# Curvature properties of the Vaidya metric

Absos Ali Shaikh1, Haradhan Kundu1 and Jayashree Sen1 A. A. Shaikh, H. Kundu
and J. Sen,
Department of Mathematics,
University of Burdwan, Golapbag,
Burdwan-713104,
West Bengal, India<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Dedicated to Professor
Prahalad Chunnilal Vaidya on his 100${}^{\mbox{th}}$ birthday

###### Abstract.

As a generalization of the Schwarzschild solution, Vaidya presented a
radiating metric to develop a model of the exterior of a star including its
radiation field, named later Vaidya metric. The present paper deals with the
investigation on the curvature properties of Vaidya metric. It is shown that
Vaidya metric can be considered as a model of different pseudosymmetric type
curvature conditions, namely, $C\cdot C=\frac{m}{r^{3}}Q(g,C)$, $R\cdot
R-Q(S,R)=\frac{m}{r^{3}}Q(g,C)$ etc. It is also shown that Vaidya metric is
Ricci simple, vanishing scalar curvature and its Ricci tensor is Riemann-
compatible. As a special case of the main result, we obtain the curvature
properties of Schwarzschild metric. Finally, we compare the curvature
properties of Vaidya metric with another radiating metric, namely, Ludwig-
Edgar pure radiation metric.

###### Key words and phrases:

Einstein Field equations, Schwarzschild metric, Vaidya metric, Ludwig-Edgar
pure radiation metric, quasi-Einstein manifold, Einstein manifold, Ricci flat
manifold, Ricci-simple manifold, pseudosymmetry type curvature condition

###### 2010 Mathematics Subject Classification:

53B20, 53B25, 53B30, 53B50, 53C15, 53C25, 53C35, 83C15

Published in: Indian Journal of Mathematics, Vol. 61, No. 1, 2019, 41-59

##  1\. Introduction

Let $M$ be a semi-Riemannian manifold of dimension $n$ endowed with a semi-
Riemannian metric $g$ and let $\nabla$, $R$, $S$ and $\kappa$ be respectively
the Levi-Civita connection, the Riemann-Christoffel curvature tensor, the
Ricci tensor and the scalar curvature of $M$. The additional restriction(s) on
the curvature tensor(s) of a specific manifold help us to realize the geometry
of that manifold. For example, if on a manifold $R=0$ (resp., $\nabla R=0$)
then it is flat (resp., locally symmetric). Hence it is very important to
investigate the curvature restricted geometric structures on a manifold.
In the literature of differential geometry we find various generalizations of
locally symmetric manifolds ([5], [6]) in several directions, such as
recurrent manifold by Ruse ([49], [50], [51], [86]), some generalized
recurrent manifolds by Shaikh and his coauthors ([25], [68], [67], [69], [53],
[71]), semisymmetric manifolds by Cartan [7] and Sinyukov ([72], [73]) (which
were latter classified by Szab$\acute{\mbox{o}}$ ([75], [76], [77])),
generalized semisymmetric manifold by Mikeš ([43], [44], [45], [46], [47],
[48]), pseudosymmetric manifolds by Chaki [8], pseudosymmetric manifolds by
Deszcz ([1], [14], [20], [21], [30], [31], [82], [83], [84]), weakly symmetric
manifolds by Tam$\acute{\mbox{a}}$ssy and Binh ([79], [80]), manifolds of
recurrent curvature 2-forms ([3], [35]) etc.
In his theory of general relativity, Einstein made a bridge between the
geometrical and physical quantities of a spacetime (a connected 4-dimensional
Lorentzian manifold) by presenting the famous field equation

$S-\frac{\kappa}{2}g+\Lambda g=\frac{8\pi G}{c^{4}}T,$

where $\Lambda$ is the cosmological constant, $G$ is Newton’s gravitational
constant, $c$ is the speed of light in vacuum and $T$ is the energy-momentum
tensor. For instance, a spacetime represent perfect fluid if and only if it is
quasi-Einstein, i.e., its Ricci tensor satisfies the condition $S=\alpha
g+\beta\Pi\otimes\Pi$, where $\alpha,\beta$ are some scalars and $\Pi$ is a
1-form ([9], [10], [16], [57]). Therefore the study of curvature restricted
geometric structures become essential both physically and geometrically in the
investigation of a spacetime.
According to Birkhoff’s theorem [4], the vacuum spherically symmetric
spacetime with zero cosmological constant is the asymptotically flat solution
given by Schwarzschild [28]

$ds^{2}=-\left(1-\frac{2m}{r}\right)du^{2}-2drdu+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (1.1)

where $m$ is an arbitrary parameter. To develop a model of the exterior of a
star, which includes its radiation field, there arose an important
generalization of the Schwarzschild solution by Vaidya [81]. In terms of
$(r,t,\theta,\phi)$-coordinates Vaidya metric [81] can be written as

$ds^{2}=\left(1-\frac{2m}{r}\right)^{-1}\left(dr^{2}-\frac{(m_{t})^{2}}{(m_{r})^{2}}dt^{2}\right)+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$

where $m=m(r,t)$. In fact, the above metric can be expressed in a much more
useful form (for outgoing radiation) by introducing a null coordinate $u$, as
follows:

$ds^{2}=-\left(1-\frac{2m(u)}{r}\right)du^{2}-2drdu+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}).$ (1.2)

If $m(u)=constant$, then the Vaidya metric (1.2) reduces to the Schwarzschild
metric (1.1). Hence (1.2) is a generalization of the Schwarzschild metric
(1.1). In this case, $m(u)$ is an arbitrary non-increasing function of the
retarded null coordinate $u$.
It is known that the Schwarzschild metric is a pseudosymmetric metric [24]
(Proposition 2). The Schwarzschild metric, the Kottler metric, the Reissner-
Nordström metric, as well as some Friedmann-Lemaître-Robertson-Walker metrics
are the “oldest” examples of pseudosymmetric non-semisymmetric warped product
metrics (see, e.g., [19], Section 1, [21], Chapter 6). In [34] (Section 5) it
was stated that the Vaidya metric $g$ is a non-pseudosymmetric metric, rank
$S\leq 1$ and the tensors $R\cdot R-Q(S,R)$ and $Q(g,C)$ are linearly
dependent. Further, in view of Theorem 2 of [13], the tensors $C\cdot C$ and
$Q(g,C)$ of the metric $g$ are linearly dependent. These properties of $g$, in
details, are presented in Example 7.1 (ii)-(iii) of [19] (eq. (7.14)). In
addition, in that example it was mention that $g$ satisfies also the following
conditions: $S^{2}=0$, $\kappa=0$ and $S\cdot R=2mr^{-3}g\wedge S$.
The main object of the present paper is to investigate the geometric
structures admitted by the Vaidya metric (1.2). It is interesting to note that
such a metric admits several important geometric structures, such as, it is
Ricci simple, manifold of vanishing scalar curvature, its Ricci tensor is
Riemann-compatible and its conformal curvature 2-forms are recurrent. Moreover
Vaidya metric satisfies various pseudosymmetric type curvature conditions,
such as pseudosymmetric Weyl conformal curvature tensor, the difference tensor
$R\cdot R-Q(S,R)$ is linearly dependent with $Q(g,C)$. As a special case of
the main result, we obtain the curvature properties of Schwarzschild metric.
It is shown that Schwarzschild metric is pseudosymmetric in the sense of
Deszcz.
The paper is organized as follows. Section 2 deals with the different
geometric structures due to various curvature tensors. Section 3 is concerned
with the calculations of components of various tensors of Vaidya spacetime.
Section 4 is devoted to the conclusion on the geometric structures admitted by
Vaidya metric and Schwarzschild metric. Finally, in section 5, we compare the
curvature properties of Vaidya metric with Ludwig-Edgar pure radiation metric.

