\section{Introduction} \indent

Numerous experimental studies at the LHC are a challenge to
theoretical QCD motivated approaches and models. In recent years an understanding has been obtained
that the processes at high
energies and large momentum transfer containing multiple hard
scales require using so-called unintegrated, or transverse momentum dependent
(TMD) parton density functions (PDFs), which have been used within the framework of
the phenomenological $k_T$-factorization approach\cite{1,2}
for many years. In this approach, the TMD parton
densities are among the main components that determine its
predictive power (see, for example, reviews\cite{3}
for more information).  The non-perturbative input
determines behavior of the TMD gluon density
and production cross sections at a
small gluon transverse momentum $k_T \to 0$
and plays a significant role
in the $k_T$-factorization\cite{4,5,6,7,8,9}.

In our previous papers\cite{10,11} we obtained the non-perturbative input from the
description of the inclusive spectra of hadrons produced in $pp$ collisions at the LHC energies in
the mid-rapidity region at low
transverse momenta $k_T\leq 1.5-1.6$~GeV and a starting scale $\mu_0^2 = 1$~GeV$^2$.
The proposed input is similar to the TMD gluon density
calculated within the popular color-dipole Golec-Biernat-W\"usthoff (GBW) approach\cite{12}
at large $k_T$ and differs from it at low transverse momenta.
Then, we extended this gluon density to higher $\mu^2$
using the Catani-Ciafoloni-Fiorani-Marchesini (CCFM) evolution equation\cite{13}
and considered deep inelastic $ep$ scattering at HERA.
We reasonably well described the experimental data on the proton longitudinal structure function $F_L(x,Q^2)$ and
charm and beauty contribution to the structure function $F_2(x,Q^2)$.
So, the connection between the soft LHC processes and small $x$ physics at HERA was
established.

In the present paper we continue our studies\cite{10,11} and
investigate the role of the non-perturbative input
to the TMD gluon density in description of hard processes at the LHC.
We improve the initial TMD gluon distribution
proposed earlier to describe LHC data
on the inclusive charged hadron spectra at higher transverse momenta
$2.5 < p_T < 4.5$~GeV and numerically extend it to the whole kinematical region
using the CCFM gluon evolution equation.
The CCFM equation is the most suitable tool
for our study since it smoothly interpolates between the
small-$x$ Balitsky-Fadin-Kuraev-Lipatov\cite{14} (BFKL) gluon dynamics
and the conventional
Dokshitzer-Gribov-Lipatov-Altarelli-Parisi\cite{15} (DGLAP) one.
We extract additional parameters
from a fit to the LHC data on the inclusive $b$-jet production taken by
the CMS and ATLAS Collaborations at high $p_T$ and $\sqrt s = 7$~TeV.
We supply the obtained TMD gluon density with the corresponding
TMD valence and sea quark distributions
calculated in the approximation, where
the sea quarks occur in the last gluon splitting.
Finally, we discuss several phenomenological applications
of the proposed TMD parton densities to hard LHC processes
that are most sensitive to the quark and gluon content of the proton.
We use the $k_T$-factorization approach, which is a
commonly recognized tool to investigate hard high-energy processes.
Here we see certain advantages in the fact that, even with the leading-order
(LO) matrix elements for a hard partonic subprocess, we can take into account a large
piece of higher order QCD corrections, namely all NLO + NNLO + ... terms containing
$\log 1/x$ enhancement.

\section{Starting non-perturbative TMD gluon density} \indent

As was mentioned above, the TMD gluon density was obtained\cite{10} within
the soft QCD model as a
function of the proton longitudinal momentum fraction $x$ and two-dimensional gluon
transverse momentum ${\mathbf k}_T$ at a fixed value of the scale $\mu_0^2 = 1$~GeV$^2$.
It can be
presented in the simple analytical form:
\begin{equation}
\displaystyle f_g^{(0)}(x,{\mathbf k}_T^2,\mu_0^2) = c_0 c_1 (1-x)^{b} \times \atop {
\displaystyle \times \left[R_0^2(x){\mathbf k}_T^2 + c_2\left(R_0^2(x){\mathbf k}_T^2\right)^{a/2}\right] \exp\left(-R_0(x)
|{\mathbf k}_T|-d\left[R_0^2(x){\mathbf k}_T^2\right]^{3/2}\right)},
\end{equation}

\noindent
where $R_0^2(x) = (x/x_0)^\lambda/\mu_0^2$, $c_0 = 3 \sigma_0/4\pi^2 \alpha_s$,
$x_0 = 4.21 \cdot 10^{-5}$, $\sigma_0 = 29.12$~mb, $\lambda = 0.22$, and $\alpha_s = 0.2$.
The parameters $c_1 = 0.3295$, $c_2 = 2.3$,
$a = 0.7$, $b = 12$ and $d = 0.2$
were deduced from the best fit of the LHC data on the inclusive spectra of charged
hadrons produced in $pp$ collisions in the mid-rapidity region at low $p_T \leq 1.6$~GeV.
The proposed gluon density differs from the one obtained
in the GBW model\cite{12} at $|{\mathbf k}_T| < 1$~GeV and
coincides with the GBW gluon at $|{\mathbf k}_T| > 1.5$~GeV.
Then, it was treated as a starting distribution
for the CCFM evolution equation and successfully applied to the description of the HERA data
on the proton structure functions $F_L(x,Q^2)$, $F_2^c(x,Q^2)$ and $F_2^b(x,Q^2)$\cite{11}.

However, the proposed non-perturbative gluon density~(1) is not able to describe
the LHC data on the inclusive spectrum of charged
hadrons at higher transverse momenta $2.5 < p_T < 4.5$~GeV even if
additional pertubative QCD (pQCD) corrections\cite{16,17} are taken into account
at $p_T > 2$~GeV.
Moreover, gluon density~(1) much faster decreases when ${\mathbf k}_T^2$ grows
compared to the solution of the BFKL equation outside of the saturation region\cite{14,15,18}.
Therefore, we modify the gluon density given by~(1) at $|{\mathbf k}_T| > 2 - 3$~GeV
to describe the LHC data on the charged hadron production at $2.5 < p_T < 4.5$~GeV.
Then we
match it with the TMD gluon obtained in\cite{18} as the solution of the linear BFKL equation
at low $x$, which results in flatter ${\mathbf k}_T^2$
behavior.
The modified starting TMD gluon density can be presented in the following form:
\begin{equation}
\displaystyle f_g^{(0)}(x,{\mathbf k}_T^2,\mu_0^2) = c_0 c_1 (1-x)^{b} \times \atop {
\displaystyle \times \left[R_0^2(x){\mathbf k}_T^2 + c_2\left(R_0^2(x){\mathbf k}_T^2\right)^{a/2}\right] \exp\left(-R_0^2(x) {\mathbf k}_T^2 -d\left[R_0^2(x){\mathbf k}_T^2\right]^{3/2}\right) + \atop {
\displaystyle + \, c_0 \left({x\over x_0}\right)^n \exp \left[- k_0^2 {R_0(x)\over |{\mathbf k}_T|}\right] f_g(x,{\mathbf k}_T^2), }}
\end{equation}

\noindent
where $k_0 = 1$~GeV, $\mu_0^2 = 1.1$~GeV$^2$, and
$n \simeq 0.81$. The function $f_g(x,{\mathbf k}_T^2)$
obeying the BFKL equation at not very large ${\mathbf k}_T^2$ reads\cite{18}
\begin{equation}
f_g(x,{\mathbf k}_T^2) = \alpha_s^2 \, x^{-\Delta} \, t^{-1/2} {1 \over v} \exp \left[ - { \pi \ln^2 v \over t}\right],
\end{equation}

\noindent
where $t = 14\,\alpha_s N_c \, \zeta(3) \ln(1/x)$, $v = |{\mathbf k}_T|/\Lambda_{\rm QCD}$,
and $\Delta = 4 \, \alpha_s N_c \ln 2 /\pi$.
It is important that the third term in~(2) is only non-zero
at $|{\mathbf k}_T| \ll \Lambda_{\rm QCD} \, (1/x)^{\delta}$
with $\delta = \alpha_s N_c$.
The details of the calculation and the relation
between the TMD gluon density and the inclusive hadron spectra $\rho_h(y\simeq 0,p_T)\equiv E \, d^3\sigma/d^3 p$
are given in
our previous papers\cite{10,11}.
These spectra are presented as a sum of two parts\cite{16,17}:
\begin{equation}
  \rho_h(y\simeq 0,p_T) = \rho_q(y\simeq 0,p_T) + \rho_g(y\simeq 0,p_T),
\end{equation}

\noindent
where $\rho_q$ is the quark contribution calculated within the
quark-gluon string model (QGSM)\cite{19,20,21} and $\rho_g$ is the gluon contribution,
which can be calculated using the proposed TMD gluon density~(2) and~(3).
Taking into account the energy $\sqrt s$ dependence of the $\rho_q$ and $\rho_g$ parts,
the spectrum $\rho_h(y\simeq 0,p_T)$ can be written in the following
form:
\begin{equation}
  \rho_h(y\simeq 0,p_T) = \left[\phi_q(y\simeq 0,p_T) + \phi_g(y\simeq 0,p_T) \left(1 - {\sigma_{nd} \over g (s/s_0)^{\Delta}}\right)\right] g (s/s_0)^{\Delta},
\end{equation}

\noindent
where $g = 21$~mb, $\Delta = \alpha_P(0) - 1 \simeq 0.12$,
$\alpha_P(0)$ is the Pomeron intercept and $\sigma_{nd}$
is the non-diffractive cross section given by the sum of
$n$ pomeron chain production cross sections. The
functions $\phi_q(y\simeq 0,p_T)$ and $\phi_g(y\simeq 0,p_T)$
are evaluated in\cite{16,17} and can be
presented as
\begin{equation}
\phi_q(y\simeq 0,p_T)=A_q \exp \left(-p_T/C_q\right),
\end{equation}
\begin{equation}
\phi_g(y\simeq 0,p_T)=A_g \sqrt{p_T} \exp \left(-p_T/C_g\right),
\end{equation}

\noindent
where the parameters $A_q = 3.68$~GeV$^{-2}$, $A_g = 1.7249$~GeV$^{-2}$, $C_q = 0.147$~GeV, and
$C_g= 0.289$~GeV were obtained from the combined
fit of the NA61\cite{22} and LHC\cite{23} data taken at
different energies $\sqrt s$\cite{16}.
These parameters, of course, differ from the ones obtained earlier\cite{10}
due to another form of TMD gluon density~(2) and~(3) used in the fit procedure.
The latter leads, in addition, to the values of the parameters $d = 0$ and $a = 0.3$ in~(2).
In this way the LHC data\cite{23} are
fitted with $\chi^2/n.d.f = 0.998$.
The parameters $b = 6.57$ and $\alpha_s = 0.18$ were obtained from the
best fit of the CMS data on the inclusive $b$-jet production at $|y| < 0.5$ (see below).
Let us stress here that since the parameters of nonperturbative input (2) --- (3)
were obtained from the description of the LHC and NA61 data\cite{22,23}, possible
higher-order corrections (see, for example,\cite{24,25,26})
to the leading-order BFKL motivated ${\mathbf k}_T$-dependence of the proposed
gluon input at low $x$ (as well as saturation dynamics) are effectively included.

The inclusive spectra of $\pi^-$ mesons produced in
$pp$ collisions at the initial momenta $31$ and $158$~GeV
are presented in Fig.~1 as a function of the transverse mass $m_T = \sqrt{m_\pi^2 + p_T^2}$.
Using the first part of the spectrum $\phi_q(y\simeq 0, p_{T})$, connected with
the quark contribution in the conventional string model\cite{19}, one can reasonably describe
the NA61 data\cite{22} at low $m_T < 1$~GeV.
The inclusion of the second part of the spectrum, which is due to the gluon contribution,
allows us to describe the NA61 data up to $m_T \sim 1.5$~GeV.
A similar description of the experimental data on the inclusive spectra of charged
hadrons (mainly pions and/or kaons) is achieved at the LHC (see Fig.~2).
In addition to the soft part,
we include the pQCD corrections\cite{16,17}.
The latter, made at the leading order (LO), are divergent at low
transverse momenta. Therefore, the kinematical region $p_T \sim 1.8 - 2.2$~GeV
can be treated as the matching
region of the non-pertubative QCD (soft QCD) and pQCD calculations.
One can see that the inclusive hadron spectra at the LHC
can be well described in a wide
region of transverse momenta by matching these two approaches,
and the proposed input TMD gluon density~(2) and (3) plays a crucial
role in the description of these data at low hadron transverse momenta.

\section{CCFM-evolved TMD parton densities} \indent

The average gluon transverse momentum $\langle |{\mathbf k}_T| \rangle$,
generated by the TMD gluon distribution defined above
is $\langle |{\mathbf k}_T| \rangle \sim 1.9$~GeV at
$10^{-7} < x < 1$, which is close to the non-perturbative
QCD regime.
Therefore, we can treat the proposed TMD gluon density
as a starting one and apply the CCFM
equation to extend it to the whole kinematical region.
The CCFM evolution equation resums large logarithms $\alpha_s^n \ln^n 1/(1-x)$
in addition to $\alpha_s^n \ln^n 1/x$ ones and
introduces angular ordering of initial
emissions to correctly treat gluon coherence effects.
In the limit of asymptotic energies, it is almost
equivalent to BFKL, but also similar to the
DGLAP evolution for large $x$ and high $\mu^2$\cite{13}.

In the leading logarthmic approximation, the CCFM equation with
respect to the evolution (factorization)
scale $\mu^2$ can be written as\cite{13}
\begin{equation}
\displaystyle f_g(x,{\mathbf k}_T^2,\mu^2) = f_g^{(0)}(x,{\mathbf k}_T^2,\mu_0^2) \Delta_s(\mu^2,\mu_0^2) + \atop {
\displaystyle + \int {dz\over z} \int {d q^2\over q^2} \theta(\mu - z q) \Delta_s(\mu^2,z^2 q^2)} P_{gg}(z,q^2,{\mathbf k}_T^2)
f_g(x/z,{\mathbf k^\prime}_T^2,q^2),
\end{equation}

\noindent
where ${\mathbf k^\prime}_T = {\mathbf q} (1 - z) + {\mathbf k}_T$ and the
Sudakov form factor $\Delta_s(q_1^2,q_2^2)$ describes the probability of no radiation
between $q_2^2$ and $q_1^2$.
The first term in the CCFM equation~(8), which is initial TMD gluon density
multiplied by the Sudakov form factor $\Delta_s(\mu^2,\mu^2_0)$, makes the contribution of
non-resolvable branchings between the starting scale $\mu_0^2$ and the factorization scale
$\mu^2$, the second term describes the details of the QCD evolution expressed by the
convolution of the CCFM splitting function $P_{gg}(z,q^2,{\mathbf k}_T^2)$ with the
gluon density $f_g(x,{\mathbf k}_T^2,\mu^2)$ and the Sudakov form factor $\Delta_s(\mu^2,q^2)$,
and the theta function introduces the angular ordering condition.
The evolution scale $\mu^2$ is defined by the maximum allowed angle for any gluon
emission\cite{13}.

