##  1 Introduction

The problem of ${T\bar{T}}$-deformation of field theories has sparked
significant interest in a wide range of fields, ranging from statistical
physics to high-energy physics. The unique characteristic of this deformation
is that it is irrelevant but still integrable, which is why it has garnered
special attention [1]-[3].

This deformation is implemented by inserting a composite operator,
${\mathcal{O}}_{T\bar{T}}$ constructed from the determinant of the energy-
momentum tensor. Denoting the action of the deformed theory as

$S_{\lambda}=\int d^{2}x\sqrt{h}\,{\mathcal{L}}_{\lambda}\,,$ (1)

where $\lambda$ is the parameter of deformation, the flow equation that
describes the deformation reads

$\partial_{\lambda}S_{\lambda}=\int\sqrt{h}\,{\mathcal{O}}_{T\bar{T}}\,.$ (2)

The integrability means that while this flow takes us away from the original
theory, certain quantities can still be obtained in terms of the corresponding
values in the original theory. The energy spectrum of the deformed theory is
one of those things.

The dimensionless energy of the deformed theory computed on a cylinder of
circumference $2\pi$ lies in the following spectrum (see e.g. [2] and [3])

${\mathcal{E}}_{n}=\frac{2\pi^{2}}{\lambda}\left(1-\sqrt{1-\frac{2\lambda}{\pi}M_{n}+\frac{\lambda^{2}}{\pi^{2}}J_{n}^{2}}\,\right)\
,\ \begin{cases}M_{n}=\Delta_{n}+\bar{\Delta}_{n}-\frac{c}{12}\,,\\\
J_{n}=\Delta_{n}+\bar{\Delta}_{n}\,,\end{cases}$ (3)

where $c$ is the central charge and $(\Delta_{n},\bar{\Delta}_{n})$ are the
conformal dimensions of the state of the original CFT.

Interestingly, there is an intuitive holographic picture in correspondence
with this special kind of deformation. This holographic prescription, which is
known as the _holography at finite cut-off_ , takes its main ideas from the
holographic renormalization group [4], see also [5]-[8] and [9]-[11] for a
different proposal in the context of UV/IR interpolating backgrounds in string
theory. The proposal suggests removing the asymptotic region of AdS3 and
looking at the field theory which now lives on a finite radius cut-off
(neither at infinity nor at zero, depending on the choice of the coordinate).
The proposal is that the theory living on this finite cut-off coincides with
the deformed theory with ${T\bar{T}}$-insertion. The duality implies that the
radius of this finite cut-off, $r_{c}$ will have a simple relation with the
deformation parameter, $\lambda$. A simple way to check this prescription is
to calculate the energy spectrum, holographically. One can reproduce (3),
computing the quasi-local energy of a BTZ black hole (with mass $M$ and
angular momentum $J$) at a finite cut-off $r_{c}$, [4]. This calculation
yields the proportionality $\lambda\propto r_{c}^{-2}$, as previosly assured.
The proportionality coefficient will be determined in terms of the Newton
constant or equivalently the central charge, which will be explicitly
determined later.

This proposal has been verified through the calculation of other quantities
such as thermodynamic quantities, entanglement entropy ([14]-[20]),
correlation functions ([21], and see [22] for a CFT calculation), and so on.
See also [23]-[30] for recent developments. However, the accurate description
of the duality through the matching of partition functions of field theory and
gravity remains to be explored. In this paper, we will try to explore such an
equivalence in the context of the correspondence between a Liouville theory
and its gravitational dual theory.

The paper is organized as follows:

In section 2 we examine the Liouville theory and its deformation by
${T\bar{T}}$-insertion. Section 3 focuses on the holographic reproduction of
the deformed action, where we present our holographic proposal. Finally,
section 4 concludes this study.

##  2 Liouville theory and its ${T\bar{T}}$-deformation

The simplest model that is nontrivial enough for us to test our proposal
against is the Liouville model, for earlier treatments of deformation in
scalar theory, see [3], [12] and [13].

In this paper, our focus will be on the Liouville field theory with the
following action111Here, we intentionally employed subscript $0$ for the
Lagrangian and action, because we intend to deform the theory by ${T\bar{T}}$
insertion and obtain the higher-order Lagrangian in subsequent advancements.

$S_{0}=\frac{1}{2\kappa}\int
d^{2}x\sqrt{h}{\mathcal{L}}_{0}=\frac{1}{2\kappa}\int
d^{2}x\sqrt{h}(\frac{1}{2}h^{ij}\partial_{i}\phi\partial_{j}\phi+\phi
R_{h})\,.$ (4)

We will suitably set the prefactor $\kappa$ later on.

To simplify the writing of the formulas, we define the two tensors $r_{ij}$
and $s_{ij}$ as follows

$r_{ij}=\partial_{i}\partial_{j}\phi\ ,\
s_{ij}=\partial_{i}\phi\partial_{j}\phi\,,$ (5)

so

${\mathcal{L}}_{0}=\frac{s}{2}+\phi R_{h}\,.$ (6)

We should emphasize that the metric $h_{ij}$ is auxiliary and will choose
properly to give us the action in its appropriate form. We will construct all
contractions of $r_{ij}$ and $s_{ij}$ using this metric, for instance,
$s=h^{ij}s_{ij}$. The equation of motion fixes the curvature tensor as
$R_{h}=r$.

For this theory, the energy-momentum tensor with definition

$T^{0}_{ij}=-\frac{2}{\sqrt{h}}\frac{\delta S_{0}}{\delta
h^{ij}}=h_{ij}{\mathcal{L}}_{0}-2\frac{\delta{\mathcal{L}}_{0}}{\delta
h^{ij}}\,,$ (7)

would be determined as

$T^{0}_{ij}=2(r_{ij}-h_{ij}r)-(s_{ij}-\frac{1}{2}h_{ij}s)\,.$ (8)

In continuation, we work in conformal gauge with the Liouville field as the
rescaling parameter that sets the stage for our future objectives. Covering
the two dimensional base manifold, ${\mathcal{M}}$ with the complex
coordinates $(z,\bar{z})$, the metric reads

$ds^{2}_{\mathcal{M}}=h_{ij}dx^{i}dx^{j}=e^{-\phi(z,\bar{z})}dzd\bar{z}\,.$
(9)

In this gauge

$R_{h}=r=4e^{\phi}\partial\bar{\partial}\phi\ ,\
s=4e^{\phi}\partial\phi\bar{\partial}\phi\,,$ (10)

and defining

$\begin{cases}t=r_{zz}-\frac{1}{2}s_{zz}=(\partial\phi)^{2}-\frac{1}{2}\partial^{2}\phi\\\
\bar{t}=r_{\bar{z}\bar{z}}-\frac{1}{2}s_{\bar{z}\bar{z}}=(\bar{\partial}\phi)^{2}-\frac{1}{2}\bar{\partial}^{2}\phi\,,\end{cases}$
(11)

one explicitly finds that

$T^{0}_{zz}=t\ ,\ T^{0}_{\bar{z}\bar{z}}=\bar{t}\ ,\
T^{0}_{z\bar{z}}=T^{0}_{z\bar{z}}=-\frac{1}{4}re^{-\phi}\,.$ (12)

