# Functional dependencies with null markers

Antonio Badia<EMAIL_ADDRESS>CECS Department, University of
Louisville, Louisville KY 40292, USA  Daniel Lemire  LICEF, Université du
Québec, 5800 Saint-Denis, Montreal, QC, H2S 3L5 Canada

###### Abstract

Functional dependencies are an integral part of database design. However, they
are only defined when we exclude null markers. Yet we commonly use null
markers in practice. To bridge this gap between theory and practice,
researchers have proposed definitions of functional dependencies over
relations with null markers. Though sound, these definitions lack some
qualities that we find desirable. For example, some fail to satisfy
Armstrong’s axioms—while these axioms are part of the foundation of common
database methodologies. We propose a set of properties that any extension of
functional dependencies over relations with null markers should possess. We
then propose two new extensions having these properties. These extensions
attempt to allow null markers where they make sense to practitioners. They
both support Armstrong’s axioms and provide _realizable_ null markers: at any
time, some or all of the null markers can be replaced by actual values without
causing an anomaly. Our proposals may improve database designs.

###### keywords:

Functional Dependencies; Database Design; Missing Information

##  1 Introduction

Functional dependencies (hereafter, FDs) are the basis of good relational
database design. Database designers use FDs to define and enforce consistency;
for example, if each user account has only one corresponding primary email
address, we say that the user account determines the email address
($\textrm{user-account}\to\textrm{email}$). Though database designers may not
always work directly with FDs, almost all of them work with normal forms which
exist to support FDs.

Meanwhile, relational databases use null markers to cope with incomplete
information. Indeed, Codd introduced null markers in the relational model
along with a 3-value logic: a comparison between a value and a null marker has
an unknown truth value [1]. A null marker can indicate anything from an
applicable but unknown value to a non-applicable attribute; it may even
indicate that we have _no information_ [2, 3, 4]. We refer the reader to
Libkin [5] for an early survey on incomplete information within databases.

Unfortunately, the concept of functional dependence, even though central to
relational database design, is not part of the SQL standard. Hence, FDs are
not _directly_ supported by relational databases. Admittedly, given the
procedural extensions of the SQL standard, SQL is computationally complete and
therefore can enforce FDs using checks, assertions or triggers. However, the
problem is more fundamental: FDs are not _defined_ in the presence of null
markers. Thus, there is no clear semantics to enforce when we have FDs and
null markers.

We believe that the fact that FDs are not defined in the presence of null
markers undermines the role of FDs in database design [6]. While there are
proposed definitions of FDs in relations with null markers, such proposals may
not be practical.

Our work is organized as follows. In the next section, we argue for a minimal
set of desirable properties that FDs in relations with null markers should
have to be of practical use. After briefly introducing our formal notation in
§ 3, we review two existing extensions of FDs to null markers in § 4: weak and
strong FDs. We show that neither extension is satisfactory in our context, as
they do not have the desired properties. We introduce two novel extensions in
§ 5: literal and super-reflexive functional dependencies, and show that they
that possess our properties. In § 6 we compare our extensions to weak and
strong FDs, in order to give the reader a context for interpretation. Next in
§ 7 we address the issue of whether the newly proposed concepts can be
efficiently supported. In § 8, we show how to extend logical database design
to include null markers using one of the proposed extensions. Finally, we
close in § 9 with some comments and a discussion of further research.

##  2 Desirable properties

The SQL standard allows null markers, but does not allude to FDs [7]. One
could point out that the concept of _key_ is explicitly present in the SQL
standard; given that the concepts of key and FD are related, it would seem
that we support FDs by enforcing the _key constraint_ [8]. However, when we
enforce FDs by key constraints, we assume that all the tables in the database
have attained Boyce-Codd normal form (BCNF). Yet such a normal form excludes
the commonly used null markers.

Codd was well aware of the perceived problems with null markers. Yet he was
unconcerned by our inability to enforce FDs in the presence of null markers.
He believed that only when the null markers where replaced by actual values
would concepts such as keys, normalization and FDs be applicable:

> It should be clear that, because nulls (or, as they are now called, marks)
> are NOT database values, the rules of functional dependence—and of multi-
> valued dependence—do not apply to them. (E. F. Codd [1])

However, consider the relation in Table 1 subject to the FDs
$\text{professor}\to\text{chair}$ and $\text{department} \to\text{chair}$.
Given that Jill and Arthur are distinct people, it is not possible to replace
the null marker by an actual value. Going back to Codd’s vision, we have that
keys, normalization and FDs are never going to be applicable to all tuples of
such relation. This might reasonably be considered anomalous.

To avoid such problems, we establish as a first goal that if a FD is enforced
in a relation with null markers, it should always be possible to replace some
or all of the null markers by actual values without violating the FD (G1). In
such a case, we say that such FDs are _realizable_ under null markers.
Moreover, and since we consider _sets_ of FDs for design (as opposed to single
FDs in isolation), we posit that FDs should be defined so that if a _set_ of
FDs is enforced in a relation with null markers, it should be possible to
replace some or all of the null markers without violating any of the FDs in
the set (strong G1).

Table 1:  Example relation with attributes professor, chair and department professor | chair | department
---|---|---
Joe | null | Mathematics
Joe | Jill | Computer Science
Bill | Arthur | Mathematics

When defining FDs in the presence of null markers, we are also interested in
Armstrong’s three axioms, especially transitivity:

Reflexivity: If $Y\subseteq X$, then $X\to Y$.

Augmentation: we have that $X\to Y$ implies $XZ\to YZ$.

Transitivity: If $X\to Y$ and $Y\to Z$, then $X\to Z$. E.g., we might say that
your social security number (SSN) determines your income
($\text{SSN}\to\text{Income}$), and that your income determines your tax
bracket ($\text{Income}\to\text{Taxation}$). If transitivity holds, then your
SSN determines your tax bracket ($\text{SSN}\to\text{Taxation}$).

Table 2:  Example relation with attributes SSN, income and taxation SSN | income | taxation
---|---|---
1112233 | null | 15%
1112233 | null | 25%

Codd’s interpretation, that FD do not apply when there are null markers fails
to enforce transitivity in the following sense: given the FD
$\textrm{SSN}\to\textrm{Income}$ and $\textrm{Income}\to\textrm{Taxation}$,
both tuples in Table 2 are allowable, even though one would expect not to see
such data in the database. It implies that $\textrm{SSN}\to\textrm{Taxation}$
does not hold (whereas it should under transitivity) even though there is no
null marker over attributes SSN and Taxation.

Codd would no doubt reply that there is no violation of transitivity since FDs
do not apply in the presence null markers. But we wish to consider null
markers as an integral part of the database.

Failing to enforce Armstrong’s axioms has significant consequences. For one
thing, without these axioms, normalization is no longer sufficient to enforce
FDs. In practical terms, any redefinition of the FDs that fails to satisfy
Armstrong’s axioms cannot be enforced through normalization. Indeed, given a
database $D$ and a set of FDs ${\cal F}$ on $D$, before we can use ${\cal F}$
to determine normal forms for the relations in $D$, we need to make sure that
${\cal F}$ is in minimal or canonical form [9]. But minimizing ${\cal F}$
depends _crucially_ on FDs respecting transitivity, as covered in standard
database textbooks.

We might be willing to forgo normalization and enforce FDs through other
means. In such a case, it might seem like Armstrong’s axioms are no longer
required. For example, we might think that it is possible to build a database
design without assuming that FDs are transitive. However, such watered-down
FDs might be impractical for other reasons. The first problem that we
encounter is that standard database design methodologies, like the entity-
relationship model, implicitly assume Armstrong’s axioms and transitivity in
particular [10].

We believe that database designers would have a hard time coping with the lack
of transitivity (see, for instance, the example of Table 2, where intuitively
one would expect to see, for the same SSN, the same Taxation), and hence we
require that Armstrong’s axioms hold, even though such a requirement could be
seen as too strong. Of course, we could help designers with additional tools
and methodologies [11] to compensate for the added constraint. Nevertheless,
everything else being equal, we view as desirable that a new definition of FDs
in the presence of null markers should respect Armstrong’s axioms (G2), as
well as enforce realizable null markers.

Of course, our goals so far can be accomplished by being very restrictive on
the use of null markers. One might even take the stance that null markers
should always be forbidden [12]. But we also want to allow common uses seen in
production-quality applications. For example, we have observed that many
database schemas allow null markers on attributes that do not determine other
attributes. Thus, it is another objective (G3) of this research to allow null
markers when this can be done without violating other goals. At a minimum, we
should allow null markers without having to come up with contrived examples.

Finally, any enforcement of FDs is going to be considered, from the point of
view of transaction or query processing, as overhead—just like enforcing
primary and foreign key constraints. Thus, any definition should be
computationally inexpensive (G4). In practice, this means that we exclude
elegant but challenging models such as v-tables [13]. For example, we should
be able to determine whether the FD $X\to Y$ holds by considering _only_ the
attributes in $X$ and $Y$.

To summarize, we seek to extend FDs to include null markers in such a way
that:

G1: FDs enforce realizable null markers. Further, this should hold for sets of
FDs (strong G1).

G2: Armstrong’s axioms are satisfied.

G3: FDs should not restrict the use of null markers unnecessarily.

G4: Enforcing FDs should be computationally practical.

##  3 Basic concepts

Let a _relation_ $r$ be as in SQL: a finite multiset of tuples over a given
schema $\mathop{\mathrm{sch}}(r)$, with the caveat that a tuple may contain
null markers. Two tuples are considered _duplicates_ if all non-null
attributes are equal and any null marker in one tuple is matched by a null
marker in the other tuple; otherwise the tuples are _distinct_.

We assume that there is an infinite set $V$ of values, from where all the
constants in any relation are drawn. These values have a relation ’=’ defined
on them, which is reflexive, symmetric and transitive: given any $x,y,z\in V$,
we have than $x=x$, $(x=y)\Rightarrow(y=x)$ and $(x=y,y=z)\Rightarrow x=z$.
Hence, the relation “=” is an equivalence relation.

Given an attribute $A$ in the schema of relation $r$ and a tuple $t$, we use
$t[A]$ as is customary, to denote the value of $t$ for $A$. This is extended
to sets of attributes $X\subseteq\mathop{\mathrm{sch}}(r)$ as usual. We then
say, for two tuples $t$, $t^{\prime}$, that $t[A]=t^{\prime}[A]$ is _true_ if
both $t[A]$ and $t^{\prime}[A]$ are equal non-null values; _false_ if both
$t[A]$ and $t^{\prime}[A]$ are values, but they are different; and _unknown_
otherwise (i.e., if either one of $t[A]$ or $t^{\prime}[A]$, possibly both,
are null markers). Again, this is extended to sets of attributes
$X\subseteq\mathop{\mathrm{sch}}(r)$ in the usual way: for two tuples $t$,
$t^{\prime}$, $t[X]=t^{\prime}[X]$ is _true_ if $t[A]=t^{\prime}[A]$ is true
for every $A\in X$; _false_ if $t[A]=t^{\prime}[A]$ is false for some $A\in
X$, and _unknown_ otherwise. As a shorthand, we write $t=t^{\prime}$ for
$t[\mathop{\mathrm{sch}}(r)]=t^{\prime}[\mathop{\mathrm{sch}}(r)]$. We write
$\pi_{X}(r)$ for the projection of all tuples in $r$ on $X$: starting from
$\\{t[X]|t\in r\\}$, all duplicates are removed.

