# Time, Simultaneity, and Causality in Wireless Networks with Sensing and
Communications

Petar Popovski P. Popovski is with the Department of Electronic Systems,
Aalborg University, Denmark.

###### Abstract

Wireless systems beyond 5G evolve towards embracing both sensing and
communication, resulting in increased convergence of the digital and the
physical world. The existence of fused digital-physical realms raises critical
questions regarding temporal ordering, causality, and the synchronization of
events. This paper addresses the temporal challenges arising from the fact
that the wireless infrastructure becomes an entity with multisensory
perception. With the growing reliance on real-time interactions and
applications such as digital twins, extended reality, and the metaverse, the
need for accurate timestamping and temporal forensics becomes crucial. The
paper introduces a model that incorporates Temporal Windows of Integration
(TWI) to emulate human multisensory perception and discusses the implications
for setting timing constraints in real-time applications and enabling temporal
forensics. The analysis explores trade-offs, probabilities, and bounds for
simultaneity and causality violation in the context of wireless systems
evolving towards perceptive networks. This work underscores the significance
of timestamping in the evolving wireless landscape, provide insights into
system-level implications, and points out new research avenues for systems
that combine sensing and communications.

##  I Introduction

Time discrepancy and manipulation has been a continuous inspiration to science
fiction authors [1, 2]. The physical time has its order and arrow [3, 4], such
that its manipulation is still in the domain of science fiction. Nevertheless,
the perception of time in digital interconnected systems is dependent on the
way information is processed and transported. Our reality becomes increasingly
a fusion of physical and digital worlds, which brings in the central question
of perception of time and ordering of events.

With the advent of 5G wireless systems, the domain of digital communication
was expanded to embrace real-time interaction among humans and machines,
bringing into focus latency and other timing-related measures as some of the
main design objectives [5, 6]. The low latency constraints from 5G are set to
evolve towards more general timing requirements in 6G [7], reflecting the way
information is processed and used. There is a growing number of applications
that rely on real-time interaction over digital communication links, such as
digital twins [8], eXtended Reality (XR) [9] or the metaverse [10]. The
evolution vector of wireless systems beyond 5G points towards increased fusion
of the digital and the physical world. An early consensus reached about 6G has
been the integration of communication with two other modalities for
acquisition of information, sensing and localization [11]. This gives rise to
the concept of _perceptive mobile networks_ [12], in which the mobile network
infrastructure is used to map the physical into the digital world via sensing.

The physical-digital convergence brings in the central questions of
_chronology_ , or temporal ordering, and _causality_ among the intertwined
physical and digital events. Specifically, the devices that use digital links,
sensors, as well as Integrated Sensing and Communication (ISAC) [11] can be
treated as entities with _multisensory perception_. Digital links also
contribute to the multisensory perception by carrying sensory data from the
mobile devices or from dedicated, passive target monitoring terminals [12]. In
the 3GPP terminology [13], sensing using radio resources is termed 3GPP-
sensing, while carrying sensory data through digital links is referred to as
transparent sensing. Similar to the multisensory perception for humans [14],
it is critical to have the proper temporal ordering and definition of
simultaneity for the events coming from different sensor and links.

Figure 1: An example of chronological discrepancy. Acoustic event
${\cal{E}}_{1}$ causes the event ${\cal{E}}_{2}$ at a later time. $B$ receives
${\cal{E}}_{1}$ as ${\cal{D}}_{B,1}$, communicated by an acoustic sensor
(microphone) $S$, and later on receives ${\cal{E}_{2}}$ as ${\cal{S}}_{B,2}$
by using its radar sensing. $R$ receives at first ${\cal{E}}_{2}$ as
${\cal{D}}_{R,2}$, an image sent by $Z$, and after that receives
${\cal{E}}_{1}$ as ${\cal{S}}_{R,1}$ via its own acoustic sensor.

Fig. 1 provides a motivation for the ideas discussed in this paper. The
acoustic event ${\cal{E}}_{1}$ causes the event ${\cal{E}}_{2}$, which can be
a change in person’s position or a change of the object’s shape. The sensor
$S$ has a microphone that registers ${\cal{E}}_{1}$ and transmits the data of
this event to the Base Station (BS) $B$. This is an instance of transparent
sensing in 3GPP; throughout the paper we will refer to it as _digital input_ ,
in contrast to _sensory input_ obtained directly from a sensor. Hence, $B$
detects ${\cal{E}}_{1}$ as a reception of digital data ${\cal{D}}_{B,1}$.
Furthermore, $B$ has a radar sensing capability and detects ${\cal{E}}_{2}$ as
a sensing event ${\cal{S}}_{B,2}$. Here $B$ perceives the correct
chronological and causal order of ${\cal{E}}_{1}$ and ${\cal{E}}_{2}$. The
robot $R$ has an acoustic sensor and detects the event ${\cal{E}}_{1}$.
Assuming that $R$ is at much larger distance to ${\cal{E}}_{1}$ as compared to
$S$, it will detect ${\cal{E}}_{1}$ as a sensing event ${\cal{S}}_{R,1}$ with
a significant delay. $Z$ sends an image of ${\cal{E}}_{2}$ to $R$, which
corresponds to transparent sensing and is received by $R$ as an event
${\cal{D}}_{R,2}$. Thus, the chronological/causal order observed by $R$ is
different from the one observed by $S$, that is, there is a _chronological
discrepancy_ between $S$ and $R$. Considering the original causal order of
${\cal{E}}_{1}$ and ${\cal{E}}_{2}$, the robot $R$ experiences _causality
violation_. These occurrences can have significant impact in combined
physical-digital worlds. For instance, if $Z$ is a malicious agent, she may
preemptively send a synthetic, fake image to $R$ in order to affect the
chronology/causality of the events. This may have dire consequences both in
real-time, but also post-hoc, if forensics needs to be done after, say,
traffic accidents that involve autonomous vehicles.

Figure 2: Illustration of light cones and the impact of information
processing. The event ${\cal E}_{i}$ for $i=A,B,C$ occurs at the location
$x_{i}$. ${\cal E}_{A}$ is received as ${\cal E}^{\prime}_{A}$ via digital
link, while ${\cal E}_{C}$ is detected as ${\cal E}^{\prime}_{C}$ as a sensing
event. The Base Station (BS) is placed at $x_{B}$. (a) Reception with
instantaneous processing. The chronology observed by the BS is ${\cal
E}^{\prime}_{A}\rightarrow{\cal E}^{\prime}_{C}\rightarrow{\cal E}_{B}$. (b)
Reception of ${\cal E}_{A}$ with data size $D_{A}$ at a rate $R_{A}$ and
detection of ${\cal E}_{C}$ with a sensor detection time $T_{s}$. The
chronology observed by the BS is ${\cal E}^{\prime}_{C}\rightarrow{\cal
E}_{B}\rightarrow{\cal E}^{\prime}_{A}$.

In physics, causal relations are related to the chronology of events, such
that if the event ${\cal E}_{1}$ causes event ${\cal E}_{2}$, then it must be
that ${\cal E}_{1}$ happened before ${\cal E}_{2}$, that is, at an earlier
time. Conversely, if ${\cal E}_{1}$ happened before ${\cal E}_{2}$, then
${\cal E}_{1}$ does not necessarily cause ${\cal E}_{2}$. Thus, a temporal
ordering of events only tells which causality relations are possible. However,
two events at different spatial points may not be causally related in any way,
such that one cannot determine which event happened earlier or whether the
events have occurred simultaneously. This is one of the takeaways of the
Special Theory of Relativity [15], which is built upon the key assumption that
the speed of light sets an upper bound on the speed by which information
propagates through the spacetime and, in that way, it defines the possible
causal connection between various events. Nevertheless, the Special Theory of
Relativity does not consider the aspect of the finite rate of information
transmission and processing, which results in additional time components that
affect the perceived chronology of events. Fig. 2 shows this impact using the
light cones defined in Special Theory of Relativity. If the time required for
information processing is ignored, then the chronology of the events observed
at location $x_{B}$ is ${\cal E}^{\prime}_{A}\rightarrow{\cal
E}^{\prime}_{C}\rightarrow{\cal E}_{B}$. However, if the data transmission
rate from $A$ is low, the event ${\cal E}^{\prime}_{A}$ arrives later and the
chronology of the events at $x_{B}$ is changed to ${\cal
E}^{\prime}_{C}\rightarrow{\cal E}_{B}\rightarrow{\cal E}^{\prime}_{A}$.

Inspired by relativistic time, Lamport brought the notion of _logical clock_
in networked computers and distributed systems [16]. Logical clocks are used
to attain temporal ordering of events in a distributed system, where different
processes interact through communication links. The logical clock defines a
consistent happened-before relation in the distributed system based on
timestamps/counters, which can be used to define causal relations and
histories. Looking at Fig 2(b), happened-before relations are defined by the
BS at $x_{B}$ based on the order of detection of the digital/sensing events.
One may argue that we can easily go around the problem of variable data rate
from $A$, as $A$ can be perfectly synchronized to the BS and timestamp the
transmitted data. However, the sensing event cannot be timestamped by the
sender (nature) and it can only be timestamped after it enters into the
digital systems. There are other tricks one can try to arrive to the “true”
chronology, but there remains the principle of relativity that the chronology
depends on the observer. Regarding causality, determining it cannot be always
based on the times of arrival, as the example on Fig 1 illustrates and one may
need additional causal inference to determine the causal order of events.

In this paper we consider the problem of temporal ordering and timestamping of
events in a wireless infrastructure that involves communication and sensing.
This infrastructure contains entry points for the sensing the physical into
the digital world and, vice versa, for actuating commands from the digital
into the physical realm. Hence, the future Base Stations and Access Points
will have a _timestamping functionality_ that determines the chronology of the
events, simultaneity, and causality. This functionality is used for two
generic purposes:

  * •

Facilitate real-time applications by setting timing constraints that are
dependent on the multi-sensory contributions rather than fixed and predefined
latency constraints. This is done in analogy with human multisensory
perception, in which the Temporal Window of Integration (TWI) are used to
determine which events occur simultaneously and thus determine the pace of
reality at which humans reason and operate.

  * •

Enable temporal forensics, not necessarily applied in real-time, to determine
the chronology and causality of events. This may increase in importance as
more synthetic data gets generated through Artificial Intelligence (AI);
indeed, in a world where manipulated data can impact the order of the events,
with large societal and economic consequences, the wireless infrastructure
will have the responsibility to, as consistently as possible, timestamp the
physical reality.

The specific contributions of this work are the following. _(i)_ Bring
attention to the important problem of timestamping as wireless networks evolve
towards being perceptive, by including sensing and communication. _(ii)_
Define a simple model that enables us to introduce the TWI and the notions of
simultaneity/causality violation. _(iii)_ Carry an elementary analysis for the
case of two events to highlight the main tradeoffs in determining simultaneity
and causality. This is followed by analysis for a sequence of $N$ events and
derivation of probabilities and their bounds for simultaneity/causality
violation. _(iv)_ System-level implications in terms of determining the
latency requirements based on the TWIs and multisensory setup, as well as the
resource allocation in ISAC-based systems. This is supplemented with new
avenues for future research.