The CCFM equation~(8) describes only
the emission of gluons, while quark emissions are left aside.
In order to calculate the TMD valence quark denisities, we have to replace in~(8)
the gluon splitting function $P_{gg}(z,q^2,{\mathbf k}_T^2)$
by the quark one $P_{qq}(z,q^2,{\mathbf k}_T^2)$ \cite{27,28}.
The starting TMD valence quark distribution can
be parameterized using standard collinear PDFs $xq_v(x,\mu^2)$ as
\begin{equation}
  f_{q_v}^{(0)}(x,{\mathbf k}_T^2,\mu_0^2) = xq_v(x,\mu_0^2) \, {2\over \mu_0^2} \exp \left[ - { {\mathbf k}_T^2 \over \mu_0^2/2} \right].
\end{equation}

\noindent
Numerically, we applied the LO
parton densities from the MSTW'2008 set\cite{29}.
The exact analytical expressions for the splitting functions $P_{gg}(z,q^2,{\mathbf k}_T^2)$,
$P_{qq}(z,q^2,{\mathbf k}_T^2)$,
and Sudakov form factor can be found, for example, in\cite{30}.
Concerning the TMD sea quark density, we calculate it using the approximation where the sea quarks
occur in the last gluon-to-quark splitting.
At the next-to-leading logarithmic accuracy $\alpha_s (\alpha_s \ln x)^n$ the TMD sea
quark distribution
can be written\cite{7} as:
\begin{equation}
  f_{q_s}(x,{\mathbf k}_T^2,\mu^2) = \int \limits_x^1 {dz \over z} \int d{\mathbf q}_T^2
    {1\over {\mathbf \Delta}^2} {\alpha_s \over 2\pi} P_{qg}(z,{\mathbf q}_T^2,{\mathbf \Delta}^2) f_g(x/z,{\mathbf q}_T^2, \bar \mu^2),
\end{equation}

\noindent
where $z$ is the fraction of the gluon light cone momentum carried out by
the quark, and $\mathbf \Delta = {\mathbf k}_T - z{\mathbf q}_T$.
The sea quark evolution is driven by the off-shell gluon-to-quark
splitting function $P_{qg}(z,{\mathbf q}_T^2,{\mathbf \Delta}^2)$\cite{31}:
\begin{equation}
  P_{qg}(z,{\mathbf q}_T^2,{\mathbf \Delta}^2) = T_R \left({\mathbf \Delta}^2\over {\mathbf \Delta}^2 + z(1-z)\,{\mathbf q}_T^2\right)^2
    \left[(1 - z)^2 + z^2 + 4z^2(1 - z)^2 {{\mathbf q}_T^2\over {\mathbf \Delta}^2} \right],
\end{equation}

\noindent
where $T_R = 1/2$. The splitting function $P_{qg}(z,{\mathbf q}_T^2,{\mathbf \Delta}^2)$
was obtained by generalizing to finite transverse momenta of
the two-particle irreducible kernel expansion\cite{32}.
It takes into account the small-$x$ enhanced transverse momentum dependence
up to all orders in the strong coupling constant, and reduces to the conventional splitting
function at the lowest order for $|\mathbf q_T| \to 0$.
The scale $\bar \mu^2$ was defined\cite{33} from the angular ordering condition which is natural
from the point of view of the CCFM evolution: $\bar \mu^2 = {\mathbf \Delta}^2/(1-z)^2 + {\mathbf q}_T^2/(1-z)$.

The CCFM evolution equation with the starting TMD gluon and quark distributions
given by~(2) --- (3) and (9)
was solved numerically\footnote{Authors are very grateful to Hannes Jung for providing
us with the appropriate numerical code.} in the leading logarithmic approximation
using the \textsc{updfevolv} routine\cite{30}.
Thus, the TMD gluon and valence quark densities were
obtained for any $x$, ${\mathbf k}_T^2$, and $\mu^2$ values.
The TMD sea quark distributions can be evaluated according to~(10) and~(11).

The gluon density $f_g(x,{\mathbf k}_T^2,\mu^2)$
obtained according to~(2),~(3) and~(8),
labeled below as {\it Moscow-Dubna 2015}, or MD'2015,
is shown in Fig.~3 as a
function of ${\mathbf k}_T^2$ for different values of $x$ and $\mu^2$.
Additionally, we plot the TMD gluon distribution\cite{34} (namely, the set A0)
which is widely discussed in the literature and commonly used in the applications.
One can observe some difference in the absolute normalization and shape between
both TMD gluon distributions. Below we will consider the corresponding phenomenological
consequences for several LHC processes.

\section{Phenomenological applications} \indent

We are now in a position to apply the proposed TMD parton densities
to some processes studied at hadron colliders.
In the present paper we consider the inclusive production
of $b$-jets, $B^+$ and $D^*$ mesons as well as the associated production of $W^\pm$ or $Z/\gamma^*$ bosons
and hadronic jets at the LHC conditions. We also study the charm and beauty contribution to the
proton structure function $F_2(x,Q^2)$ and the longitudinal proton structure function $F_L(x,Q^2)$.
These processes are known to be strongly sensitive to the gluon and/or quark content of
the proton.

In according to the $k_T$-factorization prescription\cite{1,2},
the cross sections of the processes under consideration can be written as
\begin{equation}
  \displaystyle \sigma = \int dx_1 dx_2 \int d {\mathbf k}_{1T}^2 d {\mathbf k}_{2T}^2 \, f_{q/g}(x_1,{\mathbf k}_{1T}^2,\mu^2) f_{q/g}(x_2,{\mathbf k}_{2T}^2,\mu^2) \times \atop {
  \displaystyle  \times d\hat \sigma(x_1, x_2, {\mathbf k}_{1T}^2,{\mathbf k}_{2T}^2, \mu^2) },
\end{equation}

\noindent
where $\hat \sigma(x_1, x_2, {\mathbf k}_{1T}^2,{\mathbf k}_{2T}^2, \mu^2)$ are relevant
off-shell (depending on the transverse momenta of incoming particles)
partonic cross sections. The detailed description of the
calculation steps (including the evaluation of the off-shell
amplitudes) can be found in our previous papers\cite{35,36,37,38}.
Here we only specify the essential numerical parameters.
Following\cite{39}, we set the charmed and beauty quark masses $m_c = 1.4$~GeV and
$m_b = 4.75$~GeV,
$D^*$ and $B^+$ meson masses $m_{D^*} = 2.01$~GeV and $m_{B^+} = 5.28$~GeV, masses of
gauge bosons $m_W = 80.403$~GeV and $m_Z = 91.1876$~GeV,
$Z$ boson decay width $\Gamma_Z = 2.4952$~GeV, and
Weinberg mixing angle $\sin^2 \theta_W = 0.231$. We use the two-loop formula for
the strong coupling constant (as it is implemented in the \textsc{updfevolv} routine)
with $n_f= 4$ active quark flavors at
$\Lambda_{\rm QCD} = 200$~MeV and apply the running QCD and QED coupling constants.
We use the factorization and renormalization scales $\mu_F$ and $\mu_R$
according to the process under consideration\cite{35,36,37,38}. Additionally, we
estimate the theoretical uncertainty coming from the
renormalization scale.

The multidimensional integration was performed
by the Monte Carlo technique, using the routine \textsc{vegas}\cite{40}.
The corresponding C++ code is available from the authors on request.

\subsection{Proton structure functions $F_2^c$, $F_2^b$ and $F_L$} \indent

The charm and beauty contributions to the proton structure
function $F_2(x,Q^2)$ and the longitudinal structure function $F_L(x,Q^2)$
are directly connected to the gluon content of the
proton. The structure function $F_L(x,Q^2)$ is equal to zero in the parton model
with spin $1/2$ partons and only has nonzero values within the pQCD.
The experimental data on these structure functions were obtained\cite{41,42,43,44,45}
by the H1 and ZEUS Collaborations at HERA.
Our consideration is based on the formalism\cite{35},
and here we present the main results only.

The results of our calculations are presented in Figs.~4 --- 6.
The solid curves correspond to the predictions obtained with the
MD'2015 gluon distribution, the upper and lower dashed curves represent
the estimate of the corresponding theoretical uncertainties (as given by the usual
scale variations) and the dash-dotted curves correspond to the results
obtained with the A0 gluon density.
Additionally, we plot the data/theory ratios for the longitudinal structure function $F_L(x,Q^2)$.
One can see that the predictions corresponding to the proposed MD'2015
gluon distribution agree well with the H1
and ZEUS data in the whole kinematical region of $x$ and $Q^2$
within the theoretical and experimental uncertainties.
It only tends to slightly overestimate the $F_L$ data\cite{41} at very low $Q^2$
but still agrees with them within the theoretical and experimental uncertainties.
Therefore, the main conclusion of\cite{10,11}, where the link
between soft processes at the LHC and low-$x$ physics at HERA was pointed out, is confirmed.
At the same time, the A0 gluon density
does not reproduce the
shape of the structure functions $F_2^c$ and $F_2^b$ at low $Q^2$.
It means that the influence of the shape and other parameters of the initial
non-perturbative gluon distribution on the description of experimental
data is significant for a wide region of $x$ and $Q^2$.
We conclude that the best description of the HERA data\cite{41,42,43,44,45} is
achieved with the MD'2015 gluon density.

\subsection{Inclusive $b$-jet production}\indent

As is well known, beauty quarks at the LHC energies are produced mainly via
standard QCD gluon-gluon fusion subprocess; therefore,
the corresponding total and differential cross sections
are strongly sensitive to the gluon content of the proton.
In the $k_T$-factorization approach this was demonstrated
in our previous paper\cite{36}.
Below we probe the MD'2015 gluon density in the
inclusive $b$-jet production at the LHC.
The CMS Collaboration measured the
$b$-jet cross sections in five $b$-jet rapidity regions, namely,
$|y| < 0.5$, $0.5 < |y| < 1$, $1 < |y| < 1.5$, $1.5 < |y| < 2$,
and $2 < |y| < 2.2$ as a function of the jet transverse momentum at
$\sqrt s = 7$~TeV\cite{46}.
The ATLAS Collaboration performed the measurements
at central rapidities $|y| < 2.1$\cite{47}.

Our main results are presented in Figs.~7 and~8,
where we plot the calculated transverse momentum distributions
of $b$-jets compared to the LHC data as well as the corresponding data/theory ratios.
We obtained a good description of the data using the MD'2015 gluon
distribution. The shape and absolute normalization of the measured $b$-jet
cross sections are reproduced well.
Moreover, the differential cross section as a function of
the angular separation $\Delta \phi$ between
two $b$-jets measured at $p_T > 40$~GeV and $M > 110$~GeV,
where $M$ is the invariant mass of the produced $b\bar b$ pair,
is also well described. This observable is known to be very
sensitive to the ${\mathbf k}_T^2$-behavior of the TMD gluon
distribution.
The predictions based
on the A0 gluon density lie below the
data at high $p_T > 40$ --- $50$~GeV.
Once again, we conclude that the best description of the LHC data\cite{46,47}
is achieved with the proposed MD'2015 gluon density.

\subsection{$B^+$ meson production}\indent

Besides the $b$-jet production, we probe the proposed MD'2015 gluon density in the
inclusive $B^+$ meson production at the LHC, which was measured
by the CMS\cite{48}, ATLAS\cite{49}, and LHCb\cite{50} Collaborations.
Our basic formulas and the description of the calculation details
are collected in\cite{36,37}.
Here we only note that
we convert beauty quarks produced in the hard subprocess
into $B^+$ mesons using the Peterson
fragmentation function\footnote{Of course, the predicted transverse momentum distributions are
sensitive to the quark-to-hadron fragmentation function.
This dependence was studied earlier\cite{36} and not
considered in the present paper.}
with the usual shape parameter $\epsilon_b = 0.006$\cite{51}.
Following\cite{39}, we set the branching fraction $f(b \to B^+) = 0.398$.
The CMS Collaboration reported the total and differential $B^+$
cross sections measured at central rapidities $|y| < 2.4$ for $p_T > 5$~GeV and
$\sqrt s = 7$~TeV\cite{48}.
The ATLAS Collaboration presented $B^+$ meson cross sections
for $|y| < 2.25$ and $9 < p_T < 120$~GeV\cite{49}.
The LHCb Collaboration measured $B^\pm$ meson cross sections
in the forward rapidity region $2 < y < 4.5$ at $p_T < 40$~GeV\cite{50}.

Our main results are presented in Figs.~9 --- 11 compared with the LHC data
and corresponding data/theory ratios.
One can see that a good description of the $B^+$ meson
transverse momentum and rapidity distributions
is achieved using both TMD gluon densities under consideration.
The results obtained with the MD'2015 gluon density
lie somewhat above the A0 ones and tend to slightly overestimate the measured
cross sections at high transverse momenta but agree with the
data within the uncertainties.
However, both considered TMD gluon densities give similar behavior
of the rapidity distributions.
We would like to point out the remarkable
description of the LHCb data in the forward
rapidity region that extends essentially the applicability area
of the proposed MD'2015 gluon distribution.

\subsection{$D^*$ meson production}\indent

Similar to beauty quarks, charmed quarks
are mainly produced at the LHC via the gluon-gluon fusion subprocess, and therefore,
the charm production cross section is also sensitive to the gluon density function.
The transverse momentum distributions of several charmed mesons
($D^*$, $D^\pm$, $D^0$, $D_s$) were measured by the LHCb Collaboration at
forward rapidities, $2 < y < 4.5$\cite{52}.
As a representative example, below we consider $D^*$ meson production.
To
convert $c$-quarks into $D^*$ mesons, we apply the
non-perturbative
fragmentation function\cite{53,54,55},
which is often used in the collinear QCD calculations.
We set the branching fraction $f(c \to D^*) = 0.255$\cite{39}.

Our numerical results are presented in Fig.~12.
We find reasonably good agreement between
our predictions obtained using the MD'2015 gluon distribution and the LHCb data\cite{52},
which demonstrates again a wide area of its applicability.
The A0 gluon density predicts flatter $D^*$ transverse momentum
distributions and does not contradict the data.

