initialized to be 8–12 Gyr. The best-fitting result
compared to Hunt et al. (2019) shows that considering this SP (the best
matched age is 8 Gyr) can better match the optical observations. The derived
SFR from the SED fitting is 1.17 $\pm$ 0.47 _M ⊙_yr-1. Furthermore, the AGN
component has a derived bolometric luminosity ($L_{\rm bol}$) of
1.25$\times$1042 erg s-1, implying it is a low luminosity nucleus. It is worth
noting that we also explored fitting the SED without the X-ray data and the
measurement of SFR and the AGN luminosity were not affected, thanks to the
constraint from high quality data from UV/optical to FIR. This is similar to
the findings in Koutoulidis et al. (2022), who used X-CIGALE to study the host
galaxy properties of a sample of X-ray AGN. The inclusion of hard X-ray data
point is crucial to determine the contribution from AGN robustly in the longer
wavelength range (Yang et al., 2020).

We also utilize the [C ii] 158 $\rm{\mu m}$ and [N ii] 205 $\rm{\mu m}$
emission line measurements to further calculate the SFR. The ionization
potential of neutral carbon (11.3 eV) is lower than that of neutral hydrogen
(13.6 eV), so the origins of [C ii] may be multifold (Stacey et al., 1991).
Furthermore, when considering large IR luminosity, the emissivity of [C ii]
will decrease (e.g., Sargsyan et al. 2012; De Looze et al. 2014). However,
Sutter et al. (2019) found that there was not deficit of [C ii] in the neutral
hydrogen regions. Therefore, the [C ii] from the neutral regions can be used
for SFR calibration regardless of the lower emissivity. Fortunately, [N ii]
205 $\rm{\mu m}$ emission only arises from the H ii regions, and its ratio to
[C ii] is almost constant, independent of the electron density ($n_{\rm e}$)
(Croxall et al., 2017). Meanwhile, their critical densities are similar, so [N
ii] 205 $\rm{\mu m}$ can be used to separate the [C ii] emission originated
from the neutral and ionized gas (e.g., Oberst et al. 2006; Walter et al.
2009; Decarli et al. 2014).

The line flux ratio of [N ii] 122 $\rm{\mu m}$ to [N ii] 205 $\rm{\mu m}$ is
an excellent probe for low-density ionized gas (e.g., Tayal 2011; Herrera-
Camus et al. 2016; Croxall et al. 2017). When the electron temperature is
$\sim 8000$ K, it is very sensitive to the variation of $n_{\rm e}$ (10 $\sim$
300 cm-3) (e.g., Herrera-Camus et al. 2016; Croxall et al. 2017), enabling
measurement of the $n_{\rm e}$. With the theoretical relationship between the
line flux ratio of [C ii] to [N ii] 205 $\rm{\mu m}$ and $n_{\rm e}$ from
Croxall et al. (2017), the line flux ratio of [C ii] to [N ii] 205 $\rm{\mu
m}$ in the ionized region can be obtained. Finally, the proportion of [C ii]
emission from the H ii regions can be estimated. The result shows that the
$n_{\rm e}$ is 120 $\pm$ 6 cm-3 in NGC 1266, and the corresponding ratio ([C
ii]/N ii] 205 $\rm{\mu m}$) is 3.7 $\pm$ 0.2. That is, 12.0 $\pm$ 3.5 $\%$ of
the total [C ii] emission comes from the H ii regions, and the remaining 88.0
$\pm$ 3.5 $\%$ comes from the neutral regions. As shown in Table 3, the SFR
derived from the total [C ii] 158 $\rm{\mu m}$ is 1.74 $\pm$ 0.18 _M ⊙_yr-1,
whereas the SFR derived from the [C ii] 158 $\rm{\mu m}$ that is associated
with the neutral gas phase is 1.38 $\pm$ 0.14 _M ⊙_yr-1.

In addition, Zhao et al. (2013) showed that there is a nearly linear
relationship between the [N ii] 205 $\rm{\mu m}$ luminosity and the total IR
luminosity in galaxies, indicating that this emission line may be used for SFR
tracing. The 60-to-100 $\rm{\mu m}$ flux density ratio observed by _IRAS_ for
NGC 1266 is 0.78 ($f_{60}/f_{100}$), which is the “warm” case as defined by
Zhao et al. (2016). The corresponding formula (listed in Table 3, and
references therein) can be adopted for [N ii] 205 $\rm{\mu m}$ calculation,
which leads to a derived SFR as 3.87 $\sim$ 6.46 $M_{\odot}$yr-1. There are
two groups of different fitting coefficients for the relationship between [N
ii] 205 $\rm{\mu m}$ and SFR (Zhao et al., 2016), so this result is a range.

Also, using the mid- and high-$J$ CO emissions (Liu et al., 2015; Lu et al.,
2015), we can trace SFR, especially CO(7-6). The result shows that the
derivation result must be overestimated (listed in Table 3) because the CO
emissions are severely affected by the shock (Figure 1).

Table 3: Independently derived SFR for NGC 1266

Spectral |  | Luminosity
---
($L_{\odot}$)
Calculation formulas |  |  SFRa
---
(_M ⊙_yr-1)
IR |  1.30 $\pm$ 0.39 $\times$ 1010 |  SFR(_M ⊙_yr-1) = 1.73 $\times$ 10-10 $L_{\rm IR}$($L_{\odot}$)b |  2.25 $\pm$ 0.68
[C ii] 158$\rm{\mu m}$ |  0.30 $\pm$ 0.03 $\times$ 108 |  log SFR(_M ⊙_yr-1) = a log $L_{\rm[C_{II}]}$(erg s-1) $-$ bc |  1.74 $\pm$ 0.18
[N ii] 205$\rm{\mu m}$ |  0.28 $\pm$ 0.09 $\times$ 107 |  log SFR(_M ⊙_yr-1) = a log $L_{\rm[N_{II}]}$(L⊙) - bd |  3.87 $\sim$ 6.46
CO(7-6) |  1.36 $\pm$ 0.15 $\times$ 106 |  SFR(_M ⊙_yr-1) = 1.31 $\times$ 10-5.00±0.12($L_{\rm CO(7-6)}$/$L_{\odot}$)e |  $\sim$ 17.82
X-CIGALE | – | – |  1.17 $\pm$ 0.47

  * a

Values calculated directly from the calibration formulas. The results derived
by the others are summarized in Table 4.

  * b

Kennicutt (1998).

  * c

From Sutter et al. (2019). For the total emission, a and b are 1.02, $-$41.64;
For neutral emission, a and b are 0.98, $-$40.05.