##  2\. Curvature Restricted Geometric Structures

A curvature restricted geometric structure on a semi-Riemannian manifold $M$
is a geometric structure obtained by imposing some restriction(s) on some
curvature tensor(s) of $M$ by means of covariant derivatives of first order or
higher orders. In this section we will explain some useful notations and
definitions of various curvature restricted geometric structures.
Let $A$ and $E$ be two symmetric $(0,2)$-tensors. The Kulkarni-Nomizu product
$A\wedge E$ of $A$ and $E$ is defined by (see e.g. [17], [26])

$\displaystyle(A\wedge E)(X_{1},X_{2},X_{3},X_{4})$ $\displaystyle=$
$\displaystyle A(X_{1},X_{4})E(X_{2},X_{3})+A(X_{2},X_{3})E(X_{1},X_{4})$
$\displaystyle-$ $\displaystyle
A(X_{1},X_{3})E(X_{2},X_{4})-A(X_{2},X_{4})E(X_{1},X_{3}),$

where $X_{1},X_{2},X_{3},X_{4}\in\chi(M)$, the Lie algebra of all smooth
vector fields on $M$. Throughout the paper we will consider
$X,Y,X_{1},X_{2},\cdots\in\chi(M)$. Again for a symmetric $(0,2)$-tensor $E$,
we can define two endomorphisms $\mathcal{E}$ and $(X\wedge_{E}Y)$ as ([11],
[17])

$g(\mathcal{E}X_{1},X_{2})=E(X_{1},X_{2})\ \mbox{ and }\
(X\wedge_{E}Y)X_{1}=E(Y,X_{1})X-E(X,X_{1})Y.$

Now in terms of Kulkarni-Nomizu product $\wedge$ and the endomorphism
$(X\wedge_{E}Y)$, the Gaussian curvature tensor $\mathfrak{G}$, the projective
curvature tensor $P$, the conharmonic curvature tensor $K$, the concircular
curvature tensor $W$ and the conformal curvature tensor $C$ are given by
([32], [87])

$\mathfrak{G}=\wedge_{g},\ \ \ \ P=R-\frac{1}{n-1}(\wedge_{S}),\ \ \ \
K=R-\frac{1}{n-2}(g\wedge S),$ $W=R-\frac{r}{n(n-1)}\mathfrak{G},\ \ \ \
C=K+\frac{r}{(n-2)(n-1)}\mathfrak{G}.$

For a $(0,4)$-tensor $D$ we can define the corresponding $(1,3)$-tensor
$\mathcal{D}$ and the endomorphism $\mathscr{D}(X,Y)$ as follows:

$D(X_{1},X_{2},X_{3},X_{4})=g(\mathcal{D}(X_{1},X_{2})X_{3},X_{4}),\ \
\mathscr{D}(X,Y)X_{1}=\mathcal{D}(X,Y)X_{1}.$

Now operating $\mathscr{D}(X,Y)$ and $(X\wedge_{A}Y)$ on a $(0,k)$-tensor $H$,
$k\geq 1$, we get two $(0,k+2)$-tensors $D\cdot H$ and $Q(A,H)$ respectively
as follows (see [15], [16], [22], [55], [59], [78] and also references
therein)

$D\cdot
H(X_{1},X_{2},\cdots,X_{k},X,Y)=-H(\mathcal{D}(X,Y)X_{1},X_{2},\cdots,X_{k})-\cdots-H(X_{1},X_{2},\cdots,\mathcal{D}(X,Y)X_{k}).$

and

$\displaystyle Q(A,H)(X_{1},X_{2},\ldots,X_{k},X,Y)=((X\wedge_{A}Y)\cdot
H)(X_{1},X_{2},\ldots,X_{k})$
$\displaystyle=A(X,X_{1})H(Y,X_{2},\cdots,X_{k})+\cdots+A(X,X_{k})H(X_{1},X_{2},\cdots,Y)$
$\displaystyle-A(Y,X_{1})H(X,X_{2},\cdots,X_{k})-\cdots-A(Y,X_{k})H(X_{1},X_{2},\cdots,X).$

######  Definition 2.1.

$($[1], [7], [14], [29], [59], [63], [64], [70], [75]$)$ A semi-Riemannian
manifold $M$ is said to be $H$-semisymmetric type if $D\cdot H=0$ and it is
said to be $H$\- pseudosymmetric type if
$\left(\sum\limits_{i=1}^{k}c_{i}D_{i}\right)\cdot H=0$ for some scalars
$c_{i}$’s, where $D$ and each $D_{i}$, $i=1,\ldots,k$, $(k\geq 2)$, are (0,4)
curvature tensors.

In particular, if $D=R$ and $H=R$ (resp., $S$, $C$, $W$ and $K$), then $M$ is
called semisymmetric (resp., Ricci, conformally, concircularly and
conharmonically semisymmetric). Again, if $i=2$, $D_{1}=R$,
$D_{2}=\mathfrak{G}$ and $H=R$ (resp., $S$, $C$, $W$ and $K$), then $M$ is
called Deszcz pseudosymmetric (resp., Ricci, conformally, concircularly and
conharmonically pseudosymmetric). Especially, if $i=2$, $D_{1}=C$,
$D_{2}=\mathfrak{G}$ and $H=C$, then $M$ is called a manifold of
pseudosymmetric Weyl conformal curvature tensor.
We note that the pp-wave metric [54] is semisymmetric and the Robinson-
Trautman metric [52] is pseudosymmetric.

######  Definition 2.2.

A semi-Riemannian manifold $M$ is said to be quasi-Einstein if $S=\alpha
g+\beta\Pi\otimes\Pi$ for some scalars $\alpha$ and $\beta$, and a 1-form
$\Pi$. In particular, if $\beta\equiv 0$ (resp., $\alpha\equiv 0$), then a
quasi-Einstein manifold is called Einstein [3] (resp., Ricci simple [9]).

We mention that the Robertson-Walker spacetimes are quasi Einstein [2],
Kaigorodov metric [33] is Einstein, and Gödel spacetime [23] is Ricci simple.
Quasi-Einstein manifolds were investigated in several papers, see, e.g., [11],
[16], [17], [19], [22], [55].
In 1983, Derdzinski and Shen [12] showed that if $E$ is a Codazzi tensor [27]
such that $V_{\lambda}$ and $V_{\mu}$ are two eigenspaces corresponding to the
eigenvalues $\lambda$ and $\mu$ of the operator $\mathcal{E}$, then the
subspace $V_{\lambda}\wedge V_{\mu}$ is invariant under the action of the
curvature operator $\mathscr{R}(X,Y)$. Recently, Mantica and Molinari [37]
extended their theorem by replacing the condition ‘Codazzi type’ by ‘Riemann-
compatibility’.

######  Definition 2.3.

Let $D$ be a $(0,4)$-tensor and $E$ be a symmetric $(0,2)$-tensor on $M$. Then
$E$ is said to be $D$-compatible ([18], [38], [39]) if

$D(\mathcal{E}X_{1},X,X_{2},X_{3})+D(\mathcal{E}X_{2},X,X_{3},X_{1})+D(\mathcal{E}X_{3},X,X_{1},X_{2})=0$

holds, where $\mathcal{E}$ is the endomorphism corresponding to $E$. Again a
1-form $\Pi$ is said to be $D$-compatible if $\Pi\otimes\Pi$ is
$D$-compatible.