Let $r$ be a fixed relation. If we disallow null markers in $r$, then a FD
$X\to Y$ is satisfied if $t[Y]=t^{\prime}[Y]$ when two tuples $t,t^{\prime}$
are such that $t[X]=t^{\prime}[X]$. In this context (where null markers are
forbidden), FDs satisfy Armstrong’s axioms. We can also formalize the concept
of key if there is no null marker in $r$. A set of attributes $K$ is a
_superkey_ iff $K\to A$ holds for any attribute
$A\in\mathop{\mathrm{sch}}(r)$; and a _key_ if it is a minimal superkey.
_Primary keys_ in SQL are keys with attributes where null markers are
forbidden; SQL allows null markers in other keys.

We say that an attribute $A$ is _non- null_ in a relation $r$ if for all $t\in
r$ we have that $t[A]$ is non-null. Given a set of attributes $X$, we say that
the null marker appearing in a tuple $t$ at attribute $A$
($t[A]=\texttt{null}{}$) is _in $X$_ if $A\in X$.

##  4 Strong and weak functional dependencies

Levene and Loizou [14] propose one of the few extensions of FDs over null
markers. We formalize their definitions as follows. A _valuation_ $\varphi$
for a relation $r$ assigns to each null marker in a tuple of $r$ a value from
$V$—each null marker may receive a different value—while leaving non-null
values from $V$ unchanged. Given a relation $r$, each $\varphi(r)$ is called a
_possible world_ for $r$. The semantics of FDs with null markers, as defined
by Levene and Loizou, follows the idea of modal logic [15]: we have two
distinct readings, depending on whether the FD holds in some or all possible
worlds.

###### Definition .

A FD $F$ holds weakly in relation $r$ iff $F$ holds in a possible world for
$r$—i.e., there exists a valuation $\varphi$ such that $F$ holds in
$\varphi(r)$. $F$ is called a _weak FD_ (WFD).

###### Definition .

A FD $F$ holds strongly in relation $r$ iff $F$ holds in all possible worlds
for $r$—i.e., for every valuation $\varphi$ for $r$, $F$ holds in
$\varphi(r)$. $F$ is called a _strong FD_ (SFD).

Looking back at Table 1, consider the following FDs:
$\textrm{chair}\to\textrm{professor}$ and
$\textrm{professor}\to\textrm{chair}$. Both hold weakly, while neither holds
strongly (see Table 3).

Table 3: Various FDs applied to the relation of Table 1 and whether they hold strongly, weakly, super-reflexively or literally |  SFD |  WFD |  SRFD |  LFD
---|---|---|---|---
$\textrm{chair}\to\textrm{professor}$ |  no |  yes |  no |  yes
$\textrm{professor}\to\textrm{chair}$ |  no |  yes |  yes |  no

A strong FD is always also a weak FD. Strong FDs satisfy Armstrong’s axioms
while weak FDs do not. When a FD holds strongly, we can replace any null
marker by a value, and the FD still holds. In fact, strong FDs enforce
realizable null markers.

The Levene and Loizou model has a substantial formal appeal, but it also has
some drawbacks from a pragmatic point of view:

  * •

Weak FDs might be too weak. Even if each FD in a set $\mathcal{F}$ of FDs
holds weakly, there might be no single possible world $\varphi(r)$ where all
of the FDs in the set $\mathcal{F}$ hold. See Table 1 for a counterexample:
both FDs ($\textrm{professor}\to\textrm{chair}$ and
$\textrm{chair}\to\textrm{professor}$) hold weakly, but there is no possible
world where they both hold. That is, we can substitute some value for the null
marker to satisfy the first FD, and substitute _another_ value to satisfy the
second FD, but no single value makes both FDs true at the same time (thus,
$\mathcal{F}$ does not have property strong G1). This is true even when each
attribute appears only once on the right-hand-side of a FD: given the schema
$A,B,C$ and the FDs $A\to B$ and $B\to C$, the set of tuples
$(a,\texttt{null}{},b)$ and $(a,\texttt{null}{},c)$ satisfy both FD weakly,
but there is no world where they both hold. This last example also illustrates
that weak FDs are not transitive: $A\to B$ and $B\to C$ can hold weakly while
$A\to C$ may not. That is, weak FDs do not satisfy Armstrong’s axioms (thus
failing G2).

We could _fix_ weak FDs to ensure that they enforce, for example, realizable
null markers by requiring that there exists a valuation corresponding to the
set of all FDs. However, this may prove computationally challenging (hence
failing G4).

  * •

Strong FDs might be too strong (thus failing G3). Indeed, it seems unnecessary
to always require that FDs should hold in all possible worlds. For example,
consider the schema $A,B,C$ and the FDs $A\to B$ and $B\to C$, and the set of
tuples $(a,b,\texttt{null}{})$ and $(c,b,\texttt{null}{})$. Though it appears
like a reasonable relation, the FD $B\to C$ fails to hold strongly.

In some sense, the weak and strong FDs are two extremes, whereas the right
solution might be somewhere in between. Indeed, any form of FD that supports
realizable null markers (G1) on a per FD basis, is going to be equivalent to
or stronger than weak FDs. Meanwhile, strong FDs have the properties we seek,
except that they are too restrictive (G3).

##  5 Literal and Super-reflexive functional dependencies

We propose two alternative definitions of the concept of FD in the presence of
null markers. The first one is a natural extension of the 3-value logic
proposed by Codd and used by SQL.

###### Definition .

A FD $X\to Y$ holds super-reflexively if, for any two tuples $t,t^{\prime}$,
when $t[X]=t^{\prime}[X]$ is not false (i.e., it is either true or unknown),
then $t[Y]=t^{\prime}[Y]$ is also not false. We say $X\to Y$ is a _super-
reflexive_ FD (SRFD).

In this first definition, the null marker is effectively equal to any other
value (i.e., $\texttt{null}{}=a$ is treated as true for any value $a$), hence
the term _super-reflexive_ (SR).

As an illustration, consider Table 1. We have that
$\textrm{professor}\to\textrm{chair}$ hold super-reflexively whereas
$\textrm{chair}\to\textrm{professor}$ does not (see Table 3).

Our second definition is reminiscent of how languages such as JavaScript
handle null markers. As a first approximation, they consider null to be
effectively a regular value, with $\texttt{null}{}=\texttt{null}{}$ is always
true (i.e., “=” remains a reflexive relation111In fact, ’=’ remains an
equivalence relation, something that does not hold for Codd’s 3-value logic if
you consider null markers to be part of the value domain.), but
$\texttt{null}{}=a$ always false for $a$ non-null.

We say that $t[A]$ and $t^{\prime}[A]$ are _identical_ if both contain null or
both contain the same value; this is also extended to set of attributes $X$
and to whole tuples as usual: $t[X]$ is identical to $t^{\prime}[X]$ if and
only if $t[A]$ and $t^{\prime}[A]$ are identical for all $A\in X$.

###### Definition .

A FD $X\to Y$ holds literally if, for any two tuples $t,t^{\prime}$, when
$t[X]$ and $t^{\prime}[X]$ are identical then $t[Y]$ and $t^{\prime}[Y]$ are
also identical. We say $X\to Y$ is a _literal_ FD (LFD).

Consider again Table 1. In contrast with the super-reflexive case, we have
that the FDs $\textrm{chair}\to\textrm{professor}$ holds literally whereas
$\textrm{professor}\to\textrm{chair}$ does not. (See again Table 3.)

There are alternative definitions that we could have used. For example, we
could have defined FDs $X\to Y$ to hold if when $t[X]=t^{\prime}[X]$ is true
(as per Codd’s 3-value logic) then $t[Y]=t^{\prime}[Y]$ must be true. Or, we
could have defined an FD $X\to Y$ to hold if when $t[X]=t^{\prime}[X]$ is not
false then $t[Y]=t^{\prime}[Y]$ must be true; or if when $t[X]=t^{\prime}[X]$
is true then $t[Y]=t^{\prime}[Y]$ must be not false. However, these
alternative definitions are unsatisfying: they fail to satisfy Armstrong’s
axioms (G2), or do not allow null markers where they are commonly used (G4).
By contrast, Definitions 5.1 and 5.2 have the properties that we required in §
2 starting with Armstrong’s axioms (which follows by inspection). For example,
regarding transitivity, consider the two FDs $X\to Y$ and $Y\to Z$ that hold
super-reflexively (resp. literally). Given two tuples $t,t^{\prime}$ such that
$t[X]=t^{\prime}[X]$ is not false (resp. $t[X]$ and $t^{\prime}[X]$ are
identical), then $t[Y]=t^{\prime}[Y]$ is not false (resp. $t[Y]$ and
$t^{\prime}[Y]$ are identical) which implies that $t[Z]=t^{\prime}[Z]$ is not
false (resp. $t[Z]$ and $t^{\prime}[Z]$ are identical). Thus we have that
$X\to Z$, proving transitivity.

###### Lemma .

Super-reflexive and literal FDs respect Armstrong’s axioms (G2).

Before we proceed to establish other properties, we need a technical result
that makes other proofs easier. First, we can verify that all FDs can be
decomposed into FDs where the right-hand-side contains a single attribute. For
example, the FD
$\textrm{professor}\to\\{\textrm{chair},\textrm{department}\\}$ is equivalent
to the following two FDs:

  * •

$\textrm{professor}\to\textrm{chair}$, and

  * •

$\textrm{professor}\to\textrm{department}$.

###### Lemma .

We have that $X\to Y$ for $Y=\\{A_{1},\ldots A_{n}\\}$ is equivalent to
$(X-\\{A_{1}\\}\to\\{A_{1}\\})\land\cdots\land(X-\\{A_{n}\\}\to\\{A_{n}\\})$
whether we consider weak, strong, literal or super-reflexive FDs.

The proof of this lemma follows by inspection. Hence, it is enough to consider
FDs $X\to Y$ where $Y$ is a singleton disjoint from $X$.

First, we must show that SRFDs and LFDs satisfy condition G1: null markers are
realizable. We begin with SRFDs. We show that given the SRFD $X\to Y$, all
null markers in $X\cup Y$ are realizable with respect to $X\to Y$. To
illustrate this result, consider Table 1 where $\textrm{professor}
\to\textrm{chair}$ holds super-reflexively: we can substitute Jill for the
null marker without violating the FD.

###### Lemma .

Consider a SRFD $X\to Y$ in a relation such that $X$ and $Y$ are disjoint and
$Y$ is a singleton.

  1. 1.

We can replace any null marker in $X$ by any actual value without violating
the SRFD $X\to Y$.

  2. 2.

We can replace any null marker in $r$ in $Y$ by at least one actual value
without violating the SRFD $X\to Y$, assuming that attributes in $X$ are non-
null.

###### Proof .

Assume that the SRFD is initially satisfied over $r$.

(1) Suppose we replace a null marker in attribute $B\in X$. Regarding the SRFD
$X\to Y$, the following might happen for two tuples $t,t^{\prime}$:

  * •

if $t[X]=t^{\prime}[X]$ was false, then it is still false: this cannot affect
the SRFD;

  * •

if $t[X]=t^{\prime}[X]$ was true, then it is still true: this cannot affect
the SRFD;

  * •

if $t[X]=t^{\prime}[X]$ was unknown, then it might become true or false.
Because of the SRFD $X\to Y$, and because $t[X]=t^{\prime}[X]$ was not false,
we have that $t[Y]=t^{\prime}[Y]$ is not false. Therefore, if the tuples
$t,t^{\prime}$ satisfied the SRFD before the update, they must satisfy it
after the update as well.

Because all pairs of tuples satisfy the conditions of SRFD after the update,
the SRFD still hold, proving the first part of the result.