  * d

From Zhao et al. (2016), when the $f_{60}/f_{100}$ of the target source is
between 0.6$\sim$0.9 (warm). For this formula, there are different fitting
coefficients: a is 1, b is 5.64, or a is 0.98, b is 5.53. The SFR obtained in
the two cases is 6.64 $M_{\odot}$yr-1 and 3.87 $M_{\odot}$yr-1 respectively.

  * e

Lu et al. (2015), CO lines are seriously affected by shock, so we do not give
error in the calculation result.

Table 4: Summary of SFR for NGC 1266 derived in the literature

Methods |  | SFR
---
(_M ⊙_yr-1)
Referencesd
PAH+FUV | 0.3 | 1
[Ne ii] | 1.5 | 1
IRa | 2.2 | 1
CIGALE model |  1.46 $\pm$ 1.12 | 2
GRASIL model |  1.40 $\pm$ 1.20 | 2
MAGPHYS model |  0.45 $\pm$ 1.15c | 2
FUV+TIRb | 1.79 | 2
Hα \+ 24 $\rm{\mu m}$ | 1.98 | 2
TIRb |  5.37 $\pm$ 1.23 | 3

  * a

Assuming that IR is all from SF, $L_{\rm IR}$ is calculated using the IRAS
four–band fluxes.

  * b

$L_{\rm TIR}$ is the total infrared-submm energy budget of galaxies, its range
is 8–1000 $\rm{\mu m}$.

  * c

The reason for the large error is that the model does not consider the AGN
component.

  * d

1 is Alatalo et al. (2015), 2 is Hunt et al. (2019), 3 is De Looze et al.
(2014)

##  4 DISCUSSION

NGC 1266 has been investigated previously (e.g., Alatalo et al. 2011, 2015;
Glenn et al. 2015; Hunt et al. 2019). Combining the existing observations
(e.g., _CARMA, ALMA, XMM-Newton_), Alatalo et al. (2015) not only gave the
range of SFR in NGC 1266, but also corrected the mass loss rate reported
previously (Alatalo et al., 2011), and discussed the possible nuclear
activity. Glenn et al. (2015) studied the characterization of nuclear
molecular gas in NGC 1266 combined with _Herschel_ and literature archival
data. By fitting to the CO lines, they gave strong evidence that the warm
outflow gas is driven by AGN and excited by shock, because the SF is too weak.
Also, Pellegrini et al. (2013) used _Herschel_ and the ground-based
observational CO data to obtain evidence that shocks dominate the gas
excitation with the help of existing models (e.g., PDR and Shock).

###  4.1 Cross-check of SFR derivation

The derived SFR using the [C ii] 158 $\rm{\mu m}$ associated with neutral
regions is 1.38 $\pm$ 0.14 _M ⊙_yr-1, and the derivation result of the total
[C ii] 158 $\rm{\mu m}$ emissions is 1.74 $\pm$ 0.18 _M ⊙_yr-1. Compared to
the SFR given by the SED fitting ($\sim$1.2 _M ⊙_yr-1) and the values reported
in the literature (Table 4), these are in good agreement, especially
considering the scatter of the empirical formula (listed in Table 3, and
references therein).

Obviously, the derived SFR from the measured [N ii] 205 $\rm{\mu m}$ flux is
significantly larger, three times more than that of [C ii] 158 $\rm{\mu m}$.
For the relationship between [N ii] 205 $\rm{\mu m}$ and SFR, Zhao et al.
(2016) fitted each sample using a least-squares, geometrical mean functional
relationship with a linear form (listed in Table 3). The formula listed in
Table 3 has two groups of fitting coefficients according to the definition of
Zhao et al. (2016), the SFR obtained is 6.46 $M_{\odot}$yr-1 and 3.87
$M_{\odot}$yr-1 respectively, and the scatter is 0.25 dex. We suggest that the
reason is because the dependence of [N ii] 205 $\rm{\mu m}$ on FIR color
($f_{60}/f_{100}$) has the significant contribution to the total scatter of
the [N ii] 205 $\rm{\mu m}$-SFR relation (Zhao et al., 2016). For
mid-/high-$J$ CO emissions, taking CO(6-5) as an example, the shock
contributes up to $\sim$ 76$\%$ to its emissions, which is also shown in
Figure 1. Moreover, the ratio of CO luminosity simulated by the shock model
with the best parameters to the IR luminosity is 0.46 $\times$ 10-3 ($L_{\rm
CO}/L_{\rm IR}$), which is very close to the observed ratio (0.54 $\times$
10-3). Therefore, shocks can well stimulate the mid-/high-$J$ CO emission to
the observed level. In fact, the result of CO SLED analysis shows that the
mid-_J_ CO transitions are severely affected by the shocks (Fig 1), and
consequently its calculation result must be overestimated (Table 3). If the
presence of the AGN and shocks in NGC 1266 is fully ignored, and all the IR
flux is attributed to SF, the derived SFR from IR (2.25 $\pm$ 0.68
$M_{\odot}$yr-1) can be viewed as the upper limit. This is consistent with the
conclusions in the literature (e.g., De Looze et al. 2014).

As shown in Table 4, previous work have used various methods (emission lines
or existing models) to calculate the SFR of NGC 1266. Alatalo et al. (2015)
suggested that due to the combined effect of AGN and shock in NGC 1266, the
derived SFR (about 0.3 $M_{\odot}$yr-1) using PAHs + UV should be the lower
limit of its SFR. Moreover, we can find that all the results are all around
2.0 $M_{\odot}$yr-1 in tables 3 and 4, regardless of the tracer used (e.g.,
continuums or emission lines). So we think that the SFR range in NGC 1266 may
be between 0.3 $M_{\odot}$yr-1 – 2.0 $M_{\odot}$yr-1. Although the derived
values of some methods are obviously large in these two tables, as mentioned
in Section 3.2: UV, MIR and H ii regions emission lines are affected by AGN in
different levels, and shock will affect the PAHs and gas emission lines, so
these results are expected (Davis et al., 2012; Alatalo et al., 2015). In our
work, we perform calculations with [C ii] from the neutral hydrogen regions
after correcting for deficit. Comparing the results in Table 3 and combining
our SED fitting result, we suggest that the [C ii] calculation result may be
more reliable.

###  4.2 Evidence for a low luminosity AGN

Although it is postulated that the AGN in NGC 1266 has strong mechanical
energy to drive the massive molecular outflow (Alatalo et al., 2011, 2014;
Nyland et al., 2013), it is inconclusive whether the compact continuum source
resolved by _ALMA_ is dominated by an AGN or an ultra-compact starburst
(Alatalo et al., 2015). The observations of dense gas tracers (e.g., CS(2-1)
and HCN(1-0)) by _ALMA_ and _CARMA_ had shown that its nuclear region lies
behind a large gas column density ($N_{\rm H_{2}}\thicksim 3\times 10^{24}$
cm-2, or $N_{\rm H}\thicksim 6\times 10^{24}$ cm-2, Alatalo et al. 2015). Such
a large gas column density is considered to be Compton-thick, which can
obscure an intrinsically bright AGN as well as attenuate X-rays.

