# On the external forcing of global eruptive activity in the past 300 years

J-L. Le Mouël  Université Paris Cité, Institut de Physique du globe de Paris,
CNRS UMR 7154, F-75005 Paris, France  D. Gibert  LGL-TPE - Laboratoire de
Géologie de Lyon - Terre, Planètes, Environnement, Lyon, France  V. Courtillot
Université Paris Cité, Institut de Physique du globe de Paris, CNRS UMR 7154,
F-75005 Paris, France  S. Dumont<EMAIL_ADDRESS>Universidade de Lisboa,
Faculdade de Ciências, Instituto Dom Luiz (IDL), Lisboa, Portugal
Universidade da Beira Interior, Instituto Dom Luiz (IDL), Covilhã, Portugal
J. de Bremond Ars  Université de Rennes, CNRS, Géosciences Rennes - UMR 6118,
Rennes, France  S. Petrosino  Instituto Nazionale di Geofisica e Vulcanologia,
Sezione di Napoli – Osservatorio Vesuviano, Naples, Italy  P. Zuddas  Sorbonne
Université, CNRS, METIS, F75005, Paris, France  F. Lopes  Université Paris
Cité, Institut de Physique du globe de Paris, CNRS UMR 7154, F-75005 Paris,
France  J-B. Boulé  CNRS UMR7196, INSERM U1154, Museum National d’Histoire
Naturelle, Paris, F-75005, France  M.C. Neves  Universidade do Algarve, FCT,
Campus de Gambelas, 8005-139, Faro, Portugal  S. Custódio  Universidade de
Lisboa, Faculdade de Ciências, Instituto Dom Luiz (IDL), Lisboa, Portugal  G.
Silveira  Universidade de Lisboa, Faculdade de Ciências, Instituto Dom Luiz
(IDL), Lisboa, Portugal  Instituto Superior de Engenharia de Lisboa, Rua
Conselheiro Emídio Navarro 1, 1959-007 Lisboa, Portugal  V. Kossobokov
Institute of Earthquake Prediction Theory & Mathematical Geophysics, Russian
Academy of Sciences (IEPT RAS), Moscow, Russian Federation  L. Coen  Muséum
National d’Histoire Naturelle, /CNRS laboratoire PHYMA, UMR7221, F-75005,
France  M. Geze  Muséum National d’Histoire Naturelle, Sorbonne Université,
CEMIM, Paris, F-75005, France

(August 28, 2024)

###### Abstract

The decryption of the temporal sequence of volcanic eruptions is a key step in
better anticipating future events. Volcanic activity is the result of a
complex interaction between internal and external processes, with time scales
spanning multiple orders of magnitude. We review periodicities that have been
detected or correlated with volcanic eruptions/phenomena and interpreted as
resulting from external forces. Taking a global perspective and longer time
scales than a few years, we approach this interaction by analyzing three time
series using singular spectral analysis: the global number of volcanic
eruptions (NVE) between 1700 and 2022, the number of sunspots (ISSN), a proxy
for solar activity, the polar motion (PM) and length of day (lod), two proxies
for gravitational force. Several pseudo-periodicities are common to NVE and
ISSN, in addition to the 11-year Schwabe cycle that has been reported in
previous work, but NVE shares even more periodicities with PM. These quasi-
periodic components range from 5 to 130 years. We interpret our analytical
results in light of the Laplace’s paradigm and propose that, similarly to the
movement of Earth’s rotation axis, global eruptive activity is modulated by
commensurable orbital moments of the Jovian planets, whose influence is also
detected in solar activity.

##  1 Introduction

Volcanic eruptions represent the ultimate expression of a long series of
physico-chemical processes initiated in the mantle with the generation of
magma. Isotopic studies, mainly based on U-series, have shown that the
formation, accumulation, and evolution of magma are processes that occur over
long time scales, typically ranging from a few hundred years to a few
millennia. On the other hand, magma transport, ascent to the surface, and
other processes leading to eruption, as identified by geophysical and petro-
geochemical studies, are much faster, occurring over days to decades (see
[1]). Closely linked to plate tectonics and mantle dynamics, volcanic activity
can also be seen as a response to changes in stress fields resulting from
regional or global processes, such as large tectonic earthquakes or tidal
effects from the Earth and oceans. Thus, volcanic eruptions reflect the
internal dynamics of our planet, often acting as a ”regulator” of accumulated
stresses in the lithosphere over a wide range of time scales, from decades to
centuries and beyond, as illustrated by rifting episodes along divergent plate
margins (eg [2]) or the effect of variations in ice cap thickness on magma
plumbing systems (eg [3, 4]). Deciphering the temporal pattern of volcanic
activity is therefore crucial for assessing triggers of volcanic eruptions.
This would not only help understand the dynamics of their internal plumbing
system, but also unravel the interaction with potential external processes
that may contribute to the destabilization of volcanoes on variable time
scales (eg [5, 6, 7, 3, 8, 9, 10, 11, 12, 13, 4]). Furthermore, the
recognition of specific temporal patterns in eruptive activity is a crucial
objective in volcanology, as it would allow for anticipation of future events
and thus better mitigation of volcanic hazards.

The coverage of volcanic eruptions has become increasingly systematic and
comprehensive over the past two centuries (eg [14, 15]). However, for most
individual volcanic systems, only a few dozen events are recorded, making any
statistical study of local structures at this timescale particularly
challenging. Nevertheless, when considered collectively for regional or global
analysis, volcanic eruptions can reveal specific spatial and temporal patterns
that can ultimately be correlated and linked to global processes. This
approach has already contributed to demonstrating that the spatial and
temporal distribution of eruptive events can be related to sea level
variations (eg [16, 7]). Recently, Dumont et al. ([13]) showed that the major
decadal periodicities present in both the global mean sea level and the
monthly number of eruptions are also present in the movement of Earth’s
rotation axis. This has led the authors to hypothesize that there could be a
causative link between pole motion (i.e., movement of Earth’s rotation axis),
a global process, and local events, i.e., volcanic eruptions occurring
worldwide. The redistribution of fluid masses in the crust would act as the
coupling link between these two processes; it would enable the transfer of
stresses, a link already suggested at seasonal scale (eg [6, 16]), and over
several thousand years (eg [7, 17, 18]).

Some studies (some dating back over a century) have highlighted an apparent
correlation between global, regional, and local volcanic activity and solar
cycles (eg [19, 20, 21]). Similar observations have been made for large
earthquakes as well (eg [20, 22, 23, 24]). Furthermore, it appears that the
occurrence of some of these major events, whether seismic or volcanic (such as
the eruption of Pinatubo in 1991), is associated with specific planetary
alignments (eg [25, 26]).However, most of these correlations/associations
between local geological phenomena and global processes such as solar activity
or planetary alignment are widely debated (eg [27]), primarily due to a major
challenge: the lack of a credible physical mechanism (eg [20, 21, 24]). How do
these interactions work and why do only certain geological structures seem to
be sensitive to such external influences?

In the following, we revisit these correlations between volcanic eruptions and
global processes from a global perspective, in light of recent studies ([28,
29, 30, 31, 32, 33, 34]). We address the interaction between local eruptive
phenomena occurring worldwide on planet Earth, i.e., the global number of
volcanic eruptions ([35]), and global processes using a two-step approach.
First, we analyze two time series: the global number of volcanic eruptions
(NVE) and solar activity recorded by the international sunspot number (ISSN).
Then, we perform a similar analysis/comparison with Earth’s pole motion (PM),
considering that a link between the coordinates of the rotation pole, sea
level variations, and global volcanic eruptions has already been introduced in
Dumont et al. ([13]). These analyses involve extracting, identifying, and
comparing the common quasi-periodic components present in the different time
series since 1700. This two-step procedure first allows us to assess whether
volcanic eruptions on Earth are consistent with Laplace’s global theory, the
main principles of which are summarized in Section 3; and second, to shed
light on a possible common external force that could influence processes
occurring in the Sun and in the Earth. In the latter case, the common external
force could participate in the processes leading to volcanic eruptions and act
as a trigger through a global mechanism of mass redistribution and stress. It
is important to note that we do not hypothesize a solar force on volcanic
activity, unlike a number of previous studies (eg . [22, 23, 21, 36, 37, 38]).