###  I-A Related Work

Integrated Sensing and Communication (ISAC) has been seen as one of the
hallmarks of the 6G wireless systems [11]. Having different information
acquisition modalities, as well as gathering transparent sensing data [13]
through digital transmissions, the BSs and the edge/mobile infrastructure
evolves into an entity with multisensory perception. This has been the
motivation for perceptive mobile networks from [17], where another layer of
passive target monitoring terminals (TMTs) are added to the conventional
cellular networks. this concept is further extended in [12] where, among the
other issues, the problem of clock synchronization is discussed, which is
related to the present work.

The related concepts from the literature on multisensory perception can be
found in [18, 14], where the theme is inter-sensory sychnrony, which
corresponds to the concept of simultaneity discussed in this paper.

Another related body of work comes from the literature on logical clocks in
distributed systems, applied to cyber-physical systems. The work [19]
considers timing in cyber-physical systems, the problem of synchronization and
outlines the importance of chronology, simultaneity, and causality in these
systems. A recent related work [20] shows how availability, a real-time
property of a cyber-physical system, and consistency, a logical property,
relate to clock synchronization and latencies introduced by networks and
computation. They generalize the notion of consistency to include physical
state rather than just variables in software.

###  I-B Paper Organization

The introduction is followed by a System Model, which introduces the models
for sensing and digital inputs, the timestamping function and the notions of
simultaneity/causality violation. Section III analyzes the simultaneity and
causality when only two inputs are considered, sensing or digital. This is
generalized to $N$ inputs in Section IV. Section V provides a numerical
illustration of the analysis presented before. This is followed by system-
level perspective in Section VI and discussion of timestamping, simultaneity
and causality in practical wireless systems and Base Stations with ISAC.
Section VII discusses the possible generalization of the assumptions from this
work and outlines avenues for future work. The last section concludes the
paper and this is followed by appendices that contain proofs and supplementary
analysis.

TABLE I: Table of Symbols Symbols | Description
---|---
${\cal E}$ | generic event, of any type
${\cal P}$ | physical event
${\cal S}$ | sensing event
${\cal D}$ | digital event
${\cal A}$ | actuation event
$\tau_{a}$ | action time
$T_{s}$ | sensor integration window
$D_{s}$ | data size produced by a sensor
$T_{p,s}$ | time between a physical event and its sensing event
$\tau_{s}$ | propagation time for a sensing event
$\phi_{s}$ | time offset for a sensing event
$T_{AB}$ | transmission time for digital data from $A$ to $B$
$\tau_{d}$ | propagation time for digital data
$D_{d}$ | data size of a digital event
$W$ | size of the temporal window of integration
$s(t)$ | timestamping function

##  II System Model

###  II-A Processes and Events

Three event types are considered: physical, sensing, and digital. A _physical
event_ ${\cal P}$ happens in nature and, in the context of this work, where a
physical event is any occurrence that can result in a detection by a sensor. A
_sensing event_ ${\cal S}$ has a physical cause and happens at a digital
system when a physical event/parameters are turned into digital data. Finally,
a _digital event_ ${\cal D}$ has a digital cause and happens at a
digital/computing system when a digitally encoded data is decoded and
interpreted. The encoded data that causes a digital event may be, for example,
sent over a communication link or generated locally at the computer. We use
${\cal E}$ as a generic notation of an event that can be of any type. A
digital system, such as a mobile device or a wireless infrastructure,
perceives the physical world through sensing and digital inputs. We note that
there can be a fourth type of event, actuation event ${\cal A}$ that happens
in nature by turning a digital data into a physical event/action; however,
this is not discussed in details, as the focus of this work is on multi-input
perception within the digital systems.

The fundamental difference between a sensing and a digital event is that the
sensing events are not labelled, while the digital events can be labelled. To
illustrate, when a data is received over a communication link, it can be
labelled by the ID of the sender and a timestamp when it was sent, for
example, according to the local clock of the sender. However, a sensing event
cannot be labelled, as nature does not add labels; the ID and the timing of
the sensing event are subject to a process of inference and decision. For
instance, an acoustic event does not carry a label about who or when created
that sound; this can be inferred, with a certain confidence, through the data
acquired by sensing that event. It should be noted that the possibility for
labeling and timestamping of the digital causes of the digital events is the
basic premise for defining Lamport’s logical clocks.

A receiver, such as a BS can have multiple inputs, where each input is either
a sensory or a digital communication link. The events that come from the same
input are received in a First In First Out (FIFO) manner, such that the
chronology of the events within a single input is preserved.

We define _action time_ $\tau_{a}$ as the time required for an event at one
location to cause an event at another location. For instance, in case of a
lightning, $\tau_{a}$ is the time of light propagation plus the time for a
sensor to detect the lighting. The action time for a digital event at device
$A$ to cause a digital event at a device $B$ is the time needed to prepare the
data packet at $A$, transmit it to $B$, and having $B$ decode the packet.

####  II-A1 Physical Process

A physical process consists of a sequence of physical events. If not stated
otherwise, throughout the paper it will be assumed that the events of a given
physical process are causally related, such that a physical event affects
causally all subsequent physical events from the same physical process.

####  II-A2 Sensing Process

A sensing process takes place at a sensor, defined here as a subsystem that
converts physical parameters into a digital data. The sensing process consists
of a sequence of sensing events, denoted by ${\cal S}_{i}$. A sensing event
occurs after a sensor integrates the observation of the physical parameters
for a duration equal or greater than the _sensor integration window_ , denoted
by $T_{s}$. At the end of the window the sensor produces a data of size
$D_{s}$ bits. This is a simplified model of how sensors work, as it represents
the sensor as an entity that produces data after a certain integration time.
This model is inspired by the _dwell time_ in radar systems [21], which is the
time required by the radar to detect the target reliably. Our model idealizes
the radar model by assuming a perfect target detection after a dwell time of
$T_{s}$. The impact of the uncertainty and imperfection in the sensing process
is discussed in Section VII.

Figure 3: Illustration of temporal ordering in various processes. (a)
Synchronous sensor. (b) Asynchronous sensor. (c) Communication between digital
processes. (d) Integrated process of sensing and communication. (e) Temporal
window of integration (TWI) applied to the integrated process from (d).

Fig. 3(a) depicts a _synchronous sensor_ , in which the sensor integration
windows are consecutively ordered and the start of a window is not related to
a physical event. This is a special case, as it assumes that the sensor
integration window $T_{s}$ is equal to the periodicity of synchronous sensing
and there is no idle/deaf time for the sensor. In principle, the periodicity
window can be larger than $T_{s}$, such that the sensor has some idle time;
this is further discussed in Section VI-B. Assuming no idle time, the total
time between the occurrence of the physical event and the detection by the
sensor is:

$T_{p,s}=\tau_{s}+\phi_{s}+T_{s}$ (1)

where $\tau_{s}$ is the time required to start the detection of the sensory
event, not controllable by the sensor. In the simplest case it is equal to the
_propagation time_ and this is how we will refer to it. In a more general
case, it is the time until the event is detectable with the sensor available;
for instance, an event becomes detectable by radar sensing upon causing motion
or change of a position. In (1), the time offset $\phi_{s}\in[0,T_{s})$
between the instance at which the effect of the physical event becomes
detectable by the sensor and the start of the integration window. We use
$\phi_{s}$ to model the fact that the sensor is not event-driven, such that
the detection needs to wait for new sensing window to start. It therefore
takes longer time for the sensor to detect ${\cal P}_{2}$ as ${\cal S}_{2}$ as
compared to detecting ${\cal P}_{1}$ as ${\cal S}_{1}$. If the offset
$\phi_{s}$ is uniformly randomly distributed in $[0,T_{s})$, then $T_{p,s}$ is
uniformly distributed in the support set $[\tau_{s}+T_{s},\tau_{s}+2T_{s})$.

As illustrated in Fig. 3(a), the sensor observes the physical process at a
maximal rate of $R_{s}=\frac{D_{s}}{T_{s}}$ [bps] (bits per second); this
happens when every event starts to be detected at the start of an integration
window. This model does not capture analog sensors, such as human senses, but
one can still define a data processing rate; see “Bandwidth of senses” in
[22]. Note that, if a physical event causes a sensing event, then the action
time is $\tau_{a}=T_{p,s}$.

The model can also be applied to an _asynchronous sensor_ , where the
integration window is triggered by the arrival of a physical event. An
asynchronous sensor needs to have a reliable mechanism that can trigger the
start of a sensor integration window. As a result, we can model the
asynchronous sensor by setting $\phi_{s}=0$ and thus there is a deterministic
$T_{p,s}=\tau_{s}+T_{s}$. This is depicted on Fig. 3(b) for the same physical
events from Fig. 3(a). Note that ${\cal S}_{3}$ is detected earlier as
compared to the synchronous sensor on Fig. 3(a). However, if the same
interface is used to trigger a new integration window and perform detection
during an integration window, then the arrival of new event cannot trigger a
new window while another window is running; this is why the event ${\cal
S}_{2}$ on Fig. 3(b) is not detected at all. Such operation is captured by the
simplified sensor model; in practice, the arrival of new events during an
integration window would interfere with the detection of the original event.

The model can be used for both active and passive sensors. Specifically, the
radar capability considered in ISAC is an active sensing, where a radio wave
is transmitted and its reflections are processed during the integration
window. Nevertheless, the model does not include the possible deaf time,
during which the beam is not pointed to the direction of the event; see
Section VI.VI-B.

####  II-A3 Digital Process

A digital process consists of a sequence of events at a receiver of a digital
communication link, as depicted on Fig. 3(c). A _digital event_ , denoted by
${\cal D}_{i}$, at the receiver $B$ happens when a data packet, received over
a digital input/link, is decoded. For simplicity, at the receiver $B$ we do
not consider digital events that are locally generated by $B$’s computing
system. At the sending device $A$ we denote the process simply as process $A$
and the events ${\cal E}_{1},{\cal E}_{2}$ that initiate a transmission can be
of two types: (1) physical event that is sensed by $A$ and transmitted
digitally, i.e., a transparent sensing data. (2) digital event at $A$, created
by decoding a command from an operating system, AI module, human input to a
digital interface, or even a data receive through another digital link111In
this case, if $A$ retransmits the same data received previously, we have a
multihop transmission.. As mentioned before, unlike the sensing events at $B$,
the digital events sent by $A$ are labelled; for instance, the sender $A$ can
label the sensing data with the timestamp when the data was sensed.