In one of the scenarios, Alatalo et al. (2015) assumed that the compact FIR
continuum source (within 30 pc of the nucleus) is heated by an AGN, the IR
emission associated with AGN will account for 66$\%$ of the total flux
density. Their spectral modeling using Chandra and XMM-Newton indicates the
absorption corrected 2–10 keV luminosity of the putative AGN is $7\times
10^{43}$ erg s-1. However, Ho (2008) indicated that the bolometric luminosity
of nearby galactic nuclei span $\thicksim$1038 – 1044 erg s-1, with a median
value of $L_{\rm bol}$ = 3 $\times$ 1040 erg s-1 and half of the nearby
galactic nuclei lying between 3 $\times$ 1039 and 3 $\times$ 1041 erg s-1.
Therefore, the 2–10 keV luminosity of the putative AGN obtained by Alatalo et
al. (2015) exceeds the IR luminosity in the nucleus. Noting that the column
density is poorly constrained due to limited energy range of Chandra and XMM-
Newton, and the hard X-ray may include scattered and reflected components,
this is suggested to be the upper limit of the hard X-ray luminosity (Alatalo
et al., 2015). Alternatively, scaling from the bolometric luminosity of the
central source, they further estimate an AGN with 2–10 keV luminosity $\sim
10^{42}$ erg s-1 obscured by $N_{H}=3\times 10^{24}$ cm-2 can match the
observed X-ray flux.

In contrast, according to our best-fit multi-wavelength SED (AGN emission in
Figure 3, yellow curve), it implies that the fraction of IR luminosity from
AGN is lower than the assumption above. The derived AGN bolometric luminosity
$L_{\rm bol}$ is $1.25\times$1042 erg s-1, so its $L_{\rm bol}/L_{\rm X}$ =
27.80, which is very close to the value ($L_{\rm bol}/L_{\rm X}$ = 28.0) given
by Ho (2008). Assuming $N_{\rm H}=6\times 10^{24}$ cm-2, with an estimated
intrinsic 2–10 keV luminosity of $10^{42}$ erg s-1 (Alatalo et al., 2015), one
would expect to find a _NuSTAR_ count rate (2.6 $\pm$ 0.6 $\times$ 10-2) at
least one order of magnitude higher than the observed (1.8 $\pm$ 0.6 $\times$
10-3). Therefore the hidden AGN is a low luminosity one, with an Eddington
ratio of $L_{\rm bol}/L_{\rm Edd}=6\times 10^{-3}$ (for a black hole mass
$M_{\rm BH}=1.7\times 10^{6}$M⊙, Alatalo et al. 2015). The presence of a radio
jet (Nyland et al., 2013) is also consistent with the theoretical regime for
the low luminosity AGN (Shi et al., 2021) and Seyfert where hot accretion flow
becomes important (Yuan & Narayan, 2014; Mosallanezhad et al., 2019). We
conclude that it is more likely the central compact emission is dominated by
starburst in NGC 1266.

##  5 CONCLUSIONS

Comparing with previous work: we adopt the combination of SHOCK+RADEX.
Although the final results are not significantly different. Furthermore, we
also confirm this result qualitatively ($R_{\rm mid\ CO}$ value). For the
construction of galaxy SED, we use X-CIGALE which contains more AGN templates
(e.g., CIGALE), and we consider different SPs to describe the bump clearly
present in the optical band in the fitting process. Most importantly, we
include the X-ray data in the fitting process. In addition, we also calibrate
the galaxy SFR using different methods, especially the [C ii] only from the
neutral regions.

In this paper, we present our results of the inner activity in a S0 galaxy,
NGC 1266. With the help of existing models as well as the RmidCO value, we
confirm the mechanism responsible for gas excitation in the system. Due to the
special case that NGC 1266 exists, we provide different SFR calibration
results, especially [C ii] from the neutral regions. The most difference is
that we include X-ray data from _NuSTAR_ in the SED fitting process and use it
to differentiate the activity inside the galaxy. The main conclusions of our
work are as follows:

(1) The ${\rm R_{mid\ CO}}$ value in NGC 1266 is larger than that in SFGs,
which is consistent with the result of shock heating. The fitting results of
CO SLED also show that shocks dominate the gas excitation in the galaxy.

(2) The SED of the galaxy is constructed and modeled by the X-CIGALE, and a
series of physical parameters are obtained, indicating that the SFR is 1.17
$\pm$ 0.47 $M_{\odot}$yr-1. In the fitting we consider different parameter
settings to better describe the bump clearly present in the optical band.
Meanwhile, the [C ii] only from neutral regions also show consistent result
1.38 $\pm$ 0.14 $M_{\odot}$yr-1. The higher measurement from [N ii] 205
$\rm{\mu m}$ is affected by the uncertainty of empirical calibration and
presence of shocks. We also provide the related results from others in Table
4.

(3) Based on our fitted multi-wavelength SED, we find that the compact
emission related to AGN does not account for 66$\%$ of the total flux density
as previously expected. If the gas column density is indeed $\thicksim 6\times
10^{24}$ cm-2, the hidden Compton-thick AGN could be detected by hard X-ray
observations that are sensitive to $>10$ keV. We used the 3–79 keV archival
data from _NuSTAR_ hard X-ray observation to constrain this and found a
marginal 3$\sigma$ detection. An obscured low luminosity AGN is more
favorable. Therefore, the central compact emission is probably dominated by
starburst in NGC 1266.

We are grateful to the anonymous referee for her/his thoughtful review and
very constructive suggestions, which greatly improved this paper. We thank Dr.
Guang Yang for helpful instruction on using X-CIGALE. This work is supported
by the National Natural Science Foundation of China (NSFC grant Nos. 12033004,
U1831205, 12173079 and 12221003). J.W. acknowledges the science research
grants from the China Manned Space Project with No. CMS-CSST-2021-B02 and CMS-
CSST-2021-A06. This work is based in part on observations made with _Herschel_
, an European Space Agency Cornerstone Mission with significant participation
by the National Aeronautics and Space Administration (NASA). This research has
made use of the NASA/IPAC Extra-galactic Database (NED), operated by the Jet
Propulsion Laboratory of the California Institute of Technology, under
contract with NASA.