We note that the Ricci tensor of Gödel metric [23] and Som-Raychaudhuri metric
[62] are Riemann-compatible.
The curvature 2-forms $\Omega_{(D)l}^{m}$ ([3], [35]) associated to a
$(0,4)$-curvature tensor $D$ and the 1-forms $\Lambda_{(Z)l}$ [74] associated
to a symmetric (0, 2)-curvature tensor $Z$ are defined as follows:

$\Omega_{(D)l}^{m}=D_{jkl}^{m}dx^{j}\wedge dx^{k}\mbox{ and
}\Lambda_{(Z)l}=Z_{lm}dx^{m},$

where $\wedge$ indicates the exterior product. Now $\Omega_{(D)l}^{m}$ (resp.,
$\Lambda_{(Z)l}$) is recurrent if

$\mathfrak{D}\Omega_{(D)l}^{m}=\Pi\wedge\Omega_{(D)l}^{m}\mbox{ (resp.,
}\mathfrak{D}\Lambda_{(Z)l}=\Pi\wedge\Lambda_{(Z)l}),$

where $\mathfrak{D}$ is the exterior derivative and $\Pi$ is the associated
$1$-form. Recently Mantica and Suh ([40], [41], [42]) showed that
$\Omega_{(D)l}^{m}$ are recurrent if and only if

$\displaystyle(\nabla_{X_{1}}D)(X_{2},X_{3},X,Y)+(\nabla_{X_{2}}D)(X_{3},X_{1},X,Y)+(\nabla_{X_{3}}D)(X_{1},X_{2},X,Y)=$
(2.1) $\displaystyle\hskip
72.26999pt\Pi(X_{1})D(X_{2},X_{3},X,Y)+\Pi(X_{2})D(X_{3},X_{1},X,Y)+\Pi(X_{3})D(X_{1},X_{2},X,Y)$

and $\Lambda_{(Z)l}$ are recurrent if and only if

$(\nabla_{X_{1}}Z)(X_{2},X)-(\nabla_{X_{2}}Z)(X_{1},X)=\Pi(X_{1})Z(X_{2},X)-\Pi(X_{2})Z(X_{1},X)$
(2.2)

for a 1-form $\Pi$.

##  3\. Various tensors of Vaidya metric

In terms of $(u,r,\theta,\phi)$-coordinates ($r>0$), the Vaidya metric (1.2)
is given by

$\displaystyle g=\left(\begin{array}[]{cccc}-1+\frac{2m}{r}&-1&0&0\\\
-1&0&0&0\\\ 0&0&r^{2}&0\\\ 0&0&0&r^{2}\sin^{2}\theta\end{array}\right).$

Then the non-zero components of its Riemann-Christoffel curvature tensor $R$,
Ricci tensor $S$ and scalar curvature $\kappa$ of (1.2) are given by

$\left.\begin{array}[]{c}R_{1212}=-\frac{2m}{r^{3}},\ \
R_{1313}=-\frac{2m^{2}+r^{2}m^{\prime}-mr}{r^{2}},\ \
R_{1424}=\frac{m\sin^{2}\theta}{r},\\\ R_{1323}=\frac{m}{r},\ \
R_{1414}=-\frac{\left(2m^{2}+r^{2}m^{\prime}-mr\right)\sin^{2}\theta}{r^{2}},\
\ R_{3434}=2mr\sin^{2}\theta,\end{array}\right\\}$ (3.2)

$S_{11}=\frac{2m^{\prime}}{r^{2}}\ \mbox{and }\kappa=0,$ (3.3)

where $m=m(u)$ and $m^{\prime}=\frac{dm}{du}$. The non-zero components of its
conformal curvature tensor $C$ are given by

$\left.\begin{array}[]{c}C_{1212}=-\frac{2m}{r^{3}},\ \
C_{1313}=-\frac{m(2m-r)}{r^{2}},\ \ C_{1323}=\frac{m}{r},\\\
C_{1414}=-\frac{m(2m-r)\sin^{2}\theta}{r^{2}},\ \
C_{1424}=\frac{m\sin^{2}\theta}{r},\ \
C_{3434}=2mr\sin^{2}\theta.\end{array}\right\\}$ (3.4)

now the non-zero components of its $\nabla R$, $\nabla S$ and $\nabla C$ of
(1.2) are given by

$\left.\begin{array}[]{c}R_{1212,1}=-\frac{2m^{\prime}}{r^{3}},\ \
R_{1212,2}=\frac{6m}{r^{4}},\ \
R_{1213,3}=\frac{6m^{2}+r^{2}m^{\prime}-3mr}{r^{3}},\\\
R_{1214,4}=\frac{\sin^{2}(\theta)\left(6m^{2}+r^{2}m^{\prime}-3mr\right)}{r^{3}},\
\ R_{1223,3}=R_{1323,2}=-\frac{3m}{r^{2}},\\\
R_{1224,4}=R_{1424,2}=-\frac{3m\sin^{2}(\theta)}{r^{2}},\ \
R_{1313,1}=-\frac{r^{2}m^{\prime\prime}-rm^{\prime}+4mm^{\prime}}{r^{2}},\\\
R_{1313,2}=\frac{6m^{2}+2r^{2}m^{\prime}-3mr}{r^{3}},\ \
R_{1323,1}=\frac{m^{\prime}}{r},\ \
R_{1414,1}=-\frac{\sin^{2}(\theta)\left(r^{2}m^{\prime\prime}-rm^{\prime}+4mm^{\prime}\right)}{r^{2}},\\\
-R_{1334,4}=r\sin^{2}(\theta)m^{\prime},\ \
R_{1434,3}=r\sin^{2}(\theta)m^{\prime},\ \
\frac{1}{2}R_{3434,1}=r\sin^{2}(\theta)m^{\prime},\\\
R_{1414,2}=\frac{\sin^{2}(\theta)\left(6m^{2}+2r^{2}m^{\prime}-3mr\right)}{r^{3}},\
\ R_{1424,1}=\frac{\sin^{2}(\theta)m^{\prime}}{r},\\\
R_{2334,4}=-R_{2434,3}=-\frac{1}{2}R_{3434,2}=3m\sin^{2}(\theta),\end{array}\right\\}$
(3.5)
$\left.\begin{array}[]{c}S_{11,1}=\frac{2\left(r^{2}m^{\prime\prime}+2mm^{\prime}\right)}{r^{4}},\
\ S_{11,2}=-\frac{4m^{\prime}}{r^{3}},\\\ S_{13,3}=-\frac{2m^{\prime}}{r},\ \
S_{14,4}=-\frac{2\sin^{2}(\theta)m^{\prime}}{r},\end{array}\right\\}$ (3.6)
$\left.\begin{array}[]{c}C_{1212,1}=-\frac{2m^{\prime}}{r^{3}},\ \
C_{1212,2}=\frac{6m}{r^{4}},\ \
C_{1213,3}=C_{1313,2}=\frac{3m(2m-r)}{r^{3}},\\\
C_{1214,4}=C_{1414,2}=\frac{3m\sin^{2}(\theta)(2m-r)}{r^{3}},\\\
C_{1223,3}=C_{1323,2}=-\frac{3m}{r^{2}},\ \
C_{1224,4}=C_{1424,2}=-\frac{3m\sin^{2}(\theta)}{r^{2}},\\\
C_{1313,1}=\frac{(r-2m)m^{\prime}}{r^{2}},\ \
C_{1323,1}=\frac{m^{\prime}}{r},\ \
C_{1414,1}=\frac{\sin^{2}(\theta)(r-2m)m^{\prime}}{r^{2}},\\\
C_{1424,1}=\frac{\sin^{2}(\theta)m^{\prime}}{r},\ \
C_{2334,4}=-C_{2434,3}=-\frac{1}{2}C_{3434,2}=3m\sin^{2}(\theta),\ \
C_{3434,1}=2r\sin^{2}(\theta)m^{\prime}.\end{array}\right\\}$ (3.7)

Now from above we can easily evaluate the non-zero components of $R\cdot R$,
$C\cdot R$, $R\cdot C$, $C\cdot C$, $Q(g,R)$, $Q(S,R)$, $Q(g,C)$ and $Q(S,C)$
as follows:

$\left.\begin{array}[]{c}-2R\cdot R_{121313}=R\cdot
R_{131312}=\frac{4mm^{\prime}}{r^{3}},\ \ R\cdot R_{121323}=-R\cdot
R_{122313}=\frac{3m^{2}}{r^{4}},\\\ -2R\cdot R_{121414}=R\cdot
R_{141412}=\frac{4mm^{\prime}\sin^{2}\theta}{r^{3}},\ \ R\cdot
R_{121424}=-R\cdot R_{122414}=\frac{3m^{2}\sin^{2}\theta}{r^{4}},\\\ -R\cdot
R_{133414}=R\cdot
R_{143413}=\frac{m\left(6m^{2}+4r^{2}m^{\prime}-3mr\right)\sin^{2}\theta}{r^{3}},\\\
R\cdot R_{133424}=-R\cdot R_{143423}=R\cdot R_{233414}=-R\cdot
R_{243413}=\frac{3m^{2}\sin^{2}\theta}{r^{2}},\end{array}\right\\}$ (3.8)
$\left.\begin{array}[]{c}R\cdot C_{121313}=-\frac{3mm^{\prime}}{r^{3}},\ \
R\cdot C_{121323}=-R\cdot C_{122313}=\frac{3m^{2}}{r^{4}},\\\ R\cdot
C_{121414}=-\frac{3mm^{\prime}\sin^{2}\theta}{r^{3}},\ \ R\cdot
C_{121424}=-R\cdot C_{122414}=\frac{3m^{2}\sin^{2}\theta}{r^{4}},\\\ -R\cdot
C_{133414}=R\cdot
C_{143413}=\frac{3m\left(2m^{2}+r^{2}m^{\prime}-mr\right)\sin^{2}\theta}{r^{3}},\\\
R\cdot C_{133424}=-R\cdot C_{143423}=R\cdot C_{233414}=-R\cdot
C_{243413}=\frac{3m^{2}\sin^{2}\theta}{r^{2}},\end{array}\right\\}$ (3.9)
$\left.\begin{array}[]{c}4C\cdot R_{121313}=C\cdot
R_{131312}=\frac{4mm^{\prime}}{r^{3}},\ \ C\cdot R_{121323}=-C\cdot
R_{122313}=\frac{3m^{2}}{r^{4}},\\\ 4C\cdot R_{121414}=C\cdot
R_{141412}=\frac{4mm^{\prime}\sin^{2}\theta}{r^{3}},\ \ C\cdot
R_{121424}=-C\cdot R_{122414}=\frac{3m^{2}\sin^{2}\theta}{r^{4}},\\\ -C\cdot
R_{133414}=C\cdot
R_{143413}=\frac{m\left(6m^{2}+r^{2}m^{\prime}-3mr\right)\sin^{2}\theta}{r^{3}},\\\
$$C\cdot R_{133424}=-C\cdot R_{143423}=C\cdot R_{233414}=-C\cdot
R_{243413}=\frac{3m^{2}\sin^{2}\theta}{r^{2}},\end{array}\right\\}$ (3.10)
$\left.\begin{array}[]{c}C\cdot C_{121323}=-C\cdot
C_{122313}=\frac{3m^{2}}{r^{4}},\ \ C\cdot C_{121424}=-C\cdot
C_{122414}=\frac{3m^{2}\sin^{2}\theta}{r^{4}},\\\ -C\cdot C_{133414}=C\cdot
C_{143413}=\frac{3m^{2}(2m-r)\sin^{2}\theta}{r^{3}},\\\ C\cdot
C_{133424}=-C\cdot C_{143423}=C\cdot C_{233414}=-C\cdot
C_{243413}=\frac{3m^{2}\sin^{2}\theta}{r^{2}},\end{array}\right\\}$ (3.11)
$\left.\begin{array}[]{c}2Q(g,R)_{121313}=-Q(g,R)_{131312}=2m^{\prime},\ \
Q(g,R)_{121323}=-Q(g,R)_{122313}=\frac{3m}{r},\\\
2Q(g,R)_{121414}=-Q(g,R)_{141412}=2m^{\prime}\sin^{2}\theta,\ \
Q(g,R)_{121424}=-Q(g,R)_{122414}=\frac{3m\sin^{2}\theta}{r},\\\
-Q(g,R)_{133414}=Q(g,R)_{143413}=\left(6m^{2}+r^{2}m^{\prime}-3mr\right)\sin^{2}\theta,\\\
Q(g,R)_{133424}=-Q(g,R)_{143423}=Q(g,R)_{233414}=-Q(g,R)_{243413}=3mr\sin^{2}\theta,\end{array}\right\\}$
(3.12)
$\left.\begin{array}[]{c}-2Q(S,R)_{121313}=Q(S,R)_{131312}=\frac{4mm^{\prime}}{r^{3}},\
\
-2Q(S,R)_{121414}=Q(S,R)_{141412}=\frac{4mm^{\prime}\sin^{2}\theta}{r^{3}},\\\
$$-Q(S,R)_{133414}=Q(S,R)_{143413}=\frac{4mm^{\prime}\sin^{2}\theta}{r},\end{array}\right\\}$
(3.13)
$\left.\begin{array}[]{c}Q(g,C)_{121323}=-Q(g,C)_{122313}=\frac{3m}{r},\ \
Q(g,C)_{121424}=-Q(g,C)_{122414}=\frac{3m\sin^{2}\theta}{r},\\\
-Q(g,C)_{133414}=Q(g,C)_{143413}=3m(2m-r)\sin^{2}\theta,\\\
Q(g,C)_{133424}=-Q(g,C)_{143423}=Q(g,C)_{233414}=-Q(g,C)_{243413}=3mr\sin^{2}\theta,\end{array}\right\\}$
(3.14)
$\left.\begin{array}[]{c}-2Q(S,C)_{121313}=Q(S,C)_{131312}=\frac{4mm^{\prime}}{r^{3}},\
\
-2Q(S,C)_{121414}=Q(S,C)_{141412}=\frac{4mm^{\prime}\sin^{2}\theta}{r^{3}},\\\
-Q(S,C)_{133414}=Q(S,C)_{143413}=\frac{4mm^{\prime}\sin^{2}\theta}{r}.\end{array}\right\\}$
(3.15)

Now from Einstein’s field equations with zero cosmological constant, the
energy momentum tensor is given by

$T=\frac{c^{4}}{8\pi G}\left(S-\frac{\kappa}{2}g\right),$

where $c=$ speed of light in vacuum, $G=$ gravitational constant. Then the
only non-zero component (upto symmetry) of $T$ is given by

$T_{11}=\frac{2c^{4}m^{\prime}}{8\pi Gr^{2}}.$ (3.16)

Then the non-zero components of $\nabla T$ are given by

$\left.\begin{array}[]{c}T_{11,1}=\frac{c^{4}\left(r^{2}m^{\prime\prime}+2mm^{\prime}\right)}{4\pi
Gr^{4}},\ \ T_{13,3}=-\frac{c^{4}m^{\prime}}{4\pi Gr},\\\
T_{11,2}=-\frac{c^{4}m^{\prime}}{2\pi Gr^{3}},\ \
T_{14,4}=-\frac{c^{4}\sin^{2}(\theta)m^{\prime}}{4\pi
Gr}.\end{array}\right\\}$ (3.17)

##  4\. Curvature properties of Vaidya metric

From above we can conclude that the metric (1.2) fulfills the following
curvature restricted geometric structures.

######  Theorem 4.1.

The Vaidya metric given in (1.2) possesses the following curvature properties:

  1. (i)

neither Ricci flat nor Ricci symmetric but its scalar curvature is zero and
hence $R=W$ and $C=K$.

  2. (ii)

neither conformally semisymmetric nor Deszcz pseudosymmetric but $C\cdot
C=\frac{m}{r^{3}}Q(g,C)$.