##  2 Periodicities detected in volcanic activity and other geophysical
phenomena

Periodic behaviors have been detected in short-term studies on volcanoes,
using either physical parameters or recorded visual observations during
periods of unrest, eruptions, and/or relative quiescence (see [13] and therein
references). Some of these periodicities correspond to those of terrestrial
and oceanic tides and have been observed in various records, such as changes
in eruptive behavior and explosion rates (eg [39]), seismic tremor (eg [40,
41, 42, 43]), seismicity (eg [44]), deformation (eg [45]), gas fluxes (eg [46,
42]), flow of lava at surface – energy radiated and volume emitted (eg [41,
42]). These observations show that magmatic and hydrothermal fluids, as well
as the volcanic structure, primarily respond to tidal forces with oscillatory
behavior ranging from hours to months. However, so far, the available
observations have been too scattered to deduce the main conditions that make a
volcano more sensitive or capable of responding to lunisolar tides.
Nevertheless, a critical state, open or fluid-rich systems represent the main
characteristics of volcanoes for which tidal influence has been reported.

On longer timescales, the triggering of volcanic eruptions, typically
considered through the eruptive history of individual volcanoes or analyzed in
the context of global or regional catalogs, may also be influenced by lunar
cycles, particularly the fortnightly cycle (eg [47, 48, 49, 5]). The signature
of gravitational forces induced by the relative motion of the Moon and the Sun
at these timescales is also present in global parameters recorded on Earth,
such as pole motion (eg [50, 51]), and local parameters that respond to global
processes on Earth, such as the geomagnetic field ([52, 33]), atmospheric
pressure and sea level ([53, 54, 55, 56, 57]), or tectonic plate motion ([58,
59]).

Seasonal variations correspond to oscillations that occur over a year or half
a year. They are associated with the Earth’s revolution around the Sun.
Seasonal variations have been observed in eruptive sequences, both at the
level of individual volcanoes and regionally ([6, 16]). Their correlation with
sea level variations has led to the interpretation that some volcanic
eruptions are the result of redistribution of oceanic mass. The influence of
the Sun (through variations in sunspot numbers) on eruptive activity has also
been suggested over longer timescales. Some eruptive events or episodes seem
to occur during periods of solar minimum or maximum activity, with cycles
mainly of $\sim$11 yr and $\sim$22 yr, but also $\sim$80 yr and more (eg [20,
19, 47, 60, 21, 36, 61, 62, 63, 37, 64, 38, 13]). Similarly to short-term
timescales and tidal periodicities, these multi-annual to centennial
variations are present in other geophysical parameters. Most of them record
both tidal and solar periodicities, as illustrated by the magnetic field (eg
[65, 66, 52, 67, 68, 33]), polar motion (eg [69, 30, 70]); sea level observed
from local and global data (eg [71, 72, 29, 56]), global surface air
temperatures (eg [73, 74, 75, 34]) and more generally, climate (eg [76, 77,
78, 73, 79, 80, 81]). Moreover, these links also seem to exist 1) on much
longer timescales of millennia, governed by astronomical configurations and
impacting climate and sedimentation (eg [82, 7, 9, 17, 83, 18]); and 2) also
for earthquakes, as suggested by numerous studies (eg [84, 22, 24]).

In this section, we sought to highlight the fact that among the many
geophysical observables associated with the various processes occurring on and
within the Earth (see also [30]), volcanic phenomena are no exception.
Although still poorly understood, volcanic phenomena are influenced by
periodicities across a wide range of time scales, as is the case for many
other geophysical observables measured on Earth, as suggested by various
methodologies - computational approach (eg [66, 67]), statistical (eg [16]),
spectral analysis (eg [78]), or decomposition approach (eg [69, 51]). This
first-order analogy between the periodic components detected in volcano-
related parameters and other geophysical parameters only pertains to the
observations of these periodicities and not the interpretations drawn from
them. In fact, although lunar-solar gravitational forces and solar activity
have both been considered as external forcings acting on the dynamics of our
planet and particularly its fluid envelopes, the way they interact is still
debated, as we will see in the next section.

##  3 On Laplace’s theory

In his treatise on celestial mechanics, Laplace ([85]) asserted that all
masses on the surface and interior of the Earth are influenced by orbital
moments, also known as angular moments, generated not only by the Moon and the
Sun, but also by other celestial bodies present in the solar system. The same
applies to the complete motion of the Earth’s rotation axis. This theory has
been revisited and validated in light of recent analyses based on time series
data spanning approximately 300 years of the Earth’s rotation axis motion (cf
[30, 70, 86]). This recent theory contradicts the currently most accepted
theory (eg [55]), which attributes a key role to atmospheric dynamics ([87]).
In this latter theory, seasonal oscillations present in many datasets,
including polar motion (eg [55, 29, 88, 57]), have generally been considered
in terms of angular momentum and interpreted as resulting from mass
redistribution of the atmosphere on the rotating Earth. This current theory
has not yet been quantitatively validated (see [55], chapter 7 introduction).
Both of these theories can be considered as a transfer of energy from an
undefined energy source to the fluid and solid envelopes of the Earth, as
described by the Liouville-Euler equations system (cf system 1, eg [55]
chapter 3). The direct implication of this description is that any changes in
the motions of the Earth’s rotation axis and the length of day (lod)
inevitably lead to a reorganization of mass on the surface and interior of the
Earth, and thus a redistribution of stresses on the Earth’s surface.

$\displaystyle j(\dot{\textbf{m}}/\sigma_{r})+\textbf{m}$
$\displaystyle=\bm{\psi}$ (1a) $\displaystyle\dot{m}_{3}$
$\displaystyle=\psi_{3}$ (1b)

In the system of Liouville-Euler equations (1), $j$ represents the complex
number such $j^{2}=-1$, $\sigma_{r}=\dfrac{C-A}{A}\Omega$ is the Euler’s free
wobble frequency, $A$ ($\sim 8.010*10^{37}kg.m^{2}$) and $C$ ($\sim
8.036*10^{37}kg.m^{2}$) are respectively the equatorial and axial Earth’s
momenta, $\Omega$ ($\sim 7.292115*10^{-5}rad.s^{-1}$) is the instantaneous
rotation velocity. The $\textbf{m}=m_{1}+j*m_{2}$ and $m_{3}$ parameters, are
dimensionless numbers defining the motion of the rotation pole (Figure 1); the
functions $\bm{\psi}=\psi_{1}+j*\psi_{2}$ and $\psi_{3}$ corresponding to
excitation functions (eg [55], chapter 3). The reorganization of the masses
and stresses on the Earth’s surface intervene through the functions
$\bm{\psi}$ as illustrated by the description of the isostasy theory (eg
[89]).

Figure 1: Geophysical reference system where the different parameters of the
system of Liouville-Euler equations are presented (see system 1 in main text).

This is about identifying common periodic components present in the various
envelopes of the Earth that enable the evaluation of these two theories. These
periodicities can be divided into two main categories. First, periodicities
shorter than 1 year (except for lunar nutation which has a period of 18.6
years). These periods have been attributed to tides induced by luni-solar
gravitational forces. Second, periodicities longer than 1 year. The most
frequently detected periods in this group are approximately $\sim$11,
$\sim$22, $\sim$30, $\sim$90 and $\sim$160 years, many of which are common to
solar physics (eg [82, 90, 91, 92, 93, 52, 94, 95, 96, 97, 98, 28]). In
particular, they have been associated with the Schwabe, Hale, an unnamed
30-year cycle linked to Saturn, Gleissberg, and José solar cycles,
respectively (cf [99, 100, 90, 92]).

Mörth and Schlamminger ([101]) show that these different solar cycles can also
be recognized in the orbital motions (specifically, the orbital moments) of
pairs and groups of pairs of the four Jovian planets; results that have been
confirmed by several recent studies (eg [31, 86]).

Let us explore this idea by first considering the Sun. Located at the center
of the solar system, it directly exerts on all planets of the solar system the
strongest gravitational attraction, yet it does not carry any orbital
momentum. According to the law of action-reaction in a Galilean referential,
we can convert dimensionally the value of Earth’s force of attraction exerted
by the Sun ($F=3.57*10^{22}kg.m.s^{-2}$) into an angular momentum ($F$*Sun-
Earth distance*Earth’s revolution period), which corresponds to the angular
momentum exerted by the Sun on the Earth, and which is of
$1.68*10^{41}kg.m^{2}.s^{-1}$. This value is smaller by 1-3 orders of
magnitude than that of the Jovian planets which are of
$1.93*10^{43}kg.m^{2}.s^{-1}$ for Jupiter (revolution: $\sim$11 years),
$7.82*10^{42}kg.m^{2}.s^{-1}$ for Saturn (revolution: $\sim$30 years),
$1.69*10^{42}kg.m^{2}.s^{-1}$ for Uranus (revolution $\sim$90 years) and
$2.50*10^{42}kg.m^{2}.s^{-1}$ for Neptune (revolution: $\sim$165 years). Mörth
and Schlamminger ([101]) studied this physical quantity through the
”commensurability” of different pairs and groupings of planet pairs that make
up our solar system. This quantity, which should be rational, represents the
ratio of the average orbital period of one or more pairs of planets. It
highlights their ability to transmit energy through angular momentum and is
associated with periodicities, expressing specific orbital configurations,
such as alignment or quadrature. For example, the neighboring pair of planets
Jupiter and Saturn induces periodic variations in planetary angular moments of
about $\sim$12, $\sim$29, $\sim$18 and $\sim$41 years and the Uranus-Neptune
pair induces variations of about $\sim$84, $\sim$165, $\sim$81 and $\sim$250
years. These pairs can be grouped together to obtain another set of
periodicities, and so on (see [101, 30, 28]). Thus, a 270-year oscillation can
be associated with the Uranus-Neptune pair, the Jose cycle (cf [92]) with
Uranus, the 30-year cycle (unnamed, [97]) with Saturn, the Hale cycle (cf
[100]) with the Jupiter-Saturn pair, and finally, the famous Schwabe cycle (cf
[99]) with Jupiter. Hence, these Jovian planets and their combinations act
through their angular momentum on the solar spots of our star, as illustrated
by Courtillot et al. ([28]) who used these results to predict the upcoming
solar cycle. These commensurabilities result in closely grouped periodicities
(cf [98]), explaining in particular the variations of the Schwabe cycle
between 9 and 14 years since 1700, with an average of 10.8 years. These
orbital moments act as an external force on the Sun, and possibly on the Earth
itself, and on volcanic activity in particular, as we will examine in this
article.