The transmission time between the event at the sender $A$ that initiated the
digital transmission and the digital event at the receiver $B$ is:

$T_{AB}=T_{A}+\tau_{d}+T_{B}$ (2)

where $T_{A}$ is the time used by the sender $A$ until the data is encoded and
transmission started, $\tau_{d}$ is the propagation time of the digital data,
and $T_{B}$ is the time used by the receiver $B$ for decoding. The time
$T_{A}$ depends on the type of event that triggers the transmission. If $A$
transmits sensing information to $B$, then $T_{A}$ consists of the time needed
for $A^{\prime}$s sensor to detect the event and the time for processing the
data before the transmission. Denoting the processing time after sensing by
$T_{\mathrm{proc}}$, we have222It is interesting to relate this analysis to
the popular measure of Age of Information (AoI) [23, 24]. Assuming that the
sensing data gets timestamped after sensing, AoI can be expressed as
$T_{\mathrm{AoI}}=T_{\mathrm{proc}}+\tau_{d}+T_{B}$. Note that, for correct
interpretation of the AoI, the clocks of $A$ and $B$ needs to be synchronized,
otherwise $B$ cannot correctly interpret the timestamp added by $A$.:

$T_{A}=\tau_{s}+\phi_{s}+T_{s}+T_{\mathrm{proc}}$

Differently from this, if $A$ transmits data that is synthetically generated
by an AI algorithm, then ${\cal E}_{i}$ is a digital event, with zero
propagation time, and $T_{A}=T_{\mathrm{proc}}$ is the processing time used
for generating that data with the AI algorithm.

For the digital sender/receiver we will not deal with the detailed operations
and only assume that the digital event outputs a data of size $D_{d}$. Since
we consider communication from devices to the wireless infrastructure, the
propagation time $\tau_{d}$ corresponds to the light propagation time. In the
more general case of multihop transmission, $\tau_{d}$ can be random due to
jitter. For simplicity, we will assume that $D_{d}$ corresponds to a single
packet sent over that link; generalizations of this setup are given in Section
VII.

The component $T_{A}$ is partially controlled by $A$, while $T_{B}$ is
partially controlled by both $A$ and $B$. To see this assume that the encoding
at $A$ and decoding at $B$ are instantaneous. Then $T_{A}=0$ while
$T_{B}=\frac{D_{d}}{R}$ where $R$ is the transmission rate over the digital
link between $A$ and $B$ and this rate is subject to capacity constraints and
codebook agreement by $A$ and $B$. For the subsequent analysis it will be more
convenient to assume the equivalent setup in which the entire digital
processing time $T_{d}=T_{A}+T_{B}$ takes place at the receiver $B$, such that
an event at the digital process of the sender is immediately followed by the
propagation time.

For the subsequent analysis, we can model $T_{AB}$ as a random variable with
probability density $p_{AB}(t)$ and non-negative support set
$[T_{\min},T_{\max}]$, such that $p_{AB}(t)=0$ outside of this set.

###  II-B Temporal Windows for Timestamping

We now turn to the way in which the receiving device perceives the temporal
ordering of the events. As illustrated on Fig. 3(a) and (b), the sensing
events coming from the same physical process or digital events based on the
transmissions from the same sender preserve the causality of events. A more
interesting situation occurs when there are multiple sensing and digital
processes at a single device, as in this case it becomes questionable whether
the causality from the physical world can be preserved for the events in the
digital world. Fig. 3(c) depicts the _integrated process_ observed at $B$,
where it is assumed that $B$ harbors both the sensor from Fig. 3(a) and the
digital receiver from Fig. 3(b). It can be seen that ${\cal S}_{1}$ occurs
before ${\cal D}_{1}$, while from the temporal relation between ${\cal E}_{1}$
and ${\cal P}_{1}$ it could be the case that ${\cal E}_{1}$ causes ${\cal
P}_{1}$.

This situation can be remedied by taking inspiration from multisensory
perception in humans [14]. For instance, looking at my hand and listening
while I am knocking on a wood gives an impression of simultaneity, that is,
the audio, visual, and tactile inputs from the knocking are perceived as
simultaneous. This is because the brain uses a _Temporal Window of Integration
(TWI)_ [14, 18] and all the sensing events that have occurred during a single
window are treated as simultaneous. Similar to this, one can define a temporal
integration window $W$ for a digital device that has multiple physical and
digital inputs. This is depicted on Fig. 3(d), where all events that happened
during a temporal integration window are considered to have happened
simultaneously. For the example, ${\cal S}_{1},{\cal D}_{1},{\cal S}_{2}$
happen simultaneously and are followed by the simultaneous occurrence of
${\cal D}_{2},{\cal S}_{3}$ at the end of the next temporal integration
window.

We use the concept of TWI to define a timestamping function
$s:\mathbb{R}\mapsto\mathbb{Z}$ that assigns an integer timestamp to each
received event:

$s(t)=\left\lceil\frac{t}{W}\right\rceil$ (3)

Here $W$ is the temporal integration window. From a system perspective, a
temporal integration window resides on top of the modules that receive the
sensing and communication data and delivers timestamped events to the higher
layers. Thus, if two events arriving at times $t_{1}$ and $t_{2}$ are received
within the same temporal integration window, they have $s(t_{1})=s(t_{2})$,
such that they are simultaneous from application perspective.

###  II-C Logical Clocks and Causality Violation

In [16], Lamport defined the _happened-before_ relation between two events.
Based on this, there are three possible relations between two events ${\cal
E}_{1}$ and ${\cal E}_{2}$ in a distributed system [25]: (1) ${\cal E}_{1}$
happened before and may have caused ${\cal E}_{2}$, denoted as ${\cal
E}_{1}\rightarrow{\cal E}_{2}$; (2) ${\cal E}_{2}$ happened before and may
have caused ${\cal E}_{1}$, denoted as ${\cal E}_{2}\rightarrow{\cal E}_{1}$;
(3) ${\cal E}_{1}$ and ${\cal E}_{2}$ occurred concurrently, that is, it
cannot be determined which one happened before, denoted as ${\cal
E}_{1}||{\cal E}_{2}$. With the introduction of the temporal integration
window, we need to expand the set of possible relations to include the
_happens-simultaneously-with_ relation; this is equivalent to the notion of
logical simultaneity from [20], where two events are simultaneous in a device
if their logical timestamps are identical333The work [20] uses the term tag as
more general that the timestamp, which enables consideration of a superdense
time system; this is not in the scope for the current paper and we use only
_timestamp_.. Thus, a receiver can perceive events in two ways:

  * •

If the receiver registers the event ${\cal E}_{i}$ before ${\cal E}_{j}$, then
${\cal E}_{i}$ _happened before_ ${\cal E}_{j}$ and this is denoted by ${\cal
E}_{i}\rightarrow{\cal E}_{j}$.

  * •

If the receiver registers the event ${\cal E}_{i}$ simultaneously with ${\cal
E}_{j}$, then the event ${\cal E}_{i}$ _happens simultaneously with_ ${\cal
E}_{j}$ and this is denoted by ${\cal E}_{i}\leftrightarrow{\cal E}_{i}$.

We can thus describe the events on Fig. 3(d) as ${\cal S}_{1}\rightarrow{\cal
D}_{1}\rightarrow{\cal S}_{2}\rightarrow{\cal D}_{2}\rightarrow{\cal S}_{3}$,
while for Fig. 3(e) we have ${\cal S}_{1}\leftrightarrow{\cal
D}_{1}\leftrightarrow{\cal S}_{2}\rightarrow{\cal D}_{2}\leftrightarrow{\cal
S}_{3}$. Based on these definitions, two events registered at the same
receiver cannot be concurrent.

Let ${\cal E}_{1}$ and ${\cal E}_{2}$ denote two events and assume that ${\cal
E}_{1}$ causes ${\cal E}_{2}$. Here ${\cal E}_{i}$ can be a physical, sensing,
or digital event. In the following we assume that, whenever ${\cal E}_{1}$
happened before ${\cal E}_{2}$ then ${\cal E}_{1}$ caused ${\cal E}_{2}$, such
that we can write ${\cal E}_{1}\rightarrow{\cal E}_{2}$ to denote causality in
the physical world. Let ${\cal E}^{\prime}_{A,i}$ denote the perceived event
at the receiving device $A$ that corresponds to ${\cal E}_{i}$, where ${\cal
E}^{\prime}_{A,i}$ can be either a sensing or a digital event. We can then
define the following:

######  Definition II.1.

Causality violation ${\cal V}_{c}$ for device $A$ occurs when two events are
causally related as ${\cal E}_{1}\rightarrow{\cal E}_{2}$, while their
respective per events at $A$ are related as ${\cal
E}^{\prime}_{A,2}\rightarrow{\cal E}^{\prime}_{A,1}$. Simultaneity violation
${\cal V}_{s}$ for device $A$ occurs when two events ${\cal E}_{1},{\cal
E}_{2}$ should be perceived as simultaneous, while their respective received
events at $A$ are related as ${\cal E}^{\prime}_{A,2}\rightarrow{\cal
E}^{\prime}_{A,1}$ or ${\cal E}^{\prime}_{A,1}\rightarrow{\cal
E}^{\prime}_{A,2}$.

When the receiver is clear from the context, we will omit the subscript $A$.
According to this definition, there is no causality violation if ${\cal
E}_{1}\rightarrow{\cal E}_{2}$, while ${\cal
E}^{\prime}_{A,1}\leftrightarrow{\cal E}^{\prime}_{A,2}$. As an example,
${\cal E}_{1}$ can be an explosion ${\cal P}_{1}$, ${\cal
E}^{\prime}_{A,1}={\cal S}_{1}$ can be a sensing event at $A$ that detects the
explosion acoustically. Another device, $B$ is much closer to the explosion
and detects the sensing event ${\cal E}^{\prime}_{B,1}$ earlier than $A$ and
sends a digital image of the explosion to $A$. Then the event ${\cal E}_{2}$
can be the reaction of a device $B$ to that explosion and ${\cal
E}^{\prime}_{A,2}={\cal D}_{2}$ the received digital image of that reaction. A
case where causality violation could be a problem is the one in which $B$ is a
malicious device that, upon detecting the explosion, sends a fake image. If
this fake image is generated and sent sufficiently quickly, a causality
violation can occur and thus a doubt about the cause of the explosion.

The following remark is in order. It is clear that causality/simultaneity
violations are possible when there is at least one sensing event. When all the
events are digital, they can be labeled at the senders $\\{A_{i}\\}$ and this
can avoid causality violation, as the times of the initiating events will be
adjusted. However, if the transmission at a device $A_{i}$ is initiated by a
sensing event, then the true timing of the physical event that caused it still
remains elusive. If, on the other hand, transmissions are initiated by digital
events that are precisely timestamped at the senders, then causality violation
can be avoided. However, this puts high demands on the required
synchronization that may not be feasible for all devices. For instance,
synchronization can be markedly improved by using the Precision Time Protocol
(PTP) [26]; however, as stated in [27], this _“affects utility, but not
correctness”_. If the clocks of the sender and the receiver are not
synchronized perfectly and there is an offset of $\Delta$, then the
probability of causality/simultaneity violation depends on this $\Delta$. We
will analyze the violation probabilities by not assuming any timestamping at
the sending device; this can be further generalized, as discussed in Section
VII.