(2) Assume that attributes in $X$ are non-null. We want to show that we can
replace any null marker in $Y$ by an actual value. Pick a tuple $t$ in $r$
where $t[Y]$ contains a null marker. Consider the set $\tau$ of $t^{\prime}$
such that $t^{\prime}[X]=t[X]$ is not false. (Because attributes in $X$ are
non-null, we have that “$t^{\prime}[X]=t[X]$ is not false” is equivalent to
“$t^{\prime}[X]=t[X]$ is true”.) We have that the projection of $\tau$ over
$Y$ contains at most one actual value. (Suppose it does not, then you can find
tuples $t^{\prime\prime}$ and $t^{\prime\prime\prime}$ in $\tau$ such that
$t^{\prime\prime}[X]=t^{\prime\prime\prime}[X]$ is not false but
$t^{\prime\prime}[Y]=t^{\prime\prime\prime}[Y]$ is false.) If there is one
actual value, set $t[Y]$ to this value; if not, pick a value at random. This
modification clearly does not violate the SRFD $X\to Y$ but it eliminates one
null marker.

To see why Lemma 5.5 implies that SRFDs satisfy G1, consider any SRFD $X\to Y$
over a given relation. We can substitute actual values for any null marker in
an attribute of $X$ by the first part of the lemma. As a second step, since
attributes in $X$ have become non-null, we can substitute actual values for
any null marker in $Y$.

As for LFDs, we are going to prove something stronger: that they support
strong G1.

###### Lemma .

LFDs strongly enforces realizable null markers (strong G1).

###### Proof .

To prove G1, it suffices to replace all the null markers with a single $v\in
V$ not already in the relation. By inspection, this extends to sets of FDs, so
we get also strong G1 with this method.

We have that both LFDs and SRFDs enforce realizable null markers. That is,
given a relation with a set of FDs, we can always replace null markers with
some actual values without violating the FDs. In fact, if we add an extra
constraint on SRFDs, they both _strongly_ enforce realizable null markers (in
the sense of strong G1). In this context, we adopt the practical convention
that some attributes are allowed to contain null markers while others may not:
this is motivated by the SQL standard. Given a set of FDs $\mathcal{F}$, we
say that an attribute $B$ _determines_ another attribute $A$ under
$\mathcal{F}$ if there is a FD $B\in X\to Y\ni A$ in the transitive closure of
$\mathcal{F}$. Naturally, this property is transitive: if $A$ determines $B$
and $B$ determines $C$ then $A$ determines $C$. (By convention, we omit loops
in $\mathcal{F}$: $X\to X$.)

###### Condition 1RHS.

Consider a set of FDs $\mathcal{F}$ over a relation. Consider any attribute
$A$ allowed to contain null markers. Then $A$ must appear on the right-hand-
side of at most one FD in the set of FDs $\mathcal{F}$ of the relation.
Moreover, given two distinct attributes allowed to contain null marker, $A$
and $B$, if $A$ determines $B$, $B$ cannot determine $A$.

We stress that Condition 1RHS only applies to attributes allowed to contain
null markers: no constraint is required on other attributes.

Fig. 1 gives an example of a set of FDs satisfying the condition 1RHS even if
all attributes are allowed to contain null markers:
$\\{E,D\\}\to\\{A\\},\\{A\\}\to\\{F\\},\\{A,B\\}\to\\{F\\}$. However, if we
replaced the single FD $\\{E,D\\}\to\\{A\\}$ by two FDs such as
$\\{E\\}\to\\{A\\}$ and $\\{D\\}\to\\{A\\}$, we would need to add the
requirement that $A$ is non-null to satisfy 1RHS. Similarly, if we added the
FD $\\{F\\}\to\\{A\\}$ in addition to the existing FD $\\{A\\}\to\\{F\\}$, we
would need to require that both $A$ and $F$ are non-null since $F$ would
determine $A$ while $A$ determines $F$.

###### Lemma .

SRFDs strongly enforces realizable null markers whenever the 1RHS condition is
satisfied (strong G1).

###### Proof .

Assume without loss of generality that all SRFDs in the set are of the form
$X\to Y$ where $Y$ is a singleton and $X,Y$ are disjoint.

$A$$B$$C$$D$$E$$F$$\\{A\\}\to\\{F\\}$$\\{A,B\\}\to\\{F\\}$$\\{E,D\\}\to\\{A\\}$$\\{E,D\\}\to\\{A\\}$$\\{A\\}\to\\{F\\}$
Figure 1: Illustration used by the proof of Lemma 5.9 for the set of FDs
$\\{\\{E,D\\}\to\\{A\\},\\{A\\}\to\\{F\\},\\{A,B\\}\to\\{F\\}\\}$.

Construct a graph where each attribute allowed to contain null markers is a
node, and there is an edge between two attributes $A,B$ if and only if $A$
determines $B$. We illustrate such a graph in Fig. 1 where, for simplicity, we
omitted some of the edges that are implied by transitivity.

Temporarily assume that the graph is not empty. Because of condition 1RHS, the
graph must be cycle-free and, therefore, some of the nodes must have a zero
in-degree (e.g., $E$ and $D$ in Fig. 1). Call this set of nodes/attributes
$\mathcal{A}_{0}$.

As per Lemma 5.5, we can substitute actual values for any null marker they
contain. Indeed, consider such an attribute $A\in\mathcal{A}_{0}$. This
attribute appears on the right-hand-side of at most one FD in the set (call it
$F_{A}$), however $A$ may appear on the left-hand-side of several FDs. These
FDs are not a concern: replacing a null marker in attribute $A$ may never
violate a super-reflexive FD as per the first part of Lemma 5.5. Meanwhile,
because $F_{A}$ is such that no attribute on its left-hand-side contains a
null marker, then the second part of Lemma 5.5 tells us that the null markers
of $A$ are realizable.

After substituting actual values for any null marker in the attributes of
$\mathcal{A}_{0}$, remove these nodes from the graph. There must again be
nodes with zero in-degree (e.g., node $A$ in Fig. 1) or the graph is empty.
Repeat the process until the graph is empty.

Attributes that either do not appear as part of any FD, or that are not
allowed to contain null markers, are not a concern.

##  6 Comparing functional dependencies

We are now in a position to characterize LFDs and SRFDs properly. We can
relate LFDs and SRFDs to each other, and with Levene and Loizou’s definitions.
All are _conservative_ extensions of the classical concept: in a table without
any null, they coincide with classical FDs. However, in general, LFDs and
SRFDs are _incomparable_ as illustrated by Table 3.

We show that $\mbox{strong}\Rightarrow\mbox{super-
reflexive}\Rightarrow\mbox{weak}$ and
$\mbox{strong}\Rightarrow\mbox{literal}\Rightarrow\mbox{weak}$ (see Fig. 2).
That is, both LFDs and SRFDs are stronger than weak FDs, whereas strong FDs
are stronger than both LFDs and SRFDs.

Figure 2: Venn diagram illustrating Lemma 6.1.

###### Lemma .

The following holds:

  1. 1.

If the FD $X\to Y$ holds literally, then it holds weakly.

  2. 2.

If the FD $X\to Y$ holds super-reflexively, then it holds weakly.

  3. 3.

If the FD $X\to Y$ holds strongly then it must hold literally and super-
reflexively.

###### Proof .

Assume without loss of generality that $X$ and $Y$ are disjoint and that $Y$
is a singleton.

(1) If the FD $X\to Y$ holds literally then we can replace any null marker by
any one $v\in V$ not already present in the relation, and the FD still holds
by inspection. This shows that literal FDs are stronger than weak FDs.

(2) By Lemma 5.5, we can construct a valuation such that the super-reflexive
FD is valid in the conventional sense. Because of the existence of the
valuation, we have that $X\to Y$ holds weakly.

(3) We first prove that strong implies super-reflexive. Suppose that $X\to Y$
holds strongly. Consider two tuples $t,t^{\prime}$. If $t[X]=t^{\prime}[X]$ is
not false (in Codd’s 3-value logic), then in some possible world,
$t[X]=t^{\prime}[X]$ must hold. In such a possible world $t[Y]=t^{\prime}[Y]$
must hold which implies that $t[Y]=t^{\prime}[Y]$ must be not false (in Codd’s
3-value logic). This prove that a strong FD implies a super-reflexive FD.

We prove that strong implies literal. Suppose that the FD $X\to Y$ holds
strongly and that $t[X]$ and $t^{\prime}[X]$ are identical. (We assume that
$X$ and $Y$ are disjoint and $Y$ is a singleton as previously stated.) There
are some worlds where $t[X]=t^{\prime}[X]$ is true. If $t[Y]$ and
$t^{\prime}[Y]$ are not identical, then in at least one of these worlds, they
would differ. That is, if one has a null marker in $t[Y]$ and a value $a$ in
$t^{\prime}[Y]$ (or vice versa) we can replace the null marker by a value that
differs from $a$. Hence, we must have that $t[Y]$ and $t^{\prime}[Y]$ are
identical given that $t[X]$ and $t^{\prime}[X]$ are identical. Therefore
strong FDs imply literal FDs. This completes the proof.

We can verify that LFDs and SRFDs are strictly weaker than SFD. Recall that
given the schema $A,B,C$ and the FDs $A\to B$ and $B\to C$, the set of tuples
$(a,b,\texttt{null}{})$ and $(c,b,\texttt{null}{})$ violates the strong FD
$B\to C$. However, it satisfies the FD $A\to B$ and $B\to C$ literally and
super-reflexively. The fact that LFDs and SRFDs allow this use of null markers
support our claim that they do not unnecessarily forbid null markers where
they might make sense (G3).

##  7 Computational efficiency

The remaining question is the efficient implementation of our proposed
concepts (G4). Here we show that both LFDs and SRFDs can be enforced with a
cost similar to enforcing _regular_ FDs.

Since SQL does not enforce FDs directly, we compare with a closely associated
property, the cost of enforcing the key constraint in a relation. This assumes
that the database is in an appropriate normal form such that enforcing keys is
equivalent to enforcing FDs. Given relation $r$, if $X\subseteq r$ is declared
as the primary key for $r$, any insertion $t\in r$ is checked to make sure
that $X$ remains a key. In practice, relational database systems forbid the
occurrence of two distinct tuples $t,t^{\prime}$ agreeing on $X$
($t[X]=t^{\prime}[X]$ must be false). This can be achieved by creating an
index (traditionally, a tree-based index) on $X$: on inserting $t$, we search
for any $t^{\prime}\in r$ with $t[X]=t^{\prime}[X]$ (note that null markers
are forbidden in a primary key). Call the set of tuples resulting from the
search $S$. When inserting, if $S=\emptyset$, the insertion can proceed; else,
for each $t^{\prime}\in S$ we check whether $t^{\prime}$ is identical to $t$.
If this is so, the insertion can proceed; else it is forbidden. Updates are
handled in a similar manner. The complexity of this procedure is considered
$\mathcal{O}(\log\mid r\mid)$ when using a tree-based index, since it is
expected that $\mid S\mid\leq 1$. (Of course, we can get an expected constant
time complexity by using a hash-based index.)

We can also enforce FDs directly, whether they are literal or super-reflexive.
Given relation $r$ and arbitrary FD $X\to Y$ on it, we create an index on
$X$—for concreteness, we assume a $B^{*}$-tree index since these are available
on almost any database system.

To enforce FDs with an index, it suffices to check whether the insertion of a
new tuple $t$ is allowed. Indeed, an update can be viewed as a deletion
followed by an insertion, and deletions cannot violate a FD.

