# An Extension Of Combinatorial Contextuality For Cognitive Protocols

Abdul Karim Obeid, Axel Bruns, Daniel Angus
ARC Centre Of Excellence for Automated
Decision-Making & Society (ADM+S)
Brisbane, QLD, Australia
{Abdul Karim<EMAIL_ADDRESS>&Peter Bruza, Catarina Moreira
School of Information Systems
Queensland University of Technology
Brisbane, QLD, Australia

###### Abstract

This article extends the combinatorial approach to support the determination
of contextuality amidst causal influences. Contextuality is an active field of
study in Quantum Cognition, in systems relating to mental phenomena, such as
concepts in human memory [Aerts et al., 2013]. In the cognitive field of
study, a contemporary challenge facing the determination of whether a
phenomenon is contextual has been the identification and management of
disturbances [Dzhafarov et al., 2016]. Whether or not said disturbances are
identified through the modelling approach, constitute causal influences, or
are disregardableas as noise is important, as contextuality cannot be
adequately determined in the presence of causal influences [Gleason, 1957]. To
address this challenge, we first provide a formalisation of necessary elements
of the combinatorial approach within the language of canonical causal models.
Through this formalisation, we extend the combinatorial approach to support a
measurement and treatment of disturbance, and offer techniques to separately
distinguish noise and causal influences. Thereafter, we develop a protocol
through which these elements may be represented within a cognitive experiment.
As human cognition seems rife with causal influences, cognitive modellers may
apply the extended combinatorial approach to practically determine the
contextuality of cognitive phenomena.

_K_ eywords Contextuality $\cdot$ Combinatorics $\cdot$ Cognition $\cdot$
Disturbance $\cdot$ Causality

##  1 Introduction

Under the assumption that the properties of a system have well established,
pre-existing values prior to measurement, contextuality is when the result of
a property’s measurement is not independent of the co-properties that are
measured along with it [Peres, 1991]. However, a key characteristic of
contextuality is its inability to be explained by any causal relationship.
Consequently, any experiment that declares the presence of contextuality must
remove all doubt that the phenomenon is the result of some causal influence.
In Quantum Information Science, this requirement is solely fulfilled by the
‘No-Disturbance’ (ND) condition [Ramanathan et al., 2012], which can be
experimentally verified by measurement of consistent marginal probabilities,
correspondent to the necessary properties. The ND condition was first
described in the work of Gleason [1957], who determined the basis of the
condition from the physical nature of quantum states. Based on Gleason’s work,
Kochen and Specker [1975] were able to prove a hypothetical system of
orthonormal bases in which no deterministic model of outcomes could be
realised that was non-contextual.

It cannot be as easily claimed that the ND condition, let alone ’quantum-like’
contextuality is inherent to cognitive processes studied within Quantum
Cognition. This is because unlike in Quantum Physics, the emergent theories in
which the field is grounded do not have any immediate basis in physical
properties (i.e., the necessary measurements are mapped to the cognitive
experiment by interpretation). For this reason, cognitive modellers have
resorted directly to the examination of the associated probabilistic model,
for which numerous works have suggested possible frameworks [Aerts et al.,
2013; Asano et al., 2014; Bruza et al., 2015]. What has been consistently
demonstrated is that disturbance is unavoidably inherent in the probabilistic
outcomes of cognitive experiments, and this causes the failure of the ND
condition [Dzhafarov et al., 2016]. This is not to say that quantum-like
contextuality does not appear within cognitive experiments, as only causal
influences falsify the determination of contextuality, and disturbances are
not always reduced to causal influences [Atmanspacher and Filk, 2019]. But
rather, cognitive modellers require a method to adequately distinguish
disturbances that are due to causal influences from those that are due to
noise, which constitutes the first challenge of this article.

The second challenge addressed in this article is that the vast majority of
literature published on the determination of quantum-like contextuality does
not consider the convex decomposition of an associated probabilistic model
into a set of deterministic models. In turn, this prevents the identification
of causal influences within deterministic models that cancel each other out
when aggregated into the combined probabilistic model, as articulated in
Yearsley and Halliwell [2019]’s criticism of Cervantes and Dzhafarov [2018].

This article develops an experimental protocol that addresses both of the
forementioned challenges by the combinatorial approach of Acín et al. [2015].
This is realised by two theoretical elements: the Foulis Randall (F-R)
product, and the Weighted Fractional Packing Number (WFPN).

Specifically, the F-R product is relevant to the first challenge, as it
expresses all causal constraints of the ND condition. Here it is combined with
a method developed by Chaves et al. [2015] for assessing the exact amount of
causal influence observed in any causal relationship between two observables.
In doing so, we realise a process for determining all the disturbances that
are only due to causal influences (and not noise) within arbitrary
experimental settings, which in turn addresses the first challenge of the
article.

With regard to the WFPN, this directly concerns how (non)contextuality is
determined in terms of the combinatorial approach, as the cliques enumerated
on its graph structure correspond to the deterministic models of the necessary
probabilistic model. Acín et al. [2015] relate constraints by Shannon [1956]
to said cliques, ensuring that their definition of contextuality remains
faithful to the convex decomposition of the relative experiment’s
probabilistic model, and this overcomes the previously mentioned criticisms of
Yearsley and Halliwell [2019]. However so, this definition does not yet
anticipate the determination of contextuality under the pretense of
disturbances, for which it is here adapted, and this addresses the second
challenge.

In addressing both of the forementioned challenges, this article produces a
novel experimental protocol within the combinatorial approach, for the
adequate determination of quantum-like contextuality given the presence of
disturbances.

The article proceeds with the following structure: in Section 2, we discuss
related work; in Section 3, we relate the EPR framework, a seminal example for
modelling contextuality that will assist in the understanding of proceeding
sections. In Section 4, we detail relevant definitions of the combinatorial
approach; in particular, this section motivates the graph structure of the
Weighted Fractional Packing Number (WFPN) as the solution to the previously
mentioned challenge concerning convex decompositions of probabilistic models.
In Section 5, we introduce key aspects of causal models and diagrams, and
substantiate that the ND condition is highly restrictive, as well as that
Chaves et al. [2015]’s method is necessary for its relaxation in determining
quantum-like contextuality. In Section 6, we declare the necessary mappings to
integrate the previously mentioned causal modelling techniques with the
combinatorial approach. In Section 7, we prove the main result of the article:
a theorem necessary to determine contextuality in the presence of experimental
disturbances.

##  2 Related Work

This work continues upon previous research undertaken by Obeid [2021] in the
manner of the combinatorial approach, for determining quantum-like
contextuality amidst causal influences. To the best of our knowledge, the only
other line of research that has addressed the forementioned issues for
determining of quantum-like contextuality is that of Jones [2019]. We perceive
that our work is similar in that we consider causal modelling techniques to
remedy the issues concerning disturbance, and assume the ‘no-hidden-influence’
principle, as has been articulated in Jones [2021]. However, we distinguish
our work in that our approach is based within the combinatorial approach of
Acín et al. [2015], while Jones [2019]’s approach is based on the
probabilistic causal models which are shown to be equivalent to the
‘Contextuality-by-Default’ framework Dzhafarov and Kujala [2016].

##  3 The ‘Einstein-Podolsky-Rosen’ Framework

In Quantum Cognition, the Einstein-Podolsky-Rosen (EPR) framework [Clauser et
al., 1969] is one that is largely applied among cognitive modellers in
investigations of quantum-like contextuality; for this reason, the framework
will be used to convey the results of this paper. The framework involves a
system of two parties ($A$ and $B$), in which each have an input measurement
that may be configured to one of two settings (${}^{\mathrm{ipt}}\\!A\,=\,+1$
and ${}^{\mathrm{ipt}}\\!A\,=\,-1$ for party $A$;
${}^{\mathrm{ipt}}\\!B\,=\,+1$ and ${}^{\mathrm{ipt}}\\!B\,=\,-1$ for party
$B$). In either setting, an outcome is observed as
${}^{\mathrm{opt}}\\!A\,=\,+1$ or ${}^{\mathrm{opt}}\\!A\,=\,-1$ for party
$A$, or ${}^{\mathrm{opt}}\\!B\,=\,+1$ or ${}^{\mathrm{opt}}\\!B\,=\,-1$ for
party $B$,. Thereafter, a series of experimental trials are sampled to produce
four pair-wise distributions (defined by the input measurements) that
communicate the probabilistic outcomes of the experiment:

###### Definition 1

Four pair-wise joint distributions generated by the EPR experimental
framework:

$\displaystyle P(\,{}^{\mathrm{ipt}}\\!A=$
$\displaystyle+1,\,{}^{\mathrm{ipt}}\\!B=+1\,),$ $\displaystyle
P(\,{}^{\mathrm{ipt}}\\!A=$ $\displaystyle+1,\,{}^{\mathrm{ipt}}\\!B=-1\,),$
$\displaystyle P(\,{}^{\mathrm{ipt}}\\!A=$
$\displaystyle-1,\,{}^{\mathrm{ipt}}\\!B=+1\,),$ $\displaystyle
P(\,{}^{\mathrm{ipt}}\\!A=$ $\displaystyle-1,\,{}^{\mathrm{ipt}}\\!B=-1\,)$

In the adaption of Bruza et al. [2015], the systems corresponded to concepts
in a bi-amgiguous conceptual combination, such as “APPLE CHIP”. Each concept
would have two senses e.g., “APPLE” has a ‘fruit’ or ‘computer’ sense.
Measurements corresponded to priming words given to human subjects who then
had to interpret the sense of the associated concept: $+1$ would indicate that
the interpretation aligns with the prime. For instance, if the prime was
‘banana’, the human subject interpreted ‘APPLE’ in the ‘fruit’ sense.
Conversely, $-1$ would denote the dis-alignment between prime and
interpretation.

|  | CHIP
---|---|---
|  | $\begin{matrix}[cc]{}^{\mathrm{ipt}}\\!B=+1&{}^{\mathrm{ipt}}\\!B=-1\\\ \mathrm{(\,potato\,)}&\mathrm{(\,circuit\,)}\end{matrix}$
|  | $\begin{matrix}[cccc]+1\;\,&-1\;\,&\;\,+1&\;\,-1\end{matrix}$
APPLE | $\begin{matrix}[lc]{}^{\mathrm{ipt}}\\!A=+1&+1\\\ \mathrm{(\,banana\,)}&-1\\\ {}^{\mathrm{ipt}}\\!A=-1&+1\\\ \mathrm{(\,computer\,)}&-1\end{matrix}$ | $\begin{pmatrix}[cc|cc]0.94&0.06&0.00&0.75\\\ 0.00&0.00&0.25&0.00\\\ \hline\cr 0.00&0.35&0.47&0.00\\\ 0.65&0.00&0.00&0.53\end{pmatrix}$
Table 1: Pair-Wise Joint Distributions Of Conceptual Combination “APPLE CHIP”
[Bruza et al., 2015]

Contextuality could be determined by a set of inequalities known as the ‘Bell-
CHSH inequalities’, that had been previously conceived by Bell [1964]. The
inequalities summated the statistical correlations of the pair-wise joint
distributions.

###### Equation 1

The Bell-CHSH inequalities define a violation of the linear system of
constraints on the correlations of a probabilistic model:

$\displaystyle
0\,-\,\mathrm{corr}_{\\_}{\,\scalebox{0.75}{\mbox{$\displaystyle\begin{matrix}{}^{\mathrm{ipt}}\\!A\,=\,+1\\\\[-2.5pt]
{}^{\mathrm{ipt}}\\!B\,=\,+1\end{matrix}$}}}\,+\,\mathrm{corr}_{\\_}{\,\scalebox{0.75}{\mbox{$\displaystyle\begin{matrix}{}^{\mathrm{ipt}}\\!A\,=\,-1\\\\[-2.5pt]
{}^{\mathrm{ipt}}\\!B\,=\,+1\end{matrix}$}}}\,+\,\mathrm{corr}_{\\_}{\,\scalebox{0.75}{\mbox{$\displaystyle\begin{matrix}{}^{\mathrm{ipt}}\\!A\,=\,+1\\\\[-2.5pt]
{}^{\mathrm{ipt}}\\!B\,=\,-1\end{matrix}$}}}\,+\,\mathrm{corr}_{\\_}{\,\scalebox{0.75}{\mbox{$\displaystyle\begin{matrix}{}^{\mathrm{ipt}}\\!A\,=\,-1\\\\[-2.5pt]
{}^{\mathrm{ipt}}\\!B\,=\,-1\end{matrix}$}}}\,\leq\,2$ $\displaystyle
0\,+\,\mathrm{corr}_{\\_}{\,\scalebox{0.75}{\mbox{$\displaystyle\begin{matrix}{}^{\mathrm{ipt}}\\!A\,=\,+1\\\\[-2.5pt]
{}^{\mathrm{ipt}}\\!B\,=\,+1\end{matrix}$}}}\,-\,\mathrm{corr}_{\\_}{\,\scalebox{0.75}{\mbox{$\displaystyle\begin{matrix}{}^{\mathrm{ipt}}\\!A\,=\,-1\\\\[-2.5pt]
{}^{\mathrm{ipt}}\\!B\,=\,+1\end{matrix}$}}}\,+\,\mathrm{corr}_{\\_}{\,\scalebox{0.75}{\mbox{$\displaystyle\begin{matrix}{}^{\mathrm{ipt}}\\!A\,=\,+1\\\\[-2.5pt]
{}^{\mathrm{ipt}}\\!B\,=\,-1\end{matrix}$}}}\,+\,\mathrm{corr}_{\\_}{\,\scalebox{0.75}{\mbox{$\displaystyle\begin{matrix}{}^{\mathrm{ipt}}\\!A\,=\,-1\\\\[-2.5pt]
{}^{\mathrm{ipt}}\\!B\,=\,-1\end{matrix}$}}}\,\leq\,2$ $\displaystyle
0\,+\,\mathrm{corr}_{\\_}{\,\scalebox{0.75}{\mbox{$\displaystyle\begin{matrix}{}^{\mathrm{ipt}}\\!A\,=\,+1\\\\[-2.5pt]
{}^{\mathrm{ipt}}\\!B\,=\,+1\end{matrix}$}}}\,+\,\mathrm{corr}_{\\_}{\,\scalebox{0.75}{\mbox{$\displaystyle\begin{matrix}{}^{\mathrm{ipt}}\\!A\,=\,-1\\\\[-2.5pt]
{}^{\mathrm{ipt}}\\!B\,=\,+1\end{matrix}$}}}\,-\,\mathrm{corr}_{\\_}{\,\scalebox{0.75}{\mbox{$\displaystyle\begin{matrix}{}^{\mathrm{ipt}}\\!A\,=\,+1\\\\[-2.5pt]
{}^{\mathrm{ipt}}\\!B\,=\,-1\end{matrix}$}}}\,+\,\mathrm{corr}_{\\_}{\,\scalebox{0.75}{\mbox{$\displaystyle\begin{matrix}{}^{\mathrm{ipt}}\\!A\,=\,-1\\\\[-2.5pt]
{}^{\mathrm{ipt}}\\!B\,=\,-1\end{matrix}$}}}\,\leq\,2$ $\displaystyle
0\,+\,\mathrm{corr}_{\\_}{\,\scalebox{0.75}{\mbox{$\displaystyle\begin{matrix}{}^{\mathrm{ipt}}\\!A\,=\,+1\\\\[-2.5pt]
{}^{\mathrm{ipt}}\\!B\,=\,+1\end{matrix}$}}}\,+\,\mathrm{corr}_{\\_}{\,\scalebox{0.75}{\mbox{$\displaystyle\begin{matrix}{}^{\mathrm{ipt}}\\!A\,=\,-1\\\\[-2.5pt]
{}^{\mathrm{ipt}}\\!B\,=\,+1\end{matrix}$}}}\,+\,\mathrm{corr}_{\\_}{\,\scalebox{0.75}{\mbox{$\displaystyle\begin{matrix}{}^{\mathrm{ipt}}\\!A\,=\,+1\\\\[-2.5pt]
{}^{\mathrm{ipt}}\\!B\,=\,-1\end{matrix}$}}}\,-\,\mathrm{corr}_{\\_}{\,\scalebox{0.75}{\mbox{$\displaystyle\begin{matrix}{}^{\mathrm{ipt}}\\!A\,=\,-1\\\\[-2.5pt]
{}^{\mathrm{ipt}}\\!B\,=\,-1\end{matrix}$}}}\,\leq\,2$

Note:While there are many expressions of the Bell-CHSH inequalities, the
version provided here most closely resembles that taken from Fine [1982].

In the literature, it is common for the maximal statistical correlation of the
Bell inequalities to be referred to as the “Bell parameter”, and is typically
denoted as $\mathcal{B}$. For the EPR framework, the Bell parameter would
simply be largest L.H.S. value of any of the lines of Equation 1.

###### Definition 2

For the EPR framework, the Bell parameter $\mathcal{B}$ is defined as the
maximal statistical correlation recorded for all pair-wise joint distributions
of its input measurements.

$\displaystyle\mathcal{B}\;=\;\underset{\begin{matrix}a\,\in\,\\{\,+1,\,-1\,\\}\\\
b\,\in\,\\{\,+1,\,-1\,\\}\end{matrix}}{\mathrm{max}}\raisebox{-8.00003pt}{\scalebox{2.0}{\mbox{$\displaystyle|$}}}$
$\displaystyle\;\mathrm{corr}_{\\_}{\scalebox{0.75}{\mbox{$\displaystyle\begin{matrix}{}^{\mathrm{ipt}}\\!A\,=\,+1\\\\[-2.5pt]
{}^{\mathrm{ipt}}\\!B\,=\,+1\end{matrix}$}}}\;+\;\mathrm{corr}_{\\_}{\scalebox{0.75}{\mbox{$\displaystyle\begin{matrix}{}^{\mathrm{ipt}}\\!A\,=\,+1\\\\[-2.5pt]
{}^{\mathrm{ipt}}\\!B\,=\,-1\end{matrix}$}}}$
$\displaystyle\;\;\;\;\;\;\;+\;\mathrm{corr}_{\\_}{\scalebox{0.75}{\mbox{$\displaystyle\begin{matrix}{}^{\mathrm{ipt}}\\!A\,=\,-1\\\\[-2.5pt]
{}^{\mathrm{ipt}}\\!B\,=\,+1\end{matrix}$}}}\;+\;\mathrm{corr}_{\\_}{\scalebox{0.75}{\mbox{$\displaystyle\begin{matrix}{}^{\mathrm{ipt}}\\!A\,=\,-1\\\\[-2.5pt]
{}^{\mathrm{ipt}}\\!B\,=\,-1\end{matrix}$}}}\;-\;2\mathrm{corr}_{\\_}{\scalebox{0.75}{\mbox{$\displaystyle\begin{matrix}{}^{\mathrm{ipt}}\\!A\,=\,a\\\\[-2.5pt]
{}^{\mathrm{ipt}}\\!B\,=\,b\end{matrix}$}}}\;\raisebox{-8.00003pt}{\scalebox{2.0}{\mbox{$\displaystyle|$}}}$

Furthermore, the R.H.S. of Equation 1 (which is 2 for the EPR framework) is
known as the classical bound on the statistical correlations of the Bell
inequalities. It is the largest value that can be obtained without violating
noncontextual hidden variable theories.

###### Definition 3

The value $\mathcal{B}_{\\_}0$ is defined as the maximal Bell parameter that
can be obtained without violating noncontextual hidden variable theories.

Then, Equation 1 can be simplified to the following expression.

###### Equation 2

A simplification of Equation 1 that integrates the usage of the Bell
parameter.

$\displaystyle\mathcal{B}\leq\mathcal{B}_{\\_}0$

For the previously mentioned experiment of Bruza et al. [2015], a violation of
any of the Bell inequalities would constitute evidence that the concepts are
‘quantum-like’ contextual. While true in principle, this depth of analysis did
not identify if the violation of the inequalities was the effect of some
causal influence, as pointed out by Dzhafarov et al. [2016]; later in this
article, a set of techniques will be introduced in the manner of the
combinatorial approach to remediate this challenge. Nevertheless, the elements
described in the EPR framework demonstrate an experimental specification
necessary to investigate quantum-like contextuality, and will consequently be
recalled when introducing various preliminaries used to communicate the
findings.

##  4 The Combinatorial Approach

The combinatorial approach of Acín et al. [2015] introduces contextuality
scenarios as a hypergraph-based abstraction of a given experiment. The
Weighted Fractional Packing Number (WFPN) constitutes the method for
determining contextuality within the approach, and it is demonstrated here
that the WFPN must be extended in order to support experimental results that
exchange disturbances.

###  4.1 Contextuality scenarios

For any experiment involving the determination of contextuality, contextuality
scenarios offer a hypergraph-based abstraction on which all further procedures
are conducted.

###### Definition 4

For an experiment of $n$ parties, $H$ denotes a system of contextuality
scenarios that correspond to the experiment. The relative contextuality
scenario of any party $i$ is then a hypergraph $H_{\\_}i$, in which its
measurements are hyperedges $E(H_{\\_}i)$, and the possible outcomes of said
measurements are vertices $V(H_{\\_}i)$.

$\displaystyle
H=\\{H_{\\_}1,\ldots,H_{\\_}n\\},\;\;\;\mathrm{such\;that}\;\;\;E(H_{\\_}i)\subset
2^{V(H_{\\_}i)}\;\;\;\mathrm{and}\\!\\!\\!\\!\\!\bigcup_{\\_}{\;\;\;\;\;e\,\in\,E(H_{\\_}i)}\\!\\!\\!\\!\\!\\!\\!\\!e\;=\;V(H_{\\_}i)$

Furthermore, for a hyperedge $e\in E(H_{\\_}A)$, a vertex $v\in e$ describes
an outcome for the measurement, and is notated as $v|e$. For any party, all
outcomes and measurements are conventionally reduced to numerical shorthands,
which is reflected within their verices and edges - the same also applies for
joint outcomes among arbitrary parties. Some shorthands relative to the EPR
framework are communicated in Table 2.

Outcomes & Measurements | Shorthand
---|---
${}^{\mathrm{opt}}\\!A=+1$ | $0\,|$
${}^{\mathrm{opt}}\\!A=-1$ | $1\,|$
${}^{\mathrm{ipt}}\\!A=+1$ | $|\,0$
${}^{\mathrm{ipt}}\\!A=-1$ | $|\,1$
${}^{\mathrm{opt}}\\!A=+1\;|\;{}^{\mathrm{ipt}}\\!A=+1$ | $0\,|\,0$
${}^{\mathrm{opt}}\\!A=-1,\,{}^{\mathrm{opt}}\\!B=+1\;|\;{}^{\mathrm{ipt}}\\!A=+1,\,{}^{\mathrm{ipt}}\\!B=+1$ | $1,0\,|\,0,0$
${}^{\mathrm{opt}}\\!A=-1,\,{}^{\mathrm{opt}}\\!B=+1\;|\;{}^{\mathrm{ipt}}\\!A=-1,\,{}^{\mathrm{ipt}}\\!B=+1$ | $1,0\,|\,1,0$
Table 2: Conventional Shorthands Applied Within The Combinatorial Approach of
Acín et al. [2015]

Then, the contextuality scenarios of the parties $A$ and $B$ are hypergraphs
$H_{\\_}A$ and $H_{\\_}B$.

###### Definition 5

The contextuality scenarios $H_{\\_}A$ and $H_{\\_}B$ for the parties $A$ and
$B$.

$\displaystyle
H_{\\_}A=\\{\,(\,v_{\\_}{0|0},\;v_{\\_}{1|0}\,),\;(\,v_{\\_}{0|1},\;v_{\\_}{1|1}\,)\,\\}\;\;\;\;\;\;H_{\\_}B=\\{\,(\,v_{\\_}{0|0},\;v_{\\_}{1|0}\,),\;(\,v_{\\_}{0|1},\;v_{\\_}{1|1}\,)\,\\}$

###  4.2 Probabilistic models

Probabilistic models define the probabilities of experimental outcomes taking
place, and correspond directly to the vertices of the relative contextuality
scenarios.

###### Definition 6

For any party $i$, all possible outcomes that can be generated have weightings
attributed to a probabilistic model $p$ that coincides with the relative
contextuality scenario $H_{\\_}i$: Formally, the probabilistic model of any
hypergraph $H_{\\_}i$ is an assignment of a probability to each vertex $v\in
V(H_{\\_}i)$.

$\displaystyle
p:V(H_{\\_}i)\rightarrow[0,1]\;\;\;\mathrm{such\;that}\;\;\;\forall_{\\_}{e\,\in\,E(H_{\\_}i)}\sum_{\\_}{v\,\in\,e}p(v)=1$

###  4.3 Compositional products

Compositional products are operations executed on the hypergraphs of one or
more contextuality scenarios, in order to assist the determination of
contextuality. Further to this detail, the outcome of a compositional product
is also a contextuality scenario.

####  4.3.1 The Cartesian product

The first compositional product detailed here is the Cartesian product.

###### Definition 7

For any system of $n$ contextuality scenarios $H$, the Cartesian product
$\bigtimes^{n}_{\\_}{i=1}H_{\\_}i$ is defined as having the following vertices
and hyperedges.

$\displaystyle\times^{n}_{\\_}{i=1}V(H_{\\_}i)\;=\;V(H_{\\_}1)\times\ldots\times
V(H_{\\_}n)$
$\displaystyle\times^{n}_{\\_}{i=1}E(H_{\\_}i)\;=\;E(H_{\\_}1)\times\ldots\times
E(H_{\\_}n)$

It follows that any edge of the Cartesian product represents one of the
possible combinations of joint measurements that can be conducted between the
respective parties of the system. In the EPR experiment, the Cartesian product
$H_{\\_}A\times H_{\\_}B$ is visualised in Figure 1.

Figure 1: Cartesian Product of Contextuality Scenarios $H_{\\_}A$ & $H_{\\_}B$

####  4.3.2 Measurement protocols

In certain cases, it may be necessary to describe the measurements of a given
party $j$ as the result of the outcomes of another party $i$. For this
purpose, the combinatorial approach defines measurement protocols.

###### Definition 8

A measurement protocol $E_{\\_}{H_{\\_}i\rightarrow H_{\\_}j}$ is a hypergraph
generated from a function that maps one or more measurements (as hyperedges)
of a contextuality scenario $H_{\\_}i$ to all the measurements (as hyperedges)
of another contextuality scenario $H_{\\_}j$.