The introduction of TWI mitigates the causality violation, as the events are
treated as simultaneous. If the events are timestamped as simultaneous, the
causality is determined by a certain form of inference over the set of
detected events, that is, the time component is insufficient to determine
causality. Interestingly, the human brain also uses compensatory and inference
mechanisms to determine sequence of events [14]. The TWI introduces the
following tradeoff. The larger the window $W$ is, the lower the chances of
causality violation. However, larger $W$ decreases the temporal resolution of
events, leading to a lower _event throughput_. In fact, we can identify an
_event throughput loss_ for a certain input as $\frac{W}{T_{0}}$, where
$T_{0}$ is the minimal detectable time between two events in a given sensory
or digital input. For instance, the minimal detectable time for a sensor is
$T_{0}=T_{s}$, the sensory integration time, while the minimal detectable time
for a digital link is the minimal duration of a transmission $T_{\min}$.

##  III Simultaneity and Causality Violation with Two Inputs

In this section we analyze several selected elementary cases. The analysis is
done for simultaneity and causality violation according to the Definition
II.1. Note that for each elementary case both simultaneity and causality
violation can be analyzed; however, for simplicity, only one type of analysis
is shown per elementary case. Specifically, we analyze the probability of
occurrence of these events under a certain stochastic model. A higher
probability of simultaneity/causality violation is an indicator for a higher
probability of incorrect operation of the overall system. Based on the
presented analysis, the reader can apply similar analysis to more general
cases.

###  III-A Simultaneity with Two Sensory Inputs

Figure 4: A physical event that is detected by two asynchronous sensors. (a)
Use of asynchronous temporal window of integration that starts with the start
of the detection of the first sensor. (b) Synchronous windows of integration
with $W<t_{S_{2}}-t_{S_{1}}$. (c) Synchronous windows of integration with
$W\geq t_{S_{2}}-t_{S_{1}}$.

At first we consider two sensory inputs, with integration times $T_{s1}$ and
$T_{s2}$, respectively. The arrival times of the sensory events are:

$\displaystyle t_{S_{i}}=\tau_{s,i}+T_{s,i}$ (4)

where $i\in\\{1,2\\}$. The simplest case is the one of asynchronous sensors,
where the detection starts at the instant at which the sensory event becomes
detectable, as depicted on Fig. 3(b). This is reminiscent of multisensory
perception in humans for, say, visual and audio inputs [18], and, following
this line of thought, we want to find the condition for selecting the window
size $W$ that guarantees correct simultaneity/synchrony of the received
events.

Fig. 4 illustrates detection of a physical event with two sensors. Fig. 4(a)
shows an asynchronous TWI and, without loss of generality, assume that
$\tau_{s,1}<\tau_{s,2}$, such that sensor $1$ gets activated first. In order
to perceive the two events ${\cal S}_{1}$ and ${\cal S}_{2}$ as simultaneous,
the following two conditions have to be met:

$W\geq T_{s,1}\qquad\tau_{s,1}+W\geq\tau_{s,2}+T_{s,2}$ (5)

leading to:

$W=\max\\{T_{s,1},\tau_{s,2}-\tau_{s,1}+T_{s,2}\\}$ (6)

To make a parallel with human audio-visual perception, assume that sensor 1 is
visual with $T_{s,1}=50$ ms and sensor 2 is acoustic with $T_{s,2}=10$ ms
[14]. This is consistent with $\tau_{s,1}<\tau_{s,2}$ as light propagates
faster than sound. The human window of integration is chosen such that
simultaneity is preserved within $10-15$ m, also termed _horizon of
simultaneity_ [28]. This indicates that the TWI for asynchronous digital
sensors has to be chosen to satisfy the simultaneity requirement within a
certain spatial setup. Regarding the event throughput loss, if
$T_{s,2}<T_{s,1}$ then the loss is:

$\frac{W}{T_{s,2}}\geq\frac{\tau_{s,2}-\tau_{s,1}+T_{s,2}}{T_{s,2}}=1+\frac{\tau_{s,2}-\tau_{s,1}}{T_{s,2}}$
(7)

that is, the propagation times have an impact on the event throughput loss.

Consider now synchronous TWI, as in Fig. 4(b) and (c). If the TWI is of a size
$W<(t_{S_{2}}-t_{S_{1}}))$, then simultaneity violation occurs with
probability $P({\cal V}_{s})=1$ since, as seen from Fig. 4(b), the events
${\cal S}_{1}$ and ${\cal S}_{2}$ always have different timestamps,
$s(t_{S_{1}})<s(t_{S_{2}})$. If $W\geq t_{S_{2}}-t_{S_{1}}$ then simultaneity
violation occurs when an edge of the synchronous TWIs falls between ${\cal
S}_{1}$ and ${\cal S}_{2}$. Assuming that the start of the synchronous windows
is chosen uniformly the probability of simultaneity violation is $P({\cal
V}_{s})=\frac{t_{S_{2}}-t_{S_{1}}}{W}$. In summary, this can be written as:

$\Pr[{\cal V}_{s}]=\min\left\\{1,\frac{t_{S_{2}}-t_{S_{1}}}{W}\right\\}$ (8)

###  III-B Causality Violation with a Sensing and a Digital Input

The main difference introduced by the digital input is the variable duration
of the reception period $T_{d}=\frac{D_{d}}{R}$ which depends both on the data
size and the data rate. Both $D_{d}$ and $R$ are, at least partially, subject
to choice by the sender $A$, which can have an impact on the
causality/simultaneity of the received events. For instance, $A$ can choose
$D_{d}$ to be smaller and thus represent a certain source in a coarser, more
distorted way. It thus becomes interesting to consider the case in which at
least one of the events at the receiving device $B$ is digital due to the
possibility to have an external entity that has an impact on the rate of
information processing, unlike content of the sensory inputs, decided by
physics/nature.

In this subsection we consider the temporal ordering and causality violation
at a device $B$ with a synchronous sensor and a digital receiver; that is, $B$
observes an integrated sensing and digital process. Two different cases are
treated. In the first case, depicted on Fig. 5, a physical event triggers a
digital data transmission at device $A$. In the second case from Fig. 6,
device $A$ triggers the physical event and at the same time it transmits
digital data to $B$; it can be interpreted as the digital transmission
triggers a physical event that is detected by $B$’s sensor. Regarding the
synchronous TWI, it is defined and run at a higher system levels.

Figure 5: Physical event triggers a digital transmission. Example of causality
violation at device $B$ when the event ${\cal D}$ of decoding the data from
$A$ precedes the sensing event ${\cal S}$ and the events are timestamped such
that $s(t_{D})=s(t_{S})-1$.

####  III-B1 Case 1: Physical Event Triggers A Digital Transmission

From Fig. 5, the physical event ${\cal P}$ arrives at the sensor
$\phi_{s}\in[0,T_{s})$ seconds before the start of a sensor integration
window. Recall that, in our model, the event is not detected reliably if the
sensor collects the data for time $\phi_{s}<T_{s}$. This, in combination with
the assumption about predefined sensor integration windows, leads to the
detection of the physical event as a sensing event ${\cal S}$ at the end of
the subsequent sensor integration window. The event ${\cal P}$ causes a
digital transmission event at device $A$ after time $\tau_{a}$ and the data is
received by device $B$ after time $T_{AB}$, defined in (2), resulting in a
digital event ${\cal D}$. The times at which the sensing and the digital event
are registered are:

$t_{D}=\tau_{a}+T_{AB}\qquad t_{S}=\tau_{s}+\phi_{s}+T_{s}$ (9)

Considering the timestamping with the TWI, causality violation occurs if at
device $B$ the event ${\cal D}$ happens before ${\cal S}$, that is
$s(t_{D})<s(t_{S})$. Following a similar line of thought as in the case with
two sensors, causality violation occurs if $t_{D}<t_{S}$ and there is at least
one TWI edge between $t_{D}$ and $t_{S}$. Assuming that the TWIs are not
synchronized with the sensing or digital process, then the probability of
causality violation can be expressed as follows:

$\Pr[{\cal V}_{c}]=\min\left\\{1,\frac{\max\\{0,t_{S}-t_{D}\\}}{W}\right\\}$
(10)

Causality violation never occurs if $t_{S}-t_{D}<0$. Substituting the
expressions from (9) this becomes:

$\phi_{s}<T_{AB}+\tau_{a}-T_{s}-\tau_{s}$ (11)

Since $\phi_{s}<T_{s}$ it follows that causality violation never occurs if the
right-hand side of (11) is larger than $T_{s}$, or equivalently:

$T_{AB}>2T_{s}+\tau_{s}-\tau_{a}$ (12)

Differently from this, causality violation is certain whenever
$t_{S}-t_{D}\geq W$ which can be written as:

$\phi_{s}\geq W+T_{AB}+\tau_{a}-T_{s}-\tau_{s}$ (13)

Since $\phi_{s}$ is nonnegative, the condition for certain causality violation
can be written as:

$W<T_{s}+\tau_{s}-T_{AB}-\tau_{a}$ (14)

On the other hand, to be able to mitigate the causality violation, from (13)
it follows:

$W\geq T_{s}+\tau_{s}+\phi_{s}-T_{AB}-\tau_{a}$ (15)

In order to be satisfied for all values of $\phi_{s},T_{AB},\tau_{s},$ and
$\tau_{a}$, the minimal window size is:

$W_{\min}\geq 2T_{s}+\tau_{s,\max}-T_{\min}-\tau_{a,\min}$ (16)

where it is set $\phi_{s}=T_{s}$ and $T_{AB}=T_{\min}$, assuming that $T_{AB}$
is a random variable with a support set in $T_{\min},T_{\max}$, while
$\tau_{s,\max}$ and $\tau_{a,\min}$ are the maximal value of $\tau_{s}$ and
minimal value of $\tau_{a}$ in a certain physical-computational setup. Note
that this does not guarantee that causality violation will not occur, the
choice $W\geq W_{\min}$ only makes the probability of causality violation to
be less than one.

Example 1. Consider a wireless cellular system where the device $B$ is a Base
Station that uses integrated communication and sensing. Assume that both the
physical process, where ${\cal P}$ happens, and the device $A$ are at a
distance of $3$ km from device $B$. Assume that the distance between device
$A$ and ${\cal P}$ is $30$ m and the effect of ${\cal P}$ propagates to $A$ at
the speed of sound. Then $\tau_{a}\approx 100$ ms, while
$\tau_{s}=\tau_{d}=10$ $\mu$s, as both are based on radio propagation. Let
$T_{s}$ be in the range of ms. If ${\cal P}$ causes the digital transmission
of $A$, then for these values we have that $2T_{s}+\tau_{s}-\tau_{a}<0$, such
that (12) is always satisfied and causality violation does not occur. However,
assume that $A$ can accurately predict ${\cal P}$ and or generate a fake data
to be transmitted; this can decrease $\tau_{a}$ significantly and may lead to
causality violation.