Given an index, enforcing LFDs is not difficult. We build a single index on
$X$ by considering a concatenation of all attributes—lexicographic order is
followed, with null markers in individual attributes kept together at the
beginning or end of their positions. That is, in an index for attributes
$ABC$, tuple $(a_{1},b_{1},\texttt{null}{})$ would go before or after all
tuples $(a_{1},b_{1},c)$, for $c$ any value of $C$; tuple
$(a_{1},\texttt{null}{},b_{1})$ would go before or after any tuples
$(a_{1},c,b_{1})$ for $c$ any value of $C$; and so on. This can be achieved by
considering null markers as either strictly larger than any other value, or
strictly smaller. Suppose that we insert a new tuple $t$. We can find in time
$\mathcal{O}(\mid S\mid+\log\mid r\mid)$ the set $S$ of all tuples
$t^{\prime}$ such that $t[X]$ and $t^{\prime}[X]$ are identical:

$\displaystyle S=\bigcap_{A\in X}\\{t^{\prime}\in r|t^{\prime}[A]=t[A]
\enskip\lor\enskip\text{$t^{\prime}[A]$ and $t[A]$ are {null}{}}\\}$

We can then check whether $t[Y]$ and $t^{\prime}[Y]$ are identical for all
$t^{\prime}$ in linear time $\mathcal{O}(\mid S\mid)$. Note that we consider
the cardinality of the set $X$ ($\mid X\mid$) to be a small constant.

We can also enforce $X\to Y$ as an SRFD efficiently, though the total cost is
probably larger. Index each one of the attribute $A\in X$ using, as before,
the convention that y considering null markers as either strictly larger than
any other value, or strictly smaller. Given $t$, first we check whether $t[Y]$
contains a null marker (assume $Y$ is a singleton disjoint from $X$), in which
case no work is needed. Otherwise, we find all $t^{\prime}$ such that “$t[X]
=t^{\prime}[X]$ is not false” by computing

$\displaystyle S=\bigcap_{A\in X|t[A]\text{~{}not {null}{}}}\\{t^{\prime}\in
r|t^{\prime}[A]=t[A] \lor t^{\prime}[A]\text{~{}is {null}{}}\\}$

with the convention that the result is $r$ if $t[X]$ only contains null
markers. Computing the intersection $S$ between several sets
$S_{1},S_{2},\ldots,S_{\mid X\mid}$ requires only complexity
$\mathcal{O}(\sum_{i=1}^{\mid X\mid}|S_{i}|+\mid S\mid)$ or better [16]. Each
set $S_{i}$ can be generated in time $\mathcal{O}(\mid S_{i}\mid+\log\mid
r\mid)$ for an overall complexity of $\mathcal{O}(\sum_{i=1}^{\mid
X\mid}|S_{i}|+\mid S\mid+\log\mid r\mid)$. We then consider $t^{\prime}[Y]$
for all $t^{\prime}\in S$: if there is an actual value, check that $t[Y]$ is
equal to $t^{\prime}[Y]$. That is, we return

$\displaystyle\bigwedge_{t^{\prime}\in S}t^{\prime}[Y]\
=t[Y]\enskip\lor\enskip t^{\prime}[Y]\text{~{}is {null}{}}.$

This can be computed in time $\mathcal{O}(\mid S\mid)$.

To sum up, enforcing a LFD or SRFD under an update or insertion can be done in
time

  * •

$\mathcal{O}(\mid S\mid+\log\mid r\mid)$ or

  * •

$\mathcal{O}(\sum_{i=1}^{\mid X\mid}|S_{i}|+\mid S\mid+\log\mid r\mid)$.

Our sketched implementations only require tree look-ups and set intersections,
both of which are well supported by all database systems. This is sufficient
to conclude that they are computationally practical (G4).

##  8 Extending logical database design to include null markers

As we discussed in § 5, it is possible to enforce FDs with null markers using
either SRFDs or LFDs—without any particular effort on the part of the database
designer. However, we commonly enforce FDs using logical database design. That
is, we decompose relations into normal forms and identify keys.

We would like to remain as close as possible to the spirit of SQL. Thus, we
ask whether we can use logical design with SRFDs and LFDs. This would allow an
extension of logical database design to include null markers. Unfortunately,
while both LFDs and SRFDs fulfill all our desiderata (G1 to G4), SRFDs are not
compatible with conventional logical design. But we have better luck with
LFDs.

Recall that in a relation $r_{1}$, a set of attributes
$X\subseteq\mathop{\mathrm{sch}}(r_{1})$ is a _key_ if the (standard) FD
$X\to\mathop{\mathrm{sch}}(r_{1})$ holds and if $X$ is minimal. Moreover, in
relation $r_{2}$, a set of attributes $Y\subseteq\mathop{\mathrm{sch}}(r_{2})$
is a _foreign key_ for $r_{1}$ if $\pi_{Y}(r_{2})\subseteq\pi_{X}(r_{1})$ for
all extensions of $r_{1}$ and $r_{2}$.

A join $r=r_{1}\Join_{X=Y}r_{2}$ is a new relation made by combining all
tuples $t_{1}\in r_{1}$ and all tuples $t_{2}\in r_{2}$ such that
$t_{1}[X]=t_{2}[Y]$ into a new tuple $t$ equal to $t_{1}$ on
$\mathop{\mathrm{sch}}(r_{1})$ and equal to $t_{2}$ on
$\mathop{\mathrm{sch}}(r_{2})-Y$. A join $r=r_{1}\Join_{X=Y}r_{2}$ is lossless
when $\pi_{\mathop{\mathrm{sch}}(r_{1})}(r)=r_{1}$ and
$\pi_{\mathop{\mathrm{sch}}(r_{2})}(r)=r_{2}$.

We extend the concepts of keys and joins in the context of LFDs as follows.

  * •

We say that $X\subseteq\mathop{\mathrm{sch}}(r)$ is a _literal superkey_ for
$r$ if $X\to Y$ holds literally for any $Y\subseteq\mathop{\mathrm{sch}}(r)$
and a _literal key_ iff it is a minimal literal superkey. A _literal foreign
key_ is an integrity constraint between two relations: a set of attributes $X$
in relation $r_{1}$ must match a set of attributes $X$ in a relation $r_{2}$
such that for every tuple $t$ in $r_{1}$, there must be a tuple $t^{\prime}$
in $r_{2}$ such that $t[X]$ and $t^{\prime}[X]$ are identical and such that
$X$ contains a literal key in $r_{2}$.

  * •

As in SQL, each literal foreign key constraint supports a corresponding join.
The literal join of $r_{1}$ and $r_{2}$ on $X$, a foreign key in $r_{1}$,
noted $\widehat{\Join}{}$ is defined as follows: given any two tuples
$t_{1},t_{2}$ from $r_{1},r_{2}$ such that $t_{1}[X]$ and $t_{2}[X]$ are
identical, we generate the tuple $t$ such that $t[A]=t_{1}[A]$ for all
$A\in\mathop{\mathrm{sch}}(r_{1})$ and $t[A]=t_{2}[A]$ for all
$A\in\mathop{\mathrm{sch}}(r_{2})-\mathop{\mathrm{sch}}(r1)$.

With these definitions, we have that lossless joins are supported, even with
null markers. Formally, given relation $r$ with
$\mathop{\mathrm{sch}}(r)=Z\cup W$ and such that $Z\cap W\to W$ holds
literally, $\widehat{\Join}{}$ is a _lossless join_ :
$r=\pi_{Z}(r)\widehat{\Join}{}\pi_{W}(r)$.

###### Proposition .

(Lossless join) If $\mathop{\mathrm{sch}}(r)=Z\cup W$ and $Z\cap W\to W$ holds
literally then $r=\pi_{Z}(r)\widehat{\Join}{}\pi_{W}(r)$.

###### Proof .

For the purpose of the literal join, the null marker can be treated like any
other value. That is, the set of values $V$ extended with the null marker is
effectively reflexive, symmetric and transitive. To conclude the proof, we
have to show that $r=\pi_{Z}(r)\widehat{\Join}{}\pi_{W}(r)$.

  1. 1.

Suppose that $t\in r$. There will be a tuple $t^{(Z)}$ in $\pi_{Z}(r)$ such
that $t[Z]$ and $t^{(Z)}$ are identical. Similarly, there will be a tuple
$t^{(W)}$ in $\pi_{W}(r)$ such that $t[W]$ and $t^{(W)}$ are identical. We
have that $t^{(W)}[Z\cap W]$ is identical to $t^{(Z)}[Z\cap W]$. Thus we have
that $t\in\pi_{Z}(r)\widehat{\Join}{}\pi_{W}(r)$.

  2. 2.

Suppose that $t\in\pi_{Z}(r)\widehat{\Join}{}\pi_{W}(r)$. There must be
$t^{\prime}\in r$ such that $t^{\prime}[Z]=t[Z]$. We have that $t[Z\cap W]$
and $t^{\prime}[Z\cap W]$ must be identical because $Z\cap W\subset Z$.
Because $Z\cap W\to W$, we have that $t[W]$ and $t^{\prime}[W]$ are identical.
Thus we have that $t[Z\cup W]$ and $t^{\prime}[Z\cup W]$ which shows that
$t\in r$.

This concludes the proof.

In fact, we can show that logical design is sound over LFDs: as long as we can
normalize a relation in the conventional sense, then we can normalize it in
the sense of LFDs.

##  9 Conclusion and Further Research

We have reviewed the concept of FD in the presence of null markers, and have
specified a set of properties that we believe any definition of the concept
should satisfy. We have proposed two new definitions of what it means for an
FD to hold in this situation for which the properties do hold: our definitions
satisfy Armstrong’s axioms for relations with null markers, allow null markers
to be used in practice (not only with contrived examples), and at the same
time allows those null markers to be updated to real values consistently.
These FDs can also be enforced efficiently in computational terms despite null
markers. Both definitions have slightly different properties (LFDs enforce
lossless join, in addition to database consistency), but they both satisfy our
axioms.

Clearly, our requirements are tied to our goal of obtaining a definition that
can be used in practical situations. An open question is whether the
properties put forth here as desirable are the only ones—or at least the
_important_ ones, in some sense of _important_. We believe that our properties
satisfy intuitions that make the concept of FD usable in practical situations.
However, additional properties should be explored to get a better
understanding of the _desired_ or _expected_ behavior of FDs in the presence
of null markers; perhaps a set of alternative properties, each giving raise to
a concept of FD, can be developed for different scenarios. Agreeing on some
set or sets of basic requirements would provide researchers with an explicit
milestone by which to judge different formalizations.

A narrower question is whether the definitions proposed here are the only ones
that can satisfy all the requirements put forth, or whether alternatives
exist. While the authors have considered many alternative definitions, and
found them missing some requirement, it is not known whether other definitions
could exist that would still satisfy all properties, and if so, what would the
relationships between different definitions be.

##  10 Funding

This work was supported by the Natural Sciences and Engineering Research
Council of Canada [26143].

##  11 Acknowledgements

We thank Carles Farré and Andre Vellino for their helpful comments.

## References

  * [1] Codd, E. F. (1986) Missing information (applicable and inapplicable) in relational databases.  SIGMOD Rec., 15, 53–53.
  * [2] Zaniolo, C. (1984) Database relations with null values.  J. Comput. Syst. Sci., 28, 142–166.
  * [3] Atzeni, P. and Morfuni, N. M. (1984) Functional dependencies in relations with null values.  Inf. Process. Lett., 18, 233–238.
  * [4] Hartmann, S. and Link, S. (2012) The implication problem of data dependencies over SQL table definitions: Axiomatic, algorithmic and logical characterizations.  ACM Trans. Database Syst., 37, 13:1–13:40.
  * [5] Libkin, L. (1994) Aspects of partial information in databases.  PhD thesis University of Pennsylvania.
  * [6] Badia, A. and Lemire, D. (2011) A call to arms: revisiting database design.  SIGMOD Rec., 40, 61–69.
  * [7] ISO 9075-1:2008 (2008) Information technology: database languages – SQL– Part 1 Framework, 3rd edition. ISO, Geneva.
  * [8] Hartmann, S., Leck, U., and Link, S. (2011) On Codd families of keys over incomplete relations.  Comput. J., 54, 1166–1180.
  * [9] Bernstein, P. (1976) Synthesizing third normal form relations from functional dependencies.  ACM Trans. Database Syst., 1, 277–298.
  * [10] Chen, P. P.-S. (1976) The entity-relationship model—toward a unified view of data.  ACM Trans. Database Syst., 1, 9–36.
  * [11] Le, V., Link, S., and Ferrarotti, F. (2013) Effective recognition and visualization of semantic requirements by perfect sql samples.  Conceptual Modeling, Lecture Notes in Computer Science, 8217, pp. 227–240. Springer, Berlin Heidelberg.
  * [12] Date, C. J. (2009) SQL and Relational Theory: How to Write Accurate SQL Code. O’Reilly Media, Inc., Sebastopol, California.
  * [13] Imieliński, T. and Lipski, W., Jr. (1984) Incomplete information in relational databases.  J. ACM, 31, 761–791.
  * [14] Levene, M. and Loizou, G. (1998) Axiomatisation of functional dependencies in incomplete relations.  Theor. Comput. Sci., 206, 283–300.
  * [15] Chellas, B. F. (1980) Modal logic: an introduction. Cambridge Univ Press, Cambridge.
  * [16] Ding, B. and König, A. C. (2011) Fast set intersection in memory.  Proc. VLDB Endow., 4, 255–266.