We are now fully equipped to investigate the ${T\bar{T}}$-deformation of the
theory. Let us start with the basic definition of the deformation operator,
${\mathcal{O}}_{T\bar{T}}$ which is a composite operator defined as

${\mathcal{O}}_{T\bar{T}}=\frac{1}{2}\epsilon^{ik}\epsilon^{j\ell}T^{\lambda}_{ij}T^{\lambda}_{k\ell}\,,$
(13)

where

$T^{\lambda}_{ij}=-\frac{2}{\sqrt{h}}\frac{\delta S_{\lambda}}{\delta
h^{ij}}=h_{ij}{\mathcal{L}}_{\lambda}-2\frac{\delta{\mathcal{L}}_{\lambda}}{\delta
h^{ij}}\,,$ (14)

is the energy momentum tensor of deformed theory. Using this definition one
explicitly finds [12]

${\mathcal{O}}_{T\bar{T}}={\mathcal{L}}_{\lambda}^{2}-2{\mathcal{L}}_{\lambda}\,h^{ij}\frac{\delta{\mathcal{L}}_{\lambda}}{\delta
h^{ij}}+2\epsilon^{ik}\epsilon^{j\ell}\frac{\delta{\mathcal{L}}_{\lambda}}{\delta
h^{ij}}\frac{\delta{\mathcal{L}}_{\lambda}}{\delta h^{k\ell}}\,.$ (15)

Expanding the deformed Lagrangian in powers of parameter of deformation as

${\mathcal{L}}_{\lambda}=\sum_{k=0}\lambda^{k}{\mathcal{L}}_{k}\,,$ (16)

one can perturbatively track the deformation. To the first order, one must
substitute ${\mathcal{L}}_{0}$ in the right-hand side of (15), then solve (2).
Doing so one finds

${\mathcal{L}}_{1}=\frac{1}{2\kappa^{2}}\left[r^{ij}(s_{ij}-r_{ij})+r^{2}-\frac{1}{2}rs-\frac{1}{8}s^{2}\right]+\frac{1}{\kappa^{2}}\left[\phi(r^{ij}-\frac{1}{2}s^{ij})-\frac{1}{2}\phi^{2}R_{h}^{ij}\right]G^{h}_{ij}\,,$
(17)

where $G^{h}_{ij}$ is the Einstein tensor on the two-dimensional base
manifold. since the Einstein tensor vanishes identically in two dimensions,
$G^{h}_{ij}=0$ and the second bracket drops out.

In conformal gauge, one explicitly finds

${\mathcal{L}}_{1}=\frac{1}{2\kappa^{2}}(\frac{r^{2}}{4}e^{-\phi}-4e^{\phi}t\bar{t})\,,$
(18)

and thus

$S_{1}=\frac{1}{4\kappa^{2}}\int
dzd\bar{z}\,(\frac{r^{2}}{4}e^{-2\phi}-4t\bar{t})\,,$ (19)

whereby $S_{1}$ we mean the deformed action of the Liouville theory at the
first order of perturbation in terms of the parameter of deformation.
Therefore, up to the leading order of deformation

$S_{L}=\frac{1}{4\kappa}\int dzd\bar{z}\,e^{-\phi}(\frac{s}{2}+\phi
r)+\frac{\lambda}{4\kappa^{2}}\int
dzd\bar{z}\left(\frac{r^{2}}{4}e^{-2\phi}-4t\bar{t}\right)\,.$ (20)

This is the main result on the field theory side that we seek to replicate
holographically.

##  3 Holography

A rigorous statement of the holographic principle, which is known as the
AdS/CFT correspondence, proposes an equivalence between the partition
functions of a gravitational theory in AdS spacetime and the dual conformal
field theory living on its boundary [31]-[33].

In the context of Liouville theory, holographic investigations have a rich
history; for instance, one can refer to the attempts presented in [34] to
[38]. In this study, we employ the machinery of holographic reconstruction of
spacetime to construct a three-dimensional bulk space in which evaluating the
gravitational on-shell action resembles the Liouville theory in the near-
boundary limit.

Doing so, we utilize the fact that the Liouville action appears as a result of
Weyl rescaling of the metric on the asymptotic boundary of AdS3. The Liouville
field coincides with the field of the Weyl rescaling. Referring to AdS/CFT
correspondence one needs to evaluate the gravitational action in the bulk with
a non-constant radius boundary whose profile is determined by the Liouville
field, to reproduce the Liouville action on it, see [39] for a related study.

To explicitly demonstrate this, let us run the machinery of holographic
reconstruction of spacetime in Fefferman-Graham coordinates, [43], [41] and
[42].

$ds^{2}_{\mathcal{N}}=\frac{d\rho^{2}}{4\rho^{2}}+\frac{1}{\rho}g_{ij}(x,\rho)dx^{i}dx^{j}\,,\,g_{ij}(x,\rho)=g^{(0)}_{ij}+\rho
g^{(1)}_{ij}+\rho^{2}g^{(2)}_{ij}\,.$ (21)

The asymptotic metric $g^{(0)}_{ij}$ is naturally a conformally flat metric in
which we deliberately identify the conformal field with the Liouville field.
In this way, the Liouville field contributes to the reconstruction of the bulk
spacetime. Specifically, the Einstein equation fixes the coefficient
$g^{(1)}_{ij}$ (and then indirectly $g^{(2)}_{ij}$) up to a conserved
symmetric tensor whose trace is proportional to the Ricci scalar. This tensor
serves as the energy-momentum tensor of the Liouville field, establishing the
pathway through which the Liouville field shapes the metric of the bulk space.
More accurately,

$g^{(1)}_{ij}=-\frac{1}{2}\left(g^{(0)}_{ij}R^{(0)}+\kappa
T^{(0)}_{ij}\right)\,,\,g^{(2)}_{ij}=\frac{1}{4}g^{(0)k\ell}g^{(1)}_{ik}g^{(1)}_{j\ell}\,.$
(22)

So adopting the complex coordinate system and setting
$g^{(0)}_{ij}dx^{i}dx^{j}=e^{\phi(z,\bar{z})}dzd\bar{z}$, one explicitly
arrives at

$\displaystyle ds^{2}_{\mathcal{N}}$
$\displaystyle=\frac{d\rho^{2}}{4\rho^{2}}+\frac{1}{2\rho}\left[\left(t\rho+\frac{rt}{8}e^{-2\phi}\rho^{2}\right)dz^{2}+\left(\bar{t}\rho+\frac{r\bar{t}}{8}e^{-2\phi}\rho^{2}\right)d\bar{z}^{2}\right]$
(23)
$\displaystyle+\frac{1}{\rho}\left[e^{\phi}+\frac{r}{4}e^{-\phi}\rho+\frac{1}{4}e^{-\phi}\left(t\bar{t}+\frac{r^{2}}{16}e^{-2\phi}\right)\rho^{2}\right]dzd\bar{z}\,.$
(24)

A key aspect of this holographic completion is that performing the following
change of coordinates brings us to the standard Poincaré coordinates, [44] and
[45]

$r=e^{-\frac{\phi}{2}}(\rho^{-\frac{1}{2}}e^{\phi}+\frac{s}{16}\rho^{\frac{1}{2}}e^{-\phi})\,,$
(25)

and

$(z,\bar{z})\rightarrow(z,\bar{z})+\frac{\rho}{2}\frac{1}{e^{\phi}+\frac{s}{16}\rho
e^{-\phi}}(\bar{\partial}\phi,\partial\phi)\,.$ (26)

The presence of the Liouville field on the right-hand side of the first
relation clarifies why we assert that the evaluation of the gravitational
action in a bulk space with a non-constant radius cutoff may yield the action
of the Liouville theory. We are ready now to evaluate the gravitational
action. This evaluation is presented through the following sub-sections.