  3. (iii)

neither Ricci generalized pseudosymmetric nor Weyl pseudosymmetric but
satisfies the pseudosymmetric type conditions $R\cdot
R-Q(S,R)=\frac{m}{r^{3}}Q(g,C)$ and $R\cdot C+C\cdot
R=\frac{2m}{r^{3}}Q(g,C)+Q(S,C)$.

  4. (iv)

neither conformally recurrent nor the curvature 2-forms $\Omega_{(R)l}^{m}$
nor the Ricci 1-forms $\Lambda_{(S)l}$ are recurrent but the conformal 2-forms
$\Omega_{(C)l}^{m}$ are recurrent for the 1-form
$\Pi=\left\\{\frac{m^{\prime}}{m},0,0,0\right\\}$.

  5. (v)

not Einstein but it is Ricci simple, such that $S=\beta(\eta\otimes\eta)$,
where $\beta=2m^{\prime}$ and $\eta=\\{\frac{1}{r},0,0,0\\}$ with
$||\eta||=0$. Hence $S\wedge S=0$ and $S^{2}=0$.

  6. (vi)

Ricci tensor is neither cyclic parallel nor Codazzi type but Riemann
compatible as well as Weyl compatible.

  7. (vii)

the general form of the compatible tensors for $R$ and $P$ are given by

$\left(\begin{array}[]{cccc}a_{{}_{11}}&a_{{}_{12}}&0&0\\\
a_{{}_{12}}+\frac{rm^{\prime}}{m}a_{{}_{22}}&a_{{}_{22}}&0&0\\\
0&0&a_{{}_{33}}&a_{{}_{43}}\\\ 0&0&a_{{}_{43}}&a_{{}_{44}}\end{array}\right),$

where $a_{{}_{ij}}$ being arbitrary scalars.

  8. (viii)

the general form of the compatible tensors for $C$ and $K$ are given by

$\left(\begin{array}[]{cccc}a_{{}_{11}}&a_{{}_{12}}&0&0\\\
a_{{}_{12}}&a_{{}_{22}}&0&0\\\ 0&0&a_{{}_{33}}&a_{{}_{34}}\\\
0&0&a_{{}_{34}}&a_{{}_{44}}\end{array}\right).$

###### Proof.

(i) From (3.3), (i) is obvious.
(ii) From (3.8) and (3.12), we get $R\cdot
R_{131312}=-\frac{2m}{r^{3}}Q(g,R)_{131312}$ but $R\cdot
R_{122313}=\frac{m}{r^{3}}Q(g,R)_{122313}$. Hence the Vaidya metric is not
pseudosymmetric. Again from (3.11) and (3.14), we get $C\cdot C\neq 0$ but
$C\cdot C=\frac{m}{r^{3}}Q(g,C)$.
(iii) Again from (3.8), (3.13), (3.9), (3.10), (3.14) and (3.15), one can
easily check the pseusosymmetric type conditions given in (iii).
(iv) Now putting the values of $\nabla C$ (given in (3.7)) and $C$ (given in
(3.4)) and $\Pi=\left\\{\frac{m^{\prime}}{m},0,0,0\right\\}$ in (2.1) for
$D=C$, we get (iv).
(v) It is easily follows from (3.3).
(vi) From (3.3), (3.6) and definition of compatible tensor, we get (vi).
(vii) We can easily check (vii) using components of $R$ and $P$.
(viii) Again (viii) follows from definition of compatible tensor and
components of $C$ and $K$. ∎

######  Remark 4.1.

From (3.16), we see that the energy momentum tensor of the Vaidya metric (1.2)
can be expressed as

$T=\frac{2c^{4}m^{\prime}}{8\pi Gr^{2}}(\eta\otimes\eta),\ \ \mbox{where
}\eta=\\{1,0,0,0\\}.$

Now $||\eta||=0$ and hence the Vaidya metric is a pure radiation metric [81].

######  Remark 4.2.

From the value of the local components (presented in Section 3) of various
tensors of the Vaidya metric (1.2), we can easily conclude that the metric
does not fulfill the following geometric structures:
(i) super generalized recurrent ([65], [71]) (ii) weakly symmetric ([58],
[79]) for $R$, $C$ and $P$, (iii) weakly cyclic Ricci symmetric [56], (iv)
generalized Roter type ([60], [61]), (v) $R$-space, $C$-space or $P$-space by
Venzi [85], (vi) harmonic or conformal harmonic, (vii) Weyl-pseudosymmetric or
Ricci-pseudosymmetric ([13], [14], [17], [21]).

From (1.1) and (1.2), it is clear that the Vaidya metric reduces to
Schwarzschild metric (1.1) if $m(u)$ is a non-zero constant. Hence from
Theorem 4.1, we can conclude the following about the curvature properties of
the Schwarzschild metric (1.1).

######  Corollary 4.1.

The Schwarzschild metric (1.1) possesses the following curvature properties:

  1. (i)

The metric is Ricci flat and hence $R=P=W=C=K$.

  2. (ii)

The metric is harmonic, i.e., $divR=0$.

  3. (iii)

It is Deszcz pseudosymmetric manifold satisfying $R\cdot
R=\frac{m}{r^{3}}Q(g,R)$.

  4. (iv)

The general form of the compatible tensors for $R$ is given by

$\left(\begin{array}[]{cccc}a_{{}_{11}}&a_{{}_{12}}&0&0\\\
a_{{}_{12}}&a_{{}_{22}}&0&0\\\ 0&0&a_{{}_{33}}&a_{{}_{34}}\\\
0&0&a_{{}_{34}}&a_{{}_{44}}\end{array}\right),$

where $a_{{}_{ij}}$ being arbitrary scalars.

######  Remark 4.3.

From (3.16), we see that for Schwarzschild metric (1.1), $T=0$ and hence the
Schwarzschild metric is a vacuum solution of Einstein’s field equations.

##  5\. Comparisons between Vaidya metric and Ludwig-Edgar pure radiation
metric

In 1997 Ludwig and Edgar [36] presented a pure radiation metric which is
conformally related to vacuum space or Ricci flat space. The pure radiation
metric of Ludwig and Edgar ([36], [66]) in $(u;r;x;y)$-coordinates ($r>0$ and
$x>0$) is given by

$ds^{2}=\left(xw-p^{2}\frac{r^{2}}{x^{2}}\right)du^{2}+2dudr-\frac{4r}{x}dudx-\frac{1}{p^{2}}(dx^{2}+dy^{2}),$
(5.1)

where $w$ is a smooth function of $u,x,y$, and $p$ is a non-zero constant.
Recently, Shaikh et al. [66] have studied the curvature properties of the
Ludwig-Edgar pure radiation metric (5.1). Since both the Vaidya metric (1.2)
and Ludwig-Edgar metric (5.1) represent pure radiation fields, in this section
we are mainly interested to make a comparisons between the geometric
properties of Vaidya metric (1.2) and Ludwig-Edgar pure radiation metric
(5.1).

A. Similarities:
(i) Ricci tensors of both are neither Codazzi type nor cyclic parallel but the
scalar curvatures are zero.
(ii) Both are Ricci simple.
(iii) Ricci tensors of both the metrics are Riemann compatible as well as Weyl
compatible.
(iv) conformal curvature 2-forms of both the metrics are recurrent.
B. Dissimilarities:
(i) Ludwig-Edgar pure radiation metric is semisymmetric whereas Vaidya metric
is of pseudosymmetric Weyl conformal curvature tensor.
(ii) Ludwig-Edgar pure radiation metric is weakly Ricci symmetric but Vaidya
metric is not weakly Ricci symmetric.
(iii) Ludwig-Edgar pure radiation metric is $R$-space by Venzi but Vaidya
metric is not $R$-space by Venzi.
(iv) Energy momentum tensor of Ludwig-Edgar pure radiation metric is is
parallel but energy momentum tensor of Vaidya metric is not parallel.