#  Stability analysis of a dark energy model in Rastall gravity

Shaily<EMAIL_ADDRESS>Department of Mathematics, Netaji Subhas
University of Technology, New Delhi-110 078, India  School of Computer Science
Engineering and Technology, Bennett University, Greater Noida, India  Akanksha
Singh<EMAIL_ADDRESS>Department of Mathematics, Netaji Subhas
University of Technology, New Delhi-110 078, India  J. K. Singh
<EMAIL_ADDRESS>Department of Mathematics, Netaji Subhas University of
Technology, New Delhi-110 078, India  Saddam Hussain<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology, Kanpur, Uttar Pradesh
208016, India

###### Abstract

We study a cosmological model in Rastall’s theory of gravity in the framework
of flat FLRW metric. We formulate the value of the Hubble parameter which
contains two model parameters $\alpha$ and $j$. Employing the Markov Chain
Monte Carlo (MCMC) sampling technique, we constrain the magnitude of the model
parameters with their uncertainties. Moreover, we reconstruct the effective
equation-of-state (EoS) parameter, which converges around the quintessence
region. We perform a dynamical system analysis using the linearization
technique to validate the results independently. Also, we discuss various
physical properties of the model which shows the transition to acceleration
and the violation of the Strong energy condition (SEC) in late times
evolution. Finally, we conclude that our model mimics the behavior of a dark
matter fluid during the past epoch and transitions into a quintessence dark
energy model in later times.

PACS numbers: 98.80 Cq.
Keywords: Rastall gravity, FLRW metric, Jerk parameter, Quintessence, Dark
energy, Cosmological parameters

##  I Introduction

The General Theory of Relativity is an astounding accomplishment, and it is
now widely considered to be one of the pillars of modern physics. The theory
itself is couched in the language of differential geometry and was a pioneer
for the use of modern mathematics in physical theories, leading the way for
the gauge theories and string theories that have followed. It is no
exaggeration to say that General Relativity set a new tone for what a physical
theory can be and has truly revolutionized our understanding of the Universe.
General Relativity has been established as the theory of gravitational
interactions for over a century, being consistent with all experiments and
being able to describe a huge set of observational results. Furthermore, there
are still some shortcomings in theoretical and observational aspects of
science, such as the accelerated universe and the existence of dark matter and
energy, on account of which a modification of general relativity is
inevitable.

There are possible modified and alternative theories of gravity that have been
constructed in mainly two ways. In the first method, the fundamental
assumptions of general relativity are still kept, but new additional terms are
added to the Lagrangian density of general relativity, on account of which all
field equations are modified with additional terms [1, 2, 3, 4, 5, 6, 7, 8].
In the second method, the fundamental assumptions of general relativity are
changed. Rastall gravity belongs to the latter group. One of the fundamental
assumptions of general relativity is the null covariant divergence of the
energy-momentum tensor (EMT). However, Rastall followed the second method and
argued that the vanishing divergence of the Einstein tensor does not
necessarily imply the vanishing divergence of the energy-momentum tensor [9].
In such a scenario, energy conservation must be sacrificed. Relaxation of the
condition of the vanishing of the divergence of the energy-momentum tensor
does not upset the success of general relativity. Rastall gravity attempted to
modify the theory of general relativity by assuming that the covariant
derivative of the energy-momentum tensor varies directly as the derivative of
the Ricci scalar [9]. It explicitly represents a non-minimal coupling between
matter and geometry in curved spacetimes.

The one notable negative feature of Rastall gravity is the absence of
Lagrangian. Recent progress in this direction indicates that a Lagrangian
formulation for a Rastall-type theory can be given as an MG proposal where the
Lagrangian depends on the Ricci scalar $R$ and the trace of the energy-
momentum tensor $T$. Also, there are some questions regarding the status of
Rastall gravity as an actual modification of general relativity and not only a
redefinition of the energy-momentum tensor [10, 11]. At the geometric level,
Rastall gravity is equivalent to Einstein gravity [11], unimodular theory as
well as $f(R,T)$ theory; however, the point that the physical consequences are
dramatically different has been amplified several times [12].

Dynamical system analysis is an important tool in the study of cosmology,
offering a method to qualitatively explore the behavior of solutions to cosmic
models without drowning in complex calculations. Often, this method is
employed to determine both the stability of the model and the existence of
critical points [13, 14]. Through examining the asymptotic behavior of
critical points of the dynamical system, one can explain the overarching
dynamics of the universe across different cosmological epochs [15, 16, 17].
Researchers like Kadam et al. have presented an exploration of critical points
representing different cosmic eras, including the matter-dominated, radiation-
dominated, and dark energy eras in $f(T,B)$ gravity [18]. Stachowski and
Szydłowski have tackled the dynamics of models by running cosmological
constant terms with the help of a dynamical analysis approach [19]. Roy and
Banerjee’s [20] extensive analysis of the stability of cosmological solutions
for a spatially flat homogeneous and isotropic cosmological model within an
extended Brans-Dicke theory through a phase space analysis serves as yet
another testament to the power of dynamical system methods. Moreover, the
versatility of dynamical systems extends to exploring interacting dark energy
scenarios, offering a unified framework for understanding the cosmos at both
background and perturbation levels [21, 22, 23, 24].

The dynamical systems method is applied to probe critical points, and these
points offer insights into the transitions between accelerating and
decelerating epochs, characterizing phases dominated by dark energy,
radiation, and matter [25]. Recent studies have promoted the discussion about
the stability of various cosmological models using dynamical system analysis.
For instance, Roy and Banerjee [26] scrutinized the stability criteria of
chameleon models within the framework of spatially homogeneous and isotropic
cosmology, employing the tools of autonomous system dynamical analysis.
Similarly, Das et al. [27] explored the stability of the warm quintessential
dark energy model. The collection of recent studies in modified gravity
theories underscores the pivotal role of dynamical system analysis in
advancing our understanding of the cosmos [28, 29, 30, 31, 32, 33, 34, 35, 36,
37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53].

Moreover, the intriguing theory of Rastall gravity has also been subjected to
dynamical system analysis [54, 55, 56, 57, 25, 58]. Shabani et al. utilize the
dynamical system approach to conduct a comprehensive exploration of the
universe’s evolution under the assumption of a particular form for the Rastall
parameter $\lambda$ [57]. In exploring the Rastall model within non-flat
Friedmann-Robertson-Walker (FRW) spacetime, incorporating a barotropic fluid,
researchers delve into an autonomous system analysis. The literary work by
Khyllep and Dutta [58] attempts to contribute to the ongoing debate
surrounding the equivalence between Rastall gravity and general relativity.
This is achieved through an in-depth analysis of the evolution of the Rastall-
based cosmological model, both at the background and perturbation levels, by
applying dynamical system techniques.

Over the last decades, Rastall gravity has gained the attention of scientists,
and as a result, a vast number of papers have been published [11].
Furthermore, it incorporates a parameter that weights its deviations from
general relativity. Over the last decade, the search for observational
constraints on Rastall’s parameter has received increased attention, as has
the study of applications to cosmology, black holes, and other compact
objects. The properties of the solutions of black holes in Rastall gravity
have been published [59, 60, 61, 62, 63, 64, 65, 66]. Rastall gravity offers
some interesting results; for instance, de Sitter black holes have been found
without explicitly assuming a cosmological constant [67]. It has been shown
that Rastall gravity does admit the accelerated expansion of the universe.
Various aspects of this theory, including theoretical and observational ones,
have been reported in recent literature [68, 56, 25, 69].

In this paper, we assemble our work in the following manner: In Sec. II,
formulation for the Hubble parameter in terms of jerk parameter takes place.
Afterward, in Sec. III, we obtained the Einstein field equations (EFEs) for
flat FLRW space-time metric in Rastall gravity theory. Observational analysis
has been done in Sec. IV, where three observational datasets are used to
obtain the best-fitted values of model parameters. Sec. V comes along with the
investigation of the physical and dynamical behavior of the universe. In Sec.
VI, we accomplish a meticulous dynamical system analysis to discover the
stability of the system depending on the various model parameters. In Sec.
VII, we conclude our findings. Finally, in the Appendix. A, we consider the
autonomous equation in 1D and determine the stability of the system around the
fixed point.

##  II  Formulation for the Hubble parameter in terms of jerk parameter

In physics, jerk known as jolt is the rate of change of an object’s
acceleration over time. It is a vector quantity. Jerk is denoted by the symbol
$j$ and expressed in $m/s^{3}$ (SI units) or $g_{0}/s$ (standard gravities per
second). As a vector, jerk $j$ can be expressed as the first-time derivative
of acceleration, second-time derivative of velocity, and third-time derivative
of position:

$j(t)=\frac{\mathrm{d}\mathbf{q}(t)}{\mathrm{d}t}=\frac{\mathrm{d}^{2}\mathbf{v}(t)}{\mathrm{d}t^{2}}=\frac{\mathrm{d}^{3}\mathbf{r}(t)}{\mathrm{d}t^{3}},$
(1)

where $q$, $v$, $r$, and $t$ represent acceleration, velocity, position and
time respectively.

Third-order differential equations of the form

$J\left(\overset{\mathbf{...}}{x},\ddot{x},\dot{x},x\right)=0,$ (2)

are sometimes called jerk equations. When converted to an equivalent system of
three ordinary first-order non-linear differential equations, jerk equations
are the minimal setting for solutions showing chaotic behavior. This condition
generates mathematical interest in jerk systems. Systems involving fourth-
order derivatives or higher are accordingly called hyper-jerk systems [70].

If we define the jerk parameter $j$ as the function of redshift $z$, the
relation between the jerk parameter $j$ and the deceleration parameter $q$ is
given by

$j(z)=q(z)+2q(z)^{2}+(1+z)\frac{dq(z)}{dz}.$ (3)

The constant jerk parameter as $j=1$ corresponds to a flat $\Lambda$CDM model.
Without loss of generality we consider the jerk parameter $j$ as an arbitrary
constant [71, 72] and solving the differential equation (3), we get

$\frac{\sqrt{8j+1}+4q+1}{\sqrt{8j+1}-4q-1}=\left(\beta+\beta
z\right)^{\sqrt{8j+1}},$ (4)

where $\beta$ is an integration constant.