##  4 Data and Method

###  4.1 Volcanic eruptions and sunspots

First, we analyze the datasets of global volcanic eruptions and solar sunspot
activity. This link has been studied extensively in the literature, leading to
the assumption of a causality relationship between the two phenomena (eg [21,
36, 37, 38]). Similar studies conducted with other datasets, such as polar
movements, have instead concluded common influence (cf [28, 30, 86]). By
jointly analyzing global volcanic eruptions and solar sunspots and proceeding
step by step, we aim to better understand this interaction. To do so, we focus
on a period of 322 years from January 1700 to February 2022, which begins with
the start of the time series of the ISSN.

The catalog of volcanic eruptions ([35]) lists 5829 events over the period of
interest. It includes a majority (55%) of moderately explosive events
associated with a Volcanic Explosivity Index (VEI) of 2 (see [102]), 28% of
effusive eruptions ($0\leq\textbf{VEI}\leq 1$), 14% of moderate to cataclysmic
eruptions ($3\leq\textbf{VEI}$) and 3% of unknown VEI. While these estimates
provide general information on volcanic phenomena between 1700 and 2022, we
did not consider them to calculate the number of volcanic eruptions per year.
In reality, we considered each of these events as an expression of a
disruption of the magmatic system balance at a specific location on Earth and
at a specific time. These worldwide volcanic eruptions can also be considered
as a kind of global parameter highlighting the dynamics of our planet, which
is consistent with our global-scale approach. Moreover, the respective
analyses of ”events with $\textbf{VEI}\geq 2$” and ”all events” conducted in
Dumont et al. ([13]) have led to the detection of similar decadal
periodicities.

We have taken into account a time interval for which the catalog of volcanic
eruptions is only partially complete, which is a critical aspect that must be
considered. It is estimated that for $\textbf{VEI}\geq 3$, the GVP catalog is
complete since 1960 and for $\textbf{VEI}\geq 4$, since 1820 (cf . [15]). As
highlighted by various studies (eg [103, 14, 15, 104]), a significant increase
in the number of volcanic eruptions appears in the second half of the 18th
century (Figure 2a), resulting in a general upward trend (Figure 5a). This
trend is mainly related to a more systematic reporting of eruptive events
worldwide rather than a real increase in global volcanic activity ([14]).
Furthermore, it reverses around the year 2000 (Figures 2a and 5a), suggesting
that 1) over the past two decades, the catalog of global volcanic eruptions is
likely to be nearly complete, and 2) this global eruptive activity may not be
associated with a stationary process, as generally assumed (eg [15, 105, 104].
In this study, we will focus on the oscillations composing the time series.

We used the bootstrap technique (cf [106]) to construct what we consider as
the most representative temporal evolution of volcanic activity over the past
322 years. A thousand iterations were performed to randomly sample the annual
number of eruptions listed in the catalog, thus avoiding biases that could be
introduced by a time window using a moving average. The various sub-samples
range from a minimum of 2 years to a maximum of 40 years and define a family
of curves, with the median curve presented in Figure 2a. In this study, we
considered the median curve as the reference for these temporal variations and
will refer to it as NVE.

The ISSN has been measured for more than three centuries to monitor solar
activity, with the first yearly records going back to 1700. We used the yearly
mean total sunspot number (Figure 2b) from the World Data Center SILSO, Royal
Observatory of Belgium, Brussels 111http://www.sidc.be/silso/datafiles.

Figure 2: Number of worldwide volcanic eruptions (NVE) (a) and international
sunspot number (ISSN) (b) spanning 1700-2022. (a) The black dots represent the
number of worldwide volcanic eruptions per year, connected by the gray curves
representing the bootstrapped series, the blue curve representing their
median. (b) Temporal evolution of the yearly number of sunspots since 1700.

###  4.2 Time-series analysis

The time series of ISSN and NVE were analyzed using iterative Singular
Spectrum Analysis (iSSA). For more details on the SSA method, you can refer to
Golyandina and Zhigljavsky ([107]). For the properties of Hankel and Toeplitz
matrices, you can consult Lemmerling and Van Huffel ([108]) and for the
Singular Value Decomposition (SVD) algorithm, you can refer to Golub and
Reinsch ([109]). The periodicity of each component identified by iSSA is then
evaluated using the Fourier transform, with uncertainties given by the full
width at half maximum of the peak. In the following text, we prefer to use the
term ”pseudo-period” to emphasize that these oscillations are not pure
oscillatory signals.

We also provide a comparison of the 22-year solar cycle extracted by iSSA and
by continuous wavelet transform analysis (cf [110]), a widely used technique
in the analysis of solar spot time series (eg [95]).We considered a Morse
wavelet which is particularly well-suited for signals with variable spectra
(cf [111, 112]). The scalogram of NVE is shown in Figure 3a. We perform the
comparison between the two techniques for the 22-year period which represents
an important component in the NVE time series (Figure 3). Based on the
redundancy of information of the wavelet kernels, it is possible to extract
the peak corresponding to a specific periodicity within the time scale of
interest to reconstruct the ad hoc signal (cf [110]). The difficulty in
applying such an approach lies in the spreading of energy over multiple time
scales, as illustrated in the 22-year period in Figure 3a. This effect can be
reduced using a reassignment method (eg [113]) to optimize the reconstruction
of the period of interest. This would require an additional processing step
that we have not implemented in the context of this comparison. In our case,
we relied solely on the inverse wavelet transform to reconstruct the pseudo-
oscillation corresponding to the peak’s maximum energy.

Figure 3: Extraction of pseudo-oscillations in the NVE time-series: comparison
between the continuous wavelet transform and iterative SSA with the example of
the 22-year cycle. (a) Scalogram associated with the NVE curve spanning 1700
to 2022. The gray area bordered by the white dashed curve delineates the no-
definition sector of the wavelet transform. The red rectangle points out the
wavelet coefficients for the 22-year period. (b) The 22-year pseudo-cycle
extracted by iSSA (red curve) and the same cycle extracted by continuous
wavelet transform (black). The latter corresponds to the wavelet scales boxed
in (a).

The application of iSSA to the NVE time series has allowed us to directly
extract various pseudo-cycles, including the one identified with the Scalogram
and centered around 22 years. The resulting component is shown in Figure 3b
and overlaid with the one obtained from continuous wavelet analysis. This
figure reveals excellent agreement for the pseudo-oscillations extracted by
both techniques, iSSA and continuous wavelet transform, confirming the
reliability of the results obtained by iSSA (which is not only faster and
easier to implement, but also more precise than wavelet analysis).

##  5 Pseudo-oscillations detected in the NVE and ISSN

Figure 4: Original and Reconstructed time-series of NVE (a) and ISSN (b). (a)
The NVE curve is represented by the median of bootstrapped series (blue curve
aforementioned as original time-series) and its reconstruction (red curve).
(b) The original ISSN time-series is shown in red and its reconstruction in
blue. See text for more details.

We have identified thirteen components in the NVE time series using iSSA.
These correspond to the first thirty-five components of the original time
series. In addition to the trend, twelve periodic components with durations
ranging from 4.91 $\pm$ 0.05 to 132.32 $\pm$ 37.57 years have been identified
(Table 1). They account for a total variance of 77.4% of the NVE curve,
allowing for the reconstruction of most of its variations (Figure 4a).
Similarly, we analyzed the ISSN time series and found seven pseudo-periods
ranging from 8.4 $\pm$ 0.2 to 90.9 $\pm$ 17.5 years, in addition to the trend,
using the first fifteen components (Table 1). They account for 79.5% of the
total variance (Figure 4b), as previously shown in the works of Le Mouël et
al. ([96, 98]) and Courtillot et al. ([28]). Among these seven periodic
components, four of them are common to those extracted from NVE. The four
common periodicities are all less than 30 years, with periods of approximately
$\sim$9, $\sim$11, $\sim$13 and $\sim$27 years (Table 1 and Figure 5).