##  Appendix A The X-CIGALE module usage and parameters setting

SFH: it contains several SFH forms, such as sfh2exp, sfh$\\_$buat08 and
sfhdelayed (Boquien et al., 2019). One of the most popular is “delayed” SFH
modeling, however its most obvious limitation is that it does not allow for
recent quenching of the SFR. To address this issue, Ciesla et al. (2017)
extended SFH to allow for instantaneous resent change in SFR, and called it
module sfhdelayedbq. Hunt et al. (2019) has successfully modeled the KINGFISH
galaxies using this SFH module, so here we also adopt this SFH module:
sfhdelayedbq.

Table 5: Parameters setting of each module in code X-CIGALE

Modulesa |  Parametersb | Values
---|---|---
sfhdelayedbq |  tau$\\_$main | 500,1000,2000,4000,8000 (Myr)
|  age$\\_$mainc | 8000,10000,12000 (Myr)
|  age$\\_$bq | 10,100,500,1000 (Myr)
|  r$\\_$sfr | 0.01,0.05,0.1,0.5,1.0,5.0,10.0
|  sfr$\\_$A |  1.0 ($M_{\odot}$yr-1)
bc03 | imf | 1
| metallicity |  0.02 ($M_{\odot}$)
|  separation$\\_$age | 10 (Myr)
nebular | logU | $-2.0$
distant$\\_$modified$\\_$CF00 |  A${}_{\rm V}\\_$ISM | 0.1,0.5,1.0,1.5,2.0 (mag)
dl2007d | qpah | 0.47,1.12,2.5,3.9,4.58
| umin | 0.10,0.15,0.30,0.50,0.80,1.00,1.20,1.50,2.00,4.00,10.,25. (Habing)
| umax | 1e6 (Habing)
| gamma | 0.001,0.003,0.009,0.01,0.03,0.09,0.1
fritz2006 | r | 60.0
| $\tau$ | 1.0
| $\beta$ | -0.5
| $\gamma$ | 4.0
| $\theta$ | 100.0 (°)
| $\Psi$ | 50.1 (°)
| $f_{\rm AGN}$ |  0.01,0.05,0.1,0.2,0.3,0.5 ($\%$)
| E(B-V) | 0.04,0.1
xray | $\Gamma$ | 1.8

  * a

In addition to the templates listed, there is a final module called
redshifting during setup, which first redshifts the spectrum and dims it,
multiplying the wavelength by 1+_z_ and dividing the spectrum by 1+_z_. Also,
it considers the shorter wavelength radiation absorbed by the IGM (Boquien et
al., 2019).

  * b

Parameters not listed in the module are set to default values

  * c

Age of the main stellar population in a galaxy (Boquien et al., 2019)

  * d

Draine & Li (2007)

SSPs: it is calculated using the Bruzual & Charlot (2003) SSPs, where the
metallicity _z_ is 0.02 and the Initial Mass Function is from Chabrier (2003);

Nebular emission: the CLOUDY model is used in the model to calculate the
templates, and they have the same metallicity and stellar population (SP). The
ionization parameter in this module uses the model default values ($log$U =
$-$2.0), and assumes that the fraction of Lyman continuum photons escaping the
galaxy is zero, i.e. the photon does not directly heat the dust (Hunt et al.,
2019).

Dust attenuation: the model adopts the modified starburst attenuation law to
explain the absorption of stellar and nebular radiation by interstellar dust,
which takes into account the reddening differences of SPs of different ages
(Boquien et al., 2019). Here, we adopt the Charlot & Fall (2000) attenuation
law.

Dust emission: the model includes several dust emission templates, which
assuming that the dust emission is optically thin (Draine & Li, 2007; Dale et
al., 2014; Draine et al., 2014). Also, it considers the possible variations of
the polycyclic aromatic hydrocarbon (PAH) and radiation field intensity
($U_{\rm min}$ and $U_{\rm max}$), as well as the fraction illuminated from
$U_{\rm min}$ to $U_{\rm max}$.

AGN: As discussed above, the model contains two different AGN templates –
smooth (Fritz et al., 2006) and clumpy (Stalevski et al., 2012, 2016). They
take into account three components through a radiative transfer model: a
source located in the torus, the scattered emission by dust, and the thermal
dust emission (Boquien et al., 2019). They are set by seven parameters: r the
ratio of the inner and outer radii of the dust torus, $\tau$ the optical depth
at 9.7 $\rm{\mu m}$, $\gamma$ and $\beta$ the horizontal/vertical density
distribution of the dust, $\theta$ the aperture angle of the dust torus,
$\Psi$ the angle between the AGN axis and the line-of-sight direction, and
$f_{\rm AGN}$ the AGN fraction. In our run, we adopt the AGN templates from
Fritz et al. (2006).

X-ray: X-CIGALE uses a new module, X-ray, which enables the addition of X-ray
data during the fitting process. Input data requirements are absorption-
corrected (Yang et al., 2020), but do not distinguish absorption from the
source itself, the Milky Way, or the intergalactic medium (IGM).

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# Literature Review

Alvin Tang

(22 May 2023)

##  1 Introduction

Access control is a significant component of computer systems to safeguard the
security of information by limiting the actions and operations that a user can
perform. As an important tool to manage permissions of genuine users and
processes to retrieve and manipulate data1, system architects must ensure that
the access control schemes and models work feasibly in the desired manner
efficiently. To provide a rigorous way for verifying the correctness of the
theories, formal logic is extensively utilised to describe access control
rules and policies. Given the importance of rigorousness ad comprehensiveness
in this field of study, it is worthwhile to explore the features of formal
semantics and the underlying classical logic principles upon which most
existing research outputs are established. In our proposed research, we will
discover how formal logic is applied in well-known access control models, how
the features of classical logic lead to limitations of the existent theories,
as well as how non-classical logic systems may contribute to the enhancement
thereof.

We shall hereby review the existing discoveries by computer scientists by
comparing the characteristics and purposes of different access control models,
analysing a few sets of theories formalising access control schemes and the
most significant limitations in existing access control models. In the
meantime, we will briefly explore the constraints of classical logic and the
notable features of alternative logic systems with literature by a few
computing researchers and logicians. These revelations will help us find out
the importance of improving existing models, the potential applications of
non-classical logic for access control and the aspects of which we have to be
mindful during the research.

##  2 Current Knowledge

###  2.1 Significance of Access Control for Cyber Security

The purpose of access control is to limit the range of files and programs to
which a genuine user may gain access for reading, writing and execution.
Various security components of a computer system serve distinct purposes to
shield the integrity and safety of information by providing multiple layers of
protection for programs and data. There is no exception for access control as
its implementations are generally integrated with other cyber security tools
and protocols. Their policies cannot be enforced without the use of a
reference monitor, which mediates every attempted access to resources in a
system.1 Complete mediation is achieved when architects of computer systems
designate the reference monitor as the sole point of access control decision-
making so that there is no way a legitimate or intruding user/process may
bypass the policies.