Rewriting Eq. (4) and using $\beta=1$ we get,

$q=\frac{-1}{4}+\frac{\sqrt{8j+1}}{4}-\frac{\sqrt{8j+1}}{2\left(1+\left(1+z\right)^{\sqrt{8j+1}}\right)}.$
(5)

The equation representing the deceleration parameter $q$ in terms of the
Hubble parameter $H$ and redshift $z$ is

$q(z)=-1+\frac{(1+z)}{H(z)}\frac{dH}{dz}.$ (6)

Using Eq. (5) in Eq. (6) we get,

$\frac{1}{1+z}\left[1-\frac{1}{4}+\frac{\sqrt{8j+1}}{4}-\frac{\sqrt{8j+1}}{2\left(1+\left(1+z\right)^{\sqrt{8j+1}}\right)}\right]dz=\frac{dH}{H}.$
(7)

By integrating both sides of the Eq. (7), we obtain:

$H=\alpha\left(1+z\right)^{\frac{3+\sqrt{8j+1}}{4}}\sqrt{1+\left(1+z\right)^{-\sqrt{8j+1}}},$
(8)

where $\alpha$ is an integration constant.

##  III Formulation of Einstein Field Equations in Rastall gravity

Rastall introduced a fact that the evolution of the universe does not perturb
the rate of energy-momentum transference between matter fields and geometry.
With this regard, Rastall supposed a coupling coefficient between the
divergence of EMT and the derivative of the Ricci scalar, which is a constant
and known as the Rastall parameter. This theory also describes the dark energy
section to explain the accelerated universe. The generalized field equations
for Rastall gravity theory are as follows

$R^{j}_{i}-\frac{1}{2}\delta_{ij}R=\kappa(T^{j}_{i}-\lambda\delta_{ij}R),$ (9)

or

$R_{ij}-\frac{1}{2}\left(1-2\kappa\lambda\right)Rg_{ij}=\kappa T_{ij},$ (10)

where $\lambda$ is the Rastall parameter and $\kappa$ is the gravitational
coupling constant. For flat, isotropic, and homogeneous universes the FLRW
metric is written as

$ds^{2}=-dt^{2}+a^{2}(t)(dx^{2}+dy^{2}+dz^{2}),$ (11)

where $a(t)$ stands for the scale factor of the universe. For the specific
value of $\kappa$ and $\lambda$, we can come back to the theory of general
relativity. Now, since Hubble parameter $H=\dot{a}/a$, therefore using Eqs.
(10), (11), we get the Einstein field equations for Rastall gravity theory in
terms of $H$ as

$3(1-4\kappa\lambda)H^{2}+2(1-3\kappa\lambda)\dot{H}=-\kappa p,$ (12)
$3(1-4\kappa\lambda)H^{2}-6\kappa\lambda\dot{H}=\kappa\rho,$ (13)

where $\rho$ is the energy density, $p$ is the isotropic pressure and
$\dot{H}$ indicates the derivative of $H$ with respect to cosmic time $t$.

Using the relation $\dot{H}=-(1+z)H(z)\frac{dH}{dz}$ and Eq. (8) in Eqs. (12),
(13) we get

$\rho(z)=\frac{3\left(1+z\right)^{\frac{3-\sqrt{1+8j}}{2}}\alpha^{2}\left[\left(2-5\kappa\lambda\right)\left(1+\left(1+z\right)^{\sqrt{1+8j}}\right)-\kappa\lambda\sqrt{1+8j}\left(1-\left(1+z\right)^{\sqrt{1+8j}}\right)\right]}{2\kappa}$
(14)

and

$p(z)=\frac{-\left(1+z\right)^{\frac{3-\sqrt{1+8j}}{2}}\alpha^{2}\left[3\left(1-5\kappa\lambda\right)\left(1+\left(1+z\right)^{\sqrt{1+8j}}\right)+\sqrt{1+8j}\left(1-3\kappa\lambda\right)\left(1-\left(1+z\right)^{\sqrt{1+8j}}\right)\right)}{2\kappa},$
(15)

and the Equation-of-State parameter is given by $\omega(z)=p(z)/\rho(z)$,
whose variations with respect to redshift $z$ can be seen in Fig. 4(c).

##  IV Observational analysis

Observational analysis is one of the advantageous parts of cosmology where we
find the best-fitted results for the model parameters. In this analysis, we
use the standard datasets to compute the optimal solution of the model
parameters. Here, in this paper, we are using the Hubble dataset $H(z)$ (with
$77$ points), $Pantheon$ dataset (with $1048$ points), and their joint dataset
($H(z)+Pantheon$), to calculate the best-fit value of $\alpha$ and $j$. For
this purpose, we use the Markov Chain Monte Carlo (MCMC) method and the coding
to plot the contour has been done in Python with the EMCEE library.

###  IV.1 $H(z)$ Dataset

$H(z)$ data acquired from cosmic chronometers which play an important role in
studying the dark division of the universe [73]. In literature, various models
have been discussed with different numbers of $H(z)$ data points [74, 75, 76,
77, 78]. In our model, we use $H(z)$ dataset along with $77$ data points [79].
We use the $\chi^{2}$ test to obtain the best-fit value of the model
parameters and the formula for $\chi^{2}$ is written as

$\chi_{Hub}^{2}(j,\alpha)=\sum\limits_{i=1}^{77}\frac{[H(j,\alpha,z_{i})-H_{obs}(z_{i})]^{2}}{\sigma_{z_{i}}^{2}},$
(16)

where $H(j,\alpha,z_{i})$ and $H_{obs}$ depict the theoretical and observed
values of the Hubble parameter respectively, and $\sigma_{(z_{i})}$ shows the
standard deviation for every $H(z_{i})$.

It has been observed that the analysis of the function $\chi_{Hb}^{2}$ using
the Hubble dataset is not very authentic for any cosmological models.
Therefore, we consider more reliable observation using the $Pantheon$ and
their joint datasets.

Table 1: Best fit values of model parameters. Dataset |  $j$ |  $\alpha$
---|---|---
$H(z)$ |  $0.437\pm 0.046$ |  $46.24\pm 0.43$
$Pantheon$ |  $0.95\pm 0.000094$ |  $50.000008^{+0.000076}_{-0.000097}$
$H(z)+Pantheon$ |  $1.09999^{+0.00011}_{-0.000098}$ |  $51.000001^{+0.000097}_{-0.000085}$

###  IV.2 Pantheon Dataset

From the beginning of $21^{st}$ century, Type Ia supernovae are very useful
for discussing the cosmological parameters as this is the first candidate to
evidence the accelerated expansion of the Universe. Supernovae are stars and
they they burst, they release a huge amount of energy and enlarge their outer
shell. Thus, this part is related to examining the time evolution of their
brightness and their spectrum. In our model, we use the $Pantheon$ datasets
(1048 data points) and constrain the value of model parameters [80]. Pantheon
dataset is the collection of data that have been compiled by different surveys
e.g. the CfA1-CfA4 surveys, the Supernovae Legacy Survey (SNLS), the Pan-
STARRS1 (PS1), the Carnegie Supernova Project (CSP), the Sloan Digital Sky
Survey (SDSS), various Hubble Space Telescope (HST) etc. [81, 82, 83, 84, 85].
The range of redshift is between $0.01$ to $2.26$ in these surveys. In Type Ia
supernovae analysis, we deal with redshift as well as luminosity distance
which are closely related to standard candles in the Universe. The luminosity
distance

$D_{L}(z)=(1+z)\int_{0}^{z}\frac{c}{H(z^{*})}dz^{*},$ (17)

where $c$ is the speed of light. And, the apparent magnitude can be written in
terms of luminosity distance as

$m(z)=M+5log_{10}\Bigg{[}\frac{D_{L}(z)}{1Mpc}\Bigg{]}+25,$ (18)

where $M$ is the absolute magnitude. Here, a dissipation between $H_{0}$ and
$M$ can be observe and since distance modulus $\mu=m-M$, therefore the formula
for $\chi^{2}$ is written as

$\chi_{Pan}^{2}(j,\alpha)=\sum\limits_{i=1}^{1048}\left[\frac{\mu_{th}(j,\alpha,z_{i})-\mu_{obs}(z_{i})}{\sigma_{\mu(z_{i})}}\right]^{2},$
(19)

where $\mu_{th}$ and $\mu_{obs}$ used for theoretical and observed distance
modulus respectively.

(a)

(b)

(c)

Figure 1:  Figs. a, b, and c depict the likelihood contours for $H(z)$,
$Pantheon$, and $H(z)+Pantheon$ dataset respectively corresponding to
$1\sigma$ and $2\sigma$ confidence regions.

###  IV.3 Joint Datasets

To obtain the better constraint values of the model parameter, the joint
dataset is used and the results are calculated by the $\chi_{HP}^{2}$
function, which is the sum of the functions $\chi_{Hub}^{2}$ and
$\chi_{Pan}^{2}$ as

$\chi_{HP}^{2}(j,\alpha)=\chi_{Hub}^{2}(j,\alpha)+\chi_{Pan}^{2}(j,\alpha).$
(20)

The obtained values of $j$ and $\alpha$ from the above-mentioned observational
datasets are placed in Table-1. According to the $\Lambda$CDM model, the
present value of jerk parameter $j$ is $1$ and in our model, we obtained the
best-fit value of $j$ which is close to 1 for the $Pantheon$ and joint
datasets and for $H(z)$ dataset it is approximately $0.435$.

As the $\Lambda$CDM model is one of the standard models in cosmology and by
comparing our model with the $\Lambda$CDM model we check the consistency of
our model and for this purpose, error bar plots play a significant role. In
our model, we have drawn the error bar plots for Hubble datasets and Type Ia
Supernovae datasets. In Fig. 2(a) and Fig. 2(b), we observe the similarity
between our model and the $\Lambda$CDM model with the error bar.

(a)

(b)

Figure 2:  The error bar plots for $H(z)$ and Type Ia Supernovae datasets.

##  V Physical and Dynamical Components of the model

To explore the dynamical evolution of the universe, let us start with one of
the essential parameters named as deceleration parameter ($q$). The
deceleration parameter has been calculated in Eq. (5) and now we plot the
curves for obtained best-fit values of $j$ and $\alpha$. For all three
mentioned observational datasets, we perceive that $q$ is behaving alike i.e.
in Fig. 3, at early times each trajectory is in the deceleration phase, and in
late times as well as at present our model shows the accelerating phase of the
universe. At present, the value of $q$ is $-0.25$ for $H(z)$, $Pantheon$, and
joint datasets, and the value of redshift at the time of phase transition is
$0.628$ for $H(z)$ dataset, $0.277$ for $Pantheon$ dataset and $0.239$ for
$H(z)+Pantheon$ datasets which is clearly noticeable in the subplot of Fig. 3.

Figure 3:  Plot of $q$ vs. $z$.

It is believed that after the Big Bang, the radiation-dominated era and
matter-dominated era both are the major sections in the evolution of the
Universe. The radiation-dominated era is required to anticipate primordial
nucleosynthesis whereas in the matter-dominated era, element creation takes
place. Therefore, keeping these points in mind we analyze the evolution of
various physical parameters of our model.

For the fixed values of $\kappa=1$ and $\lambda=0.12$, the energy density and
isotropic pressure have been plotted using Eqs. (14) and (15) in Fig. 4 for
the obtained best-fit values of model parameters. Fig. 4(a) illustrates the
evolution of energy density from high redshift to low redshift. As expected,
the energy density decreases from early to late times for all datasets. In
this model, the isotropic pressure is plotted in Fig. 4(b), which highlights
that in the early universe, the value of pressure is highly positive and at
present as well as in the future pressure is negative for all datasets.
According to the standard cosmology, the negative pressure results expanding
acceleration of the cosmos. The different value of EoS parameter ($\omega$)
indicates the different forms of universe for example, $\omega=0$ $\to$
pressureless matter, $\omega\in(0,\frac{1}{3})$ $\to$ hot matter,
$\omega=\frac{1}{3}$ $\to$ radiation, $\omega\in(\frac{1}{3},1)$ $\to$ hard
universe, $\omega=1$ $\to$ stiff matter, $\omega>1$ $\to$ ekpyrotic matter,
$\omega\in(0,-1)$ $\to$ quintessence universe, $\omega=-1$ $\to$ cosmological
constant and $\omega<-1$ $\to$ Phantom universe. In our model, one can observe
the different forms of the universe with respect to the changes in the value
of $\omega$ (see Fig. 4(c)). Today and in the future, our model is a
quintessence dark energy model.