###  3.1 Einstein-Hilbert action

The Einstein-Hilbert action reads

$S_{EH}=\frac{1}{2\kappa^{\prime}}\int
d^{3}X\sqrt{G}(R_{G}-2\Lambda)=-\frac{2}{\kappa^{\prime}}\int dzd\bar{z}\int
d\rho\sqrt{G}\,,$ (27)

where $\kappa^{\prime}=8\pi G_{N}$ and the integral over $\rho$ is taken from
$\rho_{c}e^{\phi}$ as mentioned before. A simple calculation yields that up
the linear order of the finite cut-off radius

$S_{EH}=-\frac{V_{2}}{2\kappa^{\prime}}\frac{1}{\rho_{c}}+\frac{1}{8\kappa^{\prime}}\int
dzd\bar{z}\,e^{-\phi}(\phi r+r\log\rho_{c})+\frac{1}{8\kappa^{\prime}}\int
dzd\bar{z}\left(\frac{r^{2}}{16}e^{-2\phi}-t\bar{t}\right)\rho_{c}\,,$ (28)

where $V_{2}$ is the volume of the two-dimensional base manifold,
${\mathcal{M}}$222We only look at the asymptotic terms and hence disregard the
IR divergences for the moment..

###  3.2 Gibbons-Hawking action and the counterterm

Locating the asymptotic boundary at $\rho_{c}e^{\phi}$, the following metric
will be induced on that

$ds^{2}_{ind}=\gamma_{ij}dx^{i}dx^{j}\ ,\
\gamma_{ij}=\frac{1}{4}s_{ij}+\frac{e^{-\phi}}{\rho_{c}}g_{ij}(x,\rho_{c}e^{\phi})\,.$
(29)

The Gibbons-Hawking action reads then

$S_{GH}=-\frac{1}{\kappa^{\prime}}\int dzd\bar{z}\sqrt{\gamma}K\,,$ (30)

where $K$ is the trace of the extrinsic curvature of the finite cut-off
surface. This boundary term should be accompanied by the following simple
counterterm, as is well known.

$S_{c.t.}=-\frac{1}{\kappa^{\prime}}\int dzd\bar{z}\sqrt{\gamma}\,,$ (31)

Up the linear order of the finite cut of radius, these boundary actions read

$\begin{split}&S_{GH}=\frac{V_{2}}{\kappa^{\prime}}\frac{1}{\rho_{c}}+\frac{1}{8\kappa^{\prime}}\int
dzd\bar{z}\,e^{-\phi}(s+2r)\\\ &+\frac{1}{64\kappa^{\prime}}\int
dzd\bar{z}e^{-2\phi}\left[\frac{1}{4}\mathrm{tr}\,(s^{2})-2\mathrm{tr}\,(rs)+2\mathrm{tr}\,(r^{2})-2r^{2}+rs\right]\rho_{c}\,,\end{split}$
(32)

and

$\begin{split}&S_{c.t.}=-\frac{V_{2}}{2\kappa^{\prime}}\frac{1}{\rho_{c}}-\frac{1}{16\kappa^{\prime}}\int
dzd\bar{z}\,e^{-\phi}(s+2r)\\\ &+\frac{1}{128\kappa^{\prime}}\int
dzd\bar{z}e^{-2\phi}\left[-\frac{1}{4}\mathrm{tr}\,(s^{2})+2\mathrm{tr}\,(rs)+2\mathrm{tr}\,(r^{2})-2r^{2}-rs\right]\rho_{c}\,,\end{split}$
(33)

which finally gives

$\begin{split}&S_{GH}+S_{c.t.}=\frac{V_{2}}{2\kappa^{\prime}}\frac{1}{\rho_{c}}+\frac{1}{16\kappa^{\prime}}\int
dzd\bar{z}\,e^{-\phi}(s+2r)\\\ &+\frac{1}{128\kappa^{\prime}}\int
dzd\bar{z}e^{-2\phi}\left[\frac{1}{4}\mathrm{tr}\,(s^{2})-2\mathrm{tr}\,(rs)+6\mathrm{tr}\,(r^{2})-6r^{2}+rs\right]\rho_{c}\,,\end{split}$
(34)

where $\mathrm{tr}\,(s^{2})=h^{ik}h^{j\ell}s_{ij}s_{k\ell}$, and so on. We
would like to highlight the sub-leading term of the boundary terms, as it will
interestingly cancel out in the subsequent steps of the proposal, as follows.

###  3.3 Bending (Willmore) energy of the finite cut-off surface

Upon revisiting the AdS/CFT correspondence one knows that moving along the
radial coordinate of the bulk corresponds to moving along the RG trajectory.
In this sense, the different radii in the bulk are energetically distinct.
Therefore, bending a surface at a finite cutoff radius requires different
energy than bending it at infinity (or at zero radius).

The bending energy of surface ${\mathcal{S}}$ will be determined by the
Willmore function which is the integral of the square of the trace of the
extrinsic curvature on ${\mathcal{S}}$, [46] and [47]

$W=\frac{1}{2}\int_{{\mathcal{S}}}K^{2}\,.$ (35)

Therefore referring to the holography at finite cut-off, one needs to subtract
the following term from the total gravitational action

$S_{W}=\frac{1}{2\kappa^{\prime}}\left(W|_{\rho=\rho_{c}e^{\phi}}-W|_{\rho=0}\right)\,.$
(36)

Interestingly enough

$S_{W}=\frac{1}{128\kappa^{\prime}}\int
dzd\bar{z}e^{-2\phi}\left[\frac{1}{4}\mathrm{tr}\,(s^{2})-2\mathrm{tr}\,(rs)+6\mathrm{tr}\,(r^{2})-6r^{2}+rs\right]\rho_{c}\,,$
(37)

and this is exactly what we get from the Gibbons-Hawking action plus the
counterterm at ${\mathcal{O}}(\rho_{c}^{1})$ order, (34).