In the meantime, authentication of user identities is critical to ensure that
all individuals and processes have no alternative way to access the system
without abiding by the permission rules. Whilst the development of access
control models has significance for discovering ways to organise and manage
permissions, research on hashing, password policies and multi-factor
authentication likewise contribute to the efficient implementation of access
control policies by preventing malicious users from being able to conduct
fraudulent activities with imposture of others’ identities. Results of our
research project, consequently, should integrate well with other layers of
security protection in computer systems.

The implementation of access control is ubiquitous in everyday computer
systems. Linux, for example, implements access control at multiple levels
based on the principle of least privilege. In addition to firewall rules and
recording of file ownership details, Linux uses permission bits to determine
whether a user may access a file or directory in line with the categorisation
of their identities.

The permission bits in Linux correspond to the user, the user’s group and
others. Each category of users may be assigned permission to read, write and
execute a file.2 Take /etc/shadow as an example. This system file documents
the list of users registered in the system and the hash of their passwords. In
most Linux installations, the file has permissions and ownership which
resembles the following.

        [alice@mysystem ~]$ ls -l /etc/shadow
        -rw------- 1 root root 900 Mar 25 16:42 /etc/shadow

The rw indicates that the owner (root) has read and write permissions. The
subsequent dashes implies that other users have no permission to read or
modify the file.

This access control list (ACL) design provides us with the assertion that an
ordinary user or anyone that has access to the computer system will not be in
the position to view the password hashes of other users, which could be
maliciously used for impersonation. With a well-implemented reference monitor,
the purpose of access control to mitigate the risks can be achieved as a
result.

In this example with the /etc/shadow file, the password hashes should be kept
protected as the use of obsolete hashing algorithms (e.g. MD5) may enable the
exploitation of login credentials with preimage or collision attacks. Even if
modern hashing functions (e.g. SHA-512) which are popular in systems we use
nowadays are currently considered safe, advancing technology and progress from
the constant scrutiny of these algorithms by mathematicians and computer
scientists may render them otherwise. Access control acts as a layer of
protection supplementary to hashing and, therefore, provides us with stronger
confidence in the security of the systems.

In summary, the significance of access control for enhancing cyber security
involves authentication of user identity, authorisation to access data and
monitoring of user activities to enforce permission policies and comply with
cyber security regulations and standards.1 Linux, as one of the most
mainstream operating systems especially for servers, demonstrates the
importance of access control to achieve the principles of cyber security.
Researchers have praised the ease of implementation with ACL. However, its
inability to specify access rights for individual users and specific groups2
has motivated us to delve into alternative access control models as there are
more sophisticated requirements for complex environments.

###  2.2 Different Access Control Models for Various Purposes

In addition to ACL, researchers have developed a vast assortment of access
control models for varying levels of security and fulfilment of security
requirements. Albeit the discretionary access control (DAC) system implemented
by Linux is simple to manage, it might not provide adequate granularity and
flexibility. The reliance of DAC on the delegation of privileges to users or
groups may be concerning in some settings, such as in the military or
intelligence organisations, when unintended authorisations to access certain
information may lead to detrimental effects. To resolve the challenges, more
sophisticated models such as the mandatory access model (MAC), the role-based
access model (RBAC) and the attribute-based access model (ABAC) are utilised.

With mandatory access control, security policies are completely controlled by
the security administrator. MAC is varied from DAC in that users do not have
control over resources or determine who has access thereto.3 With this design,
we may deduce that operations such as chmod or chown by ordinary users would
not be allowed if MAC was applied in typical Linux systems.

Role-based access control is often preferred by large entities due to its
greater adaptability and scalability. As opposed to DAC and MAC, where the
access policy is defined for each file separately, RBAC is designed around the
idea that users are assigned roles within an organisation.3 As an
illustration, suppose there are two users alice and bob in an organisation
such that they are assigned the roles as follows.

User | Roles
---|---
alice |  student, COMP2100-student, COMP2550-student
bob |  student, COMP2300-student, COMP2550-student
charlie | staff

Let the following files be assigned with their corresponding roles as follows.

File | Roles
---|---
student_info.html |  student, staff
comp2100_assignment.txt | COMP2100-student
comp2300_lecture.mp4 | COMP2300-student
research_proposal.pdf | COMP2550-student

In this scenario, alice has access to comp2100_assignment.txt whilst bob does
not. On the contrary, comp2300_lecture.mp4 is exclusively accessible by bob.
Meanwhile, both of them have permission to read research_proposal.pdf. The
webpage student_info.html is available for all three individuals.

In RBAC, both users and files may be assigned multiple roles. Furthermore, the
assignment of roles to the latter may specify the extent of permissions. For
instance, read and write permissions may be assigned to the staff role for
student_info.html while read-only permission may be assigned for the student
role.

RBAC is advantageous when the scale of a system is sizable where it is
strenuous and time-consuming to manage permission roles for every file
individually. With roles apparently indicating the rights and responsibilities
of each user, this model also thoroughly applies the separation of duties
principle1 by only giving access to resources they need to perform.

Although the clarity of role assignment for users makes RBAC easily
understandable and systematised, the level of control provided by this model
may not be adequate for dynamic environments. As a solution, decisions are
made in attribute-based access control grounded in the evaluation of
attributes associated with the user or resource.3 As a demonstration, suppose
alice, bob, charlie and dennis are students in a university. Let department
and year be attributes indicating the department to which they belong along
with the year in which they are studying.

User |  department attribute |  year attribute
---|---|---
alice | Computing | 1
bob | Social Sciences | 1
charlie | Computing | 2
dennis | Physics | 3

Suppose there are multiple files stored in the university database with the
following access policies.

File | Access policy
---|---
computing_notes.pdf | Allow access $\iff$ department is Computing
maths_notes.pdf | Deny access $\iff$ department is Social Sciences
new_students.html | Allow access $\iff$ year equals 1
returning_students.html | Allow access $\iff$ year is greater than 1

As a result, the access permissions to the files by the four users are as
follows.