\subsection{Associated $W^\pm +$~jet and $Z/\gamma^* +$~jet production}\indent

Contrary to the processes considered above, where only the
TMD gluon distribution was probed, the
associated production of gauge ($W^\pm$ or $Z$) bosons and hadronic jets at the LHC
offers high sensitivity to both quark and gluon density functions in the proton.
The experimental data on the $W^\pm +$~jet and $Z/\gamma^* +$~jet
production were obtained
by the ATLAS Collaboration\cite{56,57}.
Below we apply the TMD quark and gluon densities
from the MD'2015 set to describe the LHC data. Our consideration
is based on the off-shell amplitudes of
the quark-gluon scattering subprocesses
$q g^* \to W^\pm q^\prime$ and
$q g^* \to Z/\gamma^* q$
which include the subsequent decays $W^\pm\to l^\pm\nu$ and $Z/\gamma^* \to l^+l^-$
derived in\cite{38}. Other details of the calculations can be found there.

The ATLAS data\cite{56} on the associated $W^\pm$ and jet production refer
to the kinematical region defined as
$p_T^l > 25$~GeV, $|\eta^l| < 2.5$,
$E_T^{\rm miss} > 25$~GeV, $m_T(W) > 40$~GeV,
$p_T^{\rm jet} > 30$~GeV, and $|y^{\rm jet}| < 4.4$,
where $m_T(W)$ is the transverse mass of the produced $W^\pm$ boson,
$\eta^l$ and $p_T^l$ are the decay lepton pseudo-rapidity and transverse
momentum, $y^{\rm jet}$ and $p_T^{\rm jet}$ are the rapidity and transverse
momentum of the final hadronic jet, respectively, and $E_T^{\rm miss}$ is the
missing transverse energy. The measurements of the $Z/\gamma^* +$~jet
production were performed\cite{57} at $66 < M < 116$~GeV, $p_T^l > 20$~GeV,
$|\eta^l| < 2.5$, $p_T^{\rm jet} > 20$~GeV, and $|y^{\rm jet}| < 4.4$,
where $M$ is the invariant mass of the produced lepton pair.
Our main numerical results are shown in Fig.~13 in comparison
with the ATLAS data\cite{56,57}.
Additionally, we plot the corresponding data/theory ratios.
We obtained a good description of these data with both
TMD parton densities.
The latter, in particular, demonstrates that
the TMD quark and gluon distributions
from the MD'2015 set
are reliable at relatively large scales, up to $\mu^2 \sim m_Z^2$.
However, we note that the full hadron-level Monte-Carlo generator
\textsc{cascade}\cite{58}, which
uses the CCFM evolution equation for the initial state gluon emissions,
is needed for a more detailed analysis
of the associated production of gauge bosons and hadronic jets at the LHC.
It is connected with a more accurate jet selection algorithm
implemented in \textsc{cascade}, as was explained previously\cite{38}.

\section{Conclusion} \indent

We fitted the experimental data on the inclusive spectra of the charged particles
produced in the central $pp$ collisions at the LHC to
determine the TMD gluon density in a proton at the starting scale
$\mu_0^2 \sim 1$~GeV$^2$. We demonstrated that the best
description of these spectra can be achieved
by matching the derived TMD gluon distribution
with the exact solution of the Balitsky-Fadin-Kuraev-Lipatov (BFKL)
equation obtained at low $x$ and small gluon transverse momenta outside
the saturation region.
Moreover, we established that the parameters of this fit did not depend on the
initial energy in a wide energy interval.
The average gluon transverse momentum generated by this modified TMD gluon density
is about $1.9$~GeV in a wide region of $x$ and is
close to the non-perturbative QCD regime.
Then, we extended the derived TMD gluon density to higher $\mu^2$ using numerical
solution of the CCFM gluon evolution equation.
Additionally,
we supplied the calculated TMD gluon density with the
TMD valence and sea quark distributions. The latter was evaluated
in the approximation where the gluon-to-quark splitting occurred at the last evolution step
using the TMD gluon-to-quark splitting function. This function contains
all single logarithmic small-$x$ corrections in any order of perturbation theory.

Special attention was paid to the phenomenological applications
of the proposed MD'2015 parton distributions to the hard processes.
We considered the inclusive production
of $b$-jets, $B^+$ and $D^*$ mesons and the
associated production of $W^\pm$ or $Z/\gamma^*$ bosons
and hadronic jets at the LHC energies,
and also the charm and beauty contribution to the
proton structure function $F_2(x,Q^2)$ and the longitudinal proton structure function $F_L(x,Q^2)$.
We demonstrated significant influence of the initial
non-perturbative gluon distribution on the description of experimental
data. We showed that the LHC data could be well described
using the MD'2015.

\section{Acknowledgements} \indent

We thank H.~Jung for his extreme help in the calculation of the CCFM evolution and very
useful discussions and comments.
We also thank L.N.~Lipatov and B.I.~Ermolaev for helpful discussion
and M.A.~Malyshev for careful reading of the manuscript.
This research was supported by the FASI of Russian Federation
(grant NS-3042.2014.2).
A.V.L. and N.P.Z. are also grateful to the DESY Directorate for the
support within the framework of the Moscow --- DESY project on Monte-Carlo implementation for
HERA --- LHC.


\section{Introduction}

Search for signatures of new physics beyond the standard model (SM)
is the most important task of the particle physics at present.
The lepton family number is exactly preserved in the SM
and no lepton flavour violation (LFV) has been observed
at experiments so far.
The experimental bounds on the LFV in $Z$ decay
are given by
\begin{eqnarray}
{\rm Br}(Z \to e \mu) < 1.7 \times 10^{-6},
\nonumber \\
{\rm Br}(Z \to e \tau) < 9.8 \times 10^{-6},
\nonumber \\
{\rm Br}(Z \to \mu \tau) < 1.2 \times 10^{-5},
\end{eqnarray}
at 95\% C.L. \cite{pdg}
and those in $\tau$ decays given by
\begin{eqnarray}
{\rm Br}(\tau^- \to e^- e^+ e^-) < 3.6 \times 10^{-8},
\nonumber \\
{\rm Br}(\tau^- \to e^- \mu^+ \mu^-) < 3.7 \times 10^{-8},
\nonumber \\
{\rm Br}(\tau^- \to e^+ \mu^- \mu^-) < 2.3 \times 10^{-8},
\nonumber \\
{\rm Br}(\tau^- \to \mu^- e^+ e^-) < 2.7 \times 10^{-8},
\nonumber \\
{\rm Br}(\tau^- \to \mu^+ e^- e^-) < 2.0 \times 10^{-8},
\nonumber \\
{\rm Br}(\tau^- \to \mu^- \mu^+ \mu^-) < 3.2 \times 10^{-8},
\end{eqnarray}
and those in muon decays
\begin{eqnarray}
{\rm Br}(\mu^- \to e^- e^+ e^-) < 1.0 \times 10^{-12},
\end{eqnarray}
at 90\% C.L.
\cite{pdg}.
Although existence of the neutrino oscillations implies that
the lepton family numbers are no more preserved,
the LFV through neutrino oscillation are
too suppressed to be observed due to the smallness of neutrino mass.
Therefore observation of the LFV would be an unambiguous evidence
for deviations from the SM.

We consider the model with a separate SU(2) group
acting only on the third generation fermions,
in which the LFV generically arises.
Since different SU(2) gauge group is assigned on the third generation,
the corresponding gauge coupling constant is different
from that of the first and second generations in general,
which indicates that the electroweak gauge interaction
is nonuniversal in this model.
In the SM, the neutral current interactions are simultaneously
diagonalized with mass matrices of quarks and leptons
by a biunitary transformation with two independent unitary matrices
corresponding to left-handed and right-handed quarks and leptons,
which results in the absence of the flavour changing neutral current (FCNC)
at tree level.
However, if the SU(2) gauge couplings are nonuniversal,
the neutral current interactions cannot be diagonalized
simultaneously with mass matrices
and FCNCs exist in general.
The FCNC of the lepton sector means violation
of the lepton flavours.

The phenomenology of our model has been studied
using the $Z$-pole data and the low-energy neutral current data
\cite{topflavour1,topflavour2}.
Phenomenological implications of the FCNC interactions of this model
have been considered in Ref. \cite{topflavour2,fcnc}.
By the way the nonuniversality of the gauge couplings in this model
also leads to the violation of the unitarity
of the Cabibbo-Kobayashi-Maskawa (CKM) matrix \cite{kylee3,kylee4}.
The symmetry breaking mechanism of this model has been studied
in detail in Ref. \cite{chiang}.
This gauge group can arise as the theory
at an intermediate scale in the path of gauge
symmetry breaking of noncommuting extended technicolor
(ETC) models \cite{ETC}
and the grand unified theory based on SU(3)$^3$ or SU(15).

The LFV in this model has been discussed in Ref \cite{topflavour2}.
However they assumed the case that one mixing angle
between two flavours is dominant and just analyzed corresponding channels.
In this paper, we present a general description of the LFV
in SU(2)$_l \times$ SU(2)$_h \times$ U(1) model and
perform the complete analysis with the recent experimental data.
We show that the flavour conserving $Z \to l^-_i l^+_i$ decays
play crucial roles for constraining the model
as well as the strong limits on the LFV processes.

We consider the electroweak $SU(2)_l \times SU(2)_h \times U(1)_Y $ model,
in which the gauge group $SU(2)_l$ acts on
the first and the second generations
and $SU(2)_h$ on the third generation only.
The quantum numbers of left-handed quarks and leptons
are assigned as (2,1,Y) for the first and second generations and
(1,2,Y) for the third generation
under $SU(2)_l \times SU(2)_h \times U(1)_Y $.

The spontaneous symmetry breaking mechanism is achieved
such that the gauge group $ SU(2)_l \times SU(2)_h \times U(1) $
breaks down into the $ SU(2)_{l+h} \times U(1)_Y $
with the vacuum expectation values (VEV)
of the (2,2,0) bidoublet scalar field $\Sigma$,
$\langle \Sigma \rangle = {\rm diag}(u,u)$
and sequentially breaks down into $U(1)_{em}$
with the VEV of the scalar doublet $\Phi$,
$\langle \Phi \rangle = (0, v/\sqrt{2})$.
Since the $ SU(2)_l \times SU(2)_h $ breaking scale $u$ is
higher than the electroweak scale $v$,
we introduce a small parameter $\lambda \equiv v^2/u^2 \ll 1 $.
The masses of the heavy gauge bosons are degenerate in this model,
$m_{_{W'^{\pm}}}^2 = m_{_{Z'}}^2
= m_0^2/\lambda \sin^2 \phi \cos^2 \phi$ ,
while the ordinary $W$ boson masses given by
$ m_{_{W^{\pm}}}^2
= m_0^2 ( 1- \lambda \sin^4 \phi )
= m_{_{Z}}^2 \cos^2 \theta$
where $m_0 = ev/(2 \sin \theta)$.

We parameterize the gauge couplings by
$g_l = e/\sin \theta \cos \phi$,
$g_h = e/\sin \theta \sin \phi$, and
$g^{\prime} = e/\cos \theta$
in terms of the electromagnetic coupling $e$,
the weak mixing angle $\theta $ and the new mixing angle $\phi$
between $SU(2)_l$ and $SU(2)_h$.
The perturbativity $g_{(l,h)}^2/4 \pi < 1$ is assumed,
which results in the constraint $ 0.03 < \sin^2 \phi < 0.96 $.

We focus on the neutral current interactions in this paper.
It is advantageous to separate the nonuniversal part
from the universal part in the Lagrangian
such that
${\cal L}^{\rm NC} =  {\cal L}^{\rm NC}_{I} + {\cal L}^{\rm NC}_{3}$,
where $ {\cal L}^{\rm NC}_{I} $ denotes the universal gauge interactions
and $ {\cal L}^{\rm NC}_{3} $ the nonuniversal terms.
The universal part is given by
\begin{eqnarray}
{\cal L}^{\rm NC}_{I} =
\bar{f}_L \gamma_\mu (G_L Z^\mu + G'_L Z'^\mu) f_L
+ \bar{f}_R \gamma_\mu (G_R Z^\mu + G'_R Z'^\mu) f_R,
\end{eqnarray}
where $f = (e,\mu,\tau)^T$ for the lepton sector and
\begin{eqnarray}
G_L &=& -\frac{e}{\cos \theta \sin \theta}
         \left( T_3 - Q \sin^2 \theta - \lambda T_3 \sin^4 \phi
         \right) I,
\nonumber \\
G'_L &=& \frac{e}{\sin \theta}
           \left( T_3 \tan \phi
                   + \lambda \frac{\sin^3 \phi \cos \phi}{\cos^2 \theta}
                              (T_3 - Q \sin^2 \theta)
            \right) I,
\nonumber \\
G_R &=& \frac{e}{\cos \theta \sin \theta} Q \sin^2 \theta \cdot I,
\nonumber \\
G'_R &=& -\frac{e}{\sin \theta} \lambda Q \tan^2 \theta
                                 \sin^3 \phi \cos \phi \cdot I,
\end{eqnarray}
with the 3$\times$3 identity matrix $I$
and the electric charge defined by $ Q = T_{3l} + T_{3h} + Y/2 $.
The nonuniversal part consists of a nonzero (3,3) elements
which is given by
${\cal L}^{\rm NC}_{3} =
\bar{f}_L \gamma_\mu (Y_L Z^\mu + Y'_L Z'^\mu) f_L, $
where
\begin{eqnarray}
Y_L &=& -\frac{e}{\cos \theta \sin \theta} \lambda T_3 \sin^2 \phi \cdot M,
\nonumber \\
Y'_L &=& -\frac{e}{\sin \theta}
            \frac{T_3}{ \sin \phi \cos \phi} M,
\end{eqnarray}
with the $3\times3$ matrix $M$ defined by $M_{ij} = \delta_{3i}\delta_{3j}$.

The mass eigenstates of charged leptons $f^0_{L,R}$
by the relation $f_{L,R} = V_{L,R}^\dagger f^0_{L,R}$
with the unitary matrix $V_{L,R}$.
The left-handed neutral current interactions are derived to be
\begin{eqnarray}
{\cal L}^{NC}_L =
G_L \bar{f}^0_L \gamma_\mu
        \left( I + \epsilon V^\dagger M V \right)
            f^0_L Z^\mu
+ G'_L \bar{f}^0_L \gamma_\mu
        \left( I + \epsilon' V^\dagger M V \right)
            f^0_L Z'^\mu,
\end{eqnarray}
in terms of mass eigenstates,
where $\epsilon$ and $\epsilon'$ are given by
\begin{eqnarray}
\epsilon &=& \frac{Y_L}{G_L}
= \frac{\lambda \sin^2 \phi}{1-2 \sin^2 \theta} + {\cal O}(\lambda^2),
\nonumber \\
\epsilon' &=& \frac{Y'_L}{G'_L}
= -\frac{1}{\sin^2 \phi} + {\cal O}(\lambda),
\end{eqnarray}
while the right-handed neutral current interactions
are kept to be diagonal.
Thus $V$ is just $V_L$ here.
Since $(V^\dagger M V)_{ij} = V^*_{3i}V_{3j} \ne \delta_{ij}$
in ${\cal L}^{\rm NC}_{L}$,
the FCNC interaction terms generically arise
at tree level in this model.
Note that the matrix elements, $|V_{31}|$, $|V_{32}|$, $|V_{33}|$,
are observables in this model.
We will drop the superscript $0$ for mass eigenstates hereafter.
The FCNC parameter $\epsilon$ is of order $\lambda$
and suppress the FCNC interactions.
We note that $\epsilon'$ does not contain a suppression parameter but
the term involving $\epsilon'$ is suppressed
by the heavy mass $m_{Z'}$.