Putting things together and dropping the total derivative terms, one
ultimately gets

$\begin{split}S_{grav}&=S_{EH}+S_{GH}+S_{c.t.}-S_{W}\\\
&=\frac{1}{8\kappa^{\prime}}\int dzd\bar{z}\,e^{-\phi}(\frac{s}{2}+\phi
r)+\frac{1}{32\kappa^{\prime}r_{c}^{2}}\int
dzd\bar{z}\left(\frac{r^{2}}{4}e^{-2\phi}-4t\bar{t}\right)\,.\end{split}$ (38)

Comparing this result with (20), it becomes evident that we have reproduced
the deformed Liouville action holographically. One simply needs to take the
final steps and match the prefactors in these two structurally equivalent
actions. This is what we will address in the next sub-section.

###  3.4 Matching the actions via AdS/CFT correspondance

As mentioned above, the gravitational action, when taking the bending energy
of the boundary into account, fully replicates the deformed Liouville action.
By demanding the equality of the prefactors, one finds

$\text{At the order of $\lambda^{0}$:}\ \kappa=2\kappa^{\prime}=16\pi
G_{N}=\frac{24\pi}{c}\,,$ (39)

where in the last step we have employed the Brown-Hennaux relation,
$\frac{1}{2G_{N}}=\frac{c}{3}$ [48].

Equating the actions at the next order gives333One should note that our
definition of ${\mathcal{O}}_{T\bar{T}}$ differs from [4] by some numerical
factors, this causes a numerically different relation between $\lambda$ and
$r_{c}$.

$\text{At the order of $\lambda^{1}$:}\
\lambda=\frac{\kappa^{\prime}}{4r_{c}^{2}}=\frac{2\pi
G_{N}}{r_{c}^{2}}=\frac{3\pi}{cr_{c}^{2}}\,.$ (40)

The obtained pre-factors are consistent with the expected values and have been
presented in the literature using some alternative calculations. Presenting
these results we conclude our calculations at this point.

##  4 Conclusion

In this paper, we have developed a gravitational framework that enables the
holographic reproduction of the action of the field theory deformed by
${T\bar{T}}$-insertion. The significance of this work lies in the fact that
the gravitational action in the usual holographic reconstruction of spacetime
is scheme dependent, resulting in finite undetermined terms. However, in
pursuit of our new objective which is investigating the deformed theory within
the holographic framework of the finite cut-off proposal, these terms find
crucial roles. Therefore one needs to design a precise and unambiguous scheme
to appropriately produce the action of the deformed field theory on the
asymptotic boundary.

A very crucial point in designing this scheme is that bending the boundary in
a region with a finite radius in bulk space introduces the Willmore energy on
the surface, and we must incorporate this term into the total gravitational
action. Introducing this bending term appropriately is crucial for accurately
reproducing the action of the field theory. Adding this term provides us with
a very simple recipe for the regularization we need to impose to get the
action of deformed theory. In fact, in this scheme, all that we need is to
calculate the contribution of the volume part of the gravitational action at
the leading order of deformation. By carefully considering the Willmore
energy, all unintended boundary terms at the leading order of deformation
cancel each other out, leaving us with the deformed field theory action. To
the best of our knowledge, this procedure is entirely novel, and we have
tested it for the Liouville field theory, which is simple yet rich enough and
nontrivial.

Given that knowing the action and partition function is the first step in
generic calculations in field theory and holography, the results we have
obtained and the scheme we have fixed can be utilized in many other
calculations. For example, one could explore the modified gravitational
actions in this scheme and examine their dual deformed theories. Additionally,
further exploration of certain quantum information quantities such as the
quantum complexity, which is closely bound up with evaluation of the
gravitational action, could be among the next objectives of this study. The
next step of these calculations could be a thorough investigation of deformed
theories that, in their original form, exhibit specific symmetries, such as
non-relativistic theories. We set aside these studies for future works.

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# Utility Max-Min Fair Link Adaptation in IEEE 802.11ac Downlink Multi-User

Ali A. Khavasi, Mojtaba Aajami, Hae-Ryeon Park and Jung-Bong Suk Manuscript
received January 31, 2014 —————-.Ali A. Khavasi1, Mojtaba Aajami2, Hae-Ryeon
Park3 and Jung-Bong Suk4 are with the Department of Computer and
Telecommunications Engineering, Yonsei University, Korea (e-mail:
<EMAIL_ADDRESS>haerin203@hanmail.net).

###### Abstract

In this letter, we propose a novel model and corresponding algorithms to
address the optimal utility max-min fair link adaptation in Downlink Multi-
User (DL-MU) feature of the emerging IEEE 802.11ac WLAN standard. Herein, we
first propose a simple yet accurate model to formulate the max-min fair link
adaptation problem. Furthermore, this model guarantees the minimum utility
gain of each receiver according to its requirements. In the second step, we
show that the optimal solution of the proposed model can be obtained in
polynomial time, and then the solution algorithms are proposed and analyzed.
The simulation results demonstrate the significant achievement of the proposed
utility-aware link adaptation approach in terms of max-min fairness and
utility gain compared to utility-oblivious schemes.

###### Index Terms:

IEEE 802.11ac, DL-MU, utility max-min fairness, link adaptation

##  I Introduction

Utility max-min (UMM) fairness has been applied in numerous applications since
the initial work by Cao and Zegura [1], who attempt to find an optimal
resource allocation such that the utilities of all receivers are equal. Hence,
the social welfare in the network is equalized. More precisely, this can be
achieved by solving the following optimization problem:

$\mathbf{\mathbf{UMM}:}\max_{\rho_{r}^{i}}\min_{r\in R}U_{r}(\rho_{r}^{i})$
(1)

$s.t.\begin{cases}\sum_{r=1}^{R}p_{r}^{i}\leq P_{T}\\\ \forall
m_{r}^{j},\,j=1\ldots M\end{cases}$ (2)

where $r\,(r=1,2,..R|$$R:$ the number of receivers$)$ is the receiver index,
$\rho_{r}^{i}\in\left\\{\rho_{r}^{1},\rho_{r}^{2},\ldots\right\\}$ is defined
as the $i\textrm{th}$ policy ($i=1,2,\ldots$) of all available link adaptation
policies, which further determines the selected Modulation and Coding Scheme
(MCS) $m_{r}^{j}$ ($j=1\ldots M$$|M:$ the number of total available MCS) and
the allocated power $p_{r}^{i}$ ($p_{r}^{i}\in[0..P_{T}]\textrm{$|P_{T}:$the
total available transmission power})$, and finally $U_{r}(\rho_{r}^{i})$ is
the utility of receiver $r$ due to the applied policy $\rho_{r}^{i}$. Through
this letter, we assume each $U_{r}$ is a positive, non-decreasing and
normalized utility function of $\rho_{r}^{i}$ [2]. The max-min problem
$\mathbf{\mathbf{UMM}}$, has a unique optimal solution under a critical
assumption of strict high Signal to Interference and Noise Ratio (SINR) and
concavity of the receiver utilities $U_{r}(\rho_{r}^{i})$. However, there are
three impediments in applying the problem formulation (1). First, the
concavity assumption does not hold for real-time applications such as video
transmission and Voice over IP (VoIP). The utility of these applications can
be approximated by a step or a sigmoid function. Moreover, the high SINR
assumption is not guaranteed in WLANs. As a result, the optimization problem
(1), in general, is a non-convex problem and cannot be reduced to Geometric
Programming (GP). Indeed, it belongs to the class of problems known as
Complementary GP that is, in general, an NP-hard or that may require a very
complex algorithm to solve [3]. Second, real-time applications that need a
certain amount of transmission resources to provide the minimum required
satisfaction (acceptance) suffer from so called “bandwidth starvation”. Third,
in applying the theoretical formulas to predict the bit rate and corresponding
Frame Error Rate (FER) associated with the link adaptation policy
$\rho_{r}^{i}$, there are simply too many ways in which the observed
measurements and the actual performance fail to match the predictions of the
theory [4]. Through this letter, it is shown that the proposed scheme
efficiently addresses these three impediments.