Access granted? | alice | bob | charlie | dennis
---|---|---|---|---
computing_notes.pdf | Yes | No | Yes | No
maths_notes.pdf | Yes | No | Yes | Yes
new_students.html | Yes | Yes | No | No
returning_students.html | No | No | Yes | Yes

While roles in RBAC are discrete and enumerable, attributes in ABAC offer more
flexibility as they can be, for instance, numerical values such as the year
attribute above. Access policies in ABAC are customisable for more intricate
situations and dynamic environments. In the above example, assume that all
students proceed to a higher year every two semesters. Compared to the RBAC
where we have to reassign the roles for students every year, we just have to
increment the year attribute for all students by one in ABAC. In other words,
ABAC outperforms alternatives in terms of flexibility to define access
policies with conditions and efficiency to modify users’ permissions by
altering discrete or continuous attributes. It is suitable in situations where
the attributes of entities may be constantly changing, such as in cloud
computing or e-commerce platforms.

In summary, there are various access control models designed by researchers to
meet different security requirements and solve challenges in varying
circumstances. DAC and MAC are advantageous for their simplicity whilst RBAC
and ABAC offer more granular control over access policies with their
sophisticated design. The successive step of analysis is to discover how we
can mathematically and logically represent these models and the advantages of
doing so.

###  2.3 Use of Formal Logic in Access Control Models

Computer science researchers have been using formal logic and semantics to
discover rigorous frameworks for specifying access control policies to prove
the correctness of models, to analyse the implications of policies and to
structure proposals of new theories. RBAC, being one of the most favoured
access control models in the industry, are frequently scrutinised by academics
for verification, vulnerabilities and improvements.

Logic has been the foundation for the formal verification of the RBAC model.
For example, members of the IEEE Computer Society proposed using set theory
and sequents to not only conduct security analysis but also to explore the
computation complexity of the model implementation. In the research, it is
suggested that in reality configurations are often modified as resources may
start being shared at some point and stop being accessible sometime later.4
Taking the dynamic environment into consideration, there is a need to ensure
that permission authorisations and the revoking thereof should be free from
inadvertent effects.

In order to conduct a security analysis, we first have to define an access
control scheme formally. With reference to this research, it is defined as the
tuple $\langle\Gamma,Q,\vdash,\mathcal{A},\Sigma,\Psi\rangle$4 where

  * •

$\Gamma$ is a set of states,

  * •

$Q$ is a set of queries,

  * •

$\vdash:\Gamma\times Q\to\\{\textit{true},\textit{false}\\}$ is the function
determining whether a query is true or not in a state,

  * •

$\mathcal{A}$ is a set of principals,

  * •

$\Sigma$ is a set of actions, and

  * •

$\Psi$ is a set of state-transition rules.

We understand that in this context the researchers are viewing access control
schemes as state-transition systems, similar to finite-state automata except
start or terminal states are not explicitly defined. Using this definition, it
is evident that states correspond to information available for making access
control decisions whereas a query refers to an access request.

Let $\gamma\in\Gamma$ and $q\in Q$. Then, the function $\vdash$ is understood
as follows.

  * •

The statement $\gamma\vdash q$ means approval of the request.

  * •

The statement $\gamma\nvdash q$ equals denial thereof.

We may apply this state-transition model of verification to our RBAC system
example in Section 2.1. We can define the users as the set of principals
$\mathcal{A}=\\{\texttt{alice},\texttt{bob},\texttt{charlie}\\}$. The question
of whether alice has access to student_info.html is thus a query.

Solely with our RBAC setting in Section 2.1, we are incapable of defining the
relationships or hierarchies between varying roles. It is not sensible to
assign an individual with the COMP2100-student role but not the student role
though nothing is preventing us from doing so. The logic relation framework
advocated by this state-transition model introduces relations to form a
hierarchy of roles.

According to their definition, $RH\subseteq\mathcal{R}\times\mathcal{R}$,
where $\mathcal{R}$ is the set of roles, is an irreflexive acyclic relation.4
Suppose $r_{1},r_{2}\in\mathcal{R}$. Then, $r_{1}\succeq_{RH}r_{2}$ is
equivalent to the statement “every user who is authorised for $r_{1}$ is also
authorised for $r_{2}$”.

Back to our RBAC example. We may define $RH$ such that

$\forall r\in\mathcal{R}_{\texttt{student}}.r\succeq_{RH}\texttt{student}$

where

$\mathcal{R}_{\texttt{student}}=\\{\texttt{COMP2100-student},\texttt{COMP2300-student},\texttt{COMP2550-student}\\}.$

The roles assigned to alice and bob may then be simplified without affecting
the actual policy:

User | Roles
---|---
alice |  COMP2100-student, COMP2550-student
bob |  COMP2300-student, COMP2550-student
charlie | staff

Until now we have not yet taken into account changes in roles. Bearing the
above state-transition model definition in mind, we may consider the
assignment and revocation of roles as actions in the set $\Sigma$. For
example, $\textit{assign}(u_{a},u_{t},r_{t})$ refers to the assignment of the
role $r_{t}$ for the user $u_{t}$ by the user $u_{a}$.

In reality, such actions should only be done by some particular users (e.g.
system administrators). To verify whether the action is credible (i.e. whether
the user $u_{a}$ has the authority to perform the assignment), we check the
roles possessed by $u_{a}$ and their implications indicated in the state
against the action.

Using the state-transition model, the set of actions4 is defined as

$\chi=\\{\textit{assign}(u_{a},u_{t},r_{t})\ |\ u_{a},u_{t}\in\mathcal{U}\land
r_{t}\in\mathcal{R}\\}\cup\\{\textit{revoke}(u_{a},u_{t},r_{t})\ |\
u_{a},u_{t}\in\mathcal{U}\land r_{t}\in\mathcal{R}\\}.$

In our situation,

  * •

$\mathcal{U}=\\{\texttt{alice},\texttt{bob},\texttt{charlie}\\}$, and

  * •

$\mathcal{R}=\\{\texttt{staff},\texttt{student},\texttt{COMP2100-student},\texttt{COMP2300-student},\texttt{COMP2550-student}\\}$.

Quoting the researchers’ definition, an revocation in $\chi$ succeeds if and
only if

  1. 1.

the user $r_{t}$ is already assigned with the role $r_{t}$, and

  2. 2.

the assigner has the authority to revoke the role from a user fulfilling the
conditions.

These three requirements are further formalised by relations and tuples. For
simplicity, we can imagine that charlie is the course convenor for COMP2550.
It has the authority to assign and revoke the COMP2550-student role fulfilling
the conditions. Since a student cannot enrol in both COMP2550 and COMP4550
simultaneously, comp2550-student and comp4550-student are mutually exclusive.

For instance, suppose bob finds the coursework too demanding and wish to drop
one of the courses. We would like to specify that only charlie, the staff
member, has the authority to modify the authorisations.

To prove that
$\textit{revoke}(\texttt{charlie},\texttt{bob},\texttt{comp2550-student})$ is
valid, we may logically show that

  1. 1.

bob possess the comp2550-student role.

  2. 2.

charlie has the authority to revoke comp2550-student from an individual.