(a)

(b)

(c)

Figure 4: The graphs of $\rho$, $p$ and $\omega$ vs. $z$.

Besides phenomenal fields, there are some major points to explore such as the
classification of singularities, casual structure, energy conditions, etc. In
particular, energy conditions were originally formulated from the well-known
Raychaudhuri equation. As it is valuable to explain the contribution of
further geometrical terms in the stress-energy tensor such as effective
pressure and effective energy density we study four energy conditions. The
energy conditions are the diminution of time-like or null vector fields
concerning the energy-momentum tensor and Einstein tensor obtained from
Einstein field equations. Generally, four energy conditions are discussed in
cosmological models namely, Weak energy condition (WEC), Null energy condition
(NEC), Strong energy condition (SEC), and Dominant energy condition (DEC).
These ECs are defined as: WEC: $\rho\geq 0,~{}\rho+p>0$, NEC: $\rho+p\geq 0$,
SEC: $\rho+3p\geq 0$, and DEC: $\rho\geq|p|$.

(a)

(b)

(c)

Figure 5:  The plots for NEC, SEC, DEC.

In Fig. 5, we perceive that for all observational datasets NEC and DEC hold
for the whole range of $z$, whereas SEC violates in the future.

##  VI Dynamical system stability

In this section, we employ the dynamical system technique to assess the
stability of the system. To achieve this, we initially select a set of
dimensionless variables as follows,

$\Omega_{M}=\frac{\kappa\rho}{3H^{2}},\quad y=\frac{\kappa
p}{H^{2}},\quad\delta=\kappa\lambda,$ (21)

The variable $\Omega_{M}$ represents a time-dependent quantity, signifying the
fractional energy density budget of the universe, while $y$ captures the
dynamics of the corresponding pressure. We introduce a dimensionless constant,
denoted as $\delta$, which can take any non-zero value. Utilizing Eq. (12) and
Eq. (13), we express,

$H^{2}=\frac{\kappa(3p\kappa\lambda-\rho+3\kappa\lambda\rho)}{-3+12\kappa\lambda},$
(22)

and

$\dot{H}=\frac{-\kappa}{2}(\rho+p).$ (23)

Expressing Eq. (22) in terms of the dynamical variables that constrain $y$, we
can represent this constraint in terms of $\Omega_{M}$ as,

$y=\frac{-1+\Omega_{M}}{\delta}+4-3\Omega_{M}.$ (24)

To analyze the dynamics of the system, we construct the autonomous system by
calculating the first-order time derivative of the dynamical variable, as
follows,

$\Omega_{M}^{\prime}=\frac{d\Omega_{M}}{Hdt}=-(1-4\delta)(y+3\Omega_{M})-2\delta\left(j+\frac{3}{2}(y+3\Omega_{M})-1\right)+\Omega_{M}(y+3\Omega_{M}).$
(25)

Here, $Hdt=dN=d\log a$. We have used $j=\frac{\dddot{a}}{aH^{3}}$ as a
constant parameter. This constraint allows the evaluation of the system’s
dynamics using a single parameter, resulting in a 1D phase space. The equation
of state and deceleration parameter can then be expressed in terms of
dynamical variables as,

$\omega=\frac{p}{\rho}=\frac{y}{3\Omega_{M}},\quad
q=-1+\frac{1}{2}(y+3\Omega_{M}).$ (26)

We impose an additional constraint on the fractional energy density parameter,
necessitating it to remain positive and bounded between $0$ and $1$,

$0<\Omega_{M}\leq 1.$ (27)

The critical points that satisfy this constraint are the only ones considered
physically viable. The system produces two critical points, which are
tabulated in Tab. 2. The qualitative behavior of these points is discussed in
detail.

Table 2: The critical points and its nature. Points | $\Omega_{M*}$ | $\omega$ | $d\Omega_{M}^{\prime}/d\Omega_{M}|_{\Omega_{M*}}$
---|---|---|---
$P_{1}$ | $1-\frac{1}{2}\delta\left(\sqrt{8j+1}+5\right)$ | Fig. LABEL:fig:eos_p1 | $-\sqrt{8j+1}$
$P_{2}$ | $\frac{1}{2}\delta\left(\sqrt{8j+1}-5\right)+1$ | Fig. LABEL:fig:eos_p2 | $\sqrt{8j+1}$

(a)

(b)

Figure 6:  The plots for the constrained relation Eq. (27) and EoS $\omega$,
in the parametric space of $(j,\delta)$ for the fixed point $P_{1}$.

  * •

Point $P_{1}$: This critical point is dependent on both $j$ and $\delta$ and
becomes a real point for $j>-1/8$. However, it cannot assume arbitrary values
and must adhere to the constraint given by Eq. (27). The corresponding region
of its existence has been plotted in Fig. LABEL:fig:x_existence_p1. The nature
of the fixed point can be inferred from the equation of state $\omega$, as
illustrated in Fig. LABEL:fig:eos_p1, further constraining the values of $j$
and $\delta$. The contour plot reveals that the point can exhibit both
accelerating and non-accelerating solutions for $\delta<0.3$. However, for
$\delta\geq 0.5$, the point yields a phantom solution; nevertheless, the
fractional energy density violates the constraint relation. To assess the
stability of the fixed point, we determined the first derivative
$\frac{d\Omega_{M}^{\prime}}{d\Omega_{M}}\Big{|}_{\Omega_{M*}}$ at this point,
resulting in a negative quantity. Following the stability discussion presented
in Appendix A, we conclude that as $N\to+\infty$, the point becomes a stable
fixed point corresponding to the future epoch of the universe. Conversely, at
the past epoch, i.e., $N\to-\infty$, the point becomes unstable.

(a)

(b)

Figure 7:  The plots for the constrained relation Eq. (27) and EoS parameter
$\omega$, in the parametric space of $(j,\delta)$ for the fixed point $P_{2}$.
(a) Figure 8: The evolution of cosmological parameters for $\delta=0.04,\
j=0.8$.

  * •

Point $P_{2}$: The constraint of the point has been visualized in Fig.
LABEL:fig:x_existence_p2, illustrating its existence for $j>-1/8$. The
equation of state corresponding to this point is depicted in Fig.
LABEL:fig:eos_p2, showing positive values and indicating non-accelerating
characteristics. For $j>1.5$ and $\delta>0.4$, the point can mimic the
radiation equation of state. However, within this range, the aforementioned
point becomes unphysical. In the range $0.5<j<1.5$ and $0.01<\delta<0.2$, this
point can exhibit characteristics of a pressure-less fluid. Upon evaluating
stability, the derivative
$\frac{d\Omega_{M}^{\prime}}{d\Omega_{M}}\Big{|}_{\Omega_{M*}}$ yields a
positive value, indicating stability in the past epoch.

From the analysis of critical points and based on their characteristic
properties, we have constrained $j$ and $\delta$. One of the critical points
is found to be stable, producing an accelerating solution that represents the
present epoch of the universe. To further analyze the system’s dynamics, we
evaluate the evolution of cosmological parameters, i.e.,
$(\Omega_{M},\omega,h\equiv H/100)$ against $N=\log a$ in Fig. 8, spanning
from the past epoch to the future epoch. We select benchmark points
$(j=0.8,\delta=0.04)$ to ensure the existence of both fixed points
simultaneously. In the past epoch, the fractional energy density dominates,
and the corresponding equation of state is near $0$, indicating a matter-
dominated phase of the universe. As the system evolves toward the present
epoch, the equation of state increases towards negative values. In the future
epoch, $\omega$ saturates to $-0.70$, and energy density saturates to $0.90$.
Hence, the system exhibits quintessence-like behavior in the late-time epoch
and produces matter characteristics in the past epoch of the universe. We have
also depicted the evolution of the Hubble parameter $h(N)=H(N)/100$ using

$\frac{dh}{dN}=-\frac{3}{2}h(N)\left(\frac{y}{3}+x\right).$ (28)

The behavior indicates that in the past, the Hubble parameter increases, and
as the system progresses into the future epoch, the Hubble parameter saturates
to lower values.

##  VII Conclusion

In this work, we study Rastall’s theory of gravity, which is based on a
variational principle. In this theory, we consider a non-minimal coupling
between geometry and matter fields and discuss the recent theoretical
developments in alternative theories of general relativity. This work is
framed for the flat, isotropic, and homogeneous universe in the FLRW metric,
where we have obtained the EFEs with the calculated value of the Hubble
parameter in terms of the jerk parameter. Furthermore, we examine the
essential physical and dynamical parameters one by one. We evaluate the best-
fitted values of model parameters $\alpha$ and $j$ by the MCMC method, where
we use the Hubble dataset, $Pantheon$ dataset, and their joint dataset.

The trajectories of the deceleration parameter for these best-fit values
indicate the phase transition from deceleration to acceleration. At present,
our model shows the accelerating universe.

Also, if we look at the behavior of energy density, it is monotonically
decreased from high redshift to low redshift for all observational datasets.
The isotropic pressure $p$ is positive in the early universe and holds a
negative value at present as well as in the future, which indicates the
presence of repulsive force and an accelerating expansion of the universe.
According to the results of the EoS parameter, our model is a quintessence
dark energy model at present. In Fig. 4, $\omega$ depicts the different
dominating eras of the universe.

In this model, NEC and DEC both hold good but SEC violates which shows the
existence of exotic matter in the universe. Hence, finally, we conclude that
our model which is studied in Rastall gravity is a quintessence dark energy
model.

Finally, we conducted a dynamical stability analysis of the system, treating
the jerk parameter $j$ as a constant quantity, which imposes a vital
constraint on the system. Our findings revealed a 1D phase space, with the
system yielding two critical points. One critical point results in a stable
accelerating solution in the future epoch, resembling the behavior of a
quintessential dark energy model. The other critical point exhibits non-
accelerating characteristics, similar to dark matter fluid, and stabilizes at
the past epoch. Thus, the qualitative behavior of the system demonstrates that
Rastall’s theory of gravity can consistently account for different phases,
namely the matter-dominated and accelerating epochs of the universe.

### Acknowledgements

J. K. Singh wishes to thank Joao Luis Rosa, Institute of Physics, University
of Tartu, W. Ostwaldi 1, 50411 Tartu, Estonia for fruitful discussions.

Data Availability Statement No new data were created or analyzed in this
study.

##  Appendix A Dynamical system in 1D

Consider the autonomous equation in 1D is:

$\dfrac{dx}{dN}=\dot{x}=f(x).$ (29)

Suppose $x_{0}\in\mathbb{R}^{n}$ is a fixed point of the autonomous equation
such that $f(x_{0})=0$. To determine the stability of the system at this fixed
point, we perturb the system around it as,

$x=x_{0}+\epsilon,$ (30)

where, $\epsilon\ll 1$. We expand $f(x)$ using Taylor’s series around the
fixed point $x_{0}$ as,

$f(x)=f(x_{0})+\dfrac{df}{dx}\Big{|}_{x_{0}}(x-x_{0})+......$ (31)

Inserting it into the autonomous equation Eq. (29) and truncating the series
after the first order, we obtain,

$\begin{split}\dot{\epsilon}&=f^{\prime}(x_{0})\epsilon,\\\
\implies\epsilon&=C\exp(f^{\prime}(x_{0})\,N).\end{split}$ (32)

Here, $C$ is an integration constant. Observing Eq. (32), as
$N\rightarrow+\infty$, the perturbation increases (decreases) when
$f^{\prime}(x_{0})>0\ (<0)$, indicating the system becomes unstable (stable)
at that point.