The IEEE 802.11ac is developed by extending the air–interface techniques of
the 802.11n standard. Among the offered features, the DL-MU exploits the
beamforming technique to steer signal maxima on certain receivers while
suppressing, or at least greatly reducing the corresponding interference over
the others. The channel state information (CSI) of the target receivers needs
to be known at the transmitter, and it is provided by the channel estimation
module in the standard [5].

In this letter, we propose a novel, practical, and optimal link adaptation
algorithm in order to achieve the max-min fair utility distribution among the
receivers in the IEEE 802.11ac DL-MU.

##  II Proposed Model and Theoretical Explanations

We first start with devising a model to predict the FER corresponding to each
selected policy $\rho_{r}^{i}$ in order to address the third aforementioned
impediment. In [4], the authors propose a model, which, by using the concept
of effective SNR, performs FER prediction for each choice of $\rho_{r}^{i}$.
We further extend their model to include beamforming and to match the 802.11ac
specification [5], as follows. In accordance with the precoding scheme applied
by the transmitter, the SNR of $l$th subcarrier of the receiver $r$ (denoted
by $\varsigma_{l_{r}}$), is obtained based on the available CSI at the
transmitter. For instance, in the case of zero-forcing, the average SNR value
of the certain subcarrier $l$ is calculated as

$\varsigma_{l_{r}}=\frac{p_{r}^{i}\times||\mathbf{h}_{l_{r}}\mathbf{w}_{l_{r}}||^{2}}{\sigma^{2}}$
(3)

where $||\cdot||$ is the norm of the vector, $\sigma^{2}$ is the observed
noise variance at the receiver, and
$\mathbf{h}_{l_{k}}=(h_{l_{r1}},h_{l_{r2}},\cdots,h_{l_{rR}})$ is the vector
of channel coefficients from $R$ antennae of the transmitter to the single
antenna of receiver $r$. Additionally, $\mathbf{w}_{l_{r}}$ is the unit-norm
beamforming weight vector
($\mathbf{w}_{l_{r}}=(w_{l_{r1}},w_{l_{r2}},\cdots,w_{l_{rR}})$) for receiver
$r$. Using (3), the BER of the wideband downlink channel $b^{m_{r}^{j}}$ for
modulation of $m_{r}^{j}$ is estimated by

$b^{m_{r}^{j}}=\frac{1}{L}\sum_{l=1}^{L}BER^{m_{r}^{j}}(\varsigma_{l_{r}})$
(4)

where $L$ is the total number of data subcarriers, and the BER of the
subcarrier $l$ is denoted by $BER^{m_{r}^{j}}(\varsigma_{l_{r}})$ as a
function of the symbol SNR (here, $\varsigma_{l_{r}}$) in the narrowband
channel [6]. Note that $b^{m_{r}^{j}}$ is the uncoded bit error probability
for the modulation $m_{r}^{j}$. In order to obtain the upper bound of the
coded FER, we use the union bound on the first-event error probability
$E_{u}^{c}$. By assuming that the frame is transmitted using the convolutional
code $c$, then $E_{u}^{c}=\sum_{d=d_{free}}^{\infty}a_{d}.E_{d}$, where
$d_{free}$ is the free distance of the code $c$, $a_{d}$ is the total number
of error events of weight $d$, and $E_{d}$ is the probability of an incorrect
path of the hamming distance $d$ when it diverges from the correct path and
then re-merges with it sometime later. The value of $a_{d}$ for a specific
convolutional code is generated using its transfer function, and $E_{d}$ is
obtained by

$E_{d}=\begin{cases}\sum_{k=\frac{(d+1)}{2}}^{d}\binom{d}{k}\cdot(b^{m_{r}^{j}})^{k}\cdot(1-b^{m_{r}^{j}})^{d-k},\>\textrm{if
d is odd}&\textrm{}\\\ \\\
\frac{1}{2}\cdot\binom{d}{\frac{d}{2}}\cdot(b^{m_{r}^{j}})^{\frac{d}{2}}\cdot(1-b^{m_{r}^{j}})^{\frac{d}{2}}+\\\
\,\sum_{k=\frac{(d+1)}{2}}^{d}\binom{d}{k}\cdot(b^{m_{r}^{j}})^{k}\cdot(1-b^{m_{r}^{j}})^{d-k},\>\textrm{if
d is even}&\textrm{}\end{cases}$ (5)

Therefore, by applying the policy $\rho_{r}^{i}$, the predicted upper bound of
FER on wideband channel (here, $FER_{eff}$) for a frame of length $L_{f}$ is

$FER_{eff}\leq 1-(1-E_{u}^{c})^{L_{f}}$ (6)

Using (4) and (6), we employ a method similar to [4] in order to construct the
FER prediction model. In this method, based on the available CSI, for each
non-overlapping range of the transmit power level $p_{r}^{\Xi}$
($\Xi\in(p^{min},p^{max}]$$|min,max\in[1,2,...]$ and $min<max$), there exists
an MCS $m_{r}^{\Theta}$ ($\Theta\in[1\ldots M]$) such that we predict $\leq
FER_{eff,r}^{\Xi}$ frame error rate. Note that for each $p_{r}^{\Xi}$ (and
therefore corresponding effective SNR) only one MCS $m_{r}^{\Theta}$ results
in FER between $1-(\frac{1}{2})^{L_{f}}$ (useless) and $0$ (lossless). We
assume that $p_{r}^{\Xi}$ has the maximum value in the range, unless stated
otherwise. Subsequently, the utility value $U_{r}(\rho_{r}^{\Xi})$ associated
with each tuple $\rho_{r}^{\Xi}(p_{r}^{\Xi},m_{r}^{\Theta},FER_{eff,r}^{\Xi})$
is calculated using its specific function. On-site calibration is necessary to
obtain tuples of the table
$\Phi_{r}(p_{r}^{\Xi},m_{r}^{\Theta},FER_{eff,r}^{\Xi})$ of length $L_{\Phi}$
(the number of tuples) for each receiver $r$.