####  III-B2 Case 2: Digital Transmission Triggers a Physical Event

As Fig. 6 depicts, the event ${\cal E}$ at device $A$ causes the physical
event ${\cal P}$. At the same time with ${\cal E}$, the device $A$ starts the
transmission of a data packet and $B$ receives it after time $T_{AB}$,
corresponding to ${\cal D}$. On the other hand, the sensing event ${\cal S}$
is detected before ${\cal D}$, which is a causality violation, since it may
lead to conclusion that ${\cal P}$ happened before, and even caused, ${\cal
E}$. The times at which the sensing and the digital event are registered are
now expressed differently:

$t_{D}=T_{AB}\qquad t_{S}=\tau_{s}+\tau_{a}+\phi_{s}+T_{s}$ (17)

Figure 6: Digital transmission triggers a physical event. Example of causality
violation at device $B$ when the sensing event ${\cal S}$ is detected before
the digital data arrives at time of ${\cal D}$.

With timestamping, causality violation occurs when $s(t_{S})<s(t_{D})$. Using
the same arguments as in the previous section, the probability of causality
violation is found to be:

$\Pr[{\cal V}_{c}]=\min\left\\{1,\frac{\max\\{0,t_{D}-t_{S}\\}}{W}\right\\}$
(18)

Causality violation never occurs if $t_{D}-t_{S}<0$, leading to:

$\phi_{s}>T_{AB}-\tau_{a}-T_{s}-\tau_{s}$ (19)

Since $\phi_{s}\geq 0$, causality violation is avoided if:

$T_{AB}=T_{\max}<T_{s}+\tau_{a}+\tau_{s}$ (20)

Causality violation occurs with probability $1$ if:

$W<T_{AB}-T_{s}-\tau_{a}-\tau_{s}-\phi_{s}$ (21)

Hence, the minimal size of the TWI required to be able to mitigate the
causality violation is obtained for $T_{AB}=T_{\max}$ and $\phi_{s}=0$:

$W_{\min}\geq T_{\max}-T_{s}-\tau_{a,\min}-\tau_{s,\min}$ (22)

Example 2. In this case $B$ is a LEO satellite, while ${\cal P}$ and $A$ are
on the ground and in proximity of each other. In this case $\tau_{s}$ and
$\tau_{d}$ are in the order of milliseconds, while $\tau_{a}$ depends on how
$A$ causes detectable changes expressed by ${\cal P}$. Let $\tau_{a}$ be in
the order of milliseconds and $T_{s}$ at the satellite around $100$ ms. If the
data payload that $A$ sends to $B$ is large, while the data rate $R$ is low
due to the low link budget, then causality violation can easily happen
according to (20).

Example 3. Assume that device $A$ has an accurate learning model of the
physical process and can predict the event ${\cal P}_{1}$. In that case,
$\tau_{a}=0$ or it can even be negative (!), increasing the chances of
causality violation in the case when in reality ${\cal P}_{1}$ should happen
before and cause the transmission of $A$.

##  IV Cases with $N>2$ Inputs

We now consider a setup in which the receiver has $N$ different inputs. Two
cases are considered: (1) Simultaneity violation upon a single event that
initiates $N$ transmissions; (2) Causality violation for $N$ causally linked
events, each of them resulting in a reception of digital data. The setup is
given in Fig. 7(a), where $N$ different devices $\\{A_{i}\\}$ transmit data to
a common destination/server $B$. Note the definition of the TWI offset
$\omega$, uniformly distributed in $[0,W)$ with respect to the sole event on
Fig. 7(b) and with respect to the first event on Fig. 7(c).

Figure 7: (a) Setup with $N$ different senders $\\{A_{i}\\}$ providing inputs
to a common receiver $B$. (b) Case for analysis of simultaneity violation,
when a single event results in $N$ different digital data receptions. (c) Case
for analysis of causality violation. When no TWI is applied, there is
causality violation between ${\cal E}^{\prime}_{2}$ and ${\cal
E}^{\prime}_{3}$. The synchronous TWIs is plotted for both (b) and (c).

###  IV-A Simultaneity Violation

Referring to Fig. 7(b), a single physical or digital event ${\cal E}$ causes
reception of $N$ events ${\cal E}^{\prime}_{1},{\cal
E}^{\prime}_{2},\ldots,{\cal E}^{\prime}_{N}$ at times
$t_{1},t_{2},\ldots,t_{N}$, respectively. The model is easily applicable to
both digital and sensing events. Assuming that the event ${\cal E}$ happened
at $t=0$, then, if the $i-$th event is a digital one ${\cal
E}^{\prime}_{i}={\cal D}_{i}$ we have:

$t_{i}=\tau_{a,i}+T_{A_{i}B}$ (23)

Otherwise, if the $j-$th event is a sensing event, then ${\cal
E}^{\prime}_{j}={\cal S}_{j}$ and:

$t_{j}=\tau_{s,j}+T_{s_{j}}$ (24)

Simultaneity violation does not occur if the following is satisfied:

$s(t_{1})=s(t_{2})=\cdots=s(t_{N})$ (25)

Let us assume that synchronous TWIs of size $W$ that are not synchronized with
the arriving events. Given the set of detected event times ${\cal
T}={t_{1},t_{2},\ldots t_{N}}$ and assuming TWI offset $\omega$ is uniformly
distributed in $[0,W)$, the probability of simultaneity violation is:

$\Pr[{\cal V}_{s}|{\cal
T}]=\min\left\\{1,\frac{t_{\max}-t_{\min}}{W}\right\\}$ (26)

where $t_{\min},t_{\max}$ are the minimal/maximal values in ${\cal T}$,
respectively. Given the probability distribution $P({\cal T})$, one can
calculate the total probability of causality violation. It is reasonable to
assume that the times $T_{A_{i}B}$ are conditionally independent, given the
occurrence of the event that caused them. Based on this, all $t_{i}$ are
independently distributed, such that $P({\cal T})$ is a product distribution
$p_{1}(t_{1})\cdot p_{2}(t_{2})\cdots p_{N}(t_{N})$. However, note that, if
the $i-$th input is from an asynchronous sensor, then $\phi_{s_{i}}=0$ and
there is no randomness in $t_{i}$. A more general analysis would consider
joint distribution $p_{1}(t_{1},t_{2},\cdots t_{N})$, in which $T_{A_{i}B}$
are still dependent, even when conditioned on the event ${\cal E}$, for
example, due to the nature of the data generated by the event. This general
case is, though, out of the scope for this manuscript.

###  IV-B Causality Violation with Window Size $W=0$

Fig. 7(c) shows the setup for analysis of causality violation with $N$ inputs.
To start with, we assume that all $N$ inputs are digital such that ${\cal
E}^{\prime}_{i}={\cal D}_{i}$ for all $i$; we will revise this assumption
later on. There is a temporal/causal ordering of the physical-world events
$\\{{\cal E}_{i}\\},i\in\\{1,2,\ldots N\\}$, that result in digital
transmissions. The event ${\cal E}_{i}$ causes the event ${\cal E}_{i+1}$
after time $\tau_{i}$, where $i=1,\ldots N-1$. We consider the random time
$T_{i}$ it takes for $B$ to receive ${\cal E}_{i}$ from device $A_{i}$ as an
event ${\cal D}_{i}$. The event ${\cal E}_{1}$ occurs at time $t=0$ while the
event ${\cal D}_{i}$ is received at time (within the reference frame of the
physical process):

$t_{i}=T_{i}+\sum_{k=1}^{i-1}\tau_{k}$ (27)

for $i>1$ and $t_{1}=T_{1}$.

In this section we treat the case without a TWI, which corresponds to $W=0$.
Since ${\cal E}_{i}\rightarrow{\cal E}_{j}$ whenever $i<j$, there is no
causality violation if the following is satisfied:

$t_{1}\leq t_{2}\leq\cdots\leq t_{N}$ (28)

Let use define the indicator function:

$I(t_{i}\leq t_{j})=I_{i,j}=\left\\{\begin{array}[]{cc}1&\text{if }t_{i}\leq
t_{j}\\\ 0&\text{otherwise}\end{array}\right.$ (29)

We define the event $I_{i,j}=1$ as the correct causal ordering ${\cal
D}_{i}\rightarrow{\cal D}_{j}$ whenever $i<j$: For a set of $N$ events, there
is no causality violation if $I_{i,j}=1$ holds for any pair $i<j$. Considering
the ordering of the events in the physical process, when there is no causality
violation the ordering of the digital events is ${\cal D}_{1}\rightarrow{\cal
D}_{2}\rightarrow\ldots{\cal D}_{N}$. Since causality is a transitive
relation, this implies that ${\cal D}_{i}\rightarrow{\cal D}_{j}$ when $i>j$.
Hence, the probability of not having a causality violation is simplified to:

$\Pr\left[\prod_{i<j}I_{i,j}=1\right]=\Pr\left[\prod_{k=1}^{N-1}I_{k,k+1}=1\right]$
(30)

Considering the definition (27), it can be seen that:

$I_{k,k+1}=1\Longleftrightarrow t_{k}\leq\tau_{k}+t_{k+1}$ (31)

We proceed towards deriving an upper bound on the probability (30) for correct
causal ordering of $N$ received events. The main idea is to decouple the
pairwise probabilities since, in general,
$\Pr\left[\prod_{k=1}^{N-1}I_{k,k+1}=1\right]\neq\prod_{k=1}^{N-1}\Pr[I_{k,k+1}=1]$
since different events $I_{k,k+1}=1$ are dependent as they share common random
variables. To find the upper bound, at first we need the following lemma.

######  Lemma 1.

Let $t_{1},t_{2},$ and $t_{3}$ be independent random variables. Then:

$\Pr\left[t_{2}\leq t_{3}|t_{1}\leq t_{2}\right]\leq\Pr\left[t_{2}\leq
t_{3}\right]$

The intuition behind this is that it is more probable for $t_{2}$ to be less
or equal to $t_{3}$ when $t_{2}$ is unrestricted, rather than when it is known
that $t_{2}$ is larger than another value. The proof is given in Appendix A.

We can now show that the probability of correct causal ordering is upper
bounded by the product of the probabilities for correct causal ordering of
neighboring events; the proof is given in Appendix B.