## Acknowledgment

The authors are grateful to the referees for the remarks and comments that
helped to improve the paper. All the algebraic computations of Section $3$ are
performed by a program in Wolfram Mathematica.

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# Towards automated verification of multi-party consensus protocols

Ivan Fedotov  Moscow Institute of Physics and TechnologyRussiaInstitutskiy
Pereulok, 9Dolgoprudny141701<EMAIL_ADDRESS>,  Anton Khritankov
Moscow Institute of Physics and TechnologyRussiaInstitutskiy Pereulok,
9Dolgoprudny141701<EMAIL_ADDRESS>and  Artem Barger  Idea
Blockchain Competence CenterIsraelHaifa<EMAIL_ADDRESS>
###### Abstract.

Blockchain technology and related frameworks have recently received extensive
attention. Blockchain systems use multi-party consensus protocols to reach
agreements on transactions. Hyperledger Fabric framework exposes a multi-party
consensus, based on endorsement policy protocol, to reach a consensus on a
transaction. In this paper, we define a problem of verification of a
blockchain multi-party consensus with probabilistic properties. Further, we
propose a verification technique of endorsement policies using statistical
model checking and hypothesis testing. We analyze several aspects of the
policies, including the ability to assign weights to organizations and the
refusal probabilities of organizations. We demonstrate on experiments the work
of our verification technique and how one can use experimental results to make
the model satisfiable the specification. One can use our technique to design
enterprise applications with the Hyperledger Fabric framework.

Verification, blockchain, consensus protocols, statistical model checking

††ccs: Software and its engineering Formal software verification††ccs:
Software and its engineering Software design engineering††ccs: Information
systems Enterprise applications

##  1\. Introduction

Multi-party consensus protocols nowadays have widespread use in different
application domains: logistic area, internet of things, financial services,
and others. In blockchain applications, multi-party consensus protocol exposes
the necessity to reach a consensus between untrusted parties which need to
agree on changes in the system without the help of a third party.

Since blockchain technology has become popular just recently, frameworks and
tools for testing and verification of blockchain systems are still under
development. Vulnerabilities in blockchain applications and particularly in
multi-party protocols of Hyperledger Fabric can lead to significant losses
(b25, ).

In our work, we address a statistical model checking of multi-party consensus
protocols. Also, we show that one can apply our approach to verify policies in
the Hyperledger Fabric framework (HLF). In particular, this paper makes the
following contributions:

  1. (1)

States the verification problem for endorsement policies

  2. (2)

Provides algorithms to build a model and specification for consensus protocols

  3. (3)

Demonstrates usage of statistical model checking technique to verify consensus
protocols

The remainder of the paper is organized as follows. Section 2 states the
problem and outlines techniques that we used to solve it. In section 3 we
provide related studies on multi-party consensus protocols verification. In
section 4 we explain in detail our algorithms to build a model and
specifications from the properties of the system. We carry out experiments
with further analysis in section 5. Section 6 provides a discussion and the
concluding remarks.

##  2\. Background

In the current section, we describe the transaction confirmation process, the
statement of the problem, and the techniques that we use for model checking.

###  2.1. Transaction Confirmation

Let us consider the transaction confirmation process in Hyperledger Fabric. We
choose a blockchain network based on Fabric, as HLF is one of the most popular
blockchain frameworks among enterprise applications (b10, ). HLF allows
specifying the business logic of transactions through the chaincode. Agreement
on the chaincode proceeds among network members, which called organizations,
in the channel (b24, ). Chaincode has an endorsement policy that specifies a
set of peers on the channel that must execute the chaincode and endorse the
execution results to consider the transaction as valid in the channel.

To specify organizations that might endorse the transaction, we assign a
weight to each organization. Authors discuss that weighted multi-party
consensus protocols provide more flexibility (b13, ) (b15, ). HLF allows using
weighted members in endorsement policies (b12, ).

In weighted consensus protocols, the result depends on the ratio of weights of
organizations that reach the consensus. We formalize the endorsement policy
using a threshold value $Weight_{th}$. If the organizations in the channel
with the total weight greater than $Weight_{th}$ reach the consensus, then the
whole network either accepts or rejects the transaction, depending on the
responses of organizations that reached the consensus.

To define a probability for an organization to agree on a transaction, we
refer to the historical data that contains responses of organizations. One can
collect statistical data on how each organization agrees on a transaction.
Using well-known techniques (b8, ) (b9, ) chaincode designers can compute the
probability to give a refusal response of each organization. The refusal
probability defines the probability that an organization does not accept the
transaction. We introduce a vector of refusal responses for organizations.
Each value in the vector corresponds to the probability to reject a
transaction for a certain organization.

Endorsement policy does not take into account the probabilistic nature of
organizations to reach a consensus. We assign to each organization the
probability to give a response with which the organization does not reach a
consensus.

###  2.2. Model Checking of Probabilistic Systems

We need to use a model checking technique for probabilistic systems which do
not require expensive computations, as usually, designers shall make a
decision fast on deployment chaincode in enterprise applications. Moreover, we
need to bound an error of computation to control the accuracy of the
verification result. We consider different probabilistic model checking
techniques and choose the one that fits the most to our requirements.

Given a structure $M$ and a property $\phi$, the model-checking problem
answers whether $M$ satisfies $\phi$, which one can write in the following
form: $M\models\phi$. We consider specification in a probabilistic Linear
Temporal Logic (pLTL) form. The model-checking problem for pLTL properties
follows the same steps verification steps as with LTL formulas: model gets
composed with the Buchi automaton representation of the verified property, and
then checking that the accepting state is visited infinitely often. In pLTL
formula, a probability that the model satisfies specification is equal to the
probability of reaching a satisfying state. The model checking problem of
probabilistic systems is polynomial in the size of the model but double-
exponential in the size of the LTL formula (b18, ).

Statistical model checking refers to a series of simulation-based techniques
that help to answer two questions: (1) Qualitative: Is the probability that
the model satisfies the specification is greater than a certain threshold? and
(2) Quantitative: What is the probability of the system to satisfy the
specification? (b11, ). This technique fits better for proving the robust
satisfaction of a quantitative property, as the actual probability of
satisfying a given specification needs to be bounded away from the threshold
to which it is compared (b19, ). As we also need to check the probability
threshold to reach a consensus, we use a statistical model checking approach.

Statistical inference comes in one of two flavors: (a) hypothesis testing is
used to determine the extent to which observations “conform” to a given
specification, and (b) estimation is used to determine likely values of
parameters based on the assumption that the data is randomly drawn from a
specified type of distribution. In our modeling, we use the hypothesis testing
approach, as estimation appears to be more empiricist in form. In the
estimation technique, we need to base an estimate on the hypothesized model
that the probabilistic behavior of a system follows some specific probability
distribution. From our statistical data, we can not always argue about the
probability distribution. In the current work, we model the endorsement policy
confirmation system and verify the probability to agree on a transaction.

##  3\. Related Work

Verification using statistical model checking is well studied and numerous
statistical model checking tools are developed (b4, ). Recent researches use
statistical model checking to verify blockchain systems (b5, ), (b6, ). In
provided works, authors study different probabilistic aspects of blockchain
network: peer-to-peer system, involving multiple blockchain participants, DNS
connection. To the best of our knowledge, there is a gap in the verification
of probabilistic aspects of the consensus protocols, which we cover in the
current work.

##  4\. Modeling and verification

In this section, we describe a verification algorithm that takes as an input
parameters of the system and outputs the verification result. Verification
result gives an answer to the question: can organizations approve or reject a
transaction with a certain probability? The algorithm consists of two phases:
constructing a discrete-time Markov chain (DTMC) model and building
specification.

###  4.1. Formal Modeling of Multy-Party Consensus Protocols

The user provides an information about the system with the following
parameters:

  * •

set of organizations

  * •

weight of each organization

  * •

probability of each organization to give refusal response for the transaction.
This value we take from the vector of refusal responses that one can construct
from a historical dataset.