Now, we propose a new model to address the second aforementioned impediment.
First, we attempt to determine the policies that satisfy the user-defined
constraints. Such constraints are crucial, especially for real-time
applications that need a certain amount of transmission resources to provide
the required quantity of utility. We are interested in the policy
$\rho_{r}^{min}$ with the allocated power $p_{r}^{min}$ for each receiver $r$
such that the minimum utility requirement is satisfied. Accordingly, the
minimization problem with the object function of $U_{r}(\rho_{r}^{\Xi})$ can
be expressed as

$\rho_{r\in
R}^{min}=\min_{p_{r}^{\Xi}}(\arg\min_{\rho_{r}^{\Xi}\in\Phi_{r}}\quad
U_{r}(\rho_{r}^{\Xi}))$ (7)

$s.t.\begin{cases}\sum_{all\,receivers}p_{r}^{\Xi}\leq P_{T}\\\
p_{r}^{\Xi}\geq 0\\\ \forall m_{r}^{\Theta},\,\Theta=1\ldots M\\\
U_{min,r}\leq U_{r}(\rho_{r}^{\Xi})\leq 1\end{cases}$ (8)

where $U_{min,r}$ is the user-defined minimum value of the required utility.
Once the minimum required utility is granted to each receiver by (7), we
attempt to maximize all utilities equally using the $\mathbf{\mathbf{UMM}}$
problem definition. Therefore, we define the utility max-min fair link
adaptation problem with the object function of
$O_{r}^{\Xi}(U_{r}(\rho_{r}^{\Xi}))=(U_{r}(\rho_{r}^{\Xi})-U_{min,r})$ as

$\max_{\rho_{r}^{\Xi}\in\Phi_{r}}\min_{r\in
R}(U_{r}(\rho_{r}^{\Xi})-U_{min,r})$ (9)

$s.t.\begin{cases}\sum_{all\,receivers}p_{r}^{\Xi}\leq P_{T}\\\
p_{r}^{\Xi}\geq p_{r}^{min}\\\ \forall m_{r}^{\Theta},\,\Theta=1\ldots M\\\
U_{min,r}\leq U_{r}(\rho_{r}^{\Xi})\leq 1\end{cases}$ (10)

We state the polynomial time complexity of the optimal solution to the problem
(9) in the following theorem.

###### Theorem 1

The optimal solution of the addressed problem (9) can be obtained in $O(R\cdot
L_{\Phi}\cdot\log(L_{\Phi}))$.

###### Proof:

If the resources can be allocated in advance, the optimal max-min fair
solution is achieved by using the progressive filling algorithm [7]. In this
case, the algorithm starts with all utility gains equal to 0 and increases
them together at the same pace, until one or several limits are reached. It
continues to increase the gains for other receivers until it is not possible
to further increase any utility. If the policies are sorted descending by
their utility values (gains), in the worst case it will take $O(R\cdot
L_{\Phi})$ for the algorithm to accomplish this task. If this is not the case,
policies can be sorted in advance, and therefore the time complexity is on the
order of $O(R\cdot L_{\Phi}\cdot\log(L_{\Phi}))$.

It is worth noting that the multi-rate WLAN suffers from performance anomaly
when low-rate streams occupy most of the shared channel time. According to
[8], allocating same channel occupation time to the different streams solves
this issue and maintains channel time fairness between the streams. In the
802.11ac DL-MU transmission, all streams have the same TXOP time which implies
the channel time fairness amongst them in DL-MU transmissions. ∎

##  III Solution Algorithms

Algorithm 1 Finding the policy $\rho_{r}^{min}$

⬇

1 for r=1:R

2 $\Phi_{r}^{sel}$ = select array of tuples from $\Phi_{r}$

where: $\Phi_{r}^{\Xi}.U_{r}^{\Xi}\geq U_{min,r}$;

3 $\rho_{r}^{min}$ = min($\Phi_{r}^{sel}.p_{r}^{\Xi})$;

4 remove tuples with $U_{r}^{\Xi}\leq\rho_{r}^{min}.U_{r}^{\Xi}$ from
$\Phi_{r}^{sel}$;

5 end

6 if $\overset{R}{\underset{r=1}{\sum}}\rho_{r}^{min}.p_{r}^{\Xi}\leq P_{T}$

7 return $\Phi_{\\{r\in R\\}}^{sel}$ and $\rho_{\\{r\in R\\}}^{min}$;

8 end

Algorithm 2 Utility max-min fair solution to the problem (9)

⬇

1 $\textrm{index}_{\Phi}$(1:R)=2; $\rho_{\\{r\in R\\}}^{sel}=\rho_{\\{r\in
R\\}}^{min}$; $O_{\\{r\in R\\}}$=Inf(R,1);

2 for i=1:R

3 if $\Phi_{i}^{sel}.L_{\Phi}\geq 2$

4 $O_{i}$ = $\Phi_{i}^{sel}\\{\textrm{index}_{\Phi}(i)\\}$.$U_{i}^{\Xi}$ -
$\rho_{i}^{min}$.$U_{i}^{\Xi}$;

5 end

6 end

7 while true

8 [newmin, j] = min($O_{\\{r\in R\\}}$);

9 if newmin == Inf

10 return $\rho_{\\{r\in R\\}}^{sel}$;

11 end

12 if
$(\Phi_{j}^{sel}\\{\textrm{index}_{\Phi}(j)\\}.p_{j}^{\Xi}+\underset{r\neq
j}{\sum}\rho_{r}^{sel}.p_{r}^{\Xi})\leq P_{T}$

13 $\rho_{j}^{sel}$ = $\Phi_{j}^{sel}\\{\textrm{index}_{\Phi}(j)\\}$;

14 $\textrm{index}_{\Phi}(j)$ = $\textrm{index}_{\Phi}(j)+1$;

15 if $\textrm{index}_{\Phi}(j)\leq\Phi_{j}^{sel}.L_{\Phi}$

16 $O_{j}$ =
$\Phi_{j}^{sel}\\{\textrm{index}_{\Phi}(j)\\}.U_{j}^{\Xi}-\rho_{j}^{min}.U_{j}^{\Xi}$;

17 else

18 $O_{j}$ = Inf;

19 end

20 else

21 $O_{j}$ = Inf;

22 end

23 end

Theorem 1 states the existence of the polynomial-time solution algorithms to
problem (9) and, therefore, the first aforementioned impediment is addressed
by the proposed scheme. Here, we propose two algorithms to obtain the
solutions to the problems (7) and (9). First, we propose Algorithm 1, in
MATLAB pseudo-code, which provides the solution to the problem (7). Then,
Algorithm 2 gives the solution to the problem (9) by exploiting the previously
obtained result. In the following theorem, we state that the given solution is
optimal.

###### Theorem 2

Algorithm 2 provides the optimal solution to the problem (9) in $O(R\cdot
L_{\Phi}\cdot\log(L_{\Phi}))$.