######  Theorem 1.

The probability of correct causal ordering of $N$ digital events, without
using temporal integration window, is upper-bounded as follows:

$\Pr\left[\prod_{k=1}^{N-1}I_{k,k+1}=1\right]\leq\prod_{k=1}^{N-1}\Pr\left[I_{k,k+1}=1\right]$

The result is valid if a subset of the inputs are sensing instead of digital
links. If $i$ is a sensing input ${\cal E}^{\prime}_{i}={\cal S}_{i}$, then
$T_{i}=\phi_{s,i}+T_{s,i}$ and the distribution of $T_{i}$ can be derived from
the distribution of $\phi_{s,i}$. However, for the proof of the theorem to be
valid, we need to assume that each input comes to a different sensor. To see
why this is the case, assume that $i,j$ with $j>i$ are two sensing inputs.
Then, given the set of values $\\{\tau_{k}\\}$, the random variables
$\phi_{s,i}$ and $\phi_{s,j}$ are not independent, as they are related to the
start of the synchronous sensor integration times of the same sensor.

If the occurrence times of the original events $\\{{\cal E}_{k}\\}$ are chosen
randomly, resulting in random $\\{\tau_{k}\\}$, then the problem can be posed
as a problem of concomitants in order statistics [29]. As a general statement,
what we perceive or register with instruments is not the ordering of the
original events, but their concomitants.

We reiterate the justification of considering causality violation for digital
inputs. The devices $\\{A_{i}\\}$ may have perfectly synchronized clocks and
add timestamps to the transmitted data, such that no causality violation can
occur. This is only seemingly correct, since these timestamps are added at a
certain point within the digital devices and are concomitants for the true
order in which the events caused each other in the physical world.

###  IV-C Causality Violation with Window Size $W>0$

By applying TWIs, the condition for no causality violation turns into:

$s(t_{1})\leq s(t_{2})\leq\cdots\leq s(t_{N})$ (32)

where $t_{i}$ is defined in (28). Note that once the random TWI offset $w$ is
chosen, it is identical for all the detected events. We thus cannot use the
proof of Theorem 1, which relies on the fact that
$\Pr\left[I_{k,k+1}=1\left|\prod_{l=1}^{k-1}I_{l,l+1}=1\right.\right]=\Pr[t_{k}\leq
t_{k+1}|t_{k-1}\leq t_{k}]$. This is not necessarily valid when we consider
$s(t_{k})$ instead of $t_{k}$. We therefore derive a different upper bound for
the case $W>0$. With a slight abuse of notation, in this section we use
$I_{i,j}$ as $I(s(t_{i}),s(t_{j}))$ rather than $I(t_{i},t_{j})$.

######  Theorem 2.

The probability of correct causal ordering of $N$ digital events with temporal
integration window of size $W>0$, is upper-bounded as follows:

$\Pr\left[\prod_{k=1}^{N-1}I_{k,k+1}=1\right]\leq\prod_{k=1}^{N-1}\left(\Pr\left[I_{k,k+1}=1\right]\right)^{\frac{1}{N-1}}$

The proof relies on the use of Hölder’s inequality [30] and is given in
Appendix C. Let $P_{\max}$ denote the maximal probability
$\Pr\left[I_{k,k+1}=1\right]$ for all $k$; then the upper bound becomes rather
trivial:

$\Pr\left[\prod_{k=1}^{N-1}I_{k,k+1}=1\right]\leq
P_{\max}^{\frac{N-1}{N-1}}=P_{\max}$ (33)

which is intuitively correct.

##  V Numerical Illustration

Figure 8: Illustration of the probability of not having causality violation
$1-\Pr[{\cal V}_{c}]$ and its upper bound when the digital transmitters can
use two different rates $\frac{R_{0}}{2},R_{0}$ with equal probability.

We start by a simple verification of Theorem 1, in which $W=0$. Assume that
there is a sequence of $N$ causally related events ${\cal E}_{1},{\cal
E}_{2},\ldots{\cal E}_{N}$, as in Fig. 7(c), where the action times are
identical $\tau_{1}=\tau_{2}=\cdots=\tau_{N-1}=\tau$. Next, each device
$A_{i}$ can use only two discrete rates, low rate $\frac{R_{0}}{2}$ and high
rate $R_{0}$ and, for simplicity, the transmission time corresponding to the
low rate is $T=2T_{0}$, while the one for the high rates is $T=T_{0}$. We say
that a device $A_{i}$ is in a state $S_{i}=L/H$ if it sends at a low/high rate
respectively. Assume that $T_{0}>\tau$; then the causality for events ${\cal
E}_{k},{\cal E}_{k+1}$ is violated if the states of $A_{k}$ and $A_{k+1}$ are
$S_{k}S_{k+1}=LH$. For all other possibilities $S_{k}S_{k+1}\in\\{LL,HL,HH\\}$
it is easy to see that there is no causality violation. Assuming that the
probability of $L/H$ state is $\Pr[L]=\Pr[H]=\frac{1}{2}$ then the probability
of no pairwise causality violation, for all $k=1,\ldots N-1$ is:

$\Pr[I_{k,k+1}=1]=\Pr[S_{k}S_{k+1}\in\\{LL,HL,HH\\}]=\frac{3}{4}$

Looking at the states of the entire sequence of transmitters, it can be noted
that there a causality violation occurs if there is an $i$ for which
$S_{i}=L$, then there exists $j>i$ for which $S_{j}=H$. To see why this is the
case, note that there must be at least one $i\leq i<j$ for which the state
sequence is $S_{l}S_{l+1}=LH$, which leads to causality violation. Hence no
causality violation occurs if the entire sequence of states $S_{1}S_{2}\cdots
S_{N}$ takes values from the following sequences:

$LLL\ldots LL,HLL\ldots LL,HHL\ldots LL,\cdots$ $\cdots,HHH\ldots HL,HHH\ldots
HH$

Since $\Pr[L]=\Pr[H]=\frac{1}{2}$, then the probability of not having
causality violation is:

$1-\Pr[{\cal
V}_{c}]=\Pr\left[\prod_{k=1}^{N-1}I_{k,k+1}=1\right]=(N+1)\frac{1}{2^{N}}$

From Theorem 1, the upper bound on this probabilities based on pairwise
probability $\frac{3}{4}$ is:

$\prod_{k=1}^{N-1}\Pr\left[I_{k,k+1}=1\right]=\left(\frac{3}{4}\right)^{N-1}$

Fig. 8 shows the relation between the upper bound and the exact value of the
probability of correct causal ordering, that is, not having causality
violation, denoted as $1-\Pr[{\cal V}_{c}]$. What is not visible on the figure
is that the the upper bound becomes looser as $N$ grows; that is, the ratio
between the upper bound and the exact probability, given as
$\left(\frac{3}{2}\right)^{N-1}\frac{2}{N+1}$ is monotonously increasing. This
is because, as $N$ grows there are more combinations that can lead to
causality violation and they are not captured by the pairwise probabilities.

Figure 9: Probability of correct causal ordering for $N=10$ events, where the
action time between consecutive events is $\tau$, the TWI size is $W$ and the
transmission time for each event is exponentially distributed with
$\bar{T}=0.5\tau$. The pairwise probability of correct causal ordering, given
by $N=2$, acts as an upper bound for the case $N=10$.

Next we examine the impact of the TWI size on causality violation, with the
results shown on Fig. 9. We consider $N$ casually related events, where the
action time between two events is constant and equal to $\tau$. The
transmission times $T_{i}$ are i.i.d. random variables with exponential
distribution that has a mean value $\bar{T}=0.5\tau$. The $x-$coordinate of
Fig. 9 shows $\frac{W}{\tau}$, which corresponds to the event throughput loss.
Namely, if the transmission times $\\{T_{i}\\}$ are constant, then the event
throughput would be $\frac{1}{\tau}$, with a correct causality order; when the
window of size $W$ is used, then the event throughput is $\frac{1}{W}$. The
curve $N=10$ shows the probability of not having causality violation when all
$N=10$ transmissions are considered jointly, that is,
$\Pr\left[\prod_{k=1}^{9}I_{k,k+1}=1\right]$. The curve $N=2$ is an upper
bound, rather loose, on the pairwise probability of no causality violation,
according to (33), since, due to symmetry, here we have that
$\Pr\left[I_{k,k+1}=1\right]$ is constant for all $k$. Further analysis of the
probability of pairwise probability of correct causal ordering, in a more
general setting, is given in Appendix D.

##  VI System-Level Perspectives

In this section we look into the implications that the timestamping has on the
requirements and design of wireless communication systems, integrated with
sensing.

###  VI-A Setting the Communication Latency Requirements

Latency requirements have been the focus of mobile communication systems since
5G. Some of the earliest requirements set in 5G pointed out to a latency of
$1$ ms, while very often 6G is associated with even lower latencies (in $\mu
s$!). As discussed in details in [7], putting such maximalist requirements on
the radio links is neither always feasible nor can be efficiently supported,
such that we need to look into a set of more general timing requirements. In
this context, TWI can be used as a universal interface between a real-time
application and a set of digital/sensory inputs, such that the latency
requirements of a digital link are decided in synergy with the other inputs
and, eventually, the TWI and the timestamping requirements.

Let us assume that, in a given setting, we have $\tau_{s}-\tau_{a}=20$ ms,
while $T_{s}=10$ ms. If we want to avoid causality violation when the digital
causes a sensing event, from (20) it follows that the highest allowable value
for $T_{AB}$ is $T_{s}+\tau_{a}+\tau_{s}=30$ ms. However, from the definition
(2) it follows that these $30$ ms include processing at the sender $T_{A}$,
propagation time (which may be negligible) and the actual transmission time
$T_{B}$. Putting some timing constraints on the sender to, say, process the
data within $T_{A}<15$ ms, we have a latency requirement for the radio link to
be $15$ ms.

As a next example, assume that an application has selected the size of the
timestamping window $W$ and, based on that, we need to determine the required
latency for communication. One way in which the application can choose $W$ is
based on a control loop or sense-compute-actuate cycle, where $W$ is
determined based on which inputs are required to be able to do the necessary
inference and actuation. Note that different sensing and digital input may
belong to different systems that are not necessarily synchronized; we can thus
put the latency constraint in a way that satisfies a certain probability of
causality or simultaneity violation. We want to see how to set the latency
requirement for the transmission time $T$ by the digital transmitter $A$ based
on the edge of the current TWI. We consider two cases regarding the
synchronization of $A$ with respect to TWI:

  * •

$A$ knows the start of the TWI and prepares its transmission accordingly. Then
a violation in causality/simultaneity occurs if $A$’s transmission does not
get through by the end of the window. If the probability of causality
violation should not be larger than $\epsilon$, then we can design the
resource allocation and the transmission to satisfy $\Pr[T>W]\leq\epsilon$.

  * •

$A$ does not know the start of the TWI, that is, $\omega$ is uniformly
distributed in $[0,W)$. Let $p(t)$ denote the probability density function of
$T$. For given $T=t$, the probability of not sending before the TWI edge
equals the probability that the transmission starts later than $t$ seconds
before the end of the current window, which is $\frac{t}{W}$. Then the
probability of violation of the timestamp is found as:

$\Pr[T>W]=\int_{0}^{\infty}\frac{t}{W}p(t)\mathrm{d}t=\frac{E[T]}{W}$ (34)

where $E[T]$ is the expected value of $T$. For instance, let $T_{s}=10$ ms and
let the event throughput loss be $3$, such that $W=3\cdot 10=30$ ms. If $W=30$
ms then even with $T=3$ ms the probability of violation of the timestamp is
still $0.1$, that is, we have a little control over it.