Table 1: List of common periodicities extracted from the time-series of NVE,
ISSN and $m_{2}$ component of the PM, using iSSA for the 1700-2022 time
interval. All periodicities in the NVE are also present in the $m_{2}$ time-
series, except that of $\sim$27 and $\sim$5 years. The last two columns
indicate the combination of planet pairing and the associated periodicities as
reported in Courtillot et al., ([28]) and Lopes et al., ([30]). The planets
are listed as follows: JUP for Jupiter, NEP for Neptune, PLU for Pluto, SAT
for Saturn, URA for Uranus. The brackets separate different pairing.

We now present some of the common pseudo-cycles found in NVE and ISSN time
series. First, trends extracted from NVE and ISSN time series show a
monotonous increase from 1700 to $\sim$2000 for both datasets, followed by a
plateau starting from 2000 (Figure 5a). Although their uncertainties are about
ten years (Table 1), the amplitude modulations of the approximately 60-year
and 50-year cycles extracted from NVE and ISSN data are clearly different:
while the amplitude of the approximately 50-year component of the solar cycle
decreases over time, the amplitude of the approximately 60-year component of
NVE increases (Figure 5b). This approximately 60-year cycle is well known in
climate studies (eg [78, 114, 73]).

Figure 5: Trends and some quasi-periodic components present in both the NVE
(black) and ISSN (red) extracted using iSSA. (a) The trends of the NVE and
ISSN time-series, (b) the $\sim$50-60-year, (c) the $\sim$27-year and (d) the
$\sim$11-year pseudo-periodic components.

In Figure 5d, the well-known 11-year solar cycle, also known as the Schwabe
cycle (cf [99]), shows an increase in amplitude in both time series. This
similar behavior of amplitudes is accompanied by phases that occasionally
synchronize, such as between 1760 and 1800 or between 1910 and 1950. Finally,
we illustrate the pseudo-oscillations of 27 years extracted by iSSA from the
NVE and ISSN time series (Figure 5c). The amplitude variations for both time
series do not show as clear correlation as in the cases of approximately 50-60
year cycles and 11-year cycles.

These common periodicities could explain the numerous correlations obtained
between solar and volcanic activity that are discussed in the literature (cf
[19, 20, 60, 22, 23, 21, 36, 37, 38]).We have found that several periodicities
are common to the NVE and ISSN time series, suggesting a possible link between
the two. However, this relationship varies from one cycle to another and
evolves over time. This is illustrated with the approximately 50-60 year cycle
(Figure 5b) whose amplitude evolves in opposite directions for the components
extracted from the NVE and ISSN, or with the 11-year cycle, whose modulation
evolves similarly but whose phase apparently synchronizes every $\sim$150
years (Figure 5d). Considering that not all the pseudo-cycles detected in the
NVE are present in solar activity (Table 1) and that the common periodicities
have also been detected in other geophysical measurements (eg [77, 79, 69, 68,
98]), we hypothesize that a single common external forcing could explain both
the different and common oscillations of these two parameters, NVE and ISSN,
whose physics are completely different (eg [31, 86]).

##  6 Comparison between NVE and PM

In Laplace’s theory ([85]), it is clearly stated that celestial bodies
comprising our solar system carry energy that is transferred through exchanges
of angular momentum to the Sun and to the Earth’s rotation axis. In this
section, we examine the periods present in the polar motion (PM) that have
already been extracted by Lopes et al. ([30]), as well as detected in the
length of day (lod) by Le Mouël et al. ([51]). With the exception of the free
Chandler oscillation ($\sim$1.19 years) and the annual forced oscillation (1
year), all the periods detected in the variation of day length are also
present in the polar motion (Table 1). The periods of 1.0 year and 1.19 years
cannot be detected in the variation of day length because annual sampling has
been used. This does not question their existence; for example, the annual
component has already been suggested in the analysis of regional volcanic
activity (eg [16]).

Now let’s look at the common periodicities extracted from the variation of day
length and polar motion, specifically in the E-W component of polar motion
called m2 (Figure 6a). Although this component varies very similarly to
component m1 (the N-S component), it shows the largest variations of the two
polar motion coordinates, which is why we chose to focus on this component.
The time series of polar motion starts in 1846, and we present the components
extracted by iSSA for the variation of day length and polar motion over the
past 176 years, as well as the evolution of their phase lag over time (Figure
8). We have also considered day length to complement some observations. Since
day length is measured by satellites, this time series is much shorter than
the others, starting only in 1962 (Figure 6b). Therefore, the comparison
between the components of day length variation and day length will be
presented over the last sixty years. We draw the reader’s attention to the
inversion of the axis representing the component in Figures 8b-i. Positive
values of m2 indicate an east-west direction, a sign convention that has no
physical significance. The inversion of the axis of the components allows for
a better evaluation of the different correlations. In the following, we
present the pseudo-periodicities extracted in decreasing order.

Figure 6: Temporal evolution of the $m_{2}$ component of PM since 1846 (a) and
the lod since 1962 (b) .

As observed in Figure 8a, the two trends extracted from the NVE and PM data
show an overall increase over the 176-year interval, although the PM trend
begins after a low plateau that ends around 1880. Figure 8b represents a cycle
centered around $\sim$130-140 years, with a period of 128.86 $\pm$ 42.48 years
for NVE and 135.53 years for the $m_{2}$ component. It should be noted that
there are no uncertainties provided for the PM period (Table 1). This is due
to the relatively short time interval considered for the time series analysis,
which is not sufficient to capture such a long component. Considering the
uncertainty in the NVE data, which is approximately 40 years, it can be
suggested that this period corresponds to a cycle of approximately 160 years
cf [92]). Figure 8b shows that these two components, NVE and $m_{2}$, are in
phase, and although only two oscillations are observed for 1846-2022, their
amplitudes evolve similarly, decreasing over time.

A cycle of approximately 60 years has also been extracted, centered at 64.43
$\pm$ 9.88 years for NVE and 58.51 $\pm$ 13.14 years for $m_{2}$ (Figure 8c).
As for the cycle of approximately 130-140 years, this cycle for the NVE and PM
components is also highly correlated. For this pseudo-cycle and subsequent
ones, we calculated the instantaneous phase shift using the phase differences
between the Hilbert transforms of NVE and $m_{2}$. The phase shift for the
components of about 60 years varies smoothly, averaging about 1 year. Such a
phase shift is small compared to the pseudo-cycle of about 60 years to which
it is associated, indicating that this component for NVE and $m_{2}$ can be
considered in phase. In Figure 8d, we show a pseudo-cycle of about 40 years
extracted in both NVE and $m_{2}$, approximately 42.95 $\pm$ 5.83 and 40.96
$\pm$ 7.02 years, respectively. The phase correlation appears clearer when it
is integrated over time, a transformation for which we will provide further
explanations later in this section. We also calculated the continuous phase
shift and obtained an average value of about two years between these two
components, confirming that NVE and the integrated component of $m_{2}$ are in
phase for the considered time interval. The time series of NVE and $m_{2}$
include a component of about 25 years whose amplitudes vary in opposite
directions (Table 1 and Figure 8e). As for the periodicity of about 40 years,
we considered the integral of the component on time to compare it with NVE.
Thus, the approximately 25-year component of NVE is shifted by about 3.5 years
in 1846, a phase shift that is relatively stable until around 1960 and is in
the order of quadrature for the 25-year cycle. This suggests that these two
pseudo-cycles are mainly linked by a phase quadrature, which represents a
special characteristic that we will explain later in this section.

In Figure 8f (left), we present the pseudo-cycle of approximately 19 years
with a period estimated at 19.23 $\pm$ 2.70 years for NVE and of 18.61 $\pm$
2.71 years for $m_{2}$ component. The amplitude of both components is
characterized by a strong increase since 1846, with the increase being more
pronounced for the PM component, which has almost zero amplitude until 1900.
This cycle is well known and corresponds to the large lunar nutation. As for
the approximately 25-year cycle, the phase lag is much larger than the
uncertainties on the extracted pseudo-cycle, resulting in evolving phase lags
within the considered time interval. Moreover, this lunar periodicity is also
detected in the length of day (lod), as shown in 8f (right), although the lod
time series is much shorter, starting in 1962. By proceeding similarly to what
we have done for the previous periodicities, but this time for the lod, we
show that the phase lag between NVE and the integral of the lod over time is
small and relatively stable over time, averaging about 1 year. This suggests
that NVE and the integral of the lod over time are almost in phase. With a
similar approach for the approximately 11-year periodicity (Figure 8g and
Table 1), we observe that the amplitudes of NVE and the $m_{2}$ component vary
inversely, and their phase lag increases steadily over time (Figure 8g, left).
This component was also extracted from the lod, with a period of 11.49 $\pm$
1.57 years. Thus, NVE and the integral of the lod over time are clearly in
phase, as shown by the small variations in the phase lag over the period from
1962 to 2022 (Figure 8g, right).