$\displaystyle
E_{\\_}{H_{\\_}i\,\rightarrow\,H_{\\_}j}:=\left\\{\;\bigcup_{\\_}{v\,\in\,e}\\{v\\}\times
f(v)\;:\;e\in E(H_{\\_}i),\;\;f:e\rightarrow E(H_{\\_}j)\;\right\\}$

In terms of an experiment, this is taken to mean that for any outcomes
associated with a measurement in $E(H_{\\_}i)$, that a measurement from
$E(H_{\\_}j)$ is consequently chosen. Recalling the EPR experiment, it is
possible that the party $B$ may choose their measurements as a function of
$A$’s outcomes. The resulting measurement protocol
$E_{\\_}{H_{\\_}A\rightarrow H_{\\_}B}$ visualises this relation in Figure 2.

Figure 2: Measurement Protocol $E_{\\_}{H_{\\_}A\rightarrow H_{\\_}B}$

####  4.3.3 The Foulis-Randall product

By combining the hyperedges that correspond to all measurement protocols of
all parties mapped to all other parties, one is able to describe the
measurement of any party as the result of any other party’s outcome. These are
conveyed by hyperedges of the relative contextuality scenarios, whose vertices
correspond to the specific outcomes of such events taking place. This
comprises the Foulis-Randall (F-R) product, which is yet another compositional
product of the combinatorial approach. For a system of $n$ contextuality
scenarios $H$, the F-R product is expressed as both a hypergraph and a
contextuality scenario, and has numerous variations, all of which containing
the same vertices as the Cartesian product (see Definition 7). This article is
concerned with the _common_ F-R product, which is denoted as
${}^{\mathrm{comm}}\bigotimes^{n}_{\\_}{i=1}H_{\\_}i$.

###### Definition 9

The hyperedges $E(\,{}^{\mathrm{comm}}\bigotimes^{n}_{\\_}{i=1}H_{\\_}i\,)$ of
the common F-R product are defined as the union of all possible orderings of
the commutative, non-associative ‘$\otimes$’ operator on all contextuality
scenarios of a system $H$.

$\displaystyle
E(\,{}^{\mathrm{comm}}\bigotimes^{n}_{\\_}{i=1}H_{\\_}i\,)\;:=\;\left\\{\;e:e\in\left\\{\;\bigotimes^{n}_{\\_}{i=1}H_{\\_}i\;\right\\}\;\right\\}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathrm{where}\;\;H_{\\_}i\otimes
H_{\\_}j\;:=\;\left\\{\;E_{\\_}{H_{\\_}i\,\rightarrow\,H_{\\_}j}\;\cup\;E_{\\_}{H_{\\_}j\,\rightarrow\,H_{\\_}i}\;\right\\}$

For the EPR framework, the common F-R product would be equivalent to
$H_{\\_}A\otimes H_{\\_}B$, which is visualised in Figure 3.

Figure 3: F-R Product Of Contextuality Scenarios $H_{\\_}A$ & $H_{\\_}B$

####  4.3.4 The No Disturbance Condition

As a widely tested prerequisite for determining contextuality, the No
Disturbance (ND) condition ensures that absolutely no causal influences are
mutually exchanged between the parties of the given experiment. This satisfies
the requirement that contextuality cannot be due to causal influences. The ND
condition is achieved by imposing constraints on the probabilistic weightings
of the parties. In terms of the EPR framework, the condition is calculated as
follows.

###### Definition 10

For an EPR experiment involving two parties, $A$ and $B$, the ND condition
requires that given any input measurement by either party, that the
probabilistic outcomes must be invariant with respect the outcomes of all
other parties.

$\displaystyle\mathrm{Pr}(\;{}^{\mathrm{opt}}\\!A\,\;|\;\,{}^{\mathrm{ipt}}\\!A\;)\;$
$\displaystyle=\;\mathrm{Pr}(\;{}^{\mathrm{opt}}\\!A\,\;|\;\,{}^{\mathrm{ipt}}\\!A,{}^{\mathrm{ipt}}\\!B=+1\;)$
$\displaystyle=\;\mathrm{Pr}(\;{}^{\mathrm{opt}}\\!A\,\;|\;\,{}^{\mathrm{ipt}}\\!A,{}^{\mathrm{ipt}}\\!B=-1\;)$
$\displaystyle\mathrm{Pr}(\;{}^{\mathrm{opt}}\\!B\,\;|\;\,{}^{\mathrm{ipt}}\\!B\;)\;$
$\displaystyle=\;\mathrm{Pr}(\;{}^{\mathrm{opt}}\\!B\,\;|\;\,{}^{\mathrm{ipt}}\\!B,{}^{\mathrm{ipt}}\\!A=+1\;)$
$\displaystyle=\;\mathrm{Pr}(\;{}^{\mathrm{opt}}\\!B\,\;|\;\,{}^{\mathrm{ipt}}\\!B,{}^{\mathrm{ipt}}\\!A=-1\;)$

Within the combinatorial approach, the vertices of the corresponding
contextuality scenarios $H_{\\_}A$ and $H_{\\_}B$ convey the outcomes for the
respective marginal probabilities of all parties. As such, the probabilities
of their joint observation with the edges of all other parties restore the
necessary expressions of the ND condition.

###### Definition 11

For any outcome corresponding to a vertex $v$ of given party $H_{\\_}A$
jointly observed with a measurement corresponding to a hyperedge $e$ of a
given party $H_{\\_}B$, the probability is equivalent to that of the same
outcome observed with any other measurement corresponding to a hyperedge
$e^{\prime}$ of $H_{\\_}B$. The same applies with respect to all $H_{\\_}B$’s
outcomes when jointly observed with measurements of $H_{\\_}A$.

$\displaystyle\sum_{\\_}{w\,\in\,e}p(v,w)=\sum_{\\_}{w\,\in\,e^{\prime}}p(v,w)\;\;\;\forall
v\in V(H_{\\_}A),\;\;\;e,e^{\prime}\in E(H_{\\_}B)$
$\displaystyle\sum_{\\_}{w\,\in\,e}p(v,w)=\sum_{\\_}{w\,\in\,e^{\prime}}p(v,w)\;\;\;\forall
v\in V(H_{\\_}B),\;\;\;e,e^{\prime}\in E(H_{\\_}A)$

Furthermore, as the F-R product previously defined each of its hyperedges as
corresponding to any party’s outcome as a function of another party, it
follows that the summated probabilities of said hyperedges would need to be
collectively equivalent in order to motivate the assumption that no single
causal influence is more probable than another.

###### Corollary 1

For all hyperedges of the F-R product that corresponds to the EPR framework
$E(\,H_{\\_}A\,\otimes\,H_{\\_}B\,)$, the summation of all probabilities of
any single hyperedge are equivalent to those of any other hyperedge.

$\displaystyle\forall_{\\_}{e,\,e^{\prime}\,\in\,E(\,H_{\\_}A\,\otimes\,H_{\\_}B\,)}\;\;\sum_{\\_}{v\,\in\,e}p(v)\;=\\!\\!\sum_{\\_}{\;\;v^{\prime}\,\in\,e^{\prime}}\\!\\!\\!p(v^{\prime})$

####  4.3.5 Non-Orthogonality Graphs & The Weighted Fractional Packing Number

The WFPN is central to determining contextuality within the combinatorial
approach. In order to calculate the Weighted Fractional Packing Number (WFPN),
one must first calculate the Non-Orthogonality (NO) graph.

###### Definition 12

The Non-Orthogonality (NO) graph is defined as a simple graph
$\mathrm{NO}(\,H\,)$ of the same vertices as an input contextuality scenario
$H$, and has hyperedges for vertices that are not within the input’s common
hyperedges.

$\displaystyle V(\,\mathrm{NO}(\,H\,)\,)\;:=$ $\displaystyle\;V(\,H\,)$
$\displaystyle E(\,\mathrm{NO}(\,H\,)\,)\;:=$
$\displaystyle\;\left\\{\;e:u\sim v\,\Leftrightarrow\,\not\exists e\in
E(\,H\,)\;\;\mathrm{such\;that}\;\;\\{\,u,\,v\,\\}\,\subseteq\,e\,\right\\}$

Of significance is the NO graph of the F-R product,
$\mathrm{NO}(\,{}^{\mathrm{comm}}\bigotimes^{n}_{\\_}{i=1}H_{\\_}i\,)$, for
which all maximal cliques $C$ are enumerated.

###### Definition 13

Let $C$ denote all maximal cliques enumerated upon the the NO graph
$\mathrm{NO}(\,{}^{\mathrm{comm}}\bigotimes^{n}_{\\_}{i=1}H_{\\_}i\,)$.

$\displaystyle c\in
C\subseteq\mathrm{NO}(\,{}^{\mathrm{comm}}\bigotimes^{n}_{\\_}{i=1}H_{\\_}i\,)$

###### Definition 14

The enumeration of the maximal cliques $C$ exemplify all noncontextual
deterministic models, and are indexed by a set of weightings $q$. Then, for
any clique $c$, the corresponding model is attributed the weight $q_{\\_}c$,
which is derived from the total number of times that the model occurs within
experimentation, as a percentage of the observation of all possible
deterministic models.

$\displaystyle\forall_{\\_}{q_{\\_}i\,\in\,q}\;\;\;q_{\\_}i\,\in\,[\,0,\,1\,]$

In this respect, the WFPN is calculated by constraint of the summation of the
weightings $q$.

###### Definition 15

The WFPN $\alpha^{*}$ of the F-R product
$\mathrm{NO}(\,{}^{\mathrm{comm}}\bigotimes^{n}_{\\_}{i=1}H_{\\_}i\,)$ is
equivalent to the summation of all weightings of the set $q$, as indexed by
the cliques $C$.

$\displaystyle\sum_{\\_}{c\,\in\,C}q_{\\_}c=\alpha^{*}(\,\mathrm{NO}(\,{}^{\mathrm{comm}}\bigotimes^{n}_{\\_}{i=1}H_{\\_}i\,)\,)=1$

Finally, the probabilistic model $p$ of the system is recovered by summation
of the deterministic models.

###### Definition 16

For a system of contextuality scenarios $H$, the probability $p(v)$ of
observing any outcome associated with a vertex $v\in V(H)$ is equivalent to
the sum of the deterministic weightings corresponding to the cliques of $C$
that intersect $v$.

$\displaystyle\forall_{\\_}{v\,\in\,V(H)}\;\;p(v)=\sum_{\\_}{c\;\in\;C}\;\sum_{\\_}{c\;\cap\;v}q_{\\_}c$

While the WFPN determines the (non)contextuality of a system, it does so with
the assumption that no disturbances are in the experimental results; for this
reason, the WFPN must be extended. To do so, we refer to the clique
enumeration. As mentioned, this corresponds to all noncontextual deterministic
models of the system.

Previous attempts to classify quantum-like contextuality have not taken into
account the deterministic decompositions that form their respective
probabilistic models: as an implication, experimental disturbances may have
influenced the determination of contextuality, despite being overlooked within
probabilistic outcomes. Consider an implementation of the EPR framework in
which two experimental trials are firstly conducted. Their deterministic
models are conveyed by the pair-wise joint distributions in Table 3.

|
$\begin{matrix}[cc]{}^{\mathrm{ipt}}\\!B=+1\;\;&\;\;{}^{\mathrm{ipt}}\\!B=-1\end{matrix}$
---|---
| $\begin{matrix}[cccc]+1&-1\;\;&\;\;+1&-1\end{matrix}$
$\begin{matrix}[cc]{}^{\mathrm{ipt}}\\!A=+1&+1\\\ &-1\\\ {}^{\mathrm{ipt}}\\!A=-1&+1\\\ &-1\end{matrix}$ | $\begin{pmatrix}[cc|cc]\;1.00&0.00&0.00&1.00\;\;\\\ 0.00&0.00&0.00&0.00\\\ \hline\cr 1.00&0.00&0.00&0.00\\\ 0.00&0.00&0.00&1.00\end{pmatrix}$

(a) Pair-Wise Joint Distributions Of Individual Experimental Trial 1

|
$\begin{matrix}[cc]{}^{\mathrm{ipt}}\\!B=+1\;\;&\;\;{}^{\mathrm{ipt}}\\!B=-1\end{matrix}$
---|---
| $\begin{matrix}[cccc]+1&-1\;\;&\;\;+1&-1\end{matrix}$
$\begin{matrix}[cc]{}^{\mathrm{ipt}}\\!A=+1&+1\\\ &-1\\\ {}^{\mathrm{ipt}}\\!A=-1&+1\\\ &-1\end{matrix}$ | $\begin{pmatrix}[cc|cc]\;0.00&0.00&0.00&0.00\;\;\\\ 0.00&1.00&1.00&0.00\\\ \hline\cr 0.00&0.00&1.00&0.00\\\ 0.00&1.00&0.00&0.00\end{pmatrix}$

(b) Pair-Wise Joint Distributions Of Individual Experimental Trial 2

Table 3: Pair-Wise Joint Distributions Of Two Individual Experimental Trials

After both experimental trials, the probabilistic model is derived from the
normalisation of their summated results, as given in Table 4.

|
$\begin{matrix}[cc]{}^{\mathrm{ipt}}\\!B=+1\;\;&\;\;{}^{\mathrm{ipt}}\\!B=-1\end{matrix}$
---|---
| $\begin{matrix}[cccc]+1\;\;&\;\;-1&+1&-1\end{matrix}$
$\begin{matrix}[cc]{}^{\mathrm{ipt}}\\!A=+1&+1\\\ &-1\\\ {}^{\mathrm{ipt}}\\!A=-1&+1\\\ &-1\end{matrix}$ | $\begin{pmatrix}[cc|cc]\;0.50&0.00&0.00&0.50\;\;\\\ 0.00&0.50&0.50&0.00\\\ \hline\cr 0.50&0.00&0.50&0.00\\\ 0.00&0.50&0.00&0.50\end{pmatrix}$
Table 4: Pair-Wise Joint Distributions Of Combined Experimental Trials From
Table 3

While evaluating the ND condition (Definition 10) on the probabilistic model
of Table 4 does not reveal any disturbances, the same does not hold for the
deterministic models in Table 3. This is not to say that the observed
disturbances in Table 3 constitute causal influences (as Section 5.2.1 will
detail that the ND condition is incorrectly provisioned for this task), but
that a significant aspect of the determination of contextuality within
cognitive experiments has been so far overlooked. This claim has already been
made by Yearsley and Halliwell [2019] and Atmanspacher and Filk [2019] for
experimentation conducted by Cervantes and Dzhafarov [2018], however a
generalisation of the method to retrieve all deterministic models has not yet
been considered for cognitive experiments. In fact, this can only be achieved
by anticipating all deterministic models (such as those of Table 3) for any
arbitrary experiment. Only then can a sensitive treatment of the possible
disturbances within the corresponding experimental trials be realised. As
detailed in Definition 15, all deterministic models of an experiment are
retrieved by the enumerated cliques of the WFPN. And this holds for any
experiment in which the corresponding contextuality scenarios realise an NO
graph, demonstrating where the extension of the combinatorial approach should
be realised.