## References

  * [1] J. K. Singh, A. Singh, G. K. Goswami and J. Jena, Annals Phys. 443, 168958 (2022).
  * [2] J. K. Singh, Shaily, A. Singh, A. Beesham and H. Shabani, Annals Phys. 455, 169382 (2023).
  * [3] J. K. Singh, H. Balhara, K. Bamba and J. Jena, JHEP 03, 191 (2023) [erratum: JHEP 04, 049 (2023)].
  * [4] J. K. Singh, K. Bamba, R. Nagpal and S. K. J. Pacif, Phys. Rev. D 97, 123536 (2018).
  * [5] J. K. Singh, Shaily and K. Bamba, Chin. J. Phys. 84, 371-380 (2023).
  * [6] J. K. Singh and R. Nagpal, Eur. Phys. J. C 80, no.4, 295 (2020).
  * [7] J. K. Singh, A. Singh, Shaily and J. Jena, Chin. J. Phys. 86, 616-627 (2023).
  * [8] R. Nagpal, S. K. J. Pacif, J. K. Singh, K. Bamba and A. Beesham, Eur. Phys. J. C 78, no.11, 946 (2018).
  * [9] P. Rastall, Phys. Rev. D 6, 3357-3359 (1972).
  * [10] L. Lindblom, W. A. Hiscock, J. Phys. A: Math. Gen. 15, 1827-1830 (1982).
  * [11] M. Visser, Phys. Lett. B 782, 83-86 (2018).
  * [12] F. Darabi, H. Moradpour, I. Licata, Y. Heydarzade and C. Corda, Eur. Phys. J. C 78, 25 (2018).
  * [13] S. Santos Da Costa, F. V. Roig, J. S. Alcaniz, S. Capozziello, M. De Laurentis and M. Benetti, Class. Quant. Grav. 35, no.7, 075013 (2018).
  * [14] G. J. Olmo, Phys. Rev. D 72, 083505 (2005).
  * [15] L. K. Duchaniya, S. V. Lohakare, B. Mishra and S. K. Tripathy, Eur. Phys. J. C 82, no.5, 448 (2022).
  * [16] G. Kofinas, G. Leon and E. N. Saridakis, Class. Quant. Grav. 31, 175011 (2014).
  * [17] S. A. Kadam, B. Mishra and J. Said Levi, Eur. Phys. J. C 82, no.8, 680 (2022).
  * [18] S. A. Kadam, N. P. Thakkar and B. Mishra, Eur. Phys. J. C 83, no.9, 809 (2023).
  * [19] A. Stachowski and M. Szydłowski, Eur. Phys. J. C 76, no.11, 606 (2016).
  * [20] N. Roy and N. Banerjee, Phys. Rev. D 95, no.6, 064048 (2017).
  * [21] W. Khyllep, J. Dutta, E. N. Saridakis and K. Yesmakhanova, Phys. Rev. D 107, no.4, 044022 (2023).
  * [22] W. Khyllep, J. Dutta, S. Basilakos and E. N. Saridakis, Phys. Rev. D 105, no.4, 043511 (2022).
  * [23] R. G. Landim, Eur. Phys. J. C 79, no.11, 889 (2019).
  * [24] A. Alho, C. Uggla and J. Wainwright, JCAP 09, 045 (2019).
  * [25] A. Singh, G. P. Singh and A. Pradhan, Int. J. Mod. Phys. A 37, no.16, 2250104 (2022).
  * [26] N. Roy and N. Banerjee, Annals Phys. 356, 452-466 (2015).
  * [27] S. Das, S. Hussain, D. Nandi, R. O. Ramos and R. Silva, Phys. Rev. D 108, no.8, 083517 (2023).
  * [28] S. Chakraborty, K. Bamba and A. Saa, Phys. Rev. D 99, no.6, 064048 (2019).
  * [29] S. D. Odintsov and V. K. Oikonomou, Phys. Rev. D 96, no.10, 104049 (2017).
  * [30] A. Alho, S. Carloni and C. Uggla, JCAP 08, 064 (2016).
  * [31] S. Carloni, JCAP 09, 013 (2015).
  * [32] S. D. Odintsov and V. K. Oikonomou, Phys. Rev. D 98, no.2, 024013 (2018).
  * [33] V. K. Oikonomou and N. Chatzarakis, Nucl. Phys. B 956, 115023 (2020).
  * [34] N. Roy and N. Banerjee, [arXiv:1412.0837 [gr-qc]].
  * [35] S. Bahamonde, C. G. Böhmer, S. Carloni, E. J. Copeland, W. Fang and N. Tamanini, Phys. Rept. 775-777, 1-122 (2018).
  * [36] H. Shabani and A. H. Ziaie, Eur. Phys. J. C 77, no.5, 282 (2017).
  * [37] H. Shabani, A. De, T. H. Loo and E. N. Saridakis, [arXiv:2306.13324 [gr-qc]].
  * [38] H. Shabani, A. De and T. H. Loo, Eur. Phys. J. C 83, no.6, 535 (2023).
  * [39] V. K. Oikonomou, Int. J. Mod. Phys. D 27, no.05, 1850059 (2018).
  * [40] N. Chatzarakis and V. K. Oikonomou, Annals Phys. 419, 168216 (2020).
  * [41] L. K. Duchaniya, K. Gandhi and B. Mishra, [arXiv:2303.09076 [gr-qc]].
  * [42] S. V. Lohakare, K. Rathore and B. Mishra, Class. Quant. Grav. 40, no.21, 215009 (2023).
  * [43] S. A. Kadam, S. V. Lohakare and B. Mishra, Annals Phys. 460, 169563 (2024).
  * [44] L. Pati, S. A. Narawade, S. K. Tripathy and B. Mishra, Eur. Phys. J. C 83, no.5, 445 (2023).
  * [45] S. A. Narawade, M. Koussour and B. Mishra, Nucl. Phys. B 992, 116233 (2023).
  * [46] R. Bhagat, S. A. Narawade, B. Mishra and S. K. Tripathy, Phys. Dark Univ. 42, 101358 (2023).
  * [47] J. De-Santiago, J. L. Cervantes-Cota and D. Wands, Phys. Rev. D 87, no.2, 023502 (2013).
  * [48] J. Dutta, W. Khyllep and H. Zonunmawia, Eur. Phys. J. C 79, no.4, 359 (2019).
  * [49] S. Carloni, F. S. N. Lobo, G. Otalora and E. N. Saridakis, Phys. Rev. D 93, 024034 (2016).
  * [50] C. G. Boehmer, N. Tamanini and M. Wright, Phys. Rev. D 91, no.12, 123003 (2015).
  * [51] A. Chatterjee, S. Hussain and K. Bhattacharya, Phys. Rev. D 104, no.10, 2021 (2021).
  * [52] S. Hussain, A. Chatterjee and K. Bhattacharya, Phys. Rev. D 108, no.10, 103502 (2023).
  * [53] B. Mirza and F. Oboudiat, Int. J. Geom. Meth. Mod. Phys. 13, no.09, 1650108 (2016).
  * [54] H. Shabani, A. H. Ziaie and H. Moradpour, Annals Phys. 444, 169058 (2022).
  * [55] G. F. Silva, O. F. Piattella, J. C. Fabris, L. Casarini and T. O. Barbosa, Grav. Cosmol. 19, 156-162 (2013).
  * [56] A. Singh, R. Raushan and R. Chaubey, Can. J. Phys. 99, no.12, 1073-1081 (2021).
  * [57] H. Shabani, H. Moradpour and A. H. Ziaie, Phys. Dark Univ. 36, 101047 (2022).
  * [58] W. Khyllep and J. Dutta, Phys. Lett. B 797, 134796 (2019).
  * [59] S. Ghosh, S. Dey, A. Das, A. Chanda and B. C. Paul, JCAP 07, 004 (2021).
  * [60] M. F. A. R. Sakti, A. Suroso and F. P. Zen, Annals Phys. 413, 168062 (2020).
  * [61] M. F. A. R. Sakti, A. Suroso, A. Sulaksono and F. P. Zen, Phys. Dark Univ. 35, 100974 (2022).
  * [62] S. Guo, K. J. He, G. R. Li and G. P. Li, Class. Quant. Grav. 38, no.16, 165013 (2021).
  * [63] G. G. L. Nashed, Universe 8, no.10, 510 (2022).
  * [64] D. J. Gogoi and U. D. Goswami, Phys. Dark Univ. 33, 100860 (2021).
  * [65] C. Y. Shao, Y. J. Tan, C. G. Shao, K. Lin and W. L. Qian, Chin. Phys. C 46, no.10, 105103 (2022).
  * [66] B. Narzilloev, I. Hussain, A. Abdujabbarov and B. Ahmedov, Eur. Phys. J. Plus 137, no.5, 645 (2022).
  * [67] Y. Heydarzade, H. Moradpour and F. Darabi, Can. J. Phys. 95, no.12, 1253-1256 (2017).
  * [68] A. Singh and K. C. Mishra, Eur. Phys. J. Plus 135, no.9, 752 (2020).
  * [69] A. Singh and A. Pradhan, Indian J. Phys. 97, no.2, 631-641 (2023).
  * [70] Chlouverakis, E. Konstantinos and J. C. Sprott, Chaos, Solitons & Fractals. 28, no.3, 739–746 (2006).
  * [71] A. Al Mamon and K. Bamba, Eur. Phys. J. C 78, no.10, 862 (2018).
  * [72] N. Myrzakulov, M. Koussour, A. H. A. Alfedeel and H. M. Elkhair, Chin. J. Phys. 86, 300-312 (2023).
  * [73] L. Chimento and M. I. Forte, Phys. Lett. B 666, 205-211 (2008).
  * [74] J. K. Singh, Shaily, S. Ram, J. R. L. Santos and J. A. S. Fortunato, Int. J. Mod. Phys. D 32, 2350040 (2023).
  * [75] Shaily, A. Singh, J. K. Singh and S. Ray, [arXiv:2402.01780 [gr-qc]].
  * [76] J. K. Singh, Shaily, H. Balhara, K. Bamba and J. Jena, Astron. Comput. 46 (2024), 100790.
  * [77] J. K. Singh, H. Balhara, Shaily and P. Singh, Astron. Comput. 46 (2024), 100795.
  * [78] H. Balhara, J. K. Singh and E. N. Saridakis, [arXiv:2312.17277 [gr-qc]].
  * [79] Shaily, M. Zeyauddin and J. K. Singh, arXiv:220.05076 [gr-qc].
  * [80] D. M. Scolnic et al. [Pan-STARRS1], Astrophys. J. 859, no.2, 101 (2018).
  * [81] A. G. Riess, R. P. Kirshner, B. P. Schmidt, S. Jha, P. Challis, P. M. Garnavich, A. A. Esin, C. Carpenter, R. Grashius and R. E. Schild, et al. Astron. J. 117, 707-724 (1999).
  * [82] S. Jha, R. P. Kirshner, P. Challis, P. M. Garnavich, T. Matheson, A. M. Soderberg, G. J. M. Graves, M. Hicken, J. F. Alves and H. G. Arce, et al. Astron. J. 131, 527-554 (2006).
  * [83] M. Hicken, P. Challis, S. Jha, R. P. Kirsher, T. Matheson, M. Modjaz, A. Rest and W. M. Wood-Vasey, Astrophys. J. 700, 331-357 (2009).
  * [84] C. Contreras, M. Hamuy, M. M. Phillips, G. Folatelli, N. B. Suntzeff, S. E. Persson, M. Stritzinger, L. Boldt, S. Gonzalez and W. Krzeminski, et al. Astron. J. 139, 519-539 (2010).
  * [85] M. Sako et al. [SDSS], Publ. Astron. Soc. Pac. 130, no.988, 064002 (2018).