Figure 7: Examples of some common periodicities extracted in the NVE (black
curve) and component of the PM (blue curve, from Lopes et al., [30]) using
iSSA with (a) the trends, (b) the $\sim$130-140 year, (c) $\sim$60-year, (d)
$\sim$40-year, (e) the $\sim$25-year, (f) the $\sim$19-year, (g) the
$\sim$11-year, (h) the $\sim$13-year and (I) the $\sim$9-year pseudo-cycles.
For some periodicities (d-i), the integral of $m_{2}$ over time is also
represented by a blue dotted curve. For each periodicity, the phase shift
between NVE and $m_{2}$ component (a-c) or between NVE and the integral of
$m_{2}$ over time (d-i) was calculated and reported in years over the time
interval considered. For the $\sim$19 and $\sim$11-year pseudo-cycles, the NVE
components was similarly represented but together with that of the lod and its
integral over time (green full and dotted lines, respectively) Figure 8:
Continued

Finally, pseudo-cycles of approximately 13 and 9 years (Figure 8h-i and Table
1), are shown for NVE and for the $m_{2}$ component as well as its time
integral. For the pseudo-cycle of approximately 13 years, NVE and $m_{2}$ are
in phase, while NVE and the integral of $m_{2}$ are in quadrature phase. Their
phase lag decreases mainly from 1846 to around 2000, while that of the
approximately 9-year periodicity increases. Moreover, this phase lag, being
much larger ($>$ 20 times) than the uncertainties of the extracted
periodicity, it results in an evolving phase shift between the NVE component
and $m_{2}$.

In Figures 8, we conducted comparative analyses of the extracted periodic
components in the NVE, the E-W coordinate of the polar motion (PM) and the
length-of-day (lod), which represents the rotational component of PM, for two
periodicities. We completed this comparison by including the time integral of
PM and lod, whose phase relationships with NVE appear clearer in certain
cases. This approach stems from the Louville-Euler equations (system 1). In
reality, PM, lod (related to polar motion variations), and the excitation
functions, which are part of mass and stress redistribution, are connected by
temporal derivatives. As stated by Laplace ([85]), demonstrated again by Lopes
et al. ([30, 70]), and explained in more details by a mechanism proposed by
Lopes et al. ([86]), the lod represents the temporal derivative of polar
motion. The latter is modulated by exchanges of moments of inertia with other
planets in the solar system in a similar way to how the weight of a spinning
top interferes with its rotation ([115]). Thus, the stresses resulting from
these interactions correspond to the time integral of polar motion, resulting
in a quadrature phase relationship, similar to a differentiation link,
explaining why we integrated PM and lod over time in Figure 8.

##  7 Discussion and Conclusion

The analysis of volcanic activity over the past 300 years is challenging at
the scale of individual structures due to the low number of events. However,
when considered in the context of a global analysis, volcanic eruptions can
provide new insights into worldwide phenomena on Earth that may contribute to
triggering eruptions. We have taken a global approach to show that the
temporal evolution of the global number of volcanic eruptions is mainly
composed of cycles of approximately $\sim$9 to $\sim$130 years. Most of these
periodicities are observed in both volcanic activity and the motion of the
Earth’s rotation pole. Some of these cycles are also found in solar activity
(Table 1). By using a two-step analysis, we suggest that there is a common
external force acting on volcanic eruptions on Earth and solar activity, but
not necessarily a causal link between solar spots and volcanic eruptions. Such
a link would not be favored, as illustrated by various interpretations
suggested in the past (eg [19, 20, 60, 22, 23, 21, 36, 37, 38]).

Dumont et al. ([13]) have identified a link between global volcanic eruptions
and the global mean sea level, which is known to exhibit planetary-scale
periodicities (eg [29, 56]). Such a link can be understood in the case of
insular, submarine and coastal magmatic plumbing systems, which are the most
abundant ([13]): stress variations can be induced by pressure changes imposed
by oceanic water mass transport at different timescales. However, for volcanic
systems located further inland, the redistribution of oceanic masses likely
has a residual effect on their dynamics. Therefore, there must be another
causal link that applies to all volcanic systems to explain the common
periodicities detected in worldwide volcanic eruptions and solar activity.
Recent studies, including Courtillot et al. ([28]), Le Mouël et al. ([29]) and
Lopes et al. ([30, 70]), have shown that the temporal evolution of planetary
orbital moments is reflected in sunspots and pole motion, as evidenced by
their common periodicities. Transfer of energy through planetary orbital
moments could represent this external force that is common to volcanic and
solar activity.

In Figures 8, we show the components of iSSA that illustrate the common
periodicities in NVE and PM (we only present the results for the $m_{2}$
coordinate of the rotation pole). The most significant evidence of a shared
signature and a chain of direct or indirect causality links among the
phenomena analyzed in this study comes from Table 1. Out of the list of 12
periodicities identified in NVE, only two are apparently absent in $m_{2}$
($\sim$27 and $\sim$5 years). All the others fall within the uncertainty range
of the extracted components for $m_{2}$ and NVE. All these shared pseudo-
cycles can be associated with a commensurable periodicity that corresponds to
the Jovian planets or their combinations in pairs and pairs of pairs (see
Table 1, [101, 28, 30]).

For longer periods than $\sim$19 years, which is associated with the lunar
great nutation cycle, good correlations are obtained between the components of
the NVE and the polar motion that are either in phase or in anti-phase
(Figures 8b-e). For shorter periods (Figures 8f-i), this phase relationship
appears to be lost, and no clear phase relationship between NVE, PM and lod is
observed due to higher phase lags. However, the correlations can be improved
by considering the time integral of the length of day (lod). Therefore, the
phase relationship between NVE and $m_{2}$ indicates that the period from
which the perturbations in the length of day prevail over those of polar
motion is that of the great lunar nutation cycle.

These observations are consistent with Laplace’s theory ([85]), particularly
the system of linear differential equations of Liouville-Euler (see system 1).
Furthermore, our results suggest that the long periods resulting from the
commensurability of planet pairs affect the precession and nutation of the
Earth, which describe the major changes in the orientation of the Earth’s
rotation axis. Shorter periods appear to have a stronger influence on the
length of day, i.e., the component that measures the rotation speed around the
polar axis. The link between volcanic activity and day length has been
explored or suggested in past and recent studies ([49, 21, 116, 117, 118,
12]). Thus, two parallel causal chains of forces could be involved, one acting
directly on the tilt of the Earth’s rotation axis, and the other on its
rotation, both modulating crustal stresses and fluid movements that in turn
influence the triggering of global-scale volcanic system activity (Figure 9).
In light of this study, it is likely that the orbital moments of the Jovian
planets and more generally planetary orbits modulate other dynamic processes
on Earth, as suggested by other studies ([58, 119, 120]). The mathematical and
physical mechanisms by which energy transfer to Earth occurs through planetary
torques are discussed in Lopes et al. ([86]).

Figure 9: Cartoon summarizing the action of the Jovian planets on the Sun and
Earth dynamics. In blue, a series of other interactions and phenomena that
could possibly be affected by the passage of Jovian planets, but not
considered in this study.

As suggested for large explosive eruptions in ice core records (eg [118]), our
findings reinforce the idea that volcanic eruptions on Earth cannot be
considered as entirely random processes and are partially controlled by
external influences related to the ephemerids of celestial bodies in our solar
system, acting at multi-annual, multi-decanal, centennial, or even longer
timescales. Further investigations would help better understand this
interaction as well as the local and regional conditions that cause volcanic
systems to respond to these global stress changes, which could potentially
contribute to anticipating phases of increased volcanic activity, as suggested
by Bilham et al. ([12]).

In conclusion, we have analyzed the global number of volcanic eruptions (NVE),
as well as the number of sunspots (ISSN) and the motion of the Earth’s
rotation pole (PM) over a period of 322 years (1700-2022). We have
demonstrated the presence of a set of common periodicities in global volcanic
activity, sunspots, and polar motion. These periodicities range from
approximately 5 years to around 130-140 years. We interpret these results
within the framework of the Laplace paradigm, the same one used by Milankovic
([82]) for much longer cycles. With more shared periodicities between NVE and
PM than between NVE and ISSN, we propose that these periodic signals are the
signature of a common external forcing on solar activity and various global
phenomena on Earth, including changes in the tilt of Earth’s rotation axis.
This results in changes in stress that impact global volcanic activity. The
ultimate source of these changes lies in the energy transmitted by the orbital
moments of individual Jovian planets and/or their combinations in pairs and
pairs of pairs.