For usage later in later sections of this paper, the following definitions are
introduced.

###### Definition 17

Let all cliques enumerated by the F-R product of a system of contextuality
scenarios $H$ be denoted as ${}^{\mathrm{NC}}C$, as they correspond to the
outcomes of all noncontextual deterministic models of the relative system.

$\displaystyle{}^{\mathrm{NC}}C\;\subseteq\;\mathrm{NO}(\,{}^{\mathrm{comm}}\bigotimes^{n}_{\\_}{i=1}H_{\\_}i\,)$

###### Definition 18

Let all possible deterministic models for a system $H$ correspond to the
cliques ${}^{\mathrm{ALL}}C$ enumerated on the NO graph of the Cartesian
product of the system.

$\displaystyle{}^{\mathrm{ALL}}C\;\subseteq\;\mathrm{NO}(\,\times^{n}_{\\_}{i=1}H_{\\_}i\,)$

Note: For a system of contextuality scenarios, all its possible deterministic
models correspond to the cliques enumerated on the NO graph of its Cartesian
product.

By subtracting the cliques ${}^{\mathrm{NC}}C$ from ${}^{\mathrm{ALL}}C$, we
derive the cliques that correspond to the deterministic models that are not
noncontextual (i.e., either derived from causal influences or noise). The
result will be hereafter denoted as ${}^{\cancel{\mathrm{NC}}}C$.

###### Definition 19

The set of cliques ${}^{\cancel{\mathrm{NC}}}C$ that correspond to the
deterministic models that are not noncontextual are defined as the difference
of the set of cliques ${}^{\mathrm{ALL}}C$ and ${}^{\mathrm{NC}}C$.

$\displaystyle{}^{\cancel{\mathrm{NC}}}C\;:=\;{}^{\mathrm{ALL}}C\,\setminus\,{}^{\mathrm{NC}}C$

Note: Here it is assumed that all cliques derive from operations on the same
system of contextuality scenarios.

##  5 Causal Modelling Techniques

In this section, we detail causal models and diagrams, and argue the failure
of the ND condition to distinguish between causal influences and noise for the
adequate determination of contextuality amidst causal influences. Furthermore,
we relate the work of Chaves et al. [2015] for correctly quantifying causal
influences.

###  5.1 Causal Models & Diagrams

In certain cases, cognitive experiments may appear to determine contextuality
by probabilistic weightings, but are in fact noncontextual due to hidden
causal influences. To identify said causal influences, one must apply causal
modelling techniques, and as such, causal models are here established as
necessary. Pearl [2009] defines any causal model as follows.

###### Definition 20

A causal model $\mathrm{M}$ is an ordered triple, in which $\mathrm{U}$
represent the set of exogenous random variables, $\mathrm{V}$ represent the
set of endogenous random variables, and $\mathrm{E}$ is the set of causal
influences: formally the expressions of the values of $\mathrm{V}$ as
functions of the values within $\mathrm{U}$ and $\mathrm{V}$.

$\displaystyle\mathrm{M}\;=\;\langle\,\mathrm{U},\mathrm{V},\mathrm{E}\,\rangle$

For use in further equations, the set of exogenous and endogenous variables
will be unified into a single set.

###### Definition 21

The set $\mathrm{X}$ denotes the union of the sets $\mathrm{U}$ and
$\mathrm{V}$.

$\displaystyle\mathrm{X}:=\\{\,\mathrm{U}\,\cup\,\mathrm{V}\,\\}$

Furthermore, causal models described here assume a canonical probabilistic
model (distinct from Section 4.2) attributes weightings to all $n$ members of
the set $\mathrm{X}$.

###### Definition 22

A canonical probabilistic model assigns probabilistic weightings to all random
variables $\mathrm{X}$ of a causal model.

$\displaystyle\forall_{\\_}{\mathrm{x}\,\in\,\mathrm{X}}\;\;\;\mathrm{Pr}(\,\mathrm{x}\,)\rightarrow[\,0,\,1\,]\;\;\;\;\mathrm{such\;that}\;\;\;\mathrm{Pr}(\;\mathrm{X}_{\\_}1,\,\ldots,\,\mathrm{X}_{\\_}n\;)=1$

Causal diagrams provide a visual abstraction of causal models, by ascribing
the random variables to a simple graph.

###### Definition 23

A causal diagram ascribes the random variables within the set $\mathrm{X}$ to
nodes of a directed acyclic graph $\mathcal{G}$. Therein, the directed edges
$\mathcal{E}$ correspond to the causal influences established by the set of
expressions $\mathrm{E}$.

$\displaystyle\mathcal{G}\;=\;(\,\mathrm{X},\,\mathcal{E}\,)$

###### Definition 24

The function $f_{\\_}{\mathrm{pnt}}$ defines the causal influences, for which
any $f_{\\_}{\mathrm{pnt}}(\,\mathrm{x}\,)$ of a variable $\mathrm{x}$ returns
the parent vertex of $\mathrm{x}$ within the causal diagram. This defines the
set of directed edges $\mathcal{E}$.

$\displaystyle
f_{\\_}{\mathrm{pnt}}(\,\mathrm{x}\,)\;\subset\;\mathrm{X}\,\cup\\{\,\varnothing\,\\}\;\;\;\;\mathrm{such\;that}\;\;\;\;\forall\mathrm{x}^{\prime}\,\in\,f_{\\_}{\mathrm{pnt}}(\,\mathrm{x}\,)\;\;\exists(\,\mathrm{x}^{\prime},\,\mathrm{x}\,)\in\mathcal{E}$

###### Equation 3

The joint distribution factors of the random variables within the set
$\mathrm{X}$ are defined as the product of all probabilities, given
observation of the variables associated with their parent nodes in the
corresponding causal diagram.

$\displaystyle\mathrm{Pr}(\,\mathrm{X}\,)\;=\;\prod_{\\_}{i}\mathrm{Pr}(\;\mathrm{X}_{\\_}i\;|\;f_{\\_}{\mathrm{pnt}}(\,\mathrm{X}_{\\_}i\,)\;)$

###### Definition 25

Let the set of all exogenous variables be collected into a latent variable
$\Lambda$.

$\displaystyle\Lambda\;:=\;\\{\;\mathrm{x}:\mathrm{x}\in\mathrm{X}\;\;\mathrm{and}\;\;f_{\\_}{\mathrm{pnt}}(\,\mathrm{x}\,)=\\{\,\varnothing\,\\}\;\\}$

###  5.2 Determining Causal Influences

####  5.2.1 Association

Failures of purely probabilistic attempts to classify cognitive experiments as
determining contextuality are due to modellers relying upon the lowest level,
association, of Pearl and Mackenzie [2018]’s ladder of causation in order to
identify causal influences.

###### Equation 4

Association proposes a causal influence from a variable $\mathrm{Z}$ to
another variable $\mathrm{Y}$ by fulfilment of the following probabilistic
expression.

$\displaystyle\mathrm{Pr}(\,\mathrm{Y}\,|\,\mathrm{Z}\,)>\mathrm{Pr}(\,\mathrm{Y}\,|\,\neg\mathrm{Z}\,)$

This expression can be evaluated by probabilistic weightings alone. At best,
association only proves that $\mathrm{Y}$ _could_ be caused by $\mathrm{Z}$,
vice versa, that both are caused by some other variable, or may be due to
noise within experimental results.

It is important to note that while association does not prove any cause for
either $\mathrm{Y}$ and $\mathrm{Z}$, that it nevertheless is the primary clue
that causation may be at play. It follows that imposing constraints upon
associations between variables not only restrict all forms of causation, but
the experimental noise that is necessary to determine contextuality. Such
constraints happen to form the basis of the ND condition.

###### Proposition 1

The ND condition constitutes a system of expressions that substitute the
operator of Equation 4.

Proof: Replace the observation of $\mathrm{Y}$ with ${}^{\mathrm{opt}}\\!A$,
the observation of $\mathrm{Z}$ with
${}^{\mathrm{ipt}}\\!A,{}^{\mathrm{ipt}}\\!B=-1$, and the observation of
$\neg\mathrm{Z}$ with ${}^{\mathrm{ipt}}\\!A,{}^{\mathrm{ipt}}\\!B=+1$ in
Equation 4. Then substitute the operator with that of equivalence (i.e., ‘=’)
to recover an expression of Definition 10. The same is achieved for any other
operation within the ND condition for any combination of
inputs.$\hfill\blacksquare$

####  5.2.2 Intervention

In order to determine contextuality for cognitive experiments that may be due
to noise, it is necessary to restrict only causal influences. As such, one
must refer to the second level, intervention, of Pearl and Mackenzie [2018]’s
ladder of causation.

###### Equation 5

Intervention for a causal influence from some variable $\mathrm{Z}$ to another
variable $\mathrm{Y}$ requires that the experimenter deliberately fixes the
protocols that both exhibit and inhibit $\mathrm{Z}$, as denoted by the do
operator on a probabilistic expression.

$\displaystyle\mathrm{Pr}(\,\mathrm{Y}\,|\,\mathrm{\it{do}}(\mathrm{\,Z\,})\,)>\mathrm{Pr}(\,\mathrm{Y}\,|\,\mathrm{\it{do}}(\mathrm{\,\neg
Z\,})\,)$

This expression clearly determines whether $\mathrm{Y}$ is caused by
$\mathrm{Z}$, and cannot be conceptualised by probabilistic weightings alone.
Intervention has been detailed in the framework of Chaves et al. [2015], as
the necessary technique to differentiate causal influences from noise. And so
two equations from the framework are here recalled:

###### Equation 6

Given intervention on all $n$ variables $\mathrm{X}$ of a causal model that
are observed under pretense of their parent nodes, the do operator redefines
the joint distribution factors as follows.

$\displaystyle\mathrm{Pr}(\,\mathrm{X}\,|\,\textit{do}(\,\mathrm{X}_{\\_}i=k\,)\,)=\begin{cases}\prod^{n}_{\\_}{j\,\neq\,i}\mathrm{Pr}(\,\mathrm{X}_{\\_}j\,|\,f_{\\_}{\mathrm{pnt}}(\,\mathrm{X}_{\\_}j\,)\,)&\mathrm{if}\;\;\mathrm{X}_{\\_}i\neq
k\\\ 0&\mathrm{otherwise}\end{cases}$

The second equation111Refer to Equation 4 of Chaves et al. [2015]. simply
augments Equation 5 with respect to the latent variable $\Lambda$, to
determine the degree to which $Z$ causally influences $Y$. In this paper, the
expression is generalised.

###### Definition 26

The expression
$\mathrm{C}_{\\_}{{}_{\\_}{\mathrm{X}_{\\_}i=k,\,\mathrm{X}_{\\_}j=k^{\prime}}}$
concerns two variables, $\mathrm{X}_{\\_}i$ and $\mathrm{X}_{\\_}j$
respectively, and returns the direct causal influence that some outcome
$\mathrm{X}_{\\_}i=k$ has upon some other outcome
$\mathrm{X}_{\\_}j=k^{\prime}$.