Our model has been examined by various experimental data so far.
The electroweak precision test with the data at the $Z$-pole
has provided strong constraints on new physics models.
We have used the nonstandard electroweak precision test
with $\epsilon_i$ variables to examine the our model,
which is introduced by Altarelli \cite{sm2,altarelli}
and extended with a new parameter $\epsilon'_b$
to measure the new effects in $Z b \bar{b}$ coupling \cite{kylee5}.
The new physics effects at tree level are assumed to be
compatible to the electroweak corrections in the SM such that
$\epsilon_i = \epsilon_i^{\rm SM} + \epsilon_i^{\rm new}$.
The SM predictions $ \epsilon_i^{\rm SM} $
are calculated by ZFITTER \cite{ZFITTER}
and $\epsilon_i^{\rm new}$ can be read out
from the $Z f \bar{f}$ couplings in Eq. (7).
With the LEP and SLC data at $Z$-pole \cite{LEPEWWG},
we obtain the constraints on the model parameters
$(\lambda, \sin^2 \phi)$ in Ref. \cite{kylee3,kylee4}.

The precisely measured data
for low-energy neutral current interaction processes
such as $\nu e\rightarrow \nu e$,  $\nu N$ scattering,
and the polarized electron-hadron scattering $e_{L,R} N \rightarrow eX$
can examine our model
through corrections to the ordinary $Z$ boson couplings
and the extra $Z'$ boson contribution.
The coefficients of the effective four-fermion Hamiltonian
are generically shifted by
$C = C_{\rm SM} (1-\lambda \sin^4 \phi)$
in this model.
The atomic parity violation is described by the same Hamiltonian
as the $e_{L,R} N \rightarrow eX$ scattering.
The weak charge of an atom defined by
$Q_W = -2 \Bigl[ C_{1u} (2Z+N) + C_{1d} (Z+2N) \Bigr]$
is also corrected by $ \Delta Q_W \equiv Q_W - Q_W^{\mathrm{SM}}
= \Delta Q_W^{\rm SM} (1 - \lambda \sin^4 \phi)$,
where $Z$ ($N$) is the number of protons (neutrons) in the atom.
With the recent measurement of the weak charge for the Cs an Tl atoms
\cite{apv_exp},
and the SM predictions $ Q_W = -73.19 \pm 0.03$ (Cs),
and $ Q_W = -116.81 \pm 0.04$ (Tl),
constraints on the model parameters
have been obtained in Ref. \cite{kylee3,kylee4}.

The charged current interactions are also affected by
the nonuniversality of the SU(2) couplings
as well as the neutral currents.
For the quark sector,
the charged current couplings are measured
by the CKM matrix elements.
The `observed' CKM matrix is defined by the couplings
in the low-energy four fermion effective Hamiltonian
describing the semileptonic quark decay.
In this model,
we extract `observed' $V_{CKM}$ as
\begin{eqnarray}
V_{CKM} = V_{CKM}^0 + \epsilon^c {V_U}^\dagger M V_D
              + \left( \frac{G'^c_L}{G^c_L} \right)^2
                \frac{m_W^2}{m_{W'}^2}
     \left( V_{CKM}^0 + \epsilon'^c {V_U}^\dagger M V_D \right)
\end{eqnarray}
where $V_U$ and $V_D$ are unitary matrices
which diagonalize up- and down-type quarks,
$V_{CKM}^0 \equiv {V_U}^\dagger V_D$ is the CKM matrix in the SM,
and $M = \delta_{3i} \delta_{3j}$ are given previously.
Using the correction terms
$\epsilon^c = \lambda \sin^2 \phi + {\cal O}(\lambda^2)$
and $\epsilon'^c = 1/\sin^2 \phi + {\cal O}(\lambda)$,
we have the simple form of $V_{CKM}=V_{CKM}^0 (1 + \lambda \sin^4 \phi)$.
Since the correction term violates the unitarity of the CKM matrix,
test of the CKM unitarity can examine our model.
We define the unitarity violating term $\Delta$ by
the normalization of the first row of the CKM matrix,
$|V_{ud}|^2 + |V_{us}|^2 + |V_{ub}|^2 = 1-\Delta$,
which is the most precise test of the unitarity of the CKM matrix.
We derive $\Delta = 2 \lambda \sin^4 \phi$ in our model
while $\Delta=0$ exactly in the SM.
Recent measurements of nuclear beta decays, kaon decays and $B$ decays
result in $\Delta = 0.0009 \pm 0.0010$
implying the unitarity of the CKM matrix with high accuracy
\cite{czarnecki}.
We show that the CKM unitarity leads to the stronger
constraint than the electroweak precision test with the LEP and SLC data
and the low-energy neutral current data
on the parameter space ($\sin^2 \phi, m_{Z'}$)
as shown in Fig.1 of Ref. \cite{kylee3}.

\begin{figure}[t]
\begin{center}
\hbox to\textwidth{\hss\epsfig{file=f1.eps,width=8cm}\hss}
\caption{
Bounds on $V_{31}$, $V_{32}$, and $V_{33}$
with respect to $m_{Z'}$ from the $Z \to l^- l^+$ decays.
}
\end{center}
\end{figure}

In the lepton sector, the FCNC interactions given in Eq. (7)
yield the lepton number violating processes at tree level.
Before studying the LFV processes,
we notice that the partial decay rates of $Z$ boson into lepton pairs
are also received the flavour conserving corrections
from the diagonal elements of $V^\dagger M V$.
In this model, the partial decay rates are shifted as
\begin{eqnarray}
\Gamma(Z \to l_i^- l_i^+)
= \Gamma_{\rm SM} \left( 1 + 2 \epsilon |V_{3i}|^2 \right),
\end{eqnarray}
in the leading order of $\lambda$.
The precisely measured data of $\Gamma(Z \to l_i^- l_i^+)$
without lepton universality by LEP
is given in Table 7.1 of Ref. \cite{LEPEWWG}.
We obtain constraints on $\epsilon$ and $|V_{3i}|$ from the data.
Figure 1 shows that allowed values of $V_{31}$, $V_{32}$, and $V_{33}$
with respect to $m_{Z'}$ by LEP data varying all possible values of
$\sin^2 \phi$.
We find that conservative bounds on $V_{3i}$ can be achieved
for given value of $m_{Z'}$.

The LFV $Z$ decays are described by
\begin{eqnarray}
\Gamma(Z \to l^-_i l^+_j) = \Gamma(Z \to l_i^+ l_i^-)
\cdot \epsilon^2 |V_{3i}|^2 |V_{3j}|^2,
\end{eqnarray}
neglecting lepton masses.
Assuming that $\sin^2 \phi \sim 0.1$,
we have $\epsilon < 0.02$ which leads to
${\rm Br}(Z \to l^+_i l^-_j) < |V_{3i}|^2 |V_{3j}|^2 \times 10^{-5}$.
Since both $|V_{3i}|$ and $|V_{3j}|$ cannot be 1,
the predictions of ${\rm Br}(Z \to l^+_i l^-_j)$ in our model
is likely to satisfy the present experimental bounds.

\begin{figure}[t]
\begin{center}
\hbox to\textwidth{\hss\epsfig{file=f2.eps,width=8cm}\hss}
\caption{
Bounds on $(\sin^2 \phi, m_{Z'})$ from the CKM unitarity, the LFV bounds
and the partial $Z$ decays into lepton pairs.
}
\end{center}
\end{figure}

The LFV lepton decays occur through the $Z$ and $Z'$ exchanges.
With the FCNC couplings derived in Eq. (7),
the flavour violating $\tau$ decay rates are given by
\begin{eqnarray}
\Gamma(\tau^- \to \mu^- (e^-) \mu^+ \mu^-)
&=& \frac{m_\tau^5}{96 \pi^3} \left[
\left| G_{4LL}^{23(13)} + {G'}_{4LL}^{23(13)} \right|^2
+ \left| G_{4LR}^{23(13)} \right|^2 \right],
\nonumber \\
&=& \Gamma(\tau^- \to \mu^- (e^-) e^+ e^-),
\end{eqnarray}
where ${G^{(\prime)}}_{4 \alpha \beta}^{ij}
= \left( G_\alpha G_\beta/m_{Z^{(\prime)}}^2 \right) \epsilon^{(\prime)}
V_{3i}^* V_{3j}$
with $i,j=1,2,3$ and $\alpha,\beta = L,R$.
Note that ${{G^{(\prime)}}^{ij}_{4RR}}$ and ${{G'}^{ij}_{4LR}}$
are ${\cal O}(\lambda^2)$.
The LFV muon decay rates are described by the same formula
except for the replacement of $V_{32(31)}^* V_{33}$
by $V_{31}^* V_{32}$ neglecting electron and muon masses.

The $\tau^- \to \mu^+ e^- e^-$ and $\tau^- \to e^+ \mu^- \mu^-$ processes
include only $Z'$-exchange contribution
in the leading order of $\lambda$,
of which decay rates are given by
\begin{eqnarray}
\Gamma(\tau^- \to \mu^+ e^- e^- (e^+ \mu^- \mu^-))
= \frac{m_\tau^5}{96 \pi^3} \left| H_{LL}^{1(2)} \right|^2,
\end{eqnarray}
where $ {H^{k}_{LL}} = (G'_L/m_{Z'})^2 {\epsilon'}^2
V_{32}^* V_{33} V_{31}^* V_{3k}$.

Here we discuss a few limiting cases to get lessons on values of $|V_{3i}|$.
For simplicity, the complex phases of the unitary matrix are ignored
in this work.
First, we assume $|V_{33}| \approx 1$.
Then $|V_{31}|$ and $|V_{32}|$ should be very small
to satisfy the unitarity condition $|V_{31}|^2+ |V_{32}|^2+ |V_{33}|^2=1$
and the strong bounds of the LFV $\tau$ decays,
${\rm Br}(\tau \to l_i l_j l_k) \propto |V_{33}|^2 |V_{3i}|^2
= {\cal O}(10^{-8})$, $i,j,k=1,2$.
Second, we assume that $|V_{33}| \approx 0$.
Then the LFV $\tau$ decays are automatically suppressed.
We have to satisfy the strong limit on $|V_{31}||V_{32}|$
from the LFV muon decays
${\rm Br}(\mu \to e e e ) \propto |V_{31}|^2 |V_{32}|^2 < 10^{-12}$
and the unitarity condition $|V_{31}|^2 + |V_{32}|^2 = 1$
simulataneously.
The only possible solution is that
one of $|V_{31}|$ and $|V_{32}|$ should be close to 1
and the other is very small.
Finally, let's consider the case of $|V_{33}|={\cal O}(0.1)$.
Then either $|V_{31}|$ or $|V_{32}|$
should also be ${\cal O}(0.1)$ to satisfy the unitarity condition.
In that case, we cannot avoid that
some of the LFV $\tau$ decays involving $|V_{33}| |V_{3i}|$
exceed the present experimental bounds.
Therefore we conclude that
one of the matrix elements $|V_{31}|$, $|V_{32}|$, and $|V_{33}|$
is close to 1 and the others are sufficiently small.

When $|V_{3i}|\approx 1$ for an $i$,
the corresponding  $\Gamma(Z \to l_i^- l_i^+)$ data
provide the strong limit on $\epsilon$, and therefore
on $\lambda$ and $\sin^2 \phi$.
For instance, we find that
$|V_{33}| \approx 1$ leads to the conservative bound $m_{Z'} > 1.8 $ TeV
in Fig. 1.

Now we scan all the parameters $|V_{31}|$, $|V_{32}|$, $|V_{33}|$,
$\sin^2 \phi$ and $m_{Z'}$
with the constraints from the CKM unitarity,
the experimental bounds of Eq. (1)-(3),
and the LEP data of the partial $Z$ widths.
Allowed parameter sets $(\sin^2 \phi, m_{Z'})$
are depicted in Fig. 2.
We find that stronger constraints on the parameters
are obtained from the absence of the LFV processes
and the partial decay widths of $Z \to l_i^- l_i^+$.
We do not show the bounds from the electroweak precision data
and the low-energy neutral current interaction data in Fig. 2
since their bounds are weaker than that of the CKM unitarity.
We obtain the conservative lower bound, $m_{Z'}>1.8$ TeV
independent of $\sin^2 \phi$ and $V_{3i}$.

In conclusion,
we have studied the lepton flavour violating processes in the
extended model based on SU(2)$_l \times$SU(2)$_h \times$U(1)
gauge symmetry.
In this model the FCNC involve the matrix elements, $V_{3i}$,
as additional parameters,
which diagonalize the charged lepton mass matrix.
We show that one of $V_{3i}$'s should be close to 1
while the other two elements are sufficiently small
to accommodate all experimental bounds of the LFV processes
and unitarity condition.
When $V_{3i} \approx 1$, the corresponding partial decay width
of $Z$ boson $\Gamma(Z \to l_i^- l_i^+)$ provides
strong constraints on the model parameters.
As a result,
we obtain the stronger constraints on this model
from the present experimental data on the LFV and $Z \to l^- l^+$ decays
than those from the previous data.

\acknowledgments
This work was supported
by WCU program through the KOSEF funded by the MEST (R31-2008-000-10057-0)
and by the Korea Research Foundation Grant funded
by the Korean Government (KRF-2008-313-C00167).

\def\PRD #1 #2 #3 {Phys. Rev. D {\bf#1},\ #2 (#3)}
\def\PRL #1 #2 #3 {Phys. Rev. Lett. {\bf#1},\ #2 (#3)}
\def\PLB #1 #2 #3 {Phys. Lett. B {\bf#1},\ #2 (#3)}
\def\NPB #1 #2 #3 {Nucl. Phys. B {\bf #1},\ #2 (#3)}
\def\ZPC #1 #2 #3 {Z. Phys. C {\bf#1},\ #2 (#3)}
\def\EPJ #1 #2 #3 {Euro. Phys. J. C {\bf#1},\ #2 (#3)}
\def\JHEP #1 #2 #3 {JHEP {\bf#1},\ #2 (#3)}
\def\IJMP #1 #2 #3 {Int. J. Mod. Phys. A {\bf#1},\ #2 (#3)}
\def\MPL #1 #2 #3 {Mod. Phys. Lett. A {\bf#1},\ #2 (#3)}
\def\JPG #1 #2 #3 {J. Phys. G {\bf#1},\ #2 (#3)}
\def\PTP #1 #2 #3 {Prog. Theor. Phys. {\bf#1},\ #2 (#3)}
\def\PR #1 #2 #3 {Phys. Rep. {\bf#1},\ #2 (#3)}
\def\RMP #1 #2 #3 {Rev. Mod. Phys. {\bf#1},\ #2 (#3)}
\def\PRold #1 #2 #3 {Phys. Rev. {\bf#1},\ #2 (#3)}
\def\IBID #1 #2 #3 {{\it ibid.} {\bf#1},\ #2 (#3)}


\section{Introduction}

We concentrate on our contribution, without giving
 a broader overview of the area here.