Acknowledgement: SD acknowledges Edward Venzke from the Smithsonian Global
Volcanism Program for his support in data access. This work was co-financed by
the FEDER funds through the Programa Operacional Competitividade e
Internacionalização - COMPETE 2020 and through national funds through the
Portuguese Fundação para a Ciência e a Tecnologia (FCT - I.P./MCTES) as part
of the SHAZAM (PTDC/CTA-GEO/31475/2017 - POCI-01-0145-FEDER-031475) project
and by the FCT I.P./MCTES through national funds (PIDDAC) – UIDB/50019/2020-
IDL and PTDC/CTA-GEF/6674/2020 - RESTLESS. SD acknowledges the FCT for its
financial support through the contrat 2021.00876.CEECIND as well as the FCT
and European Union for their ﬁnancial support through the postdoctoral
fellowship SFRH/BPD/117714/2016, co-ﬁnanced by the Ministério da Ciência,
Tecnologia e Ensino Superior (MCTES), Fundo Social Europeu (FSE), and Programa
Operacional Regional Centro (Centro 2020).

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# Combining E-Graphs with Abstract Interpretation

Samuel Coward  Electrical and Electronic Engineering
Imperial College London
Email<EMAIL_ADDRESS>George A. Constantinides  Electrical and
Electronic Engineering
Imperial College London
Email<EMAIL_ADDRESS>Theo Drane  Numerical Hardware Group
Intel Corporation
Email<EMAIL_ADDRESS>
###### Abstract

E-graphs are a data structure that compactly represents equivalent
expressions. They are constructed via the repeated application of rewrite
rules. Often in practical applications, conditional rewrite rules are crucial,
but their application requires the detection – at the time the e-graph is
being built – that a condition is valid in the domain of application.
Detecting condition validity amounts to proving a property of the program.
Abstract interpretation is a general method to learn such properties,
traditionally used in static analysis tools. We demonstrate that abstract
interpretation and e-graph analysis naturally reinforce each other through a
tight integration because (i) the e-graph clustering of equivalent expressions
induces natural precision refinement of abstractions and (ii) precise
abstractions allow the application of deeper rewrite rules (and hence
potentially even greater precision). We develop the theory behind this
intuition and present an exemplar interval arithmetic implementation, which we
apply to the FPBench suite of benchmarks.

###### Index Terms:

abstract interpretation, interval arithmetic, static analysis, e-graph

##  I Introduction

Equivalence graphs, commonly called e-graphs, provide a compact representation
of equivalence classes (e-classes) of expressions, where the notion of
equivalence is with respect to some concrete semantics [1]. The recent egg
tool introduced e-class analysis, a technique to enable program analysis over
an e-graph, attaching analysis data to each e-class [2]. This paper formalises
some of the concepts required to produce e-class analyses enabling e-graph
growth via conditional rewrites. We show that partitioning expressions into
e-classes gives rise to a natural lattice-theoretic interpretation for
abstract interpretation (AI), resulting in the generation of precise
abstractions. Figure 1 provides an example, in which interval analysis of two
equivalent expressions can be combined to produce tighter enclosing intervals
than either one alone.

We develop the general theoretical underpinnings of AI on e-graphs, exploiting
rewrites to produce tight abstractions using a lattice-theoretic formalism. We
also provide a sound interpretation of cycles naturally arising in e-graphs,
corresponding to extracting abstract fixpoint equations, and show that a known
interval algorithm, the Krawczyk method, results as a special case.

This paper extends and generalises a talk abstract presented at the EGRAPHS
workshop [3] in the following ways:

  * •

formalization of AI with e-graphs in a lattice-theoretic setting

  * •

relating fixpoints to e-graph cycles to automatically discover iterative
abstract refinement methods,

  * •

an interval arithmetic implementation with associated expression bounding
results.

A short background overview is provided in §II. In §III we present the
theoretical application of AI to e-graphs and then demonstrate its viability
using an interval arithmetic implementation in §IV. Lastly, in §V we present
results generated by this implementation.

Figure 1: E-graph containing equivalent real arithmetic expressions
$\frac{x}{x+y}$ and $\frac{1}{1+(y/x)}$ in the root e-class (at the top).
Intervals are associated with each e-class. Input constraints are propagated
upwards via an e-class analysis.

##  II Background

The e-graph data structure is commonly found in theorem provers and solvers
[4, 5]. It represents multiple expressions as a graph (e.g. Figure 1), where
the nodes represent functions, grouped together into collections of e-classes.
Edges connect nodes to e-classes, as a given sub-expression may be implemented
using any of the nodes found in the child e-class. E-graphs are often combined
with an optimisation technique called equality saturation [2, 6, 7], which
deploys equivalence preserving transformations to monotonically grow the
e-graph and discover alternative equivalent expressions. A recent resurgence
of e-graph research [2] has seen the technique applied to floating point
numerical stability analysis [8], mapping programs onto hardware accelerators
[9] and optimizing linear algebra [10]. Some works make heavy use of
conditional rewrites of the form $\phi\Rightarrow\ell\to r$, for example
$x\neq 0\Rightarrow x/x\to 1$, which require the e-graph construction
algorithm to determine whether the rewrite is applicable in each case [11, 2].

AI has primarily been applied to static program analysis [12], where computer
programs are automatically analyzed without actually being executed. It uses
the theory of abstraction to consider over-approximations of the program
behaviour using alternative interpretations [13, 14]. Existing tools have
incorporated term rewriting to refine their abstractions [15]. Previous work
on abstract congruence closure [16, 17] has exploited e-graphs to combine
different abstract domains [18], but we are not aware of existing work
exploiting the tight interaction between AI and e-graph construction.

##  III Theory

###  III-A Abstraction

From a theoretical viewpoint, AI [14] is concerned with relationships between
lattices, defined via Galois connections.

######  Definition 1 (Lattice).

A lattice is a partially ordered set (poset) $\langle L,\leq\rangle$, such
that $\forall a,b\in L$ the least upper bound (join) $a\sqcup b$ and the
greatest lower bound (meet) $a\sqcap b$ both exist.

######  Definition 2 (Galois connection).

Given a poset $\langle\mathcal{C},\sqsubseteq\rangle$, corresponding to the
concrete domain, and a poset $\langle\mathcal{A},\preccurlyeq\rangle$,
corresponding to the abstract domain, a function pair
$\alpha\in~{}\mathcal{C}\rightarrow\mathcal{A}$,
$\gamma\in\mathcal{A}\rightarrow\mathcal{C}$, defines a Galois connection iff

$\forall
P\in\mathcal{C}.\;\forall\overline{P}\in\mathcal{A}.\;\alpha(P)\preccurlyeq\overline{P}\Leftrightarrow
P\sqsubseteq\gamma(\overline{P})\textrm{,}$ $\textrm{written
}\langle\mathcal{C},\sqsubseteq\rangle\overset{\alpha}{\underset{\gamma}{\rightleftarrows}}\langle\mathcal{A},\preccurlyeq\rangle.$

The pair $(\alpha,\gamma)$ define the abstraction and concretization
respectively, allowing us to over-approximate (i.e. abstract) concrete
properties in $\mathcal{C}$ with abstract properties in $\mathcal{A}$.

######  Definition 3 (Sound abstraction [14]).

$\overline{P}\in\mathcal{A}$ is a sound abstraction of a concrete property
$P\in\mathcal{C}$ iff $P\sqsubseteq\gamma(\overline{P}).$

Consider expressions evaluated over a domain $\mathcal{D}$. By imposing a
canonical ordering on the variable set, we work within a defined subset
$I\subseteq\mathcal{D}^{n}$, which encodes a precondition on the set of
(input) variable values. Now consider a (concrete) semantics of expressions
$\llbracket\cdot\rrbracket_{\cdot}\in\mathrm{Expr}\to I\to\mathcal{D}$, where
$\mathrm{Expr}$ denotes the set of expressions, so $\llbracket
e\rrbracket_{\rho}$ denotes the interpretation of expression $e$ under
execution environment (assignment of variables to values) $\rho\in I$. Let
$\llbracket e\rrbracket=\\{\llbracket e\rrbracket_{\rho}\,|\,\rho\in I\\}$.
The e-graph data structure encodes equivalence under concrete semantics, which
we shall now define precisely.