###### Proof:

The progressive filling algorithm guarantees a max-min fair distribution of
the utilities. We show that Algorithm 2 follows the problem formulation (9)
and the progressive filling algorithm. The constraint $\sum_{\textrm{all
receivers}}p_{r}^{\Xi}\leq P_{T}$ is ensured in line 12. Note that the other
constraints are already met in tables $\Phi_{r}$ and $\Phi_{r}^{sel}$, which
is previously defined in line 2 of Algorithm 1. Line 1 initiates the
progressive filling by setting all utility gains to 0
($\rho_{r}^{sel}=\rho_{r}^{min}\Longrightarrow
O_{r}^{sel}(U_{r}(\rho_{r}^{sel})))$$=U_{r}(\rho_{r}^{sel})-U_{min,r}=0$).
Lines 7-23 aim to maximize the objective function
$O_{r}^{\Xi}(U_{r}(\rho_{r}^{\Xi}))$ by increasing all gains iteratively at
the same pace. Lines 18 and 21 ensure the exclusion of the receivers that
reach the capacity limits. The iteration (lines 7-23) continues until it is
not possible to further increase the objective function (lines 9-11). ∎

###### Remark 1

Algorithm 1 is on the order of $O(R\cdot L_{\Phi})$ and Algorithm 2 runs in
$O(R\cdot L_{\Phi}\cdot\log(L_{\Phi}))$.

##  IV Simulation Results

Fig. 1: Jain’s fairness index for two compared schemes: Proposed and EPA Fig. 2: Utility average of two compared schemes: Proposed and Maximum Utility  TABLE I: Utility Functions of Different Applications Application | Utility Function | Calibration Parameters
---|---|---
VoIP | $U_{1}^{\Xi}=(1-FER_{eff,r}^{\Xi})\cdot\underset{i=0}{\overset{L_{T}}{\sum}}\alpha_{i}\cdot X(rate(m_{r}^{\theta}),l_{i})$ |  $X(x,l_{i})=\begin{cases}1&x\in l_{i}\\\ 0&x\notin l_{i}\end{cases}$, where $l_{i}:\textrm{ required rate of the quality/utility level}$
$L_{T}:\textrm{total number of levels}$, $\alpha_{i}:\textrm{scaling parameter
of }l_{i}$
Video Streaming | $U_{2}^{\Xi}=(1-FER_{eff,r}^{\Xi})\cdot\frac{1}{1+(\frac{1}{\epsilon}-1)\cdot e^{-\beta\cdot rate(m_{r}^{\theta})}}$ |  $\beta=\frac{2\cdot\ln(\frac{1}{\epsilon}-1)}{RATE_{max}}$, where $RATE_{max}:\textrm{maximal rate requirement}$
$\epsilon:\textrm{minimum of utility gain achievable by }RATE_{min}$,
$RATE_{min}=\textrm{minimal rate requirement}$
File Transfer | $U_{3}^{\Xi}=(1-FER_{eff,r}^{\Xi})\cdot\frac{\log(rate(m_{r}^{\theta})+1)}{\log(RATE_{max}+1)}$ | $RATE_{max}:\textrm{maximal rate requirement}$
Online Gaming | $U_{4}^{\Xi}=(1-FER_{eff,r}^{\Xi})\cdot\frac{1}{1+(\frac{1}{\epsilon}-1)\cdot e^{-\gamma\cdot rate(m_{r}^{\theta})}}$ |  $\gamma=\frac{1}{\underset{i=1}{\overset{N}{\sum}}\frac{t_{i}}{\gamma_{i}}}$, $\gamma_{i}=\frac{2\cdot\ln(\frac{1}{\epsilon}-1)}{RATE_{max,i}}$, where $RATE_{max,i}:\textrm{intrinsic rate of }t_{i}$
$t_{i}:\textrm{traffic proportion of applications,}$ ($1\leq i\leq
N$—$N:\textrm{number of applications}$)
$\underset{i=1}{\overset{N}{\sum}}t_{i}=1$ and $\textrm{maximal rate
requirement: }RATE_{max}=\underset{i=1}{\overset{N}{\sum}}t_{i}\cdot
RATE_{max,i}$

In this section, we numerically evaluate the performance of the proposed
optimal algorithm for the utility max-min fair link adaptation in MATLAB. The
downlink channel setup is modeled based on the TGn channel models. We use
profile B (residential) of those models in our simulations. In order to ensure
that the model is appropriate for IEEE 802.11ac scenarios, we modified some of
its parameters, including the Doppler component, the angle of arrival (AoA),
and the angle of departure (AoD) according to the IEEE 802.11ac task group
recommendations. The system has been configured to operate at the 5.25-GHz
carrier frequency with a bandwidth of $W=20$ MHz subdivided into 64
subcarriers, 52 of which are used to carry data. There are nine different
MCSs, which results in transmission rates (i.e., $rate(m_{r}^{\theta})$)
ranging from 6.5 Mbps (BPSK, 1/2) to 78 Mbps (256-QAM, 5/6). The transmitter
is equipped with four antennae, and each of the four receivers has only one
antenna. We exploit four distinct utility functions, VoIP, video streaming,
file transfer, and online gaming applications, using suggested functions in
[2] and [9]. These utility functions are described in Table I. The calibration
parameters for video streaming, file transfer and online gaming are set
according to [9]. The VoIP utility function is calibrated by the following
parameters (refer to [10] for details): $L_{T}=3$,
$\alpha_{i}\in\\{\alpha_{1}=0.92,\alpha_{2}=0.95,\alpha_{3}=1\\}$,
$l_{i}\in\\{l_{1}=\\{[21,32)\\},l_{2}=\\{[32,88)\\},l_{3}=\\{[88,\infty)\\}\\}$
(all in Kbps) and $U_{min,1}=0.7$ (for other receivers, $U_{min,2}=0.5$,
$U_{min,3}=0.4$ and $U_{min,4}=0.4$) . We have concentrated on the long-time
simulation scale of 20000 DL-MU transmissions, and for the presented results
in this section, the width of the 95% confidence interval of the true mean is
less than 5% of each plotted value.

The proposed scheme is compared with the equal power allocation (EPA) and
maximum utility schemes. The EPA scheme allocates the same (max-min fair)
power level to all receivers, while the maximum utility scheme selects the
link adaptation policies in the manner of maximizing the total utility gain
(performance gain). First, we validate the proposed scheme in terms of the
Jain’s fairness index for the objective function $O_{r}^{\Xi}$’s gap between
the minimum receiver utility $U_{min,r}$ and the increased max-min fair
utility determined by applying policy $\rho_{r}^{sel}$ in order to obtain a
quantified measure. Fig. 1 illustrates the measured results for two compared
schemes. The horizontal axis is time steps (A-MPDU transmissions) and the
vertical axis shows the corresponding fairness index. The index ranges from
$\tfrac{1}{R}$ (worst case) to $1$ (best case) and is maximized when all
receivers gain a max-min fair proportion of the utilities. We deduce that the
proposed scheme guarantees a similar utility gap among the receivers while EPA
performs worse with regard to fairness. Second, we indicate the efficiency of
the proposed scheme by presenting the average utility gain of the proposed and
maximum utility schemes. Fig. 2 demonstrates the averaged utility gains for
each compared scheme (the horizontal axis is same as in Fig. 1 and the
vertical axis is for utility score). Moreover, Table II represents the utility
of each receiver in average (to interpret the results, note that:
$U_{min,1}=0.7$, $U_{min,2}=0.5$, $U_{min,3}=0.4$ and $U_{min,4}=0.4$). We
observe that the maximum utility scheme selects the policies to maximize the
total utility, resulting in the best utility gain. In contrast, the proposed
scheme (max-min fair) achieves the lower yet relatively close (see Fig. 2 and
Table II) utility gain by paying the price of providing the max-min fairness
and utility gain efficiency combined. We conclude that the proposed scheme
achieves 95% of the best possible total utility gain. As analyzed before, the
simulation results numerically confirm and validate the max-min fair and
efficient utility gain of the receivers in the proposed scheme.