From the above examples it follows that, when TWI is used, the better the
knowledge that the transmitter has about the start of the window, the better
the guarantees for latency/timestamp violation. In this sense, the uniform
distribution of $w$ is the worst case; one can think of algorithms ran by the
devices for probabilistic tracking of the start of the TWIs, which would lead
to better results.

Regarding simultaneity violation, an illustrative case is given by a random
access protocol. Assume that a physical event happens, upon which $N$ devices
detect quantities related to the event and want to transmit to the BS. The
$i-$th device detects the event after time $\tau_{a,i}$ and generates a data
packet that is transmitted using a random access protocol. In addition, the BS
has a sensor that can detect the event after it propagates for $\tau_{s}$
seconds and the sensing time is $T_{s}$. A protocol that is particularly
suitable for batch arrival is frameless ALOHA [31]. This is a slotted
protocol, with slots of duration $T_{1}$ and after $M$ slots the fraction of
resolved users is $P_{R}$, such that the throughput is
$\frac{NP_{R}}{MT_{1}}$. In the original proposal [31], the protocol execution
is stopped when the throughput is maximal. In the considered variant with
sensing, the protocol can be stopped after the window of simultaneity has
expired; given the parameters $\\{\tau_{a,i}\\},\tau_{s},T_{s}$ and $T_{1}$,
the size of the simultaneity window can be chosen in a way that can capture
the desired fraction of the transmissions. Interestingly, using the sensing
information some of the transmissions can be predicted, such that this can be
used as a side information in the process of collision resolution and
decoding.

###  VI-B ISAC and Timestamping Base Stations

Figure 10: Synchronous slot/symbol structure in 5G NR with time-multiplexed
sensing and communication slots. The frequency division is not represented as
we are focused on the timing aspects.

We now look into ISAC, in which the BSs use radar sensing. This is seen as one
of the main developments in 6G [11, 32]. There are multiple design issues
regarding the required spectral resources for radar sensing based on the
targeted scenarios, interference between communication and sensing, or joint
waveforms for resource allocation and sensing. Yet, from the perspective of
the model used in work, the most important change that ISAC brings is the fact
that communication and sensing are time-multiplexed within a synchronous frame
used by the mobile system, with a certain numerology [32]. Fig. 10 illustrates
this time multiplexing, which brings the following three changes in the model:

  * •

Times $\\{t_{\cal{S}},t_{\cal{D}}\\}$ become discrete, according to the
structure of frame/subframe/slot/symbol. For example, the minimal discretized
time unit can correspond to the shortest slot in 5G NR numerology, which is
$62.5$ $\mu$s. Considering the frequency dimension, a group of digital
transmissions that are carrying transparent sensing data relevant for the same
phenomenon can be scheduled simultaneously; that is, they are inherently
simultaneous and not only due to the TWI.

  * •

Recall that in our previous analysis we have assumed that there is
independence between among the sensor integration windows and the digital
transmissions. However, in a frame-based system they are subject to scheduling
and resource allocation policy, such that they are not independent. In
principle, the analysis with the transmission times $T_{i}$ can be carried
over to this case, taking into account the discrete nature of the values they
can get. However, this analysis must take into account the dependency created
by the time-multiplexed structure.

  * •

Finally, the sensor model we have used is limited by the assumption that the
integration time of the sensor $T_{s}$ is equal to the period of the sensor.
For the example in [32], if there are $64$ sensing beams then the sweeping
period is $64$ ms; however, only a fraction of $\frac{1}{64}$ is given to a
specific sensing beam. This does not correspond to the radar sensing setup
with ISAC frame, as depicted in Fig. 10. Note that this can not always be
remedied by simply defining different sensor integration time $T_{s}$ and a
period of the sensor integration window $T_{s,w}>T_{s}$, since the BS may
allocate aperiodic resources for sensing, depending on the application needs.

As indicated in the introduction, there are two principal uses of the TWIs,
real-time applications and temporal forensics. Having a synchronous frame
structure, one may legitimately ask: do we need the TWI and analysis with
windowing? The answer regarding the real-time application is yes, since
different applications will have different TWIs and the scheduling/resource
allocation should be capable to gather the required inputs within the
respective windows. In the simplest case, a TWI can be a multiple of the
frame/slot size, but this does not need to be the case; the only requirement
is that the TWI is a multiple of the smallest discrete unit adopted in the
system. In terms of temporal forensics, one may argue that the devices that
are synchronized to the BS can timestamp their packets that carry transparent
sensing data. However, while this can mitigate causality violation, the true
timing of the sensory events remains unknown.

In order to illustrate the usefulness of TWI, consider a complex setup in
which the BS has radar sensing, but also other sensors (e.g. acoustic or
visual) and, in addition, gets transparent sensing data through devices
connected to it, referred to as passive TMT in [17, 12]. The works dealing
with TMTs mention only “strict latency requirements”, while the use of TWI
established meaningful timing requirements from all sensors and digital
transmissions. This leads to interesting problems of scheduling and resource
allocation. For instance, the amount of radar sensing resources required
within the frames/slots depends on the resolution/precision that needs to be
attained [32]. The sensing resources are allocated as a comb, not taking time-
adjacent resources in order not to block the channel for low-latency
transmissions. However, this is rather restrictive: we may look at the TWI
structure and allocate the resources in a way that all relevant data for the
application/phenomenon can be correctly timestamped in terms of simultaneity
and causality.

In summary, this part highlights one of the most important function that that
future BS will have and that is timestamping. Indeed, one can see the BSs and
mobile infrastructure as the entry points that put timestamps on the physical
reality as it enters the digital world, thus representing the filter that
allows trustworthy and temporally correct mapping of the physical into the
digital world.

##  VII Discussion and Future Work

As the BS and the mobile infrastructure evolve towards entities with
multisensory perception, we expect that there will be increased use of brain-
like mechanisms a the mobile edge in order to determine the temporal order of
the events. In this paper we have discussed one such mechanism, which is the
use of TWIs. In [14] this is only the first of the four mechanisms used in
determining intersensory synchrony and consistent sequence of events. We list
the other three mechanisms, with a comment that relates them to the mobile
infrastructure:

  * •

_“The brain might be “intelligent” and bring deeply rooted knowledge about the
external world into play that allows it to compensate for various external
factors”_. This corresponds to the use of causal inference at the edge/cloud,
based on the trained ML models, which is used in real time to provide the most
plausible temporal ordering beyond what is given by the timestamps.

  * •

_“The brain might be flexible and shift its criterion about synchrony in an
adaptive fashion”_. This could correspond to different states of criticality
within a set of applications; for instance, if no actuation is required within
a short time window, then the TWI could be extended.

  * •

_“In order to reduce gaps, the brain might actively shift the time at which
one information stream is perceived toward the other.”_ This case could
correspond to a compensation that the edge can apply using generative AI in
order to replace a packet/stream that has not arrived due to transmission
failure.

In the case of mobile infrastructure, the complexity increases as the inputs
to multiple BSs/edge servers are orchestrated to represent unified
timestamping of the physical world.

The sensing model that we have used in the paper is quite simplified and
limited. We have already seen the limitations in the case of modeling a radar
sensing with different beams. Another limitation comes from the idealized
assumption about the perfect detection. Once we bring imperfect detection in
the model, then a number of interesting questions arise. For instance, a
single observation within a sensory integration window may give an unreliable
detection of the event; however, the detection gets refined in the subsequent
windows, which brings the question: when did the sensory event happen and what
timestamp should be given? Throughout the paper we have worked with the
premise that the rate of information processing determines the sequence of the
events, but imperfect sensory detection implies that we should also include
the fidelity of the events in the temporal ordering. To exemplify further this
argument, in transparent sensing the sender may decide to send a small data
portion of size $D$ to describe the sensory event, e.g. image. Thus the
event/image would be detected, but with a high distortion; sending a higher
quality image would result in a different temporal instant of detection.

The consideration of timestamping and ordering of events in the mobile
infrastructure links to a rich body of literature on logical clocks in
distributed systems [16, 25]. It would be interesting to see how the ISAC-
capable BS and edge intelligence can contribute to the creation of consistent
causal histories of sensing events. For instance, the BSs can deliver data to
the cloud using _superdense time_ after discovering causal relationships among
the events with the same timestamp [27]. The fact that the sensing events are
unlabelled and can be detected by multiple devices/BSs may lead to conclusions
in which two sensing events are regarded as causally related, while the ground
truth is that is one and the same event; that is, paradoxically, the sensing
event has caused itself. As mentioned throughout the paper, timestamping and
inference can be used as tools against malicious actors that intend to create
fake events or manipulate the chronology/causal ordering of the events.

Finally, it is interesting to speculate whether the relation between
information processing, temporal ordering, and causality has deeper
implications in physics. Consider the following thought experiment. A digital
device $A$ is positioned at a center point of a circle in space and another
digital device $B$ is on a rocket that moves along the circle with high,
relativistic speeds. Due to time dilation, the time at $B$ moves at a half
pace compared to $A$, that is, while $2$ seconds have passed for $A$, only one
second has passed for $B$. Device $A$ perceives the world around it at a data
rate $R$ [bps] and has a digital link to $B$. If $B$ encodes the data at a
double rate of $2R$ [bps], its data would be received by $A$ at a rate $R$
[bps] and $A$ may conclude that the pace of change in $B$’s world is the same
as in the surrounding world.

##  VIII Conclusions

Motivated by the increasing importance of sensing and Integrated Sensing and
Communications (ISAC) in the emerging wireless systems, we have considered the
problem of temporal ordering and timestamping of events in a wireless
infrastructure. This problem is believed to grow in importance as the Base
Stations (BSs) and the mobile infrastructure are entry points for the sensing
the physical into the digital world and, vice versa, for actuating commands
from the digital into the physical realm. We have defined the problems of
causality and simultaneity and have shown how can they be modeled and
analyzed. Borrowing from the literature on multisensory perception, we have
introduced Temporal Window of Integration (TWI) in the mobile infrastructure
as an elementary tool to mitigate the problems of simultaneity/causality
violation. The paper has analyzed elementary cases of simultaneity/causality
violation and highlighted the main tradeoffs. Furthermore, the paper has
provided a system-level perspective that emerges from the timestamping and the
use of TWIs and related it to the ISAC-enabled BS. One of the most important
takeaways from the paper is that the future Base Stations and Access Points
will have a _timestamping functionality_ that determines the chronology of the
events, simultaneity, and causality and in that way ensure trustworthiness in
the physical-world events that are mapped to the digital infrastructure or
digital twins. Several potential avenues for future work have been outlined.