######  Definition 4 (Congruence).

Two expressions $e_{a}$ and $e_{b}$ are congruent, $e_{a}\cong e_{b}$, iff
$\llbracket e_{a}\rrbracket_{\rho}=\llbracket e_{b}\rrbracket_{\rho}$ for all
$\rho\in I$.

######  Lemma 1.

If $e_{a}\cong e_{b}$ and $\overline{P}$ is a sound abstraction of $\llbracket
e_{a}\rrbracket$, then $\overline{P}$ is a sound abstraction of $\llbracket
e_{b}\rrbracket$.

###### Proof.

by definition of congruence. ∎

This lemma implies that a sound abstraction of one expression in an e-class is
a sound abstraction of all expressions in the e-class. Precision refinement
relies on the following.

######  Lemma 2.

For any two sound abstractions $\overline{P}_{a}$ and $\overline{P}_{b}$ of
$P$, the meet $\overline{P}_{a}\sqcap\overline{P}_{b}$ is also a sound
abstraction of $P$.

###### Proof.

$P\sqsubseteq\gamma(\overline{P}_{a})$ (sound abstraction)
$\Rightarrow\alpha(P)\preccurlyeq\overline{P}_{a}$ (Galois connection) and
similarly
$P\sqsubseteq\gamma(\overline{P}_{b})\Rightarrow\alpha(P)\preccurlyeq\overline{P}_{b}$.
Therefore $\alpha(P)\preccurlyeq\overline{P}_{a}\sqcap\overline{P}_{b}$ (meet
definition) and hence
$P\sqsubseteq\gamma(\overline{P}_{a}\sqcap\overline{P}_{b})$ (Galois
connection). ∎

###  III-B Application to E-graphs

Consider an e-graph. Let $S$ denote the set of e-classes, and
$\mathcal{N}_{s}$ the set of nodes in the equivalence class $s\in S$. With
each e-class associate an abstraction $A\in\mathcal{A}$ and write
$\mathcal{A}\llbracket s\rrbracket=A$. Interpreting a $k$-arity node $n$ of
function $f$ with children classes $s_{1},...,s_{k}$, using the abstracted
function $\bar{f}=\alpha\circ f\circ\gamma$:

$\mathcal{A}\llbracket n\rrbracket=\bar{f}(\mathcal{A}\llbracket
s_{1}\rrbracket,...,\mathcal{A}\llbracket s_{k}\rrbracket).$ (1)

$0$-arity nodes represent either constants with exact abstractions in
$\mathcal{A}$ or variables with user specified abstract constraints.

For acyclic e-graphs, we propagate the known abstractions upwards using Eqn.
1, taking the greatest lower bound (meet) across all nodes in the e-class.

$\mathcal{A}\llbracket
s\rrbracket=\bigsqcap_{n\in\mathcal{N}_{s}}\mathcal{A}\llbracket n\rrbracket$
(2)

The propagation algorithm is described in Figure 2, where

$\texttt{make}(n)=\mathcal{A}\llbracket n\rrbracket\textrm{ and
}\texttt{meet}(A_{1},A_{2})=A_{1}\sqcap A_{2}.$

These functions are analogous to those described for an e-class analysis in
[2], but replace their join function with a meet.

⬇

workqueue = egraph.classes().leaves()

while !workqueue.is_empty():

s = workqueue.dequeue()

for n in s.nodes():

skip_node = false

for child_s in n.children():

if child_s.uninitialized:

workqueue.enqueue(s)

skip_node = true

if skip_node:

continue

elif s.uninitialized:

s.data = make(n)

s.uninitialized = false

workqueue.enqueue(s.parents())

elif !(s.data<=meet(s.data,make(n))):

s.data = meet(s.data,make(n))

workqueue.enqueue(s.parents())

Figure 2:  Pseudocode for abstract property propagation in an e-graph.

In the abstract domain the notion of equivalence is different,
$n_{a},n_{b}\in\mathcal{N}_{s}\not\Rightarrow\mathcal{A}\llbracket
n_{a}\rrbracket=\mathcal{A}\llbracket n_{b}\rrbracket$, which results in
tighter abstractions since the meet corresponds to the most precise
abstraction in $\mathcal{A}$. In the algorithm in Figure 2, by initializing
the workqueue with only the modified e-classes after application of a rewrite,
the abstract properties of the e-graph can be evaluated on the fly. On-the-fly
evaluation facilitates conditional rewrite application as more precise
properties are discovered during construction and is used in §IV.

A positive feedback loop is created by combining AI and e-graphs. A larger
space of equivalent expressions is explored as more rewrites can be proven to
be valid at exploration time. In turn, expression abstractions are further
refined by discovering more equivalent expressions, allowing even more valid
rewrites, and the cycle continues. Additionally, several equivalent
expressions may contribute to the tight final abstraction. An example is shown
in §IV.

###  III-C Cyclic E-graphs and Fixpoints

Cyclic e-graphs arise when an expression is equivalent to a sub-expression of
itself with respect to concrete semantics, for example $e\times 1\cong e$. Let
$e\cong e^{\prime}$ where $e$ appears as a subterm in $e^{\prime}$. Treating
other subterms as absorbed into the function, let
$f:\mathcal{D}\to\mathcal{D}$ be the interpretation of $e^{\prime}$ as a
(concrete) function of $\llbracket e\rrbracket_{\rho}$, so that – in
particular – $f(\llbracket e\rrbracket_{\rho})=\llbracket e\rrbracket_{\rho}$
due to the congruence. Abstracting $f$ via a sound abstraction $\bar{f}$,
yields the corresponding abstract fixpoint equation $a=a\sqcap\bar{f}(a)$
where the meet operation arises from Equation 2.

Now consider the function $\tilde{f}(a)=a\sqcap\bar{f}(a)$. The decreasing
sequence defined by $a_{n+1}=\tilde{f}(a_{n})$ corresponds to applying the
abstract property propagation around a cycle in the e-graph, given an initial
sound abstraction $a_{0}$ of $\llbracket e\rrbracket$.

######  Lemma 3.

$\alpha(\llbracket e\rrbracket)$ is a fixpoint of $\tilde{f}$.

###### Proof.

$\alpha(\llbracket e\rrbracket)=\alpha(f(\llbracket e\rrbracket))$
(congruence) $\preccurlyeq\bar{f}(\alpha(\llbracket e\rrbracket))$ (sound
abstraction). Hence $\tilde{f}(\alpha(\llbracket
e\rrbracket))=\alpha(\llbracket e\rrbracket)\sqcap\bar{f}(\alpha(\llbracket
e\rrbracket))=\alpha(\llbracket e\rrbracket)$ (meet definition). ∎

######  Lemma 4.

$a_{n}$ is a sound abstraction of $\llbracket e\rrbracket$ for all
$n\in\mathbb{N}$.

###### Proof.

By induction, $\llbracket e\rrbracket\sqsubseteq\gamma(a_{0})$ and assume
$\llbracket e\rrbracket\sqsubseteq\gamma(a_{n})$. $\llbracket
e\rrbracket=f(\llbracket e\rrbracket)\sqsubseteq\gamma(\bar{f}(a_{n}))$ (sound
abstraction of $f$). Hence $a_{n+1}=a_{n}\sqcap\bar{f}(a_{n})$ is a sound
abstraction of $\llbracket e\rrbracket$ (Lemma 2) for all $n$. ∎

Collecting these results, for some fixpoint $a^{*}$ we have

$\alpha(\llbracket e\rrbracket)\preccurlyeq
a^{*}\preccurlyeq\ldots\preccurlyeq a_{1}\preccurlyeq a_{0}.$

Thus computing abstractions around the loop refines the abstraction and is
guaranteed to terminate if the lattice
$\langle\mathcal{A},\preccurlyeq\rangle$ satisfies the descending chain
condition, as any finite abstract domain will [19]. The algorithm in Figure 2
will correctly apply abstract property propagation around loops, terminating
if the sequence $a_{n}$ converges in a finite number of steps.

##  IV Implementation

To demonstrate the theory described above, we implement interval arithmetic
(IA [20]) for real valued expressions using the extensible egg library, as an
e-class analysis [2]. We consider a concrete domain corresponding to sets of
extended real numbers, i.e.
$\mathcal{C}=\mathcal{P}(\mathbb{R}\cup\\{-\infty,+\infty\\})$ where
$\mathcal{P}$ denotes the power set. We associate each expression with a
binary64 [21] valued interval (a finite abstract domain),

$\mathcal{A}=\\{[a,b]\;|\;a\leq
b,a,b\in\texttt{binary64}\\}\cup\\{\emptyset\\}.$

In this setting the abstraction and concretization functions are as follows
(infima and suprema always exist in this setting):

$\displaystyle\alpha(X)$ $\displaystyle=\left[\textrm{round\\_down}(\inf
X),\textrm{round\\_up}(\sup X)\right]$ (3) $\displaystyle\gamma([a,b])$
$\displaystyle=[a,b]$ (4) $\displaystyle\alpha(\emptyset)$
$\displaystyle=\emptyset,\gamma(\emptyset)=\emptyset$ (5)

To ensure correctness, we use ‘outwardly rounded IA’ which conservatively
rounds upper bounds towards $+\infty$ (round_up) and lower bounds towards
$-\infty$ (round_down) [20, 22]. For particular operators, e.g. $\sqrt{x}$, we
are unable to control the rounding mode, so conservatively add or subtract one
unit in the last place for upper and lower bounds respectively. Provided NaNs
do not appear in the input expression evaluation they are not generated by the
e-graph exploration.