A remarkable result by G\"oller, Haase, Ouaknine,
Worrell~\cite{DBLP:conf/icalp/GollerHOW10} shows that model
checking a fixed CTL formula on succinct one-counter automata
(where counter increments and decrements are integers written in binary)
is EXPSPACE-hard.
Their proof is interesting and nontrivial, and uses two involved
results from complexity theory.
The technique of this proof was (a bit vaguely)
referred to by Hunter~\cite{DBLP:conf/rp/Hunter15},
by which he derived EXPSPACE-hardness of reachability games
on succinct one-counter nets
(with no zero tests).

Simulation-like equivalences on (non-succinct) one-counter nets
are PSPACE-complete
(see~\cite{DBLP:journals/corr/HofmanLMT16} for simulation equivalence
and~\cite{DBLP:journals/jcss/BohmGJ14} for bisimulation equivalence). This
immediately yields EXPSPACE-upper bounds in the ``succinct'' cases.

The PSPACE---EXPSPACE gap for the simulation problem was also
mentioned in~\cite{DBLP:journals/corr/HofmanLMT16} from where we quote:
``Another direction for further research is to establish the exact
complexity of strong/weak simulation for OCN with binary encoded
increments and decrements on the counter. Trivially, the PSPACE-lower
bound applies for this model and an EXPSPACE upper bound follows from
the results of this paper with the observation that these more
expressive nets can be unfolded into ordinary OCN with an exponential
blow-up.''

Here, in this paper, we close the complexity gap by showing EXPSPACE-hardness
(and thus EXPSPACE-completeness),
using a defender-choice technique
(cf., e.g.,~\cite{DBLP:journals/jacm/JancarS08})
to reduce reachability games to any relation between
simulation preorder and bisimulation equivalence.

But we first present a direct proof
of EXPSPACE-hardness of reachability games.
(This makes our paper self-contained
and shows that we do not need to rely on the result
from~\cite{DBLP:conf/icalp/GollerHOW10}.)
Our  direct proof
is based on the technique
from~\cite{DBLP:journals/ipl/JancarS07} used there to show
PSPACE-hardness
for emptiness of alternating finite automata with one-letter
alphabet (thus giving an alternative proof for the result by
Holzer~\cite{DBLP:conf/dlt/Holzer95});
Srba~\cite{DBLP:journals/corr/abs-0901-2068} used this result
to show PSPACE-hardness of behavioural relations for
(non-succinct) one-counter nets.
In Section~\ref{sec:addrem} we discuss a relation of our proof to
the countdown games of~\cite{DBLP:journals/lmcs/JurdzinskiSL08} and
their use in~\cite{DBLP:journals/ipl/Kiefer13}.
While the countdown games can serve as an interesting EXPTIME-complete problem,
by a slight enhancement we get an EXPSPACE-complete problem.

We stress that we do not provide an alternative proof for
the EXPSPACE-hardness result in~\cite{DBLP:conf/icalp/GollerHOW10};
the result in~\cite{DBLP:conf/icalp/GollerHOW10} is stronger,
though technically
it does not induce the hardness results for reachability and
(bi)simulation games automatically.

In Section~\ref{sec:overview} we give an informal overview
of the ideas in our paper.
Section~\ref{sec:definitions} contains formal definitions,
Section~\ref{sec:reachgames} proves the EXPSPACE-hardness of
reachability games, and Section~\ref{sec:reachtosimul} reduces
reachability games to (bi)simulation games.
Section~\ref{sec:addrem} contains some additional remarks.

\section{Informal overview}\label{sec:overview}

The mentioned strong result in~\cite{DBLP:conf/icalp/GollerHOW10}
shows that for any fixed language $L$ in EXPSPACE we can for any word $w$
(in the alphabet of $L$) construct
a succinct one-counter automaton that performs a computation which
is accepting iff $w\in L$.
Such a computation needs to access
concrete bits in the (reversed) binary presentation of the counter value.
A straightforward direct access to such bits is destructive (the counter
value is lost after the bit is read) but this can be avoided:
instead of a ``destructive
reading'' the computation just ``guesses''
the respective bits, and it is forced to guess correctly by a carefully
constructed CTL formula that is required to be satisfied by the
computation.

If we imagine that there is an opponent who can challenge the guesses
about bits, and after a challenge a destructive test follows that
either confirms the guess or exposes its invalidity, then this readily leads
to the hardness results for reachability games, and then also for
behavioural relations by using a defender-choice technique (with two
synchronously evolving copies of the respective one-counter automaton).

Performing the above sketched procedure to prove the mentioned results
rigorously would require
recalling technical details from~\cite{DBLP:conf/icalp/GollerHOW10}.
Instead we give a direct self-contained proof, which also makes clear
that our results do not rely on the involved complexity results
used in~\cite{DBLP:conf/icalp/GollerHOW10}.

\subsubsection*{EXPSPACE-hardness of reachability games}
We use a~``master'' reduction. We thus fix an arbitrary  language
$L$ in EXPSPACE, decided by a Turing machine $M$ in space $2^{p(n)}$
for a fixed polynomial $p$.
For any word $w$
in the alphabet of $L$ there is the respective computation of $M$,
which
is
accepting iff $w\in L$; the computation is a sequence
$C_0,C_1,\ldots,C_t$ of configurations, each $C_i$ being a string of
length $m=2^{p(\abs{w})}$.

Any
$k\in\Nat$ can code the $j$-th position in $C_i$
where $i=k\div m$ ($\div$ is integer division) and $j=k\bmod m$ (assuming $i\leq
t$).
Given $w$, we can construct an alternating one-counter automaton,
with Eve's and Adam's control states one of which is Eve's winning
state; there are no zero-tests but
 transitions yielding negative counter values are not allowed.
Starting in the initial configuration $(p_0,0)$ (with zero in the
counter), Eve
(who claims that there is an accepting computation
$C_0,C_1,\ldots,C_t$)
keeps
incrementing the counter until she enters a configuration
$(\langle q_+,a\rangle,k)$
by which she claims that the position coded by $k$
is the head-position in the accepting configuration $C_t$ and
contains letter $a$
($q_+$ being the accepting state of $M$).

Eve then subtracts $m$ and enters a control state corresponding to a triple of symbols that she claims
to be the $(j{-}1)$-th, the $j$-th, and the $(j{+}1)$-th symbol in
$C_{t-1}$ (where $j=k\bmod m$); the triple must be consistent with
the current symbol $\langle q_+,a\rangle$ and the transition rules of $M$.
We note that at least one symbol (of the type $b$ or
$\langle q,b\rangle$) in the triple
must be incorrect when
 $\langle q_+,a\rangle$ is incorrect (i.e., when it is
 not really the $j$-th symbol in $C_t$).

Now Adam chooses a new current symbol from the triple,
and also adds ${-}1/0/{+}1$ to the counter accordingly.
Eve then presents
 another consistent triple, etc.

In fact, Eve can also present a pair instead of a triple, claiming
that $j=0$ or that $j=m{-}1$ (where $j=k\bmod m$). Adam can challenge
this, claiming that $j>0$ or that $j<m{-}1$;
similarly he can claim that $j=0$ or that $j=m{-}1$ when Eve provides
a triple. Such claims can be easily (destructively) verified:
We first let Eve decrement the counter by $m$ repeatedly. If she
leaves a number $k'\geq m$ in the counter, then Adam
uses a transition subtracting $m$ that
enters a state precluding Eve's win.
Hence Eve is rather subtracting $m$ until
 $j=k\bmod m$ is in the counter.
Similarly we implement the respective checks
of the claims $j=0$, $j=m{-}1$, $0<j<m{-}1$, so that Eve can force her
win precisely
when her claim was correct.

Finally, Eve can claim that $C_0$ has been reached (i.e., $k\div
m=0$), which can be again punished by Adam if not true.
In the case $k\div m=0$ Eve wins
if the control
state ``claims'' the $j$-th symbol of $C_0$ (for $w$),
which is the blank tape-symbol if $j\geq |w|$;
checking this
 condition can be again
easily implemented in the game.

Hence $w\in L$ iff Eve can force reaching her winning
control state (when starting in $(p_0,0)$).

\subsubsection*{Reachability game reduces to (bi)simulation game}

Given a (succinct) one counter automaton with Eve's and Adam's control
states, one of them being Eve's winning, we first label each transition
by its unique action (action name) and
take two copies of the resulting automaton
(control state $s$ in one copy has a counterpart $s'$ in the other);
in the first copy we add a
special (``winning'') action in Eve's winning state.

We let
two players, called Attacker and Defender, to mimic the
reachability game. If there is Eve's turn, Attacker performs
a transition in one copy, and Defender must do the same in the other
copy, being obliged to use the same action as Attacker (cf.
Fig~\ref{fig:attchoice}).

The defender-choice technique is used when there is Adam's turn.
To this aim the two automaton-copies are a bit enhanced
and interconnected
(as in Fig.~\ref{fig:defendchoice}, discussed later). By performing a
``choice-action'' $a_c$ Attacker lets Defender to choose from (more
than one)
transitions labelled with $a_c$; if Attacker
does not follow Defender's choice in
the next round, then Defender installs syntactic equality
(in which case the play continues from a pair of
the same configurations of the same automaton-copy).
It is thus Defender who chooses Adam's moves.

By the above construction we achieve that if Eve has a winning
strategy in the reachability game from $(p_0,0)$, then
$(p_0,0)$ is not simulated by $(p'_0,0)$, and if
Eve has no winning
strategy, then $(p_0,0)$ and $(p'_0,0)$ are bisimilar
(in the respective labelled transition system).

\begin{figure
  \centering
  \begin{tikzpicture}[node distance=0.8cm]

    \node (dummy) {};
    \node[stav,left of=dummy,xshift=0cm] (s2) {$s_2$};
    \node[stav,right of=dummy,xshift=0cm] (s3) {$s_3$};
    \node[right of=dummy,xshift=4cm]  (dummy2) {};
    \node[stav,left of=dummy2] (s2prime) {$s'_2$};
    \node[stav,right of=dummy2] (s3prime) {$s'_3$};

  \node[stav,above of=dummy,yshift=1.5cm] (s1) {$s_1$}
  edge[pil] node[left] {$a^1_2(x)$} (s2.north)
  edge[pil] node[right] {$a^1_3(y)$} (s3.north);

  \node[stav,above of=dummy2,yshift=1.5cm] (s1prime) {$s'_1$}
  edge[pil] node[left] {$a^1_2(x)$} (s2prime.north)
  edge[pil] node[right] {$a^1_3(y)$} (s3prime.north);

  \node[stav,left of=s2,xshift=-5cm] (s2orig) {$s_2$};

  \node[stav,left of=s3,xshift=-5cm] (s3orig) {$s_3$};
  \node[stav,left of=s1,xshift=-5cm] (s1orig) {$s_1$}
  edge[pil] node[left] {$(x)$} (s2orig.north)
  edge[pil] node[right] {$(y)$} (s3orig.north);

  \node [left of=s1orig,xshift=0.2cm,yshift=0.2cm] (dummy2) {$E$};

  \node [right of=s1orig,xshift=2.3cm,yshift=0.5cm] (dummy3) {};
  \node [below of=dummy3,yshift=-2.8cm] (dummy4) {};

  \draw [thick] (dummy3) -- (dummy4) ;

 \end{tikzpicture}

	\caption{In $(s_1,s'_1)$ it is Attacker
	who chooses $(s_2,s'_2)$ or
	$(s_3,s'_3)$.
}\label{fig:attchoice}.
\end{figure}

\begin{figure
  \centering
  \begin{tikzpicture}[node distance=0.8cm]

    \node (dummy) {};
    \node[stav,left of=dummy,xshift=-0.4cm] (s2) {$s_2$};
    \node[stav,right of=dummy,xshift=0.4cm] (s3) {$s_3$};
   \node[stav,right of=s3,xshift=1.5cm] (s2prime) {$s'_2$};
\node[stav,right of=s2prime,xshift=1.5cm] (s3prime) {$s'_3$};

   \node[stav,above of=dummy,yshift=1.5cm] (s123) {$s^1_{23}$}
   edge[pil] node[left] {$a^1_2(x)$} (s2.north)
   edge[pil] node[left] {$a^1_3(y)$} (s3.north);

  \node[stav,above of=s2prime,yshift=1.5cm] (s12) {$s^1_{2}$}
  edge[pil] node[right] {$a^1_2(x'')$} (s2prime.north)
  edge[pil,bend right=25] node[right,xshift=-0.1cm,yshift=-0.1cm]
  {$a^1_3(y{-}x')$} (s3.north);

   \node[stav,above of=s3prime,yshift=1.5cm] (s13) {$s^1_{3}$}
   edge[pil, ] node[right] {$a^1_3(y'')$} (s3prime.north)
   edge[pil, out=-100, in=-60] node[right,yshift=-0.2cm]
   {$a^1_2(x{-}y')$} (s2.south);

  \node[stav,above of=s123,yshift=1.5cm] (s1) {$s_1$}
  edge[pil] node[left] {$a_c(0)$} (s123.north)
  edge[dsh, bend right=10] node[right,xshift=-0.4cm,yshift=0.5cm]
  {$a_c(x')$} (s12.west)
    edge[dsh, bend left=30] node[right,xshift=-0.9cm,yshift=0.5cm]
    {$a_c(y')$} (s13.west)
   ;

  \node[stav,above of=s12,yshift=1.5cm,xshift=1cm] (s1prime) {$s'_1$}
  edge[pil] node[left,xshift=-0cm,yshift=-0.1cm] {$a_c(x')$} (s12.north)
  edge[pil] node[right] {$a_c(y')$} (s13.north)
 ;

  \node[stav,left of=s2,xshift=-5cm] (s2orig) {$s_2$};

  \node[stav,left of=s3,xshift=-5cm] (s3orig) {$s_3$};
  \node[stav,left of=s1,xshift=-5cm] (s1orig) {$s_1$}
  edge[pil] node[left] {$(x)$} (s2orig.north)
  edge[pil] node[right] {$(y)$} (s3orig.north);

  \node [left of=s1orig,xshift=0.2cm,yshift=0.2cm] (dummy2) {$A$};

  \node [right of=s1orig,xshift=2.3cm,yshift=0.5cm] (dummy3) {};
  \node [below of=dummy3,yshift=-6.0cm] (dummy4) {};

  \draw [thick] (dummy3) -- (dummy4) ;

 \end{tikzpicture}

	\caption{In $(s_1,s'_1)$ it is Defender
	who chooses $(s_2,s'_2)$ or
	$(s_3,s'_3)$ (or a pair of equal states); to take
	the counter-changes into account correctly, we put
	$x'=\min{\{x,0\}}$, $x''=\max{\{x,0\}}$, and
 $y'=\min{\{y,0\}}$, $y''=\max{\{y,0\}}$
 (hence $x=x'{+}x''$ and $y=y'{+}y''$).}\label{fig:defendchoice}.