$\displaystyle\mathrm{C}_{\\_}{{}_{\\_}{\mathrm{X}_{\\_}i=k,\,\mathrm{X}_{\\_}j=k^{\prime}}}=\\!\\!\\!\\!\\!\\!\\!\underset{\scalebox{0.75}{\mbox{$\displaystyle\begin{matrix}\mathrm{X}_{\\_}j\,=\,k^{\prime},\\\\[1.49994pt]
f_{\\_}{\mathrm{pnt}}(\mathrm{X}_{\\_}j),\\\\[1.49994pt]
\mathrm{X}_{\\_}i\,=\,k,\;\mathrm{X}_{\\_}i\,\neq\,k\end{matrix}$}}}{\mathrm{sup}}\\!\\!\sum_{\\_}{\lambda\,\in\,\Lambda}\mathrm{Pr}(\,\lambda\,)|\mathrm{Pr}(\,\mathrm{X}_{\\_}j=k^{\prime}\,|\,do(\,\mathrm{X}_{\\_}i=k\,),\,f_{\\_}{\mathrm{pnt}}(\mathrm{X}_{\\_}j),\,\lambda\,)\;\;\;\;$
$\displaystyle-\;\mathrm{Pr}(\,\mathrm{X}_{\\_}j=k^{\prime}\,|\,do(\,\mathrm{X}_{\\_}i\neq
k\,),\,f_{\\_}{\mathrm{pnt}}(\mathrm{X}_{\\_}j),\,\lambda\,)|$

##  6 Mapping The Combinatorial Approach To Causal Models

In order for the combinatorial approach to determine contextuality in the
presence of experimental disturbances, this section formalises the necessary
causal modelling techniques described in the previous section within the
combinatorial approach.

###  6.1 Contextuality Scenarios

Contextuality scenarios are the fundamental element of the combinatorial
approach, and so require a consistent mapping to the random variables of
causal models in order to leverage causal modelling techniques. This is no
trivial task, as there is no single mapping between them. To demonstrate this
point, two separate alternatives will be detailed here in the manner of the
EPR framework. The first alternative is motivated from hidden variable
theories that precede the discovery of contextuality.

Recalling the EPR framework’s experimental settings, consider that each
configuration of both parties’ input measurements
(${}^{\mathrm{ipt}}\\!A\,=\,+1$ and ${}^{\mathrm{ipt}}\\!A\,=\,-1$ for party
$A$; ${}^{\mathrm{ipt}}\\!B\,=\,+1$ and ${}^{\mathrm{ipt}}\\!B\,=\,-1$ for
party $B$) within the experiment maps to the states of two distinct random
variables $\mathrm{X}_{\\_}1$ and $\mathrm{X}_{\\_}3$. Thereafter, the
outcomes observed (as either ${}^{\mathrm{opt}}\\!A\,=\,+1$ or
${}^{\mathrm{opt}}\\!A\,=\,-1$ for party $A$, or
${}^{\mathrm{opt}}\\!B\,=\,+1$ or ${}^{\mathrm{opt}}\\!B\,=\,-1$ for party
$B$) for either measurement are also distinct random variables
$\mathrm{X}_{\\_}2$ and $\mathrm{X}_{\\_}4$. where
$f_{\\_}{\mathrm{pnt}}(\,\mathrm{X}_{\\_}2\,)=\\{\,\mathrm{X}_{\\_}1\,\\}$ and
$f_{\\_}{\mathrm{pnt}}(\,\mathrm{X}_{\\_}4\,)=\\{\,\mathrm{X}_{\\_}3\,\\}$.
This alternative assumes that the outcomes observed by either party are
functionally assigned by the configuration of the measurements, and has a
causal probabilistic model that is visualised in Figure 4.

(a) Causal Diagram

(b) Mapping From Causal Model To Contextuality Scenarios

Figure 4: Alternative Mapping Of Causal Model To Contextuality Scenarios With
Functional Assignments

The second alternative does not assume functional assignments between random
variables, meaning that all causal influences extend from exogenous
influences. It implicates a separate interpretation of the experiment: that
each dichotomous outcome pertains to its own random variable. In turn, it
generates an entirely different system of contextuality scenarios, in which
each element of the Cartesian product (which was previously given in Figure 1)
has its own random variable.

While both of the forementioned alternatives are entirely possible, the
selection depends upon the researcher’s own preferences for how physical
measurements and outcomes should map to random variables. As measurements and
outcomes constitute the edges and vertices of contextuality scenarios, it
follows that this will determine the mapping from random variables to
contextuality scenarios.

###### Definition 27

For a system of the contextuality scenarios $H$, the function
$f_{\\_}{\mathrm{vtc}}$ defines a mapping from an outcome $X_{\\_}i\,=\,k$ to
the set of vertices $e^{\prime}$ that define it within $H$.

$\displaystyle
f_{\\_}{\mathrm{vtc}}(\,\mathrm{X}_{\\_}i=k\,)\;:\;(\,\mathrm{X}_{\\_}i=k\,)\rightarrow
w\;\;\;\mathrm{where}\;\;\;\,w\,\subseteq V(H)$

###  6.2 Causal Influences

As mentioned in Definition 24, causal influences are formalised in causal
diagrams as directed edges between random variables.

In terms of contextuality scenarios (and following on from the specification
of Section 6.1), the definition corresponds to a relation between two disjoint
subsets of the set of vertices $V(H)$ for a system of contextuality scenarios
$H$. This is further specified by the nature of the relation within the causal
diagram, for which some possibilities are visualised in the manner of the EPR
framework, as shown in Figure 5.

(a) $H_{\\_}A$ Measurement To $H_{\\_}A$ Outcome

(b) $H_{\\_}A$ Measurement To $H_{\\_}B$ Outcome

(c) $H_{\\_}A$ Outcome To $H_{\\_}B$ Outcome

Figure 5: Alternative Causal Influences

Considering that Figure 5c conveys the choice of a measurement of one party
influencing some outcomes on another party, the combinatorial approach defines
these as edges and vertices respectively. Following Figure 5c, suppose one
wishes to interrogate whether the highlighted causal influence (i.e., that
$X_{\\_}1$ influences $X_{\\_}4$), given that $X_{\\_}3=-1$. The relation is
visualised within the necessary contextuality scenarios in Figure 6.

Figure 6: Visualisation Of Relation Within Figure 5c, Given That $X_{\\_}3=-1$

Note: The relation produced by the causal influence (and its derivative
hyperedges) are highlighted in red.

In this respect, the full set of causal influences will hereafter be
attributed a set $R$, in which each member details the pair of disjoint
vertices that correspond to it.

###### Definition 28

For a causal diagram $\mathcal{G}=\\{\mathrm{X},\,\mathcal{E}\,\\}$, and a
system of contextuality scenarios $H$, let an edge $e\in\mathcal{E}$ have the
source vertex $\mathrm{X}_{\\_}{i}$, and the destination vertex
$\mathrm{X}_{\\_}{j}$. Then the outcomes $\mathrm{X}_{\\_}{i}=k$ and
$\mathrm{X}_{\\_}{j}=k^{\prime}$ defines a member $r\in R$.

$\displaystyle\forall_{\\_}{r\,\in\,R}\;\;\;r\;:=\;\\{\,(\,f_{\\_}{\mathrm{vtc}}(\,\mathrm{X}_{\\_}i=k\,),\,f_{\\_}{\mathrm{vtc}}(\,\mathrm{X}_{\\_}j=k^{\prime}\,)\,)\,\\}$

###  6.3 Probabilities

As both contextuality scenarios and random variables can share probabilistic
assignments, their respective interpretations of probabilistic models are
related here.

###### Equation 7

Any outcome $p(v)$ of the probabilistic model $p$ associated with a system of
contextuality scenarios $H$ has an equivalence to the probability of observing
one or more of the set of $n$ random variables $\mathrm{X}$.

$\displaystyle\forall_{\\_}{H_{\\_}i\,\in\,H}\;\forall_{\\_}{e\,\in\,E(\,H_{\\_}i\,)}\;\forall_{\\_}{v\,\in\,e}$
$\displaystyle\;\;\;\;\;\;\;p(\,v\,)\;=\;\mathrm{Pr}(\,\mathrm{X}_{\\_}j,\,\ldots,\,\mathrm{X}_{\\_}k\,|\,\mathrm{X}_{\\_}l,\,\ldots,\,\mathrm{X}_{\\_}m\,)\;\;\;\mathrm{where}\;\;\;j,k,l,m\in[\,1,\,n\,]$

By Definition 27 and Equation 7, it is possible to recover the probability of
observing any single outcome $\mathrm{X}_{\\_}i=k$.

###### Equation 8

The probability of observing any single outcome $\mathrm{X}_{\\_}i=k$ is
equivalent to the summation of the probabilities of all vertices within the
set returned by the function $f_{\\_}{\mathrm{vtc}}(\,\mathrm{X}_{\\_}i=k\,)$

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\mathrm{Pr}(\,\mathrm{X}_{\\_}i=k\,)\;=\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{\\_}{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;v\,\in\,f_{\\_}{\mathrm{vtc}}(\,\mathrm{X}_{\\_}i\,=\,k\,)}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!p(\,v\,)$

Similarly, the probability of observing any outcome $\mathrm{X}_{\\_}i=k$,
given the observation of any other outcome $\mathrm{X}_{\\_}j=k^{\prime}$ is
given as follows.

###### Equation 9

The probability of observing any outcome $\mathrm{X}_{\\_}i=k$, given the
observation of any other outcome $\mathrm{X}_{\\_}j=k^{\prime}$ is equivalent
to the summation of the probabilities of all vertices within the set returned
by the product of the functions,
$f_{\\_}{\mathrm{vtc}}(\,\mathrm{X}_{\\_}i=k\,)$ and
$f_{\\_}{\mathrm{vtc}}(\,\mathrm{X}_{\\_}j=k^{\prime}\,)$.

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\mathrm{Pr}(\,\mathrm{X}_{\\_}i=k\;|\;\mathrm{X}_{\\_}j=k^{\prime}\,)\;=\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{\\_}{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;v\,\in\,\\{\,f_{\\_}{\mathrm{vtc}}(\,\mathrm{X}_{\\_}i\,=\,k\,)\,\times\,f_{\\_}{\mathrm{vtc}}(\,\mathrm{X}_{\\_}j\,=\,k^{\prime}\,)\,\\}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!p(\,v\,)$

Furthermore, the joint distribution factors are also formalised, relative to
the system of contextuality scenario as follows.

###### Equation 10

Given a variable $\mathrm{X}_{\\_}i$ that has been fixed to the outcome $k$,
the joint distribution factors are expressed in terms of the relative system
of contextuality scenarios $H$. Specifically, the joint distribution only
calculates the product over any vertices within $V(H)$ that intersect the
corresponding vertices of the outcome
$f_{\\_}{\mathrm{vtc}}(\,\mathrm{X}_{\\_}i=k\,)$.

$\displaystyle\mathrm{Pr}(\,\mathrm{X}\;|\;do(\,\mathrm{X}_{\\_}i=k\,)\,)\;=\;\begin{cases}\prod_{\\_}{\,v\,\in\,V(H)}\,p(\,v\,)\;&\mathrm{if}\;\;\\{\,v\,\cap
f_{\\_}{\mathrm{vtc}}(\,\mathrm{X}_{\\_}i=k\,)\,\\}\;\not=\;\\{\,\varnothing\,\\}\\\
0&\mathrm{otherwise}\end{cases}$

##  7 Determining Quantum-Like Contextuality

In this section, we integrate the elements of the combinatorial approach. and
related causal modelling techniques in order to declare the main result of the
article: a theorem for determining contextuality within experiments that
exchange causal influences.

###  7.1 Disturbances In Deterministic Models

For any deterministic model that corresponds to a clique $c$, any edge $e$ of
the F-R product always intersects it at exactly one vertex if any only if the
model is noncontextual.

###### Lemma 1

For any deterministic model on a system of contextuality scenarios $H$ whose
corresponding clique $c$ is of the set of cliques ${}^{\mathrm{NC}}C$, the
cardinality of the set defined by the intersection of the clique and any edge
of the F-R product is always equivalent to 1.

$\displaystyle\forall_{\\_}{c\,\in\,{}^{\mathrm{NC}}C}\;\;\forall_{\\_}{e\,\in\,E(\,\bigotimes^{n}_{\\_}{i=1}H_{\\_}i\,)}\;\;\;|\\{\,c\,\cap\,e\,\\}|=1$

Proof: Firstly, it is known that all hyperedges of the Cartesian product of a
system of contextuality scenarios correspond to joint distributions of
outcomes. Since any deterministic model only records a single deterministic
weighting in any joint distribution of outcomes, it follows that any hyperedge
within the Cartesian product will only intersect the clique of the
correspondent deterministic model once. Secondly, by Corollary 1 the summated
probabilities corresponding to the vertices of any one hyperedge of the F-R
product are exactly equivalent to one another. As the F-R product is a
superset of the Cartesian product, it follows that any hyperedge of the F-R
product will also intersect any clique once, in order to remain consistent
with the subset of hyperedges that form the Cartesian product. Furthermore, to
prove that Lemma 1 only holds for noncontextual deterministic models, recall
that deterministic models that are not noncontextual must violate the ND
condition, which imbalances the equality of Corollary 1 by the F-R product.
Therefore, it holds that only noncontextual deterministic models can record a
single deterministic weighting of 1 in any hyperedge of the F-R
product.$\hfill\blacksquare$

Then, Lemma 1 can be integrated into the calculation of disturbance for all
deterministic models that are not noncontextual, by summating the weightings
of $q$ that fail the equality.