And specification parameters:

  * •

weight threshold $Weight_{th}$. If the total weight of organizations that
reach consensus is more than $Weight_{th}$, then the whole system reaches the
consensus

  * •

probability threshold $Probability_{th}$. If the whole system agrees on the
transaction with the probability of more than $Probability_{th}$, then we
consider that the multi-party protocol is valid.

We consider parameters of the system in the following form: Set(organization,
probability, weight). Each element of the set is a tuple that contains an
organization identifier, the probability to give a refusal response, and the
weight of the organization. Parameters of the specification one can provide as
two values $Weight_{th}$ and $Probability_{th}$. We build a DTMC model from
parameters of the system and pLTL formula from parameters of specification.
After that, one can check that the model satisfies the specification using
statistical model checking.

In Algorithm 1 we show the creation of a DTMC model. First, we take one
organization from the set with a corresponding probability and weight. The
first organization that we choose is the root of our DTMC model. Next, we take
the second organization and assign it to two successors of the root.
Transition to one successor happens with a probability $P$, which equals the
probability that the root’s response is refusal. The second transition happens
with a probability $1-P$, which denotes the probability of the root’s fair
response. After that, we remove the root and successor organizations from the
set and recursively apply the algorithm to each subtree. If we traverse from
the root to the state with the transition with the probability $P$, then the
root’s response is refusal, otherwise, the response is acceptance. Set $nodes$
in Algorithm 1 is a set where we save the nodes of the tree. In the input,
this set is empty, in the output it keeps all nodes of the tree. Each node
keeps the information about the assigned organization, weight, and probability
to give a refusal response of the organization and corresponding predecessor.
Also, a node keeps the information about the parent’s reply: was it refusal or
acceptance. If the reply was not a refusal, then the transition from the
predecessor happens with a probability $1-P$, otherwise, with a probability
$P$. Thus, having a set of nodes one can build a DTMC model. The order of
choosing the organizations to build the tree can be random, as variables of
the considered stochastic process are independent.

Data: Set(organization, Pr, W) orgs, root, Set nodes

Result: Set of nodes for DTMC

if  _orgs is not empty_ then

pop an organization $nextOrg$ with its probability $Pr$ and weight $W$ from
the set;

// build a subtree with a fair response:

create a node $N_{nextOrg_{c}}$ with root as a predecessor, parameters $W$,
$Pr$, and confirmation predecessor’s response;

add the node $N_{nextOrg_{c}}$ with a fair parent’s response to set nodes;

//run the algorithm recursively to build a subtree for the node with a
confirmation //predecessor’s response :

DTMC-Creation(orgs, $N_{nextOrg_{c}}$, nodes);

// build a subtree with a refusal response:

create a node $N_{nextOrg_{r}}$ with root as a predecessor, parameters $W$,
$Pr$, and refusal predecessor’s response;

add the $N_{nextOrg_{r}}$ with a refusal parent’s r response to set nodes;

//run the algorithm recursively to build a subtree for the node with a refusal
//predecessor’s response: DTMC-Creation(orgs, $N_{nextOrg_{r}}$, nodes);

else

return nodes ;

end if

Algorithm 1 DTMC-creation.

###  4.2. Statistical Model Checking of Consensus Protocols

Next, we want to check whether the consensus probability $P_{c}$ is greater
than the threshold $Probability_{th}$. $P_{c}$ is the probability to get a
fair response from organizations with a total weight more than $Weight_{th}$.
If the probability to get fair responses from a necessary number of
organizations $P_{c}$ is greater than the threshold $Probability_{th}$, then
the multi-party protocol is valid.

We assign a value to each node of the DTMC model that corresponds to the total
weight of all fair predecessors’ weights. We check, that the probability to
reach at least one node with the total weight is greater than $Weight_{th}$ is
greater than $Probability_{th}$. In other words, we check that the probability
to reach the consensus of fair organizations with a total weight more than the
weight threshold is greater than the probability threshold. If this property
holds, then the model satisfies the specification, and the multi-party
protocol is valid. Algorithm 2 takes as an input a DTMC model and outputs
nodes with the total weight assignments. The idea is similar to the depth-
first search algorithm. But additionally, we assign the total weight to each
node. The total weight of the node is a sum of the weight of all predecessors
that gave a fair response.

Data: root, nodes

Result: Set of nodes with a total sums assignment

foreach  _successor $N_{s}$ of root_ do

assign to $N_{s}$ the total weight of the root;

if  _root give a fair response_ then

add the weight of its root to the total weight of $N_{s}$

labeling-sum($N_{s}$, nodes);

end foreach

Algorithm 2 labeling-sum

Once we get the set of nodes with the total weights assignments from Algorithm
2, we can build a pLTL specification. We want to reach at least one of the
nodes whose total weight is greater than $Weight_{th}$ with a probability
greater than $Probability_{th}$. We can write a disjunction of each node with
a total weight greater than $Weight_{th}$. Further, before the disjunction we
put operator ”Finally” $F$ and the probabilistic operator
$P>Probability_{th}$. The final specification tells us that at least one node
from the desired set shall be finally reached with a probability greater than
$Probability_{th}$.

One can build a specification to compute a transaction rejection instead of
confirmation. For this instead of computing the total weight of confirmed
responses, one shall compute the total weight of refusal responses and build
the specification based on this value.

Let us illustrates the work of two algorithms with an example. We consider the
model consisting of three organizations:

  * •

Organization 1. Weight 1, Probability 0.07.

  * •

Organization 2. Weight 3, Probability 0.01.

  * •

Organization 3. Weight 2, Probability 0.02.

We also provide a specification parameters: $Probability_{th}$ = 0.95,
$Weight{th}$ = 5. Applying Algorithm 1, one can get a DTMC model from Fig. 1.
Node $O_{1}$ corresponds to organization 1, nodes $O_{2}$, $O_{2}^{\prime}$
correspond to organization 2, and nodes $O_{3}$, $O_{3^{\prime}}$,
$O_{3^{\prime\prime}}$, $O_{3^{\prime\prime}}$ correspond to organization 3.
Nodes $O_{ln}$, where n takes values from 1 to 8, are the leafs of DTMC model.
Near from each node we put the total sum that corresponds to this node. We do
not write the total sum 0.

Using Algorithm 2 we assign each node its total weight. In Fig. 1 we depict
the total weight near each node besides those which total weight is zero.
Further, we choose nodes with a total weight greater than $Weight{th}$ to
build a pLTL formula. We color these nodes green. The corresponding pLTL
specification one can write in the following form:

$P>0.95\;F\;(O_{l4}\;\;OR\;O_{l8})$

With statistical model checking techniques, one can simulate the model. Using
hypothesis testing we get the threshold probability when the model satisfies
the specification. Our model implies the following assumptions. First, the
considered multi-party consensus protocol is ergodistic. We do not consider
changing the probabilities to give a refusal reply during the verification
process. Second, as we take organization for building the DTMC model in an
arbitrary order, we assume that the probabilities of different organizations
are independent.

One can use our algorithm to build consensus protocol models with infinite
paths. For example, if the threshold weight is not reached with a certain
probability on the model, we can make a transition from this leaf to the root.
That problem refers to the optimization task. Initially, we need to introduce
a metric, the expected number of messages that pass through the system before
accepting the transaction. Then, based on this optimization function we can
change the model by adding transitions in such a way, that the model satisfies
the specification with a minimum expected number of messages. The number of
messages is important, as each message can require a fee from the organization
that sends it. This problem is interesting for future research.

Figure 1. Example of a DTMC endorsement policy model

##  5\. Experimental Evaluation

In the current section, we build models and investigate how to change the
parameters of the system to satisfy the specification. The goals of the
experiments are the following. First, we want to demonstrate the work of our
algorithms, and check that the model satisfies invariants of the system.
Second, we want to show that relying on the experimental results one can
modify the system to make it satisfiable the specifications.