##  V Conclusions

In this letter, we proposed a novel model for utility max-min fair link
adaptation in the IEEE 802.11ac DL-MU. We achieve the max-min fairness among
the receivers using the simple, accurate, and practical link adaptation model
and corresponding solution algorithms, which have been analyzed in this
letter. Via simulation experiments, the solution to the proposed model scheme
is investigated and validated by quantitative comparisons.

TABLE II: Receiver Utilities in Average for Two Compared Schemes.  Scenario\Average Utility | $U_{1}$ | $U_{2}$ | $U_{3}$ | $U_{4}$
---|---|---|---|---
Proposed Scheme | 0.9727 | 0.7706 | 0.6636 | 0.6371
Maximum Utility Scheme | 0.8772 | 0.7774 | 0.6636 | 0.8820

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be a context-closure of $M$ and $N$; hence uses of the input may cause
several evaluations of $M$ and $N$, each of which generates a new
location. In this respect, it is surprising that our second proof
avoids multiple allocations (the candidate relation $𝓢$ only mentions
one location). This is due to the massive combination of up-to
techniques whereby, whenever a new location is created, a double
application of up-to-context (the `double' is obtained from
up-to-transitivity) together with some administrative work (given by
the other techniques) allows us to absorb the location.

§ CONCLUSIONS

In this paper we have studied how to transport the rich theory of `up-to' techniques that
exists for plain (first-order) LTSs and bisimilarity, and rooted in fixed-point theory,
onto languages whose LTS and bisimilarity go beyond the first-order format.  For this
we have considered the $\pi$-calculus, the pure call-by-name $\lambda$-calculus, and a
call-by-value $\lambda$-calculus extended with imperative features.

The approach that we have proposed exhibits fully abstract translations of
the LTSs and bisimilarities of these languages onto first-order LTSs. In this way,
one can directly reuse the large corpus of up-to techniques that are available on
first-order LTSs.  The only exception to this regards basic up-to techniques that are
specific to the new languages, such as up-to-context.  Most important, the approach allows
one to take arbitrarily complex combinations of up-to techniques, whose soundness is
guaranteed by those of the corresponding first-order techniques. Direct proofs of such
combinations, on the source languages, can be long and delicate.  We have given examples
of uses of such combinations.  In particular, the second proof of the example dealt with in
Section <ref> is, in our opinion, a striking example of the benefits of the up-to
techniques. It is hard to imagine how to the example could be handled without up-to
techniques; compared to similar proofs in the literature and discussed in that section — all
of which make use of some forms of up-to techniques — the relation employed and the
proof work needed have been significantly reduced due to the large set of up-to techniques
referred to.

The work in this paper can be a further motivation for the development of a comprehensive
formalised library of up-to techniques for first-order LTSs.  Using the approach proposed
in the paper the library could then be applied to a wide range of languages.
Other directions for future work include
testing the  approach on other languages, for instance languages for mobility with explicit
notions of location (see [7] for a survey), or testing it on other forms of bisimulation
(e.g., open bisimulation).
By the time the revision of this paper has been completed,
one such application of the approach has been made,
for a calculus with delimited-control operators with dynamic prompt
generation [1].

We would also like to see if the approach proposed in this paper, based on translations to
first-order models, could be adapted to handle
the theory of
unique solutions of equations and contractions [34, 9], which allows
one to implicitly use  up-to techniques, with the goal of avoiding the development of  theories of
equations or contractions that are specific to a particular language or bisimulation.

§ ACKNOWLEDGEMENTS

We are delighted to be able to contribute to the Festschrift in  honour of Jos Baeten.
would like to take this opportunity for heartily thanking him  for having been such an
inspiring figure, both for all his many technical and scientific  contributions and for
his work in favour of the concurrency theory community.

We would like also to thanks the anonymous referees for many useful comments.

Pous was supported by the European Research
Council (ERC) under the European Union’s Horizon 2020 programme
(CoVeCe, grant agreement No 678157).
Sangiorgi  acknowledges support from the
MIUR-PRIN project `Analysis of
Program Analyses' (ASPRA, ID: [4]), and from the
European Research
Council (ERC) Grant DLV-818616 DIAPASoN.

In this paper we have studied how to transport the rich theory of `up-to' techniques that
exists for plain (first-order) LTSs and bisimilarity, and rooted in fixed-point theory,
onto languages whose LTS and bisimilarity go beyond the first-order format.  For this
we have considered the $\pi$-calculus, the pure call-by-name $\lambda$-calculus, and a
call-by-value $\lambda$-calculus extended with imperative features.

The approach that we have proposed exhibits fully abstract translations of
the LTSs and bisimilarities of these languages onto first-order LTSs. In this way,
one can directly reuse the large corpus of up-to techniques that are available on
first-order LTSs.  The only exception to this regards basic up-to techniques that are
specific to the new languages, such as up-to-context.  Most important, the approach allows
one to take arbitrarily complex combinations of up-to techniques, whose soundness is
guaranteed by those of the corresponding first-order techniques. Direct proofs of such
combinations, on the source languages, can be long and delicate.  We have given examples
of uses of such combinations.  In particular, the second proof of the example dealt with in
Section <ref> is, in our opinion, a striking example of the benefits of the up-to
techniques. It is hard to imagine how to the example could be handled without up-to
techniques; compared to similar proofs in the literature and discussed in that section — all
of which make use of some forms of up-to techniques — the relation employed and the
proof work needed have been significantly reduced due to the large set of up-to techniques
referred to.

The work in this paper can be a further motivation for the development of a comprehensive
formalised library of up-to techniques for first-order LTSs.  Using the approach proposed
in the paper the library could then be applied to a wide range of languages.
Other directions for future work include
testing the  approach on other languages, for instance languages for mobility with explicit
notions of location (see [7] for a survey), or testing it on other forms of bisimulation
(e.g., open bisimulation).
By the time the revision of this paper has been completed,
one such application of the approach has been made,
for a calculus with delimited-control operators with dynamic prompt
generation [1].

We would also like to see if the approach proposed in this paper, based on translations to
first-order models, could be adapted to handle
the theory of
unique solutions of equations and contractions [34, 9], which allows
one to implicitly use  up-to techniques, with the goal of avoiding the development of  theories of
equations or contractions that are specific to a particular language or bisimulation.

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