###### Lemma 2

Consider a system of contextuality scenarios $H$, and the set of weightings
$q$ that correspond to all deterministic models indexed by the set of cliques
${}^{\cancel{\mathrm{NC}}}C$. For any single clique
$c\,\in\,{}^{\cancel{\mathrm{NC}}}C$, its integration with Lemma 1 defines the
deterministic weight that constitutes its disturbance. Furthermore, the
summation of all such weightings define the absolute total disturbance for
exchanged within the system.

$\displaystyle\sum_{\\_}{\;\;\;c\,\in\,{}^{\cancel{\mathrm{NC}}}C}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{\\_}{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;e\,\in\,E(\,\bigotimes^{n}_{\\_}{i=1}H_{\\_}i\,)}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!q_{\\_}c\,||\\{\,c\,\cap\,e\,\\}|-1|$

Proof: It is known already from Lemma 1 that for a system of contextuality
scenarios $H$, that any member $c\,\in\,{}^{\mathrm{NC}}C$ intersects any
hyperedge of the F-R product in one vertex. Naturally, it follows that
$\forall_{\\_}{c\,\in\,{}^{\mathrm{NC}}C}$ that $|\\{\,c\,\cap\,e\,\\}|-1$
must always equal zero. In fact, irrespective of all other elements of Lemma
2, if ${}^{\cancel{\mathrm{NC}}}C$ were substituted with ${}^{\mathrm{NC}}C$,
its clear by the previous point that the entire expression would resolve to
zero, validating the fact that deterministic models corresponding to the set
${}^{\mathrm{NC}}C$ do not exchange disturbance. The same should not hold for
${}^{\cancel{\mathrm{NC}}}C$, which is supported by the fact that
$|\\{\,c\,\cap\,e\,\\}|-1$ does not equal zero for members of
${}^{\cancel{\mathrm{NC}}}C$. In a simple case, such as the system that
corresponds to the EPR framework, $|\\{\,c\,\cap\,e\,\\}|-1$ can be either
$+1$ or $-1$ for any member of ${}^{\cancel{\mathrm{NC}}}C$, which is
visualised in Figure 7.

Figure 7: Intersection Of A Relation, Two Edges Of The F-R Product, & A Clique

Note: As shown above, the clique $c\,\in\,{}^{\cancel{\mathrm{NC}}}C$ (deep
red) intersects one edge of the F-R product (light blue) twice, and the other
not once. Integrated into the expression $|\\{\,c\,\cap\,e\,\\}|-1$, both
edges produce the values $+1$ and $-1$ respectively.

If the expression $(|\\{\,c\,\cap\,e\,\\}|-1)=\pm 1$ were true for arbitrary
systems beyond the EPR framework, the expression
$\sum_{\\_}{c\in{}^{\cancel{\mathrm{NC}}}C}1$ would detail the total
disturbance. However there exist cases where $(|\\{\,c\,\cap\,e\,\\}|-1)>1$,
specifically for deterministic models that exchange disturbance for multiple
measurements. Suppose for the EPR framework that the second party were to have
three measurements instead of two. A clique
$c\,\in\,{}^{\cancel{\mathrm{NC}}}C$ could violate the ND condition for two
separate outcomes. This would mean that there are two outcomes that exchange
disturbance for this model, as visualised in Figure 8.

Figure 8: Intersection Of A Relation, An Edges Of The F-R Product, & A Clique
Within A System That Contains Three Measurements

Note: As shown above, the clique intersects the edge of the F-R product
thrice, which evaluates the expression $|\\{\,c\,\cap\,e\,\\}|-1$ to the value
$+2$.

In saying this, the assumption that the disturbance (corresponding to any
$c\,\in\,{}^{\cancel{\mathrm{NC}}}C$) is equivalent to 1 would no longer hold.
Instead, to correctly quantify disturbances for both outcomes, the ND
condition would need to be applied. This is handled by the F-R product, and so
Lemma 2 accounts for the total disturbance exchanged within a
system.$\hfill\blacksquare$

While Lemma 2 defines the expression necessary to calculate the total
disturbance exchanged within a system, consider isolating the disturbance for
only a single set of measurements from one party onto a set of outcomes of
another party. By specification of relations in Section 6.2 and Corollary 1,
this is achieved by firstly articulating the edges of the F-R product that
capture disturbance for any single relation.

###### Definition 29

For any relation $r\in R$ that corresponds to any causal influence for a
system of contextuality scenarios $H$, the edges of the F-R product necessary
for quantifying disturbances on said relation are defined as $E_{\\_}r$, and
are exactly those edges within the necessary measurement protocol that
intersect the Cartesian product of the relation.

$\displaystyle
E_{\\_}r\;:=\;\left\\{\,e\,:\,e\,\in\,E_{\\_}{\,H_{\\_}i\,\rightarrow\,H_{\\_}j\,}\;\;\mathrm{where}\;\;x\subseteq
H_{\\_}j,\;y\subseteq
H_{\\_}i,\;\;r\,:=\,\\{\,(\,x,\,y\,)\,\\},\;\;\mathrm{and}\;\;\left\\{\,e\,\cap\,\times^{|r|}_{\\_}{i=1}r\,\right\\}\neq\\{\,\varnothing\,\\}\,\right\\}$

Then by Definition 29, one can augment Lemma 2 to only quantify weightings
that intersect the relation that correspond to the measurements and outcomes
in question. Specifically for any probabilistic model, the disturbance on any
hyperedge of the F-R product can be evaluated and quantified for whether it
corresponds to a relation by the following function.

###### Lemma 3

Consider a probabilistic model $p$, a relation $r\in R$, and an edge $e\in
E(\,\otimes^{n}_{\\_}{i=1}\,)$. Let it be such that all deterministic
weightings of the set $q$ form part of the convex decomposition of $p$, as
articulated in Definition 16. Then the total disturbance exchanged between the
measurements and outcomes of the relation quantified on the said edge within
the probabilistic model are returned by the following function.

$\displaystyle
f_{\\_}{\mathrm{dtb}}(\,r,\,e,\,p\,)\;:=\begin{cases}\sum_{\\_}{c\,\in{}^{\cancel{\mathrm{NC}}}C}\;\;q_{\\_}c\\!\left|\left|\left\\{\,c\,\cap\,e\,\right\\}\right|-1\right|&\mathrm{if}\;\;\;e\,\in\,E_{\\_}r\\\
0&\mathrm{otherwise}\end{cases}$

Proof: It is known that the Cartesian product of any relation $r$ contains all
vertices of its corresponding measurements and outcomes. As any disturbances
recorded on said measurements and outcomes can only occur for weightings of
$q$ attributed to their respective vertices, it holds that imposing the
intersection of $\times^{|r|}_{\\_}{i=1}r$ by the set $E_{\\_}r$ on the
expression of Lemma 2 isolates only the disturbances for said measurements and
outcomes. Note that this may include disturbances both to and from the
intended parties, as the Cartesian product is agnostic to causal influences.
As such, the evaluating hyperedges are fixed to a subset of the F-R product,
specifically only those which fit the measurement protocol of the intended
source party to the intended destination party. This ensures that only the
intended disturbances are captured.$\hfill\blacksquare$

###  7.2 Quantifying Causal Influences

After calculating the total disturbance on a relation, it is possible to
finally formalise Definitiion 26 within the combinatorial approach.

###### Definition 30

Consider two relations $r^{\prime}$ and $r^{\prime\prime}$ which correspond to
the observation of the outcome $\mathrm{X}_{\\_}i\,=\,k$ over another outcome
$\mathrm{X}_{\\_}j\,=\,k^{\prime}$. Let it be such that when
$\mathrm{X}_{\\_}i=k$, that the relative probabilistic model is $p^{\prime}$,
and that when $\mathrm{X}_{\\_}i\neq k$, that the relative probabilistic model
is $p^{\prime\prime}$. Then the direct causal influence from
$\mathrm{X}_{\\_}i$ to $\mathrm{X}_{\\_}j$ is equivalent to the total
disturbance observed when $\mathrm{X}_{\\_}i=k$ is fixed minus the total
disturbance observed when $\mathrm{X}_{\\_}i\neq k$ is fixed.

$\displaystyle\mathrm{C}_{\\_}{{}_{\\_}{\mathrm{X}_{\\_}i\,=\,k,\,\mathrm{X}_{\\_}j\,=\,k^{\prime}}}\;=\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{\\_}{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;e\,\in\,E(\,\bigotimes^{n}_{\\_}{i=1}H_{\\_}i\,)}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left|\,f_{\\_}{\mathrm{dtb}}(\,r^{\prime},\,e,\,p^{\prime}\,)-f_{\\_}{\mathrm{dtb}}(\,r^{\prime\prime},\,e,\,p^{\prime\prime}\,)\,\right|$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\mathrm{given}\;\;\;r^{\prime}\;=\;\\{\,(\,f_{\\_}{\mathrm{vtc}}(\,do(\,\mathrm{X}_{\\_}i\,=\,k\,)\,),\,f_{\\_}{\mathrm{vtc}}(\,\mathrm{X}_{\\_}j\,=\,k^{\prime}\,)\,)\,\\}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathrm{and}\;\;\;r^{\prime\prime}\;=\;\\{\,(\,f_{\\_}{\mathrm{vtc}}(\,do(\,\mathrm{X}_{\\_}i\,\neq\,k\,)\,),\,f_{\\_}{\mathrm{vtc}}(\,\mathrm{X}_{\\_}j\,=\,k^{\prime}\,)\,)\,\\}$

Furthermore, as there is a direct correspondence between the measurements and
outcomes, $\mathrm{X}_{\\_}i\,=\,k$ and $\mathrm{X}_{\\_}j\,=\,k^{\prime}$,
within
$\mathrm{C}_{\\_}{{}_{\\_}{\mathrm{X}_{\\_}i\,=\,k,\,\mathrm{X}_{\\_}j\,=\,k^{\prime}}}$
and the relation $r^{\prime}$, it holds that the shorthand
$\mathrm{C}_{\\_}{r^{\prime}}$ can be used to denote it. Generally, the direct
causal influence for any relation $r$ can be denoted by $\mathrm{C}_{\\_}r$.

$\displaystyle\mathrm{C}_{\\_}{r}\,=\,\mathrm{C}_{\\_}{{}_{\\_}{\mathrm{X}_{\\_}i\,=\,k,\,\mathrm{X}_{\\_}j\,=\,k^{\prime}}}\;\;\;\mathrm{where}\;\;\;r\;:=\;\\{\,(\,f_{\\_}{\mathrm{vtc}}(\,\mathrm{X}_{\\_}i\,=\,k\,),\,f_{\\_}{\mathrm{vtc}}(\,\mathrm{X}_{\\_}j\,=\,k^{\prime}\,)\,)\,\\}$

Having formalised the direct causal influence for any relation within the
combinatorial approach, it is then possible to derive for any experiment the
total disturbance that constitutes causal influences.

###### Lemma 4

For any relation $r\,\in\,R$ that corresponds to any causal influence for a
system of contextuality scenarios, the total direct causal influences that
constitute all or part thereof the total disturbances are equivalent to the
following expression.

$\displaystyle\sum_{\\_}{r\,\in\,R}\,\mathrm{D}_{\\_}r-\mathrm{max}\,\\{\,0,\,\mathrm{D}_{\\_}r-\mathrm{C}_{\\_}r\,\\}\;\;\;\mathrm{where}\;\;\;\mathrm{D}_{\\_}r=\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{\\_}{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;e\,\in\,E(\,\bigotimes^{n}_{\\_}{i=1}H_{\\_}i\,)}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\,f_{\\_}{\mathrm{dtb}}(\,r,\,e,\,p\,)$

Proof: Firstly, for any relation $r$, the expression $\mathrm{C}_{\\_}{r}$
determines (for the general case), the direct causal influence exchanged by
the relation. When subtracted from the total disturbance $\mathrm{D}_{\\_}r$
of a given experiment, $\mathrm{D}_{\\_}r-\mathrm{C}_{\\_}{r}$ quantifies the
total noise. This is known because all disturbances that are not the result of
causal influences are inevidably the result of noise. In some cases, this may
be a negative value (i.e., when there is no disturbance on a relation that
anticipates disturbance comprised of causal influence), and so the value is
integrated into the expression
$\mathrm{max}\,\\{\,0,\,\mathrm{D}_{\\_}r-\mathrm{C}_{\\_}r\,\\}$. Finally,
the value is again subtracted from $\mathrm{D}_{\\_}r$ to determine the direct
causal influences in the experiment itself. It follows that the summation for
all relations quantifies the total direct causal influences for the entire
experiment.$\hfill\blacksquare$

###  7.3 Determining Contextuality By Causal Influences

At this stage, it is possible to redefine the equation used to determine
quantum-like contextuality. Prior to this work, Pironio [2003] states that for
any Bell parameter $\mathcal{B}$ that violates the Bell inequalities by means
of causal influences only, that the amount by which the violation is observed
is exactly equivalent to the exchanged causal influences.

###### Corollary 2

For an experiment that determines contextuality, let the Bell parameter
$\mathcal{B}$ generate its statistical correlation by means of causal
influences _only_ (i.e., in the full absence of noise among experimental
results). Also, let the weightings of said causal influences be totalled to
the value $\mathrm{C}_{\\_}{\mathrm{all}}$. Then it is known by Pironio [2003]
that the amount $\mathcal{B}$ violates the statistical bound
$\mathcal{B}_{\\_}0$ is equivalent to $\mathrm{C}_{\\_}{\mathrm{all}}$.