Initially, we provide parameters of the system. We need to choose the number
of organizations in our experiments. In the endorsement policy protocol, the
number of organizations can be big, i.e. dozens and even hundreds. But some
organizations can be clients of other ones, which will be responsible for the
reply on the transaction confirmation proposal. For our experiments, we take 3
organizations, similar to the recent research (b1, ).

Next, we choose weight parameters for each organization in such a way that one
organization can not make a decision alone. Also, we take small probabilities
to reject a transaction for each organization, i.e. less than 0.1. If the
probability to reject a transaction is big, then either there were too many
incorrect transactions in the past, or the organization is in an incorrect
state, i.e. the organization is not synchronized with the network. Thus, we
assume that organizations of our system mostly agree with a transaction. We
consider the model similar to one illustrated in Fig. 1.

Further, we choose a statistical model checking tool to verify the model.
Based on the survey (b4, ) we have chosen the tool for our experiments that
satisfy the following parameters:

  * •

based on the hypothesis testing approach

  * •

allows to build a DTMC model and check pLTL specifications

  * •

simulates the system at the level of the language (on-the-fly), which do not
require a memory overhead

  * •

the tool is maintained

The first two points correspond to the requirements of our algorithms. The
third and fourth points facilitate easy integration and usage of the
verification algorithms to Hyperledger Fabric enterprise applications. PRISM
tool (b2, ) satisfy all considered properties. We use PRISM to build a model
as a DTMC and a specification in a pLTL form.

First, we compare verification results based on our algorithms with analytical
computations. In the second experiment, we check that the model correctly
reflects changes in the system. Finally, we investigate probability versus
weight threshold dependency. That shows us how to change the threshold
parameters to make the system satisfiable the specification. One can provide
considered experiments for the verification of HLF endorsement policies.

We took 10000 samples for each simulation and 0.001 accuracy parameter in
PRISM. We run experiments on a 2,3 GHz Dual-Core Intel Core i5 CPU. The memory
limit is 2 GB. We did not put the time limits, as each experiment takes less
than 1 second.

###  5.1. Experimental Evaluation of Verification Algorithms

Initially, we want to prove with experiments that our algorithms for building
models and specifications are valid. We claim that the proposed method to
build a model corresponds to the theoretical computations of probability to
accept a transaction.

The probability threshold for our models is $0.95$, the weight threshold is 5.
Fig. 1 illustrates the model for the experiment. The code of the models is
available in the Github repository (b26, ). We take examples that allow
computing analytically the probability to accept a transaction. We considered
different cases:

  * •

All organizations give the same response on the transaction.

  * •

Organizations 2 and 3 with weights 3 and 2 correspondingly agree on the
transaction with probability 1, organization 1 refuses a transaction with a
probability 1.

  * •

Organization 1 with weight 1 agrees on the transaction with probability 1,
other organizations refuse a transaction with probability 1.

  * •

All organizations agree on the transaction with a probability of 0.5.

We build models for each case and compute the probability that the transaction
gets accepted. As the weight threshold is 5, desired nodes correspond to
states $O_{l4}$ and $O_{l8}$ from the model in Fig. 1.

As expected, in the first case PRISM returns the probability 1 to reach a
consensus. The second experiment also shows probability 1. That corresponds to
the analytical computation, as organizations with a total weight greater than
the threshold accept a transaction with a probability of 1. In the third case,
the probability to accept a transaction is 0, as the majority of organizations
give a refusal response with a probability of 1. The experiment shows the same
result. In the fourth experiment, the probability to accept a transaction is
0.25. We get the same result analytically by multiplying probabilities.

###  5.2. Effect of the Changing System Parameters on the Verification

In this section, we check that changing the system parameters correspond to
the changes in the model and verification result. That proves that the system
corresponds to the constructed model. We considered the following scenarios:

  * •

Invariant 1. The probability to accept transactions decreases for one
organization. We expect that the overall acceptance probability shall decrease
as well.

  * •

Invariant 2. One organization leaves the channel. The probability of the whole
system to accept the transaction shall either remain the same or decrease.

We take the model from Fig. 1 with the same probability and weight thresholds,
$0.95$ and $5$ correspondently.

####  5.2.1. Invariant 1

In the current scenario, we decrease the probability to accept the transaction
for organization $O_{3}$. We provide the dependency of the probability to
accept the transaction for organization 3 versus the probability that the
whole system agrees on the transaction. We decrease the probability to accept
a transaction for an organization iteratively on 0.01.

Figure 2. Probability Threshold versus O3 Acceptance Probability Dependency

We illustrated the experimental result in Fig. 2. The dependency is expected:
with decreasing of acceptance rate of organization 3, the probability
threshold of the whole system to accept the transaction decreases as well. We
see that the dependency is linear, and the discrepancy is less than $0.001$,
which corresponds to the experiment setup.

####  5.2.2. Invariant 2

Now we investigate how changing the number of organizations in the channel
affects the probability to accept the transaction by the whole system. We
remove organization 1, as the total sum of organizations 2 and 3 still equals
the threshold. Applying algorithms 1 and 2, we can get the new model and nodes
on this model reaching which the whole system accepts the transaction. Fig. 3
illustrates this model, the green node is a node returned by Algorithm 2.

Figure 3. DTMC model with 2 organizations

Simulating the model we get that the probability of the whole system to accept
the transaction is $0.9683$. For the system with three organizations the
probability to accept the transaction is $0.9689$. We see that the difference
between acceptance probability is less than the discrepancy in our experiment,
which is $0.001$. Thus, removal organization 1 does not affect the probability
of the whole system accepting the transaction. This is the expected result, as
the total sum of organization 1 with either organization 2 or organization 3
is less than the weight threshold of $5$. Hence, the reply of organization 1
does not affect the probability of the system accepting the transaction. We
also can see it in Fig. 1: the left and the right subtree of the node $O_{1}$
are the same and similar to the tree from Fig. 3.

###  5.3. Relation of the Specification Parameters

In this experiment, we get the dependency of the weight threshold versus the
probability threshold. Table 1 illustrates the dependency. The result is
expected: if the weight threshold is low, then the probability to accept a
transaction is also low, as in our model acceptance probability of each
organization is high. With an increase of the weight threshold, the acceptance
probability decreases. When the weight threshold is greater than 6, then the
probability to refuse transaction will be constant 0, as there are no
organizations in our model with a total weight greater than 6.

Table 1. Weight threshold versus probability threshold dependency  Weight threshold | Probability threshold
---|---
1 | 0.9990 +/-0.0004
2 | 0.9997 +/-0.0005
3 | 0.9989 +/- 0.001
4 | 0.9894 +/- 0.0003
5 | 0.9717 +/- 0.0005
6 | 0.9035 +/- 0.0009
7 | 0

###  5.4. Analysis of the Experimental Results

We proved with experiments that our algorithms for building models and
specifications are valid and invariants of the system remain the same.
Although we cannot provide experimental evaluation for all possible cases, one
can see that the experimental results based on our algorithms correspond to
the analytical computations. We also have shown that if one changes the system
parameters, then the model and verification results also change in the right
direction. Based on provided dependencies, one can choose probability and
weight thresholds in such a way that the model satisfies the specification.

##  6\. Conclusion and Future Work

In this paper, we analyzed the multi-party consensus problem. To check that
the protocol satisfies the specification we proposed algorithms to build a
DTMC model and a pLTL specification. Further, we verified the model using the
statistical model checking approach. One can use proposed algorithms and
experiments to verify endorsement policies in the Hyperledger Fabric
framework, as it allows to assign weights to organizations.

As further research, we want to investigate the scheduling of sending the
confirmation messages on the model with infinite paths. Also, we are going to
collect the data of real-world endorsement policies to provide experiments on
them.

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