There are lots of considerations we may take into account, such as the
implications of assignment or revocation of roles, as well as how we may
optimise the efficiency of the RBAC models with role hierarchies by
normalising role assignments. The analysis of RBAC systems with this state-
transition model is only one of the wide variety of methods proposed by
researchers. The abundant research into access control models demonstrates the
value of formal logic and semantics in formal verification, security analysis
and more. In our upcoming research, we will perform critical analyses
comparing different ways of logical formalisation for RBAC schemes. We shall
discover the types of logic systems used, the considerations taken into
account in various research efforts and to what extent their theories most
neatly describe and facilitate a granular control of permission management.

###  2.4 Development of Alternative Logic Systems

Classical (orthodox) logic dominates in the field of mathematics and
scientific research yet its limitations, such as the material paradoxes
arising from its definitions, motivate scholars to develop alternative logic
systems. Some prominent substitutions are namely fuzzy logic, relevant logic
and intuitionistic logic.5

Classical logic is built on Boolean algebra, where truth values are binary. A
statement is either true (1) or false (0). It is onerous to reason for
uncertain information or vague statements in this conventional logic system,
leading to the sorites paradox.5 For instance, suppose a thousand grains of
sand is a heap of sand whilst one single grain is not. It is difficult to
justify whether some amount of sand is considered a heap when the number of
grains is between one and a thousand. Fuzzy logic extends the truth values
into a continuous spectrum between 0 (absolute falsity) and 1 (definite
truth). A truth value of 0.8, for instance, indicates a higher level of truth
than 0.2.

Relevant logic resolves the material paradox from the principle of explosion
caused by the definition of implication in orthodox logic. In classical logic,
the statement “p implies q” is defined as follows:

$p$ | $q$ | $p\to q$
---|---|---
0 | 0 | 1
0 | 1 | 1
1 | 0 | 0
1 | 1 | 1

When the premise $p$ is false, the entire implication statement is true.
Logicians universally agree upon this definition with the justification that a
statement is false iff a contradiction exists. In fact, there is no
requirement for $p$ and $q$ to be interrelated. As a consequence, the
statement “if $1+1=3$, the Sun is a planet” is a totally legitimate theorem in
orthodox logic. Relevant logic is constructed upon axioms so that it requires
premises and conclusions in implication statements to be inextricably linked.
As a result, some proof techniques such as vacuous discharge in natural
deduction are no longer valid in relevant logic.6

The principle of excluded middle is a noteworthy characteristic of classical
logic. The double negation of a statement is always equivalent to itself
($\neg\neg p\equiv p$). This might not always accurately transform our
statements in natural language or common sense into rigorous logical
notations. For example, the statement “it is not that I am not happy” is
usually not regarded as interchangeable with “I am happy”. Intuitionistic
logic disallows double negation elimination in logic proofs, precluding the
principle of excluded middle.5

By understanding the features of alternative logic systems, we can critically
analyse how their characteristics can be utilised for applications in the
field of cyber security and, in particular, creating bettered access control
schemes. We shall refer to the logic theory knowledge in our third research
question (Section 4.3).

##  3 Present Lack of Knowledge and Research Gap

In spite of the multitude of access control models available, each model has
its restraints. Some of the incapabilities are attributed to the
characteristics of classical logic and the dependency of research on this
traditional logic system. Our inquiry aims to address this void in the
existing research.

###  3.1 Limitations in Existing Access Control Models

Existing access control models such as DAC, MAC, RBAC and ABAC have often been
implemented successfully in the industry. There are nonetheless drawbacks in
these models. With RBAC, it may be difficult to handle situations where
multiple roles are required to perform certain actions, or where roles have to
be dynamically assigned based on contextual information. Notwithstanding the
flexibility of ABAC to handle dynamic environments effectively, it might be
hard to handle conflicts between attribute values.

To discuss the most substantial research gaps in this field of study, we can
first refer to the definition of limitations in an access control model
summarised by academics as the failings of one or more of the following
characteristics: 7

  * •

Inescapable: inability to break security policies by circumventing access
controls set by the model.

  * •

Invisible: seamless user and administrative interaction with the model.

  * •

Feasible: cost-efficiency and practicality to implement the model.

In the following sections, we shall analyse the logic models offered by
researchers for RBAC and ABAC, and analyse how classical logic may result in
not being able to fulfil the above requirements.

####  3.1.1 Inescapability: Comprehensiveness of Logical Establishment and
Verification

In a study conducted by Fatima et al. to identify the disadvantages of
existing RBAC models, several extensions to RBAC models are developed to suit
context-sensitive and dynamic environments.8 However, they are still
inadequate to address the remaining limitations to offer fine-grained,
content-based, and multi-factored solutions.

From our analysis, we observe that these unresolved limitations are to a
considerable extent caused by the shortcomings and characteristics of
classical logic. In Section 2.3, we mentioned that in RBAC, some roles can be
defined as incompatible with others. In complex organisations or extensive
systems, there may be an immense number of roles to be managed. In the real-
world application of access control or during the design of the access control
policies, system engineers might not be capable of detect role conflicts
immediately. Meanwhile, most research depends on classical logic to perform
formal verification to demonstrate the correctness and rigorousness of these
models. When principals in RBAC have contradictory roles, we might be able to
deduce anything in accordance with the principle of explosion and, hence,
struggle to rule out mistakes efficiently when the undetected conflicts are
taken as premises of logic statements. This results in the concern that the
security policies are not inescapable, inadvertently leaving risks caused by
security vulnerabilities.8

We may refer to the publication A Modal Logic for Role-Based Access Control by
Kosiyatrakul et al., one of our references in the research proposal about the
use of modal logic, an extension to classical logic, to represent role
inheritance and role-permission associations with partial orders and sequents.
In their research, modal logic is adopted to facilitate representations of
statements about necessity and possibility.9

For example, the situation when the reference monitor makes an approval
decision for the statement “user $U$ , acting in authorised role $R$, makes a
request $q$” is logically represented with the statement

$(U\ \texttt{for}_{RA}\ R\ \texttt{says}\ q)\supset q$

where

  * •

$R$ is a role,

  * •

$U\in\texttt{authorised\\_roles}(R)$, and

  * •

$q\in\texttt{authorised\\_permissions}(R)$.

The author utilises the concepts “necessity” and “’possibility” in modal logic
to establish the role hierarchies for rigorous definitions of the functions
authorised_roles and authorised_permissions. Although their proposed proof
system efficiently achieves separation of duty, it is not well-designed for
the resolution of conflicts, such as circular references in role hierarchies.
Modal logic does not handle the situation when a role inherits from another
and vice versa concurrently.