The abstraction of a function $f$ is defined as above
$\bar{f}~{}=~{}\alpha~{}\circ~{}f~{}\circ~{}\gamma$. We will denote
$\mathcal{A}\llbracket s\rrbracket\in\mathcal{A}$, for class $s$. Under this
interpretation Eqn. 2 uses the intersection operation, the meet operation of
the lattice of intervals.

$\mathcal{A}\llbracket s\rrbracket=\bigcap_{n\in\mathcal{N}_{s}}\llbracket
n\rrbracket$ (6)

This relationship generates monotonically narrowing interval abstractions.
0-arity nodes represent either constants associated with degenerate intervals
or variables taking user defined interval constraints.

The classical problem of interval arithmetic is the so-called ‘dependency
problem’, arising because the domain does not capture correlations between
multiple occurrences of a single variables. Consider $x\in[0,1]$, under
classical IA:

$\mathcal{A}\llbracket x-x\rrbracket=[0,1]-[0,1]=[0-1,1-0]=[-1,1].$ (7)

Within the e-graph framework we discover, via term rewriting, $x-x\cong 0$ and
by Eqn. 6 the expression is now correctly abstracted by the (much tighter)
degenerate interval $[0,0]$.

TABLE I: Additional IA optimization rewrites. Class | Rewrite | Condition
---|---|---
Common Terms | $ab\pm ac\rightarrow a(b\pm c)$ | True
Binomial | $1/(1-a)\rightarrow 1+a/(1-a)$ | $0\not\in\llbracket 1-a\rrbracket$
Frac plus Int | $b/c\pm a\rightarrow(b\pm ac)/c$ | $0\not\in\llbracket c\rrbracket$
Division [20] | $a/b\rightarrow 1/(b/a)$ | $0\not\in\llbracket a\rrbracket\cup\llbracket(b/a)\rrbracket$
$a/b\rightarrow 1+(a-b)/b$ | $0\not\in\llbracket b\rrbracket$
Factorise | $a^{2}-1\rightarrow(a-1)(a+1)$ | True
Elementary | $\ln(e^{a})\rightarrow a$ | True

For this work we use a set of 39 rewrites, defining equivalences of real
valued expressions. The basic arithmetic rewrites are commutativity,
associativity, distributivity, cancellation and idempotent operation reduction
across addition, subtraction, multiplication and division. Conversion rewrites
describe the natural equivalence between the power function and
multiplication/division. Table I contains the remaining rewrites.

Conditional rewrites such as the first “Division” rewrite are only valid on a
subset of the input domain. By attaching additional information through an AI
we can prove validity of such rules. Eqn. 8 illustrates how conditional
rewrites can be applied, as IA can confirm that $0\not\in\llbracket
x+y\rrbracket$, in order to remove multiple occurrences of variables,
improving the expression bound produced by IA.

$\frac{x+y}{x+y+1}\rightarrow...\rightarrow\frac{1}{1+\frac{1}{x+y}}$ (8)

To show how multiple expressions in an e-class can independently contribute to
a tight abstraction, consider the following equivalent expressions for
variables $x\in[0,1]$ and $y\in[1,2]$.

$\displaystyle 1-\frac{2y}{x+y}$ $\displaystyle\in\left[-3,\frac{1}{3}\right]$
(9) $\displaystyle\cong\frac{x-y}{x+y}$ $\displaystyle\in[-2,0]$ (10)
$\displaystyle\cong\frac{2x}{x+y}-1$ $\displaystyle\in[-1,1].$ (11)

All three reside in the same e-class within an e-graph with associated
interval $[-3,\frac{1}{3}]\cap[-2,0]\cap[-1,1]=[-1,0]$. Thus, given the first
expression 9, we are capable of generating a tight interval enclosure using
two distinct equivalent expressions for the upper (10) and lower (11) bounds.

##  V Results

$0$$0.2$$0.4$$0.6$$0.8$$1$Relative Width

$0$$2$$4$$6$$8$Runtime (sec)

Figure 3: Relative interval width (optimized width/naive width) and runtime
boxplots to demonstrate the distribution of results on the FPBench suite.

Based on the theory described above we describe a real valued expression
bounding tool in Rust using the egg library. All test cases were run on an
Intel i7-10610U CPU.

###  V-A Benchmarks

We evaluate the implementation using 40 benchmarks from the FPTaylor [23]
supported subset of the FPBench benchmark suite [24]. We allow four iterations
of e-graph rewriting for these experiments. Figure 3 summarises the
distribution of the results, showing a modest average interval reduction over
naive IA, but a substantial improvement in particular cases. There is little
correlation between the runtime and bound improvement.

Across this benchmark set the inclusion of IA and domain specific rewrites
increased the number of e-graph nodes by 4% on average but by up to 84% in
some cases. This demonstrates the additional rewrites that have been applied
as a result of combining e-graphs and AI. The overhead of incorporating IA
into the e-graph increased runtimes by less than 1% on average.

###  V-B Iterative Method Discovery

The Krawczyk method [20] is a known algorithm to generate increasingly precise
element-wise interval enclosures of solutions of linear systems of equations
$Ax=b$, where $A$ is an $n$-by-$n$ matrix and $b$ is an $n$-dimensional
vector. Letting $X^{0}$ be an initial interval enclosure of the solutions, the
Krawcyzk method uses an update formula of the form

$X^{k+1}=\left(Yb+(I-YA)X^{k}\right)\cap X^{k},\textrm{ where
}Y=\textrm{mid}(A)^{-1}.$

mid(A) is the element-wise interval midpoint of the matrix. This sequence, via
interval extension and intersection, corresponds to a sequence of tightening
bounds on the solution $x$, which converges provided the matrix norm
$||I-YA||<1$.

We consider a specific instance of this problem,

$\begin{pmatrix}1&y\\\ y&1\end{pmatrix}\begin{pmatrix}x_{1}\\\
x_{2}\end{pmatrix}=\begin{pmatrix}b_{1}\\\ b_{2}\end{pmatrix}\textrm{, where
}y\in\left[-\frac{1}{2},\frac{1}{2}\right].$ (12)
$X_{1}^{k+1}=\left(b_{1}-yX_{2}^{k}\right)\cap
X_{1}^{k},\;X_{2}^{k+1}=\left(b_{2}-yX_{1}^{k}\right)\cap X_{2}^{k}.$ (13)

A naive solution in the concrete domain, $x=A^{-1}b$, yields,

$x_{1}=\frac{1}{1-y^{2}}(b_{1}-b_{2}y),\;x_{2}=\frac{1}{1-y^{2}}(b_{2}-b_{1}y).$
(14)

Initialising the e-graph with these expressions, the solution for $x_{1}$ can
be automatically rewritten such that Eqn. 13 arises in the abstract domain.
The “Binomial” rewrite from Table I introduces a loop into the e-graph, which
we combine with distributivity rules and “Common Terms” from Table I,

$x_{1}=b_{1}-y\left(b_{2}+(b_{2}y^{2}-b_{1}y)\frac{1}{1-y^{2}}\right).$ (15)

Applying, “Frac plus Int”, and cancelling

$x_{1}=b_{1}-y(b_{2}-b_{1}y)\frac{1}{1-y^{2}}=b_{1}-yx_{2}.$ (16)

When a cycle is introduced, the IA update procedure will continue to
iteratively evaluate the loop, taking the intersection with the previous
iteration as described in §III-C.

##  VI Conclusion

We present a combination of abstract interpretation and e-graphs,
demonstrating the natural interpretation of e-class partitions as meet
operators in a lattice, resulting in precise abstractions. Of key importance
is the positive feedback loop between e-graph exploration and abstraction
refinement, as the precision then allows the application of conditional
rewrite rules, which can be applied in many domains and may further improve
abstraction precision. An exemplar interval arithmetic implementation has
demonstrated the value of this idea, including automated discovery of a known
algorithm for iterative refinement. Future work will explore additional
abstract domains in this setting which can capture variable relations, using
the resulting tool for program optimization.

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