$\displaystyle\left|\mathcal{B}-\mathcal{B}_{\\_}0\right|=\mathrm{C}_{\\_}{\mathrm{all}}$

Finally, by rearrangement of Corollary 2 and integration of Lemma 4, we obtain
an expression that can determine contextuality in the presence of causal
influences.

###### Theorem 1

For any given experiment, let the Bell parameter $\mathcal{B}$ generate its
statistical correlation against a statistical bound $\mathcal{B}_{\\_}0$.
Then, if the total disturbances for the experiment that constitute causal
influences subtracted from the Bell parameter are greater than
$\mathcal{B}_{\\_}0$, the experiment determines contextuality.

$\displaystyle\mathcal{B}-\left(\;\sum_{\\_}{r\,\in\,R}\,\mathrm{D}_{\\_}r-\mathrm{max}\,\\{\,0,\,\mathrm{D}_{\\_}r-\mathrm{C}_{\\_}r\,\\}\right)\;>\;\mathcal{B}_{\\_}{0}$

By Theorem 1, we can determine contextuality in the presence of causal
influences. More importantly, this result allows us to correctly determine
quantum-like contextuality within cognitive experiments.

##  8 An Example Scenario

In this section, an example is provided to convey how the extension of the
combinatorial approach determines contextuality in the presence of causal
influences. The example again considers the EPR framework, however with a
separate probabilistic model that has the pair-wise joint distributions in
Table 4.

In the literature, the example is widely known as a ‘Popescu-Rohrlich’ (P-R)
box [Popescu and Rohrlich, 1998], and is known to be maximally contextual due
to the degree that it’s Bell parameter $\mathcal{B}$ violates the bound
$\mathcal{B}_{\\_}0$ of noncontextual hidden variable theories.

###### Equation 11

The Bell parameter of Table 4 violates the bound of noncontextual hidden
variable theories.

$\displaystyle\mathcal{B}_{\\_}0$ $\displaystyle\not\geq\mathcal{B}$
$\displaystyle 2\;$
$\displaystyle\not\geq\;\underset{\begin{matrix}a\,\in\,\\{\,+1,\,-1\,\\}\\\
b\,\in\,\\{\,+1,\,-1\,\\}\end{matrix}}{\mathrm{max}}\raisebox{-8.00003pt}{\scalebox{2.0}{\mbox{$\displaystyle|$}}}\;\mathrm{corr}_{\\_}{\scalebox{0.75}{\mbox{$\displaystyle\begin{matrix}{}^{\mathrm{ipt}}\\!A\,=\,+1\\\\[-2.5pt]
{}^{\mathrm{ipt}}\\!B\,=\,+1\end{matrix}$}}}\;+\;\mathrm{corr}_{\\_}{\scalebox{0.75}{\mbox{$\displaystyle\begin{matrix}{}^{\mathrm{ipt}}\\!A\,=\,+1\\\\[-2.5pt]
{}^{\mathrm{ipt}}\\!B\,=\,-1\end{matrix}$}}}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\;\mathrm{corr}_{\\_}{\scalebox{0.75}{\mbox{$\displaystyle\begin{matrix}{}^{\mathrm{ipt}}\\!A\,=\,-1\\\\[-2.5pt]
{}^{\mathrm{ipt}}\\!B\,=\,+1\end{matrix}$}}}\;+\;\mathrm{corr}_{\\_}{\scalebox{0.75}{\mbox{$\displaystyle\begin{matrix}{}^{\mathrm{ipt}}\\!A\,=\,-1\\\\[-2.5pt]
{}^{\mathrm{ipt}}\\!B\,=\,-1\end{matrix}$}}}\;-\;2\mathrm{corr}_{\\_}{\scalebox{0.75}{\mbox{$\displaystyle\begin{matrix}{}^{\mathrm{ipt}}\\!A\,=\,a\\\\[-2.5pt]
{}^{\mathrm{ipt}}\\!B\,=\,b\end{matrix}$}}}\;\raisebox{-8.00003pt}{\scalebox{2.0}{\mbox{$\displaystyle|$}}}$
$\displaystyle 2\;$
$\displaystyle\not\geq\;\left|1-1+1+1-2(-1)\right|\;\;\;\mathrm{given}\;\;\;a=+1\;\;b=-1$
$\displaystyle 2\;$ $\displaystyle\not\geq\;4$

Earlier in this article, it was mentioned that results such as the above may
be due to causal influences, and may not be truly contextual in nature. By the
relevant techniques, an experimental protocol is now detailed.

Let it be such that the experimental trials that inform the probabilistic
model of Table 4 have two equally likely deterministic models, which
correspond to the cliques seen in Figure 9.

Figure 9: The Cliques $c$ And $c^{\prime}$ Visualised On Cartesian Product Of
Contextuality Scenarios $H_{\\_}A$ And $H_{\\_}B$

Specifically, the cliques would each be attributed equal weightings by the set
$q$.

###### Definition 31

The set $q$, conveyed by the two comprising weights that correspond to the
cliques in Figure 9:

$\displaystyle q_{\\_}{c}=0.5,\;\;\;\;\;\;q_{\\_}{c^{\prime}}=0.5$

From either clique, it can be speculated that the outcomes are not invariant
with respect to the measurements, and are thus dependent upon their
selections. This is evaluated by means of mapping the measurements to a causal
model, as previously visualised in Figure 4.

Next, all relations upon the causal model are evaluated; as mentioned in
Section 6.2, these correspond to the causal influences that may be present in
the model. Of interest, the following relations are considered.

###### Definition 32

Four relations of set $R$ that are of significance to possible causal
influences within the example’s causal model:

$\displaystyle r\;$
$\displaystyle=\;\\{\,(\,f_{\\_}{\mathrm{vtc}}(\,do(\,X_{\\_}1=+1,\;\mathrm{X}_{\\_}2\,=\,+1\,)\,),\,f_{\\_}{\mathrm{vtc}}(\,\mathrm{X}_{\\_}3\,=\,-1,\,\mathrm{X}_{\\_}4\,=\,+1\,)\,)\,\\}$
$\displaystyle r^{\prime}\;$
$\displaystyle=\;\\{\,(\,f_{\\_}{\mathrm{vtc}}(\,do(\,X_{\\_}1=+1,\;\mathrm{X}_{\\_}2\,=\,-1\,)\,),\,f_{\\_}{\mathrm{vtc}}(\,\mathrm{X}_{\\_}3\,=\,-1,\,\mathrm{X}_{\\_}4\,=\,+1\,)\,)\,\\}$
$\displaystyle r^{\prime\prime}\;$
$\displaystyle=\;\\{\,(\,f_{\\_}{\mathrm{vtc}}(\,do(\,X_{\\_}1=-1,\;\mathrm{X}_{\\_}2\,=\,+1\,)\,),\,f_{\\_}{\mathrm{vtc}}(\,\mathrm{X}_{\\_}3\,=\,-1,\,\mathrm{X}_{\\_}4\,=\,+1\,)\,)\,\\}$
$\displaystyle r^{\prime\prime\prime}\;$
$\displaystyle=\;\\{\,(\,f_{\\_}{\mathrm{vtc}}(\,do(\,X_{\\_}1=-1,\;\mathrm{X}_{\\_}2\,=\,-1\,)\,),\,f_{\\_}{\mathrm{vtc}}(\,\mathrm{X}_{\\_}3\,=\,-1,\,\mathrm{X}_{\\_}4\,=\,+1\,)\,)\,\\}$

For the first relation $r$, we wish to determine the causal influence of
observing $\mathrm{X}_{\\_}3=-1$, given that $\mathrm{X}_{\\_}1$ and
$\mathrm{X}_{\\_}2$ are both fixed to the outcomes $+1$. To do this, we
firstly calculate the edges that correspond to relation $r$ by Definition 29.

###### Definition 33

For relation $r$ from Definition 32, the calculation of edges from the
necessary measurement protocol that correspond to the set $E_{\\_}r$:

$\displaystyle E_{\\_}r\;$
$\displaystyle:=\;\left\\{\,e\,:\,e\,\in\,E_{\\_}{\,H_{\\_}i\,\rightarrow\,H_{\\_}j\,}\;\;\mathrm{where}\;\;x\subseteq
H_{\\_}j,\;y\subseteq
H_{\\_}i,\;\;r\,:=\,\\{\,(\,x,\,y\,)\,\\},\;\;\mathrm{and}\;\;\left\\{\,e\,\cap\,\times^{|r|}_{\\_}{i=1}r\,\right\\}\neq\\{\,\varnothing\,\\}\,\right\\}$
$\displaystyle:=\;\\{\,e\,:\,e\,\in\,E_{\\_}{\,H_{\\_}A\,\rightarrow\,H_{\\_}B\,}\;\;\mathrm{where}\;\;\\{(\,v_{\\_}{0|0}\,)\\}\subseteq
H_{\\_}A,\;\\{(\,v_{\\_}{0|1}\,)\\}\subseteq H_{\\_}B,\;\;$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;r\,:=\,\\{\,(\,v_{\\_}{0|0},\,v_{\\_}{0|1}\,)\,\\},\;\;\mathrm{and}\;\;\left\\{\,e\,\cap\,v_{\\_}{0,0|0,1}\,\right\\}\neq\\{\,\varnothing\,\\}\,\\}$
$\displaystyle:=\;\left\\{(\,v_{\\_}{0,0|0,1},\,v_{\\_}{1,0|0,1},\,v_{\\_}{0,1|1,1},\,v_{\\_}{1,1|1,1}\,)\right\\}$

Note: In calculating set $E_{\\_}r$, the comprising values of $x$ and $y$ have
been derived by the following additional equations:

$\displaystyle x\;$
$\displaystyle:=\;f_{\\_}{\mathrm{vtc}}(\,do(\,X_{\\_}1=+1,\;\mathrm{X}_{\\_}2\,=\,+1\,)\,)$
$\displaystyle x\;$ $\displaystyle:=\;\\{(\,v_{\\_}{0|0}\,)\\}$ $\displaystyle
y\;$
$\displaystyle:=\;f_{\\_}{\mathrm{vtc}}(\,\mathrm{X}_{\\_}4\,=\,+1,\,\mathrm{X}_{\\_}3=-1\,)$
$\displaystyle y\;$ $\displaystyle:=\;\\{(\,v_{\\_}{0|1}\,)\\}$

Having determined $E_{\\_}r$ in Definition 33, only a single edge of the
measurement protocol $E_{\\_}{H_{\\_}A\rightarrow H_{\\_}B}$ within the F-R
product has been returned. This is visualised in Figure 10.

Figure 10: Intersection Of Cliques, A Hyperedge Of The F-R Product, And A
Relation

Note: The hyperedge returned by Definition 33 is highlighted in blue; the
relation $r$ is highlighted in light red.

Now it is possible to calculate the causal influence variable $C_{\\_}r$ by
Definition 30. As specified, there is a requirement that two separate
experimental tests are conducted prior to its calculation, to derive the
necessary probabilistic models that correspond to fixing the causal variables
to the outcomes of relation $r$ (i.e.,
$\mathrm{Pr}(\,\mathrm{X}_{\\_}4=+1\;|\;do(\,\mathrm{X}_{\\_}1=+1,\,\mathrm{X}_{\\_}2=+1\,),\,\mathrm{X}_{\\_}3=-1\,)$),
as well as fixing the causal variables not to the outcomes of relation $r$
(i.e.,
$\mathrm{Pr}(\,\mathrm{X}_{\\_}4=+1\;|\;do(\,\mathrm{X}_{\\_}1=+1,\,\mathrm{X}_{\\_}2=-1\,),\,\mathrm{X}_{\\_}3=-1\,)$).
Coincidentally, the latter case is exactly $r^{\prime}$, as detailed in
Definition 32. For both relations, the causal variables are significantly
fixed, and have separate probabilistic models ($p$ for $r$, and $p^{\prime}$
for $r^{\prime}$) informed by their respective experiments. The results, as
sampled under the causal interventions are given in Table 5.

$\begin{matrix}[cc]\mathrm{X}_{\\_}4=+1&\mathrm{X}_{\\_}4=-1\end{matrix}$
---
$\begin{pmatrix}[cc]\;\;0.00\;&\;1.00\;\;\end{pmatrix}$
(a) $\mathrm{X}_{\\_}4$, given
$do(\,\mathrm{X}_{\\_}1=+1,\,\mathrm{X}_{\\_}2=+1\,)$ and
$\mathrm{X}_{\\_}3=-1$

$\begin{matrix}[cc]\mathrm{X}_{\\_}4=+1&\mathrm{X}_{\\_}4=-1\end{matrix}$
---
$\begin{pmatrix}[cc]\;\;1.00\;&\;0.00\;\;\end{pmatrix}$
(b) $\mathrm{X}_{\\_}4$, given
$do(\,\mathrm{X}_{\\_}1=+1,\,\mathrm{X}_{\\_}2=-1\,)$ and
$\mathrm{X}_{\\_}3=-1$

Table 5: Probabilistic Distributions Corresponding To Observation Of
$\mathrm{X}_{\\_}4$ Under Causal Intervention

With knowledge of $p$ and $p^{\prime}$, $C_{\\_}r$ is calculated by
determining the difference in disturbance between the relations $r$ and
$r^{\prime}$, given their respective probabilistic models. It is known that
the F-R product calculates said disturbance, and that by relation $r$, the
edge given in Figure 10 is the only edge that qualifies for the relation; this
is the same for the relation $r^{\prime}$. What differs between them is that
clique $c^{\prime}$ does not intersect relation $r$, which then influences the
result of $C_{\\_}r$.