In our planned research, we should explore the different ways to resolve these
limitations with various approaches, such as allowing vagueness in the
determination of access control decisions and requiring relationships between
premises and conclusions in implication statements to avoid role explosion.

####  3.1.2 Invisibility and Complexity: Ease, Effectiveness and Practicality
of Implementations

The research, in conjunction with the comparison between RBAC and ABAC,
suggests that the driving force behind various existing ABAC-based solutions
is the same basic idea revolving around the use of attributes of the subject,
object, and environment for making access control decisions.8 They suggest a
standardisation for the theoretical foundations of ABAC to establish a more
powerful and theoretical footing. In order to achieve this, we should never
disregard the potential disadvantages when we extend the access control system
and develop logic models for them.

In ABAC, attributes can be discrete or continuous values.10 Restrictions on
access to videos on online streaming platforms based on users’ geographical
locations are examples of using a discrete attribute (country/region) for
access control. This model is more advisable for precise control over security
policies than most other models but in spite of the flexibility to integrate
continuous values as attributes (inputs), it cannot handle uncertainty or
ambiguity in its decisions (outputs). Architects of the scheme have to
precisely define the boundaries of the attributes. Being potentially arduous
to justify or reach a consensus on the boundaries, the burden to manage ABAC
may no longer be negligible.

The model examples we have considered so far have a shared characteristic of
having binary truth values for access control decisions. Referring to the
roles of a user, which are also binary in the sense that a user either has a
role or otherwise, the system must decide definitely whether a user is granted
or denied access to some resource. As an extension to the RBAC example we have
in Section 2.2, we may formulate an access control policy to deny access to a
resource iff a user has the student role (statement #1). If we are convinced
that an individual is a student if and only if not being a staff member,
designers of an access control system may assume that this policy is identical
to allowing access to the resource iff the user has the staff role (statement
#2).

In real-world scenarios, this is often not the case. Even if we have rigorous
mechanisms to disallow a user from possessing both the staff and student roles
concomitantly by defining them as mutually exclusive, we still have not
defined the situation when an individual is neither a student nor a staff
member. By the principles of classical logic, the entity would be allowed
access according to statement #1 but denied access with reference to statement
#2. A paradox arises. The definitions for mutual exclusivity of roles will
require a tedious fundamental restructuring when we expand the number of roles
in a system, which raises concerns for the feasibility of existing models in
some situations.

###  3.2 Applying Non-Classical Logic for Enhanced Access Control Models

Alternative logic systems, such as fuzzy logic, relevant logic and
intuitionistic logic, have been applied in several fields of computer science
research. However, despite the great potentiality to utilise non-classical
logic systems to enhance access control models, there is a paucity of research
regarding this topic.

Fuzzy logic is a technique to develop artificial intelligence. Given that this
unconventional logic system does not require numerical precision, the
modelling machining process in AI can be tuned up efficiently according to
varying control conditions. As a result, the resultant AI models may respond
quickly to convoluted sensory inputs.11 To our understanding, estimation with
probability is a major component of machine learning and thence fuzzy logic
can help with the creation of such models.

Relevant logic and intuitionistic logic have been applied in the analysis and
affirmation of programming language semantics. Logic programming, although not
the most applied programming paradigm in the industry, is a robust tool for
data management and natural language processing. Analysts have used these two
alternative logic systems to prove the equivalence of logic programs, taking
advantage of the fact that intuitionistic logic cannot be described by a
finite set of truth values.12

The existing research effort to apply alternative logic systems clearly
demonstrates that they truly contribute to the field of computing.
Nevertheless, there is very narrow research in academia regarding the
application of alternative logic systems in cyber security. Principally, there
is a limited amount of research on the complexity analysis and quantification
of the efficiency of access control models. Notwithstanding that alternative
logic systems have seldom been used for access control models or other
information security technologies, it can be helpful to investigate the
benefits of their application and to fuel the motivation for realising the
conceptual models into deployable systems.

The limitations of existing access control models discussed in Section 3.1
become concerning as we encounter prevailing challenges in the security,
usability and efficiency of the models. In order to advance research in this
domain, we will explore the application of these three non-classical logic
systems in access control, drawing parallels to how AI employs them for
estimation and logic programming incorporates non-binary truth values.

##  4 Research Questions and Aims

Taking into account our knowledge of access control models and the promising
prospect for their enhancement to benefit information security, the proposal
for the research is summarised with three preeminent questions to be put under
examination.

###  4.1 Formal Logic in Existing Access Control Models

Our first aim of the proposed research is to analyse the use of formal logic
in existing access control models. On top of the state-transition model for
RBAC analysed in Section 2.2, we will discover a few other logic proof systems
and modelling theories proposed by other researchers. From the research papers
we reviewed above, we recognise that academics promote the evolution of logic
models catering to different needs for access control.

By conducting in-depth analyses of existing logic establishment and
verification methods, we can find out the importance of formal logic to
confirm that the access control systems are inescapable, invisible and
feasible. When we recommend enhancements to the models with non-classical
logic, we shall be attentive to assure that these requirements are not
violated.

###  4.2 Classical Logic Features to Limitations of Existing Models

The second major research question is to investigate how features of classical
logic result in limitations of the existing models. In connection with the
flaws in existing access control models mentioned in Section 3.1, it is
observed that existent models may be unable to tackle the prevalent challenges
in dynamic or complicated environments.

Whilst researchers have scrutinised existing models a lot and applied
extensions to classical logic (e.g. modal logic, temporal logic), they do not
tackle the most elemental weaknesses of this long-established logic system. We
shall extend our investigation in Section 3 to conduct the exploration.

###  4.3 Non-Classical Logic for Enhanced Models

Our final goal of discovery is to explore the feasibility of applying non-
classical logic to form a foundation for revising access control models which
may address the issues with existing ones. The ultimate goal of this research
is to not only establish logical ways to describe and verify access control
models that tackle limitations caused by orthodox logic but also to justify
that the new models are practicable. Hence, the methodologies, including
simulations of models with Haskell programs, mentioned in the research
proposal help us certify that the models are viable and appropriate for real-
world applications.

##  5 Conclusion

The aforementioned findings highlight the crucial role of access control in
cyber security. Given the multifarious collection of access control
implementations available, careful consideration must be given to selecting
appropriate models based on the specific requirements of the application
environment. Formal logic has been an integral aspect of the research and
development process. Throughout our investigation, we maintain a constant
focus on the completeness of the logic models and ensure that any proposed
enhancements adhere to the principles of well-designed models.

## References

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## Acknowledgements

The generative AI results by ChatGPT contributed to the refinement of language
in this literature review. It also provided tips regarding LaTeX syntax and
the format of citations.