\end{figure}

\section{Definitions}\label{sec:definitions}

By $\Zset$ we denote the set of integers,
and by $\Nat$ the set of nonnegative integers
$\{0,1,2,\dots\}$.
\\
We use $[i,j]$ for denoting the set $\{i,i{+}1,\dots,j\}$, where $i,j\in\Zset$.

\paragraph{Reachability games.}
By a \emph{reachability game}, or an \emph{r-game} for short, we mean
a tuple
$\calG=(V,V_\exists, \gt{}, \calT)$,
where $V$ is the set of \emph{states} (or \emph{vertices}),
$V_\exists\subseteq V$ is the set of \emph{Eve's states},
$\mathop{\gt{}}\subseteq {V\times V}$ is the \emph{transition relation}
(or the set of \emph{transitions}), and $\calT\subseteq V$ is the set
of \emph{target states}.
By  \emph{Adam's states} we mean  the elements of
$V_\forall=V\smallsetminus V_\exists$.

We put $\winareaE=\bigcup_{\lambda\in Ord}W_\lambda$ where $Ord$ is
the class of ordinals and the sets $W_\lambda\subseteq V$ are defined
inductively as follows. We put $W_0=\calT$. For $\lambda>0$ we put
$W_{<\lambda}=\bigcup_{\lambda'<\lambda}W_{\lambda'}$, and we
stipulate:
\begin{enumerate}[a)]
	\item
if $s\not\in W_{<\lambda}$, $s\in V_\exists$, and
$s\gt{}\bar{s}$ for some $\bar{s}\in W_{<\lambda}$, then $s\in W_{\lambda}$;
	\item
if $s\not\in W_{<\lambda}$, $s\in V_\forall$,
and we have $\emptyset\neq \{\bar{s}\mid s\gt{}\bar{s}\}\subseteq
W_{<\lambda}$,
then $s\in W_{\lambda}$.
\end{enumerate}
(If a) applies, then $\lambda$ is surely a successor ordinal,
otherwise $\lambda$ can be also a limit ordinal.)

For each $s\in\winareaE$, by $\rank(s)$ we denote (the unique)
$\lambda$ such that
$s\in W_\lambda$. A transition $s\gt{}\bar{s}$ is
\emph{rank-reducing} if $\rank({s})>\rank(\bar{s})$.
We note that for any $s\in\winareaE$ with $\rank(s)>0$ we have:
if $s\in V_{\exists}$, then there is at least one rank-reducing
transition $s\gt{}\bar{s}$ (in, fact $\rank(s)=\rank(\bar{s}){+}1$ in
this case); if $s\in V_{\forall}$, then there is at least one
transition $s\gt{}\bar{s}$ and all such transitions are rank-reducing.

\medskip

\emph{Remark.}
We are primarily interested in the games that have  (at most)
countably many states and are finitely branching (the sets
$\{\bar{s}\mid s\gt{}\bar{s}\}$ are finite for all $s$). In such cases
we have $\rank(s)\in\Nat$ for each $s\in\winareaE$.
\\
We also note that $\winareaE$ is the set of states from which Eve has a winning
strategy, i.e. such that guarantees reaching (some state in) $\calT$ when Eve
is choosing a next transition in Eve's states
and Adam is choosing a next transition in Adam's states.

\paragraph{Labelled transition systems and (bi)simulations.}
A \emph{labelled transition system}, an \emph{LTS} for short, is a
tuple
$\calL = (S, Act, (\gt{a})_{a\in Act})$
  where $S$ is the set of \emph{states}, $Act$ is the set of
  \emph{actions},
  and $\mathop{\gt{a}}\subseteq S \times S$ is the set
of  \emph{$a$-transitions} (transitions labelled with $a$), for each
  $a\in Act$.

Given $\calL = (S, Act, (\gt{a})_{a\in Act})$,
a relation $R \subseteq S \times S$ is a \emph{simulation}
  if for every $(s,s')\in R$ and every
  $s \gta t$ there is $s' \gta t'$ such that $(t,t')\in R$;
  if, moreover, for every $(s,s')\in R$ and every
  $s' \gta t'$ there is $s \gta t$ such that $(t,t')\in R$, then $R$
is a \emph{bisimulation}.
  The union of all simulations (on $\calS$) is the maximal
  simulation, denoted $\simul$; it is a preorder, called
  \emph{simulation preorder}.
 The union of all bisimulations is the maximal
 bisimulation, denoted $\bisim$; it is an equivalence, called
 \emph{bisimulation equivalence} (or \emph{bisimilarity}).
 We observe that
 $\bisim\subseteq\simul$.

 \medskip

 \emph{Remark.}
We can write
 $s_1\simul s_2$ or $s_1\bisim s_2$ also for states $s_1$,
 $s_2$ from different  LTSs $\calL_1$, $\calL_2$, in which case
 the LTS arising by the disjoint union of $\calL_1$ and $\calL_2$ is
 (implicitly) referred to.
\\
It is also useful to think in terms of games here.
In the \emph{simulation game}, in a (current) pair $(s,s')$ Attacker chooses
a transition $s\gt{a}t$ and Defender responds with some
$s'\gt{a}t'$ (for the action $a$ chosen by Attacker); the play then
continues with another round, now in the current pair  $(t,t')$, etc.
If Defender has no response in a round, then Attacker wins the play.
It is standard to note that $s\notsimul s'$ iff Attacker has a winning
strategy from $(s,s')$.
\\
The case of \emph{bisimulation game} is analogous, but
in any round starting from  $(s,s')$
Attacker can choose to
play  $s\gt{a}t$ or $s'\gt{a}t'$, and Defender has to respond
with some $s'\gt{a}t'$ or $s\gt{a}t$, respectively.
Now Attacker has a winning
strategy from $(s,s')$ iff $s\notbisim s'$.

\medskip

We now define specific r-games and LTSs, presented by particular
one-counter automata.

\paragraph{(Succinct) one-counter net games.}
By a \emph{one-counter net game}, an \emph{ocn-game} for short, we
mean a tuple $\calN=(Q,Q_{\exists},\delta,p_{win})$
where $Q$ is the finite set of \emph{(control) states},
$Q_{\exists}\subseteq Q$ is the set of \emph{Eve's (control) states},
$p_{win}\in Q$ is the \emph{target (control) state}, and
$\delta\subseteq Q\times\Zset\times Q$ is the finite set of
\emph{(transition) rules}.
We often present a rule $(q,z,q')\in\delta$ as
$q\gt{z}q'$.
By \emph{Adam's (control) states} we
mean the elements of $Q_{\forall}=Q\smallsetminus Q_{\exists}$.

An ocn-game $\calN=(Q,Q_{\exists},\delta,p_{win})$ has the
\emph{associated r-game}
\begin{equation}\label{eq:gamefornet}
\calG_\calN=(Q\times
\Nat,Q_{\exists}\times\Nat,\gt{},\{p_{win}\}\times\Nat)
\end{equation}
where $(q,m)\gt{}(q',n)$ iff $q\gtl{n-m}q'$ is a rule (in $\delta$).
We often write $q(m)$ instead of $(q,m)$ for states of $\calG_\calN$.
(A rule  $q\gt{z}q'$ thus induces transitions $q(m)\gt{}q'(m{+}z)$ for
all $m\geq \max\{0,{-}z\}$.)

\medskip

We define the problem \emph{RG-SOCN} (reachability game on succinct
one-counter nets):
\begin{quote}
\emph{Instance}: an ocn-game $\calN$
with integers $z$ in rules $q\gt{z}q'$ written in binary,
\\
\hspace*{4.5em}and a control state $p_0$.
\\
\emph{Question}: Is $p_0(0)\in\winareaE$ in the game $\calG_\calN$~?
\end{quote}

\medskip

\emph{Remark.}
We have defined the target states (in $\calG_\calN$) by the control
state $p_{win}$. There are other natural variants
(e.g., one in~\cite{DBLP:conf/rp/Hunter15} defines the target set
$\{p(0)\mid p\neq p_0\}$) that are, in principle, equivalent in our
context.

\paragraph{(Succinct) labelled one-counter nets.}
A \emph{labelled one-counter net}, an \emph{OCN} for short,
is a triple $\calN=(Q,Act,\delta)$,
where $Q$ is the finite set of \emph{control states}, $Act$ the finite
set of \emph{actions},
and $\delta \subseteq Q \times Act\times \Zset \times Q$ is
the finite set of (\emph{labelled transition}) \emph{rules}.
We present a rule $(q,a,z,q')\in\delta$ as
$q\gtl{a,z}q'$ ($a\in Act$, $z\in\Zset$).
An OCN $\calN=(Q,Act,\delta)$  has the
\emph{associated LTS}
\begin{equation}\label{eq:ltsfornet}
\calL_\calN=(Q\times
\Nat,Act,(\gt{a})_{a\in Act})
\end{equation}
where $q(m)\gtl{a}q'(n)$ iff $q\gtl{a,n-m}q'$ is a rule in $\delta$.
(We again write $q(m)$ instead of $(q,m)$.
A rule  $q\gtl{a,z}q'$ thus induces transitions $q(m)\gt{a}q'(m{+}z)$ for
all $m\geq \max\{0,{-}z\}$.)

For claims on complexity (in particular for
Theorem~\ref{th:equivhard}) we define
a \emph{succinct (labelled) one-counter net},
a \emph{SOCN} for short, as an OCN where the integers in
rules $q\gtl{a,z}q'$ are written in binary.

\section{EXPSPACE-hardness of reachability games}\label{sec:reachgames}

In Section~\ref{sec:overview} we sketched a ``master'' reduction
showing EXPSPACE-hardness of reachability games on succinct
one-counter nets, which is captured by the following theorem.

\begin{theorem}\label{th:rgexpsphard}
The problem \rgsocn is EXPSPACE-hard.
\end{theorem}

As already mentioned, an idea of a proof is given
in~\cite{DBLP:conf/rp/Hunter15} by referring
to~\cite{DBLP:conf/icalp/GollerHOW10}. Here we provide a
self-contained proof, by performing the sketched master reduction  in
detail.

We first give a construction for general Turing machines,
and then implement its space-bounded variant by one-counter nets.
Hence we now fix an arbitrary (deterministic)
Turing machine~$M=(Q,\Sigma,\Gamma,\delta,q_0,\{q_+,q_-\})$,
where $Q$ is the set of (control) states, $q_0\in Q$ the initial
state,  $q_+\in Q$ the accepting
state,  $q_-\in Q$ the rejecting
state,
$\Sigma$ the input
alphabet, $\Gamma\supseteq \Sigma$ the tape alphabet, satisfying
$\blank\in\Gamma\smallsetminus\Sigma$ for the special blank
tape-symbol $\blank$, and $\delta: (Q\smallsetminus
\{q_+,q_-\})\times\Gamma\rightarrow
Q\times\Gamma\times\{{-}1,{+}1\}$ is the transition function.

Putting $\Delta=\Gamma\cup(Q\times \Gamma)$, we define the relation
$\vdash\mathop{\subseteq} {\Delta^3\times \Delta}$
in a standard way:
\\
$(\beta_1,\beta_2,\beta_3)\vdash \beta$ if $\beta_i\in Q\times\Gamma$
for at most one $i\in\{1,2,3\}$ and the following conditions
hold:
\begin{itemize}
\item if $\beta_1\beta_2\beta_3=(q,x)yz$ and
	$\delta(q,x)=(q',x',d)$, then $\beta=(q',y)$ if $d={+}1$ and
	$\beta=y$ otherwise (i.e., if $d={-}1$);
\item if $\beta_1\beta_2\beta_3=x(q,y)z$ and
	$\delta(q,y)=(q',y',d)$, then $\beta=y'$ (for any
	$d\in\{{-}1,{+}1\}$);
\item if $\beta_1\beta_2\beta_3=xy(q,z)$ and
	$\delta(q,z)=(q',z',d)$, then $\beta=(q',y)$ if $d={-}1$ and
	$\beta=y$ otherwise;
\item if  $\beta_1\beta_2\beta_3=xyz$, then $\beta=y$.
\end{itemize}
We note that $\vdash$ is a partial function, in fact.
By a \emph{configuration} of $M$ we mean a mapping
$C:\Zset\rightarrow\Delta$
where $C(j)\neq \blank$ for only finitely many $j\in\Zset$ and
$C(j)\in Q\times \Gamma$ for precisely one $j\in\Zset$, called the
\emph{head-position}; if  $C(j)=(q_+,x)$  for the head-position $j$
(and $x\in\Gamma$)
then $C$
is \emph{accepting}, and if  $C(j)=(q_-,x)$  then $C$
is \emph{rejecting}.

We put $C\vdash C'$ (thus overloading the
symbol $\vdash$) if $\big(C(j{-}1), C(j), C(j{+}1)\big)\vdash C'(j)$ for all
$j\in\Zset$. This relation $\vdash$ is again a partial function; if
$C$ is \emph{final}, i.e. accepting or rejecting, then there is no $C'$ such that $C\vdash
C'$.

Given a word $w=a_1a_2\cdots a_n\in \Sigma^*$ (hence $|w|=n$), we define the
respective initial configuration as
$C_0^w$ where $C_0^w(0)=(q_0,a_1)$ if $n\geq 1$
and $C_0^w(0)=(q_0,\blank)$ if $n=0$, $C^w_0(j)=a_{j+1}$ for all
$j\in[1,n{-}1]$, and $C^w_0(j)=\blank$ for all $j<0$ and all $j\geq
n$. If $C^w_i$ is not final,
then we define
$C^w_{i+1}$ so that $C^w_i\vdash C^w_{i+1}$.
The \emph{computation on} $w$ is either the finite sequence
$C_0^w,C^w_1,C^w_2,\dots, C_t^w$ where  $C_t^w$ is final (accepting or
rejecting), or the infinite sequence $C_0^w,C^w_1,C^w_2,\dots$;
formally we put $C^w_i(j)=\bot$ (for $\bot\not\in\Delta$) if
 there is a final $C^w_t$ and $i>t$.