###### Definition 34

The causal influence $C_{\\_}r$ determined for the relation $r$, specifically
for the causal influence that the configuration of $\mathrm{X}_{\\_}1=+1$ and
$\mathrm{X}_{\\_}2=+1$ exert upon $\mathrm{X}_{\\_}4$ when
$\mathrm{X}_{\\_}3=-1$.

$\displaystyle\mathrm{C}_{\\_}r\;$
$\displaystyle=\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{\\_}{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;e\,\in\,E(\,\bigotimes^{n}_{\\_}{i=1}H_{\\_}i\,)}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left|\,f_{\\_}{\mathrm{dtb}}(\,r,\,e,\,p\,)-f_{\\_}{\mathrm{dtb}}(\,r^{\prime},\,e,\,p^{\prime}\,)\,\right|$
$\displaystyle=\;\left|\,0\;-\;q_{\\_}{c^{\prime}}\\!\left|\left|\left\\{\,c^{\prime}\,\cap\,e\,\right\\}\right|-1\right|\right|$
$\displaystyle=\;\left|\,0\;-\;0.5\right|$ $\displaystyle=\;0.5$

The same calculation is obtained for all other relations, which for the
example reveals the following values:

###### Definition 35

The causal influence variables, as calculated for all relations of the
example:

$\displaystyle
C_{\\_}r=0.5,\;\;\;\;\;\;C_{\\_}{r^{\prime}}=0.5,\;\;\;\;\;\;C_{\\_}{r^{\prime\prime}}=0.5,\;\;\;\;\;\;C_{\\_}{r^{\prime\prime\prime}}=0.5$

By Equation 11, the Bell parameter $\mathcal{B}$ is known to equal $4$; also
by Equation 1, the statistical bound on noncontextual hidden variables
$\mathcal{B}_{\\_}{0}$ is known to equal $2$. In Theorem 1, the value
$D_{\\_}r$ is known to calculate the causal influence of each relation $r$
under the pretense of no causal intervention. For the example, this so happens
to be equivalent to the value of causal variable (i.e. $D_{\\_}r=C_{\\_}r$).
It follows that the cancellation of values is reflected in the calculation of
the theorem below:

###### Equation 12

The calculation of Theorem 1 for the example scenario given in this section.

$\displaystyle\mathcal{B}-(\;\sum_{\\_}{r\,\in\,R}\,\mathrm{D}_{\\_}r-\mathrm{max}\,\\{\,0,\,\mathrm{D}_{\\_}r-\mathrm{C}_{\\_}r\,\\})\;$
$\displaystyle\not>\;\mathcal{B}_{\\_}{0}$ $\displaystyle
4-((\mathrm{D}_{\\_}r-\mathrm{max}\,\\{\,0,\,\mathrm{D}_{\\_}r-\mathrm{C}_{\\_}r\,\\})+(\mathrm{D}_{\\_}{r^{\prime}}-\mathrm{max}\,\\{\,0,\,\mathrm{D}_{\\_}{r^{\prime}}-\mathrm{C}_{\\_}{r^{\prime}}\,\\})\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle+(\mathrm{D}_{\\_}{r^{\prime\prime}}-\mathrm{max}\,\\{\,0,\,\mathrm{D}_{\\_}{r^{\prime\prime}}-\mathrm{C}_{\\_}{r^{\prime\prime}}\,\\})+(\mathrm{D}_{\\_}{r^{\prime\prime\prime}}-\mathrm{max}\,\\{\,0,\,\mathrm{D}_{\\_}{r^{\prime\prime\prime}}-\mathrm{C}_{\\_}{r^{\prime\prime\prime}}\,\\}))\;$
$\displaystyle\not>\;2$ $\displaystyle 4-((0.5-0)+(0.5-0)+(0.5-0)+(0.5-0))\;$
$\displaystyle\not>\;2$ $\displaystyle 2\;$ $\displaystyle\not>\;2$

As can be seen from the result of Equation 12, the causal influences exchanged
between the measurements and outcomes of the respective parties cancel out any
indication that the Bell parameter is quantum-like contextual. This concludes
the example scenario of how the techniques developed in this article may be
applied to more adequately determine quantum-like contextuality.

##  9 Conclusion

This article has developed and integrated a set of modelling techniques to
address the challenges of determining quantum-like contextuality in the
presence of causal influences. It has achieved this by firstly addressing the
challenge of providing meaningful results to experimentation when disturbances
are present, by combination of the F-R product of Acín et al. [2015]’s
combinatorial approach and Chaves et al. [2015]’s causal influence formula.
The results of this article have also addressed the second challenge of
providing a sensitive treatment to the convex decomposition of the
probabilistic model associated with an experiment by means of the
combinatorial approach’s WFPN. This further ensures that hidden causal
influences are accounted for within experimental results, as previously
recommended by Atmanspacher and Filk [2019]. In addressing these challenges,
it has furthermore become possible to derive a novel theorem (see Theorem 1)
that reasonably adjusts the statistical bound of noncontextual hidden variable
theories, allowing for a theoretically consistent determination of
contextuality.

Beyond the main objectives of the article, we have also clarified incorrect
causal assumptions introduced by the usage of the ND condition within
experimentation for determining contextuality. As detailed in Section 5.2.1,
it was found that the ND condition can incorrectly classify noise within
experimental results as causal influences, which inhibits a meaningful
interpretation of experimental results. The article also details how
measurements and outcomes (as they are formalised in the combinatorial
approach) may be related to canonical causal models. This has been achieved by
the necessary mapping functions of causal variables and effects to
contextuality scenarios and relations respectively, as detailed in Section 6.
Lastly, the article has provided a comprehensive example of how the
theoretical contributions may be applied. A protocol detailing the
experimental steps has been developed, such that a cognitive modeller may
apply them to arrive at a meaningful result. While the example is communicated
in the manner of the EPR framework, the theory is generally distilled to be
applicable within systems of arbitrarily many outcomes, measurements or
parties.

In terms of limitations, it is perceived that this work has only considered
integrating the absolutely necessary causal modelling techniques required to
produce viable experimental results. We recognise that contemporary approaches
to modelling contextuality do not include rigorous causal analyses by nature,
and this was evident by the absence of such techniques within the
combinatorial approach. However it seems necessary to interrogate causal
approaches to study contextuality, which we perceive will further our
understanding of the phenomenon. Aside from this, we believe it would also be
necessary to adapt Theorem 1 to a linear program that can more generally
calculate the statistical bound for any experimental protocol, given the
presence of causal influences. It is known in the literature that the
statistical bound of the Bell inequalities form a convex polytope of all
probabilistic models that adhere to noncontextual hidden variable theories. In
saying this, while the methods described in this paper have served the
derivation of the statistical bounds necessary to achieve Theorem 1, it is
nevertheless possible to extend the findings even further to a more
straightforward solution.

## Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any
commercial or financial relationships that could be construed as a potential
conflict of interest.

## Author Contributions

In terms of contribution, Dr Obeid and Prof Bruza were responsible for the
development of the theories necessary to achieve the fundamental results of
the paper. Dr Obeid was primarily responsible for drafting the visual
materials and example scenario. All authors were responsible for the proof-
reading, grammatical structure, and communication of findings.

## Funding

This research has been partially funded by the Unitary Fund, and the Asian
Office of Aerospace Research and Development (AOARD) grant: FA2386-17-1-4016.

## Acknowledgments

In developing the main findings of this article, we’d like to thank Dr Elie
Wolfe for the wealth of guidance he has provided.

## References

  * Acín et al. [2015] Antonio Acín, Tobias Fritz, Anthony Leverrier, and Ana Belén Sainz.  A combinatorial approach to nonlocality and contextuality.  _Communications in Mathematical Physics_ , 334(2):533–628, 2015.
  * Aerts et al. [2013] Diederik Aerts, Liane Gabora, and Sandro Sozzo.  Concepts and their dynamics: A quantum-theoretic modeling of human thought.  _Topics in Cognitive Science_ , 5(4):737–772, 2013.
  * Asano et al. [2014] Masanari Asano, Takahisa Hashimoto, Andrei Khrennikov, Masanori Ohya, and Yoshiharu Tanaka.  Violation of contextual generalization of the leggett–garg inequality for recognition of ambiguous figures.  _Physica Scripta_ , 2014(T163):014006, 2014.
  * Atmanspacher and Filk [2019] Harald Atmanspacher and Thomas Filk.  Contextuality revisited: Signaling may differ from communicating.  In _Quanta and Mind_ , pages 117–127. Springer, 2019.
  * Bell [1964] John S Bell.  On the Einstein–Podolsky–Rosen paradox.  _Physics Physique Fizika_ , 1(3):195, 1964.
  * Bruza et al. [2015] Peter D Bruza, Kirsty Kitto, Brentyn J Ramm, and Laurianne Sitbon.  A probabilistic framework for analysing the compositionality of conceptual combinations.  _Journal of Mathematical Psychology_ , 67:26–38, 2015.
  * Cervantes and Dzhafarov [2018] Víctor H Cervantes and Ehtibar N Dzhafarov.  Snow queen is evil and beautiful: Experimental evidence for probabilistic contextuality in human choices.  _Decision_ , 5(3):193, 2018.
  * Chaves et al. [2015] Rafael Chaves, Richard Kueng, Jonatan Bohr Brask, and David Gross.  Unifying framework for relaxations of the causal assumptions in Bell’s theorem.  _Physical Review Letters_ , 114(14):140403, 2015\.
  * Clauser et al. [1969] John F Clauser, Michael A Horne, Abner Shimony, and Richard A Holt.  Proposed experiment to test local hidden-variable theories.  _Physical review letters_ , 23(15):880, 1969.
  * Dzhafarov and Kujala [2016] Ehtibar N Dzhafarov and Janne V Kujala.  Contextuality-by-default 2.0: Systems with binary random variables.  In _International Symposium on Quantum Interaction_ , pages 16–32. Springer, 2016.
  * Dzhafarov et al. [2016] Ehtibar N Dzhafarov, Ru Zhang, and Janne Kujala.  Is there contextuality in behavioural and social systems?  _Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences_ , 374(2058):20150099, 2016.
  * Fine [1982] Arthur Fine.  Hidden variables, joint probability, and the Bell inequalities.  _Physical Review Letters_ , 48(5):291, 1982.
  * Gleason [1957] Andrew M Gleason.  Measures on the closed subspaces of a Hilbert space.  _Journal of mathematics and mechanics_ , pages 885–893, 1957.
  * Jones [2019] Matt Jones.  Relating causal and probabilistic approaches to contextuality.  _Philosophical Transactions of the Royal Society A_ , 377(2157):20190133, 2019.
  * Jones [2021] Matt Jones.  Causal-model approach to contextuality for disturbing systems.  http://pawkur1-qcqmb.home.amu.edu.pl/wp-content/uploads/2021/11/QCQMB-Jones-Nov21.pdf, 2021\.  Accessed: 2022–02-01.
  * Kochen and Specker [1975] Simon Kochen and Ernst P Specker.  The problem of hidden variables in quantum mechanics.  In _The logico-algebraic approach to quantum mechanics_ , pages 293–328. Springer, 1975.
  * Obeid [2021] Abdul Obeid.  _Modelling contextuality amidst causal influences by means of a computationally tractable combinatorial approach_.  PhD thesis, Queensland University of Technology, 2021.
  * Pearl [2009] Judea Pearl.  _Causality_.  Cambridge university press, 2009.
  * Pearl and Mackenzie [2018] Judea Pearl and Dana Mackenzie.  _The book of why: the new science of cause and effect_.  Basic books, 2018.
  * Peres [1991] Asher Peres.  Two simple proofs of the Kochen-Specker theorem.  _Journal of Physics A: Mathematical and General_ , 24(4):L175, 1991.
  * Pironio [2003] Stefano Pironio.  Violations of bell inequalities as lower bounds on the communication cost of nonlocal correlations.  _Physical Review A_ , 68(6):062102, 2003.
  * Popescu and Rohrlich [1998] Sandu Popescu and Daniel Rohrlich.  Causality and nonlocality as axioms for quantum mechanics.  In _Causality and locality in modern physics_ , pages 383–389. Springer, 1998.
  * Ramanathan et al. [2012] Ravishankar Ramanathan, Akihito Soeda, Paweł Kurzyński, and Dagomir Kaszlikowski.  Generalized monogamy of contextual inequalities from the no-disturbance principle.  _Physical Review Letters_ , 109(5):050404, 2012\.
  * Shannon [1956] Claude Shannon.  The zero error capacity of a noisy channel.  _IRE Transactions on Information Theory_ , 2(3):8–19, 1956.
  * Yearsley and Halliwell [2019] James M Yearsley and Jonathan J Halliwell.  Contextuality in human decision making in the presence of direct influences: A comment on Basieva et al.(2019).  _arXiv preprint arXiv:1905.12570_ , 2019.