##  Appendix A Proof of Lemma 1

###### Proof.

We start by using the law of total probability:

$\displaystyle\Pr[t_{2}\leq t_{3}]$ $\displaystyle=$
$\displaystyle\Pr[t_{1}\leq t_{2}]\Pr[t_{2}\leq t_{3}|t_{1}\leq t_{2}]$ (35)
$\displaystyle+$ $\displaystyle\Pr[t_{1}>t_{2}]\Pr[t_{2}\leq
t_{3}|t_{1}>t_{2}]$

We define:

$\displaystyle f(t_{1},t_{2},t_{3})=\Pr[t_{2}\leq t_{3}|t_{2}<t_{1}]$
$\displaystyle g(t_{1},t_{2},t_{3})=\Pr[t_{2}\leq t_{3}|t_{2}\geq t_{1}]$

Note that we have switched the inequalities for the event that is conditioned
upon, but the expressions stay the same as in (35). Next, we want to show
that:

$\displaystyle f(t_{1},t_{2},t_{3})\geq g(t_{1},t_{2},t_{3})$ (36)

Since the random variables are independent, the probability density function
(p.d.f.) is given by:

$p(t_{1}=x_{1},t_{2}=x_{2},t_{3}=x_{3})=p_{1}(x_{1})p_{2}(x_{2})p_{3}(x_{3})$

where $p_{i}(x_{i})$ is the p.d.f. of $t_{i}$. Then (36) can be written
equivalently as:

$\displaystyle\int_{0}^{\infty}\int_{0}^{\infty}\left[f(x_{1},t_{2},x_{3})-g(x_{1},t_{2},x_{3})\right]p_{1}(x_{1})p_{3}(x_{3})\mathrm{d}x_{1}\mathrm{d}x_{3}\geq
0$ (37)

where we note that there is no marginalization over $t_{2}$. With $t_{2}$
remaining as a random variable we get:

$\displaystyle f(x_{1},t_{2},x_{3})-g(x_{1},t_{2},x_{3})=$
$\displaystyle\Pr[t_{2}\leq x_{2}|t_{2}<x_{1}]-\Pr[t_{2}\leq x_{2}|t_{2}\geq
x_{1}]$ (38)

If $x_{1}\leq x_{2}$ then $\Pr[t_{2}\leq x_{2}|t_{2}<x_{1}]=1$ and:

$f(x_{1},t_{2},x_{3})-g(x_{1},t_{2},x_{3})=1-\Pr[t_{2}\leq x_{2}|t_{2}\geq
x_{1}]\geq 0$ (39)

If $x_{1}>x_{2}$ then $\Pr[t_{2}\leq x_{2}|t_{2}\geq x_{1}]=0$ and:

$f(x_{1},t_{2},x_{3})-g(x_{1},t_{2},x_{3})=\Pr[t_{2}\leq
x_{2}|t_{2}<x_{1}]\geq 0$ (40)

Since this is valid for all $x_{1},x_{2}$, it follows that (37) is satisfied.
Now (35) can be lower bounded as:

$\displaystyle\Pr[t_{2}\leq t_{3}]=$ $\displaystyle\Pr[t_{1}\leq
t_{2}]g(t_{1},t_{2},t_{3})+\Pr[t_{1}>t_{2}]f(t_{1},t_{2},t_{3})$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{\geq}}\Pr[t_{1}\leq
t_{2}]g(t_{1},t_{2},t_{3})+\Pr[t_{1}>t_{2}]g(t_{1},t_{2},t_{3})$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}g(t_{1},t_{2},t_{3})=\Pr[t_{2}\leq
t_{3}|t_{2}\geq t_{1}]$

where (a) follows (36), (b) is due to $\Pr[t_{1}\leq
t_{2}]+\Pr[t_{1}>t_{2}]=1$. This proves the Lemma. ∎

##  Appendix B Proof of Theorem 1

###### Proof.

The joint probability can be written as follows:

$\displaystyle\Pr\left[\prod_{k=1}^{N-1}I_{k,k+1}=1\right]=$
$\displaystyle\Pr\left[I_{1,2}=1\right]\prod_{k=2}^{N-1}\Pr\left[I_{k,k+1}=1\left|\prod_{l=1}^{k-1}I_{l,l+1}=1\right.\right]$
(41)

For a given set ${\tau_{i}}$ and the definition (27) it follows that the value
of $I_{l,l+1}$ depends only on the random variables $t_{l}$ and $t_{l+1}$.
Since all random variables $t_{1},t_{2},\ldots t_{N}$ are independent, it
follows:

$\displaystyle\Pr$
$\displaystyle\left[I_{k,k+1}=1\left|\prod_{l=1}^{k-1}I_{l,l+1}=1\right.\right]=$
$\displaystyle\Pr$
$\displaystyle\left[I_{k,k+1}=1\left|I_{k-1,k}=1\right.\right]$
$\displaystyle=$ $\displaystyle\Pr$ $\displaystyle[t_{k}\leq
t_{k+1}|t_{k-1}\leq t_{k}]$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{\leq}}$ $\displaystyle\Pr$
$\displaystyle[t_{k}\leq t_{k+1}]=\Pr[I_{k,k+1}=1]$ (42)

where (a) follows from Lemma 1. Substituting this result in (B) proves the
Theorem. ∎

##  Appendix C Proof of Theorem 2

###### Proof.

We can write the probability as follows:

$\Pr\left[\prod_{k=1}^{N-1}I_{k,k+1}=1\right]=\int\prod_{k=1}^{N-1}I_{k,k+1}\mathrm{d}\mu$
(43)

where $\mathrm{d}\mu$ is the measure defined as

$\mathrm{d}\mu=p(t_{1})\mathrm{d}t_{1}p(t_{2})\mathrm{d}t_{2}\ldots
p(t_{N})\mathrm{d}t_{N}p(\omega)\mathrm{d}\omega$ (44)

as the $\\{t_{i}\\}\cup\omega$ are independent random variables. Then:

$\displaystyle\int\prod_{k=1}^{N-1}I_{k,k+1}\mathrm{d}\mu\stackrel{{\scriptstyle(a)}}{{=}}\left\|\int\prod_{k=1}^{N-1}I_{k,k+1}\mathrm{d}\mu\right\|_{1}\stackrel{{\scriptstyle(b)}}{{\leq}}\prod_{k=1}^{N-1}\left\|I_{k,k+1}\right\|_{N-1}$

where (a) follows from $I_{k,k+1}\geq 0$ and (b) follows from Hölder’s
inequality. We can now write:

$\displaystyle\left\|I_{k,k+1}\right\|_{N-1}=\left(\int
I_{k,k+1}^{N-1}\mathrm{d}\mu\right)^{\frac{1}{N-1}}\stackrel{{\scriptstyle(c)}}{{=}}\left(\int
I_{k,k+1}\mathrm{d}\mu\right)^{\frac{1}{N-1}}$
$\displaystyle\stackrel{{\scriptstyle(d)}}{{=}}\left(\int
I_{k,k+1}p(t_{k})p(t_{k+1})p(\omega)\mathrm{d}t_{k}\mathrm{d}t_{k+1}\mathrm{d}\omega\right)^{\frac{1}{N-1}}$
$\displaystyle\stackrel{{\scriptstyle(e)}}{{=}}\left(\Pr\left[I_{k,k+1}=1\right]\right)^{\frac{1}{N-1}}$

where (c) follows from $I_{k,k+1}\in\\{0,1\\}$, (d) follow from the fact that,
conditioned on $\omega$, $I_{k,k+1}$ depends only on $t_{k}$ and $t_{k+1}$ and
(e) is a definition of the probability, marginalized over the choices of
$\omega$. ∎

##  Appendix D Upper Bound on the Pairwise Probability of Correct Causal
Ordering

Here consider a more general setting to examine the impact of the TWI size on
the pairwise probability of correct causal ordering. In this setup we have
that the event ${\cal V}_{c}$ is equivalent to the event $I_{1,2}=0$. Assuming
that the transmission times $T_{1}/T_{2}$ have probability density functions
$p(t_{1})/p(t_{2})$, then the probability of causality violation under a given
TWI size $W$ is calculated as follows. For given values $T_{1}=t_{1}$ and
$T_{2}=t_{2}$ the probability of causality violation is calculated as follows:

$\Pr[{\cal V}_{c}|t_{1},t_{2}]=\left\\{\begin{array}[]{cc}0&\text{if
}t_{1}\leq\tau+t_{2}\\\ \frac{t_{1}-\tau-t_{2}}{W}&\text{if
}\tau+t_{2}<t_{1}\leq\tau+t_{2}+W\\\ 1&\text{if
}t_{1}>\tau+t_{2}+W\end{array}\right.$ (45)

Using this, we can write:

$\displaystyle\Pr[{\cal V}_{c}]$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}p(t_{2})\mathrm{d}t_{2}\int_{\tau+t_{2}}^{\tau+t_{2}+W}\frac{t_{1}-\tau-
t_{2}}{W}p(t_{1})\mathrm{d}t_{1}$ (46) $\displaystyle+$
$\displaystyle\int_{0}^{\infty}p(t_{2})\mathrm{d}t_{2}\int_{\tau+t_{2}+W}^{\infty}p(t_{1})\mathrm{d}t_{1}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{\geq}}$
$\displaystyle\int_{0}^{\infty}p(t_{2})\mathrm{d}t_{2}\int_{\tau+t_{2}+W}^{\infty}p(t_{1})\mathrm{d}t_{1}$

where (a) follows that the first double integral, containing
$\frac{t_{1}-\tau-t_{2}}{W}$, is nonnegative. Assume that $T_{1}$ is
distributed with an exponential tail, that is, there exists $\lambda$ such
that

$\Pr[T_{1}>\tau+t_{2}+W]=e^{-\lambda(\tau+t_{2}+W)}$

Then:

$\displaystyle\Pr[{\cal V}_{c}]$ $\displaystyle\geq$
$\displaystyle\int_{0}^{\infty}e^{-\lambda(\tau+t_{2}+W)}p(t_{2})\mathrm{d}t_{2}$
(47) $\displaystyle=$ $\displaystyle
e^{-\lambda(\tau+W)}\int_{0}^{\infty}e^{-\lambda
t_{2}}p(t_{2})\mathrm{d}t_{2}$ $\displaystyle=$ $\displaystyle
e^{-\lambda(\tau+W)}E[e^{-\lambda t_{2}}]$

where $E[e^{-\lambda t_{2}}]$ does not depend on $W$. We see that the impact
of TWI is that it exponentially decreases the lower bound on the pairwise
probability of causality violation. If we take $L=\frac{W}{\tau}$ as the event
throughput loss, then the lower bound becomes proportional to
$e^{-\lambda\tau(1+L)}$, which can be interpreted as follows: the linear event
throughput loss causes an exponential decrease of the probability of causality
violation.

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