By $L(M)$ we denote the \emph{language accepted by} $M$,
i.e. the set $\{w\in\Sigma^*\mid$ the computation on $w$ finishes with
an accepting configuration$\}$.

Given (our fixed) Turing machine $M$ and a word $w=a_1a_2\dots
a_n\in\Sigma^*$,
we define the r-game
$$\calG^M_w=(V,V_\exists, \gt{}, \calT)$$
where $V_\exists=\{s_0,s_{F}\}\cup(\Delta\times\Nat\times\Zset)$,
$V_\forall=V\smallsetminus V_{\exists}=\Delta^3\times\Nat\times\Zset$, $\calT=\{s_F\}$,
and where the transition
relation $\gt{}$ is defined as
follows:
\begin{enumerate}[1)]
	\item		Eve's moves:
		\begin{enumerate}[a)]
			\item
				$s_0\gt{}((q_+,x),i,j)$ for all
				$x\in\Gamma, i\in\Nat,j\in\Zset$;
			\item
		$(\beta,i,j)\gt{}((\beta_1,\beta_2,\beta_3), i{-}1,j)$
if $i\geq 1$ and
$(\beta_1,\beta_2,\beta_3)\vdash \beta$;
\item
$(\beta,0,j)\gt{}s_F$ if $\beta=C^w_0(j)$.
\end{enumerate}

\item Adam's moves:
$((\beta_1,\beta_2,\beta_3),i,j)\gt{}(\beta_\ell,i,j{-}2{+}\ell)$
for  $\ell\in\{1,2,3\}$.
\end{enumerate}
The next proposition shows how Eve's winning region is related to
$M$'s computation on $w$.

\begin{prop}\label{prop:abstractgame}
In $\calG^M_w$ we have
$\winareaE=X_0\cup X_1\cup X_2$
where
\begin{itemize}
	\item
		$X_0=\{s_0,s_F\}$ if $w\in L(M)$ and $X_0=\{s_F\}$ if  $w\not\in
L(M)$,
\item
$X_1=\big\{(\beta,i,j)\mid C^w_i(j)=\beta\big\}$,
\item
$X_2=\big\{((\beta_1,\beta_2,\beta_3),i,j)\mid
C^w_i(j{-}1)C^w_{i}(j)C^w_i(j{+}1)=\beta_1\beta_2\beta_3\big\}$.
\end{itemize}
\end{prop}
\begin{proof}
By definition we have $s_F\in\winareaE$. We first ignore the question
if $s_0\in\winareaE$, and show the rest of the claim by induction
on $i\in\Nat$.
The base case ($i=0$) is clear due to the transitions from the points
1c) and 2).
The induction step is also easy to check, when we note
that for any
$(\beta,i,j)$ with $i\geq 1$ we have:
\begin{itemize}
	\item
		if $\beta=C^w_i(j)$ then there is a transition
		$(\beta,i,j)\gt{}((\beta_1,\beta_2,\beta_3), i{-}1,j)$
		such that
		\\
		$\beta_1\beta_2\beta_3=C^w_{i-1}(j{-}1)C^w_{i-1}(j)C^w_{i-1}(j{+}1)$, and
	\item
if $\beta\neq C^w_i(j)$ then for every  transition
		$(\beta,i,j)\gt{}((\beta_1,\beta_2,\beta_3), i{-}1,j)$
		there is $\ell\in\{1,2,3\}$ such that
		$\beta_\ell\neq C^w_{i-1}(j{-}2{+}\ell)$.
\end{itemize}
The claim for $s_0$ follows, since
 $w\in L(M)$ iff
$C^w_t(j)=(q_+,x)$ for some $t\in\Nat$, $j\in\Zset$,  $x\in\Gamma$.
\end{proof}

If we think of mimicking the game $\calG^M_w$ by a one-counter game, it
is natural to represent a state $(\beta,i,j)$, or
$((\beta_1,\beta_2,\beta_3),i,j)$, so that $\beta$, or $(\beta_1,\beta_2,\beta_3)$, is (in)
a control state and $(i,j)$ is suitably represented by one (counter)
value $k$.
This seems manageable when we are guaranteed
that
in the computation of $M$ on
$w=a_1a_2\cdots a_n$
the head-position is never outside $[0,m{-}1]$ for a fixed $m\geq
n$; we now assume this, while also assuming $n\geq 1$ for convenience.
In such a case $\calG^M_w$ can be naturally
adjusted to yield a game $\calG^M_{w,m}$ in which
the head-position is kept inside
$[0,m{-}1]$. Our aim is to mimic $\calG^M_{w,m}$ by
(the r-game associated with) an ocn-game $\calN^M_{w,m}$; a first attempt
to built such  $\calN^M_{w,m}$ can look as follows:

\begin{equation}\label{eq:ocgamePone}
	\begin{array}{lllll}
	   p_0 \trans{+1} p_0 &
	   p_0 \trans{0} s_{(q_+,x)} & (\textnormal{where } x\in \Gamma)\\
		s_\beta\trans{-m}\AStateIII & & (\textnormal{where }(\beta_1,\beta_2,\beta_3)\vdash\beta)\\
	   \AStateIII \trans{-1} s_{\beta_1} &  \AStateIII \trans{0} s_{\beta_2}
	   &\AStateIII \trans{+1}s_{\beta_3}\\

   \end{array}
\end{equation}
By the configuration $p_0(0)$ we represent the state $s_0$ of
$\calG^M_{w,m}$.
A configuration $s_\beta(k)$ is intended to represent the state
$(\beta, k\div m, k\bmod m)$; similarly
$s_{(\beta_1,\beta_2,\beta_3)}(k)$ is intended to represent the state
$((\beta_1,\beta_2,\beta_3), k\div m, k\bmod m)$.
Hence $p_0$ and $s_\beta$ are Eve's control states, while
$s_{(\beta_1,\beta_2,\beta_3)}$ are Adam's control states.

The ocn-game given by the rules in~(\ref{eq:ocgamePone})
does not mimic the game $\calG^M_{w,m}$ faithfully,
due to possible ``cheating'' related to the boundary
head-positions.
Therefore we add

\begin{equation}\label{eq:ocgamePtwo}
	\begin{array}{lllll}
		s_\beta\trans{-m}s^L_{(\blank,\beta_2,\beta_3)}
		&		s^L_{(\blank,\beta_2,\beta_3)}\trans{0}\bar{g}_0
		&
		s^L_{(\blank,\beta_2,\beta_3)}\trans{0}s_{\beta_2}
		&
s^L_{(\blank,\beta_2,\beta_3)}\trans{+1}s_{\beta_3}
   \end{array}
\end{equation}
for the cases $(\blank,\beta_2,\beta_3)\vdash\beta$.
By entering a configuration $s^L_{(\blank,\beta_2,\beta_3)}(k)$
Eve also ``claims'' that the head-position is $0$, i.e., that $k\bmod m=0$.
The state $s^L_{(\blank,\beta_2,\beta_3)}$ is Adam's, who can believe
the claim and play accordingly, or
decide to challenge the claim by performing
$s^L_{(\blank,\beta_2,\beta_3)}(k)\gt{0}\bar{g}_0(k)$.
Symmetrically we add
\begin{equation}\label{eq:ocgamePthree}
	\begin{array}{lllll}
		s_\beta\trans{-m}s^R_{(\beta_1,\beta_2,\blank)}
		&
		s^R_{(\beta_1,\beta_2,\blank)}\trans{-1}s_{\beta_1}
				&
		s^R_{(\beta_1,\beta_2,\blank)}\trans{0}s_{\beta_2}
		&
		s^R_{(\beta_1,\beta_2,\blank)}\trans{0}\bar{g}_{m-1}
   \end{array}
\end{equation}
for the cases $(\beta_1,\beta_2,\blank)\vdash\beta$.
By entering a configuration $s^R_{(\beta_1,\beta_2,\blank)}(k)$
Eve also claims that the head-position is $m{-}1$, i.e., that $k\bmod
m=m{-}1$.
In a state $s^R_{(\beta_1,\beta_2,\blank)}$ Adam can
decide to challenge the claim by performing
$s^R_{(\beta_1,\beta_2,\blank)}(k)\gt{0}\bar{g}_{m-1}(k)$.

To complete this reasoning, by entering
$\AStateIII$ (with no superscript) Eve claims that
$0<(k\bmod m)<m{-}1$; Adam can challenge this by using one of the rules
\begin{equation}\label{eq:ocgamePfour}
	\begin{array}{lllll}
	\AStateIII\trans{0}{g}_{0}
	& & \AStateIII\trans{0}{g}_{m-1}
   \end{array}
\end{equation}
Hence control states $g_c$, for $c\in\{0,m{-}1\}$, can be viewed as
Adam's claims
``the current counter value $k$ satisfies $k\bmod m=c$''; similarly
$\bar{g}_c$ is Adam's claim
``$k\bmod m\neq c$''. We need to add some rules guaranteeing Eve's win
in the configurations $g_c(k)$ and $\bar{g}_c(k)$ precisely when
Adam's claims are incorrect. Moreover, we need that in $s_\beta(k)$
where $k<m$ Eve can force her win iff $\beta=C^w_0(k)$.
The required properties are achieved by completing
the rules~(\ref{eq:ocgamePone}),
(\ref{eq:ocgamePtwo}),~(\ref{eq:ocgamePthree}),~(\ref{eq:ocgamePfour})
with
\begin{equation}\label{eq:ocgamePfive}
   \begin{array}{lllllllll}
	   s_{\blank}\trans{0}f  & &  & s_\beta\trans{-j}\bar{e}_0
	   & \textnormal{where } 0\leq j\leq n{-}1 \textnormal{ and }
	   C^w_0(j)=\beta
   \end{array}
\end{equation}
and with the following final set:
\begin{equation}\label{eq:ocgamePsix}
   \begin{array}{lllllllll}
	   f \trans{-m} \WinA & e_c \trans{-c} e_c'  &
	   \bar{e}_c \trans{-c} \bar{e}_c' &
	   g_c \trans{-m} g_c & \bar{g}_c \trans{-m} \bar{g}_c
       \\
       f \trans{0} f' &
   e_c \trans{0} \WinE  &  \bar{e}_c' \trans{-1} \WinA &
      g_c \trans{0} g_c' &

       \bar{g}_c \trans{0} \bar{g}_c'       \\
       f' \trans{-n} \WinE &
   e_c' \trans{-1} \WinE
       &     \bar{e}_c' \trans{0} \WinE       &
      g_c' \trans{-m} \WinE &  \bar{g}_c' \trans{-m} \WinE
     \\
     & &  &
      g_c' \trans{0} e_c &  \bar{g}_c' \trans{0} \bar{e}_c &&
      \\
   \end{array}
\end{equation}
Here $c$ ranges over two values, $0$ and $m{-}1$.

Given $M$, $w=a_1a_2\cdots a_n$, and $m\geq n$,
we thus constructed the ocn-game
\begin{equation}\label{eq:constrocngame}
\calN^M_{w,m}=(Q,Q_{\exists}, \delta_{\calN}, \WinE)
\end{equation}
where
\begin{itemize}
	\item
$Q_{\exists}=\{p_0, f',e_c', \bar{e}_c, g_c', \bar{g}'_c, \WinE, \WinA\}\cup
\{s_\beta\mid \beta\in\Delta\}$,
\item
$Q_{\forall}=\{f, e_c, \bar{e}'_c, g_c, \bar{g}_c\}\cup
\{s_{(\beta_1,\beta_2,\beta_3)}\mid \beta_i\in\Delta\}\cup
\{s^L_{(\blank,\beta_2,\beta_3)}\mid\beta_i\in\Delta\}\cup
\{s^R_{(\beta_1,\beta_2,\blank)}\mid\beta_i\in\Delta\}$
\end{itemize}
(recall that 	$Q_{\forall}=Q\smallsetminus Q_{\exists}$),
and the set $\delta_\calN$ of rules is given
by~(\ref{eq:ocgamePone}), (\ref{eq:ocgamePtwo}), (\ref{eq:ocgamePthree}),
(\ref{eq:ocgamePfour}), (\ref{eq:ocgamePfive}), (\ref{eq:ocgamePsix}).

\begin{prop}\label{prop:spaceboundedocgame}
In the r-game associated with $\calN^M_{w,m}$ we have $p_0(0)\in\winareaE$ iff
the computation of $M$ on $w$ never moves the head out
of $[0,m{-}1]$ and finishes with accepting.
\end{prop}
\begin{proof}
The claim follows by Prop.~\ref{prop:abstractgame}, by the
previous discussions accompanying the construction of
$\calN^M_{w,m}$, and by the following properties that are easy to check
(recall that $c\in\{0,m{-}1\}$):
\begin{enumerate}[1.]
	\item		$f(k)\in \winareaE$ iff $n\leq k<m$;
	\item 		$e_c(k)\in\winareaE$ iff $k\neq c$, and
$\bar{e}_c(k)\in\winareaE$ iff $k=c$;
	\item 		$g_c(k)\in\winareaE$ iff $k\bmod m\neq c$, and
$\bar{g}_c(k)\in\winareaE$ iff $k\bmod m=c$;
\item
	for $k<n$, $s_{\beta}(k)\in\winareaE$ iff $\beta=C^w_0(k)$.
\end{enumerate}
Using the rules~(\ref{eq:ocgamePfive}) in
$s_{\beta}(k)$ when  $k\geq m$ is thus losing for Eve; by this
the proof is finished.
\end{proof}

We now note that the control states
in $\calN^M_{w,m}$ are determined by $M$, as well as
the rules  except of those in~(\ref{eq:ocgamePfive})
that are dependent on $w$; to be precise, the values of
counter-decrements ${-}m$,
${-(m{-}1)}$ (in ${-}c$ for $c=m{-}1$) and ${-}n$ are not determined
by $M$ but by ``parameters'' $w$ and $m$.

To finish the proof of Theorem~\ref{th:rgexpsphard}, we assume an
arbitrary
fixed language $L$ in \EXPSPACE. There is thus a Turing machine $M$ and a
 polynomial $p$ such that $M$ decides $L$ and
the head-position in the computation of $M$ on any $w$
(in the alphabet of $L$)
never moves out of the
interval $[0,m{-}1]$ where $m=2^{p(n)}$ for $n=|w|$.
Given $w$, it is straightforward to construct $\calN^M_{w,m}$,
by filling the rules~(\ref{eq:ocgamePfive})
and the parameters $n,m$ into a fixed scheme.
Since $m$ can be presented in binary by using $p(n){+}1$ bits, we can
construct $\calN^M_{w,m}$
in
logarithmic work-space (from a given $w$).

\section{Reducing reachability games to
(bi)simulation games}\label{sec:reachtosimul}

We first discuss a reduction in a general framework,
and then apply it to the case of (succinct) one-counter nets.

We assume a (general) r-game
$\calG=(V,V_\exists, \gt{}, \calT)$, and below we
define the LTS
\begin{equation}\label{eq:ltstogame}
\calL(\calG)=(S,Act,(\gt{a})_{a\in Act}).
\end{equation}
(Cf. Fig.~\ref{fig:attchoice} and~\ref{fig:defendchoice}, where we now
ignore the bracketed parts of transition-labels.)

\medskip
\noindent
The set $S$ is defined as follows (recall that
$V_{\forall}=V\smallsetminus V_{\exists}$):
\begin{itemize}
	\item every $s\in V$ and its
		``copy'' $s'$ is in $S$;
	\item
		if $s\in V_{\forall}$ and $s\gt{} \bar{s}$, then a state
		$\langle s,\bar{s}\rangle$ is in $S$
		\\
		(in Fig.~\ref{fig:defendchoice}
we write, e.g., $s^1_3$ instead of
$\langle s_1,s_3\rangle$);
	\item
if $s\in V_{\forall}$ and $X=\{\bar{s}\mid s\gt{}	\bar{s}\}$ is
nonempty,
then a state $\langle s,X\rangle$ is in $S$
\\
(in Fig.~\ref{fig:defendchoice}
		we write $s^1_{23}$ instead of
		$\langle s_1,\{s_2,s_3\}\rangle$).
\end{itemize}
We put  $Act=\{a_c,a_{win}\}\cup\{a_{\langle s,\bar{s}\rangle}\mid
s\gt{}\bar{s}\}$ and define $\gt{a}$ for $a\in Act$ as follows:
\begin{itemize}
	\item if $s\in V_{\exists}$ and $s\gt{} \bar{s}$, then
		$s\gtl{a_{\langle s,\bar{s}\rangle}} \bar{s}$ and
		$s'\gtl{a_{\langle s,\bar{s}\rangle}} \bar{s}'$
\\
		(in Fig.~\ref{fig:attchoice} we write, e.g., $a^1_3$ instead of
	$a_{\langle s_1,s_3\rangle}$);
\item
 if $s\in V_{\forall}$ and
 $X=\{\bar{s}\mid s\gt{}\bar{s}\}\neq\emptyset$,
 then
	\begin{enumerate}[a)]
		\item
		$s\gtl{a_c}\langle s,X\rangle$,
		and $s\gtl{a_c} \langle s,\bar{s}\rangle$,
		$s'\gtl{a_c} \langle s,\bar{s}\rangle$
		for all $\bar{s}\in X$
\\
	(cf. Fig.~\ref{fig:defendchoice}
	where $s=s_1$ and $X=\{s_2,s_3\}$
	and consider dashed edges as
	normal edges; the subscript $c$ in $a_c$ stands for
	``choice'');
\item
	for each $\bar{s}\in X$ we have
$\langle s,X\rangle\gtl{a_{\langle
s,\bar{s}\rangle}}\bar{s}$
and $\langle s,\bar{s}\rangle\gtl{a_{\langle
s,\bar{s}\rangle}}\bar{s}'$;
moreover, for each $\bar{\bar{s}}\in X\smallsetminus \{\bar{s}\}$
we have $\langle s,\bar{s}\rangle\gtl{a_{\langle
s,\bar{\bar{s}}\rangle}}\bar{\bar{s}}$
\\
(e.g., in Fig.~\ref{fig:defendchoice} we thus have
$s^1_2\gtl{a^1_2}s'_2$ and $s^1_2\gtl{a^1_3}s_3$).
\end{enumerate}
\item
for each $s\in\calT$ we have $s\gtl{a_{win}} s$
(for special $a_{win}$ that is not performable from $s'$).
\end{itemize}

We recall that $\bisim{\subseteq}\simul$ where
$\simul$ denotes simulation preorder and
$\bisim$ bisimulation equivalence.

\begin{prop}\label{prop:reachreduced}
	For any $s\in V$ and any relation
	$\rho$ satisfying
	$\bisim{\subseteq}\mathop{\rho}{\subseteq}\simul$
	we have:
	\begin{enumerate}[a)]
		\item
if	$s\in\winareaE$ (in $\calG$), then $s\notsimul s'$ (in
$\calL(\calG)$) and thus $(s,s')\not\in \rho$;
\item
if $s\not\in\winareaE$, then $s\bisim s'$ and thus $(s,s')\in \rho$.
 \end{enumerate}
\end{prop}
\begin{proof}
a) For the sake of contradiction suppose that there is $s\in\winareaE$
such that $s\simul s'$; we consider such $s$ with
the least rank.
We note that $\rank(s)>0$,
since $s\in\calT$ entails $s\notsimul s'$ due to the
	transition $s\gtl{a_{win}} s$.

If $s\in V_{\exists}$, then let $s\gt{} \bar{s}$ be a rank-reducing
transition. Attacker's move $s\gtl{a_{\langle s,\bar{s}\rangle}}
\bar{s}$, from the pair $(s,s')$,
must be responded with $s'\gtl{a_{\langle s,\bar{s}\rangle}} \bar{s}'$;
but we have $\bar{s}\notsimul \bar{s}'$ by the ``least-rank''
assumption,
which contradicts with the assumption $s\simul s'$.

If $s\in V_{\forall}$, then $X=\{\bar{s}\mid s\gt{}\bar{s}\}$ is
nonempty (since $s\in\winareaE$) and
$\rank(\bar{s})<\rank(s)$ for all $\bar{s}\in X$.
For the pair  $(s,s')$
we now consider Attacker's move
$s\gtl{a_{c}} \langle s,X \rangle$. Defender can choose
$s'\gtl{a_{c}} \langle s,\bar{s} \rangle$ for any $\bar{s}\in X$
(recall that $\rank(\bar{s})<\rank(s)$).
In the current pair $(\langle s,X \rangle, \langle s,\bar{s} \rangle)$
Attacker can play $\langle s,X \rangle \gtl{a_{\langle s,\bar{s}
\rangle}} \bar{s}$, and this must be responded by
$\langle s,\bar{s} \rangle \gtl{a_{\langle s,\bar{s}
\rangle}} \bar{s}'$. But we again have
$\bar{s}\notsimul \bar{s}'$ by the ``least-rank'' assumption,
which contradicts with $s\simul s'$.

\medskip

b) It is easy to verify that the following set
is a bisimulation in $\calL_\calG$:
\\
 $\{(s,s)\mid s\in S\}
 \cup\{(s,s')\mid s\in V\smallsetminus \winareaE\}
 \cup \{(\langle s,X \rangle, \langle s,\bar{s} \rangle)\mid
 s\in V_{\forall}\smallsetminus \winareaE, \bar{s}\in V\smallsetminus
 \winareaE\}$.
\end{proof}

\emph{Remark.}
We can note that the dashed edges in Fig.~\ref{fig:defendchoice} are
not necessary for the simulation game (i.e., if b) in
Prop.~\ref{prop:reachreduced} is reformulated to
``if $s\not\in\winareaE$, then $s\simul s'$'');
 they are important for the
bisimulation game.

\medskip

Now we apply the described reduction to succinct one-counter nets to
obtain:

\begin{theorem}\label{th:equivhard}
	For succinct labelled one-counter nets (SOCNs),
deciding any relation containing bisimulation equivalence and
contained in simulation preorder is EXPSPACE-hard.
\end{theorem}
\begin{proof}
Let us fix a language $L$ in EXPSPACE,
and a Turing machine $M$ and a
 polynomial $p$ such that $M$ decides $L$ and
the head-position in the computation of $M$ on any $w$
never moves out of
$[0,m{-}1]$ where $m=2^{p(|w|)}$.

Given $w$, we can construct $\calN^M_{w,m}$
as defined by~(\ref{eq:constrocngame})
in Section~\ref{sec:reachgames} (in logarithmic work-space).
By Prop.~\ref{prop:spaceboundedocgame} we have $w\in L(M)$
iff $p_0(0)\in \winareaE$ in the game $\calG$ associated with
$\calN^M_{w,m}$ (i.e., in $\calG_\calN$ defined
by~(\ref{eq:gamefornet}) for $\calN=\calN^M_{w,m}$), hence iff
$\big(p_0(0), (p_0(0))'\big)\not\in\rho$ in the LTS $\calL(\calG)$
for any $\rho$ satisfying $\bisim\subseteq\rho\subseteq\simul$
(by Prop.~\ref{prop:reachreduced}).

Therefore we will be done once we show the following claim.

\medskip

\noindent
\emph{Claim}.
For any succinct ocn-game $\calN=(Q,Q_{\exists}, \delta, \WinE)$
(where ``succinct'' refers to binary presentations of $z$ in $q\gt{z}q'$)
we can construct, in logarithmic work-space, a SOCN $\calN'$ such that
the LTS $\calL_{\calN'}$ (as defined by~(\ref{eq:ltsfornet})) is
isomorphic with $\calL(\calG_\calN)$.

\medskip

A proof of this claim is also demonstrated in Figures~\ref{fig:attchoice}
and~\ref{fig:defendchoice}, when $s_i$ are viewed as control states
and the bracketed parts of edge-labels are counter-changes (written in
binary).

First we consider
the r-game $\calN^{csg}=(Q,Q_{\exists}, \gt{}, \{\WinE\})$
(``the control-state game of $\calN$'') arising from $\calN$ by
\emph{forgetting the counter-changes}; hence $q\gt{}\bar{q}$ iff
there is a rule $q\gt{z}\bar{q}$. In fact, we will assume that
there is at most one rule  $q\gt{z}\bar{q}$ in $\delta$ (of $\calN$)
for any pair $(q,\bar{q})\in Q\times Q$; this can be achieved by
harmless modifications.

We construct the LTS $\calL(\calN^{csg})$
(as defined by~(\ref{eq:ltstogame}) for general
r-games). Hence each $q\in Q$ has the copies $q,q'$ in
$\calL(\calN^{csg})$, and other states are added
(as also depicted in Fig.~\ref{fig:defendchoice} where $s_i$ are
now in the role of control states); there are also the
respective labelled transitions in $\calL(\calN^{csg})$, with labels
$a_{\langle q,\bar{q}\rangle}$, $a_c$, $a_{win}$.

It remains to add the counter changes (integer increments and
decrements in binary), to create the required SOCN $\calN'$.
For $q\in Q_{\exists}$ this adding is simple, as
depicted in Fig.~\ref{fig:attchoice}: if $q\gt{z}\bar{q}$ (in $\calN$),
then we
simply extend the label $a_{\langle q,\bar{q}\rangle}$
in $\calL(\calN^{csg})$
with $z$;
for $q\gtl{a_{\langle q,\bar{q}\rangle}}\bar{q}$ and
$q'\gtl{a_{\langle q,\bar{q}\rangle}}\bar{q}'$ in $\calL(\calN^{csg})$
we get
 $q\gtl{a_{\langle q,\bar{q}\rangle},z}\bar{q}$
 and  $q'\gtl{a_{\langle q,\bar{q}\rangle},z}\bar{q}'$ in $\calN'$.

 For $q\in Q_{\forall}$ (where $Q_{\forall}=Q\smallsetminus
 Q_{\exists}$) it is tempting to the same, i.e. extend
the label $a_{\langle q,\bar{q}\rangle}$ with $z$ when
$q\gt{z}\bar{q}$, and extend $a_c$ with $0$.
 But this might allow cheating for Defender: she could thus
 mimic choosing a
 transition
 $q(k)\gt{x}\bar{q}(k{+}x)$ even if $k{+}x<0$. This is avoided by the
 modification that is demonstrated in Fig.~\ref{fig:defendchoice}
 (by $x=x'{+}x''$, etc.); put simply: Defender must immediately prove
 that the transition she is choosing to mimic is indeed performable.
 Formally, if   $X=\{\bar{q}\mid q\gt{}\bar{q}\}\neq\emptyset$
 (in $\calL(\calN^{csg})$), then in $\calN'$
 we put $q\gtl{a_c,0}\langle q,X \rangle$ and
 $\langle q,X \rangle\gtl{a_{\langle q,\bar{q} \rangle},z}\bar{q}$
 for each $q\gt{z}\bar{q}$ (in $\calN$);
 for  each $q\gt{z}\bar{q}$ we also define $z'=\min\{z,0\}$,
  $z''=\max\{z,0\}$ and
 put $q'\gtl{a_c,z'}\langle q,\bar{q}\rangle$,
 $\langle q,\bar{q}\rangle\gtl{a_{\langle
 q,\bar{q}\rangle},z''}\bar{q}'$.
 Then for any pair $q\gtl{\bar{z}}\bar{q}$,
 $q\gtl{\bar{\bar{z}}}\bar{\bar{q}}$
 where $\bar{q}\neq \bar{\bar{q}}$ we put
$\langle q,\bar{q}\rangle\gtl{a_{\langle
q,\bar{\bar{q}}\rangle},\bar{\bar{z}}-\bar{z}'}\bar{\bar{q}}$.

Finally, $p_{win}\gtl{a_{win}}p_{win}$ in $\calL(\calN^{csg})$
is extended to $p_{win}\gtl{a_{win},0}p_{win}$ in $\calN'$.
\end{proof}

\section{Additional remarks}~\label{sec:addrem}.

Our EXPSPACE-hardness proof of reachability games in Section~\ref{sec:reachgames}
can be easily adjusted to yield an alternative proof of
EXPTIME-hardness of countdown games
from~\cite{DBLP:journals/lmcs/JurdzinskiSL08}.
The proof in~\cite{DBLP:journals/lmcs/JurdzinskiSL08} used a reduction
from the
acceptance problem for alternating
linear bounded automata (a well-known EXPTIME-complete problem); a
crucial point was that a whole (linear) configuration can be
presented by an exponential number (presented in polynomial space),
and moving to a next configuration can be realized by adding (or,
 in another setting, subtracting) another exponential number.
The countdown
games were modified to ``count-up'' games in~\cite{DBLP:journals/ipl/Kiefer13}
(called ``hit-or-run games'' there) to show EXPTIME-hardness of
bisimilarity on BPA processes.

In the case of countdown (or count-up) games an important ingredient
is an initial (or target) exponential counter value, as a part of the
input. In our proof such a value would correspond to an upper bound on
$t$ in the computation $C_0,C_1,\dots,C_t$ on $w$. We can imagine
a modification of our game where
 Eve first sets the corresponding bound-value into the counter and then
repeatedly decrements the counter before entering a configuration
$s_{(q_+,x)}(k)$. If we restrict our attention to
Turing machines $M$ working in exponential time,
then constructing the modified game constitutes a (logspace, master) reduction
demonstrating EXPTIME-hardness.

If we only assume that $M$ works in exponential space, it can work
in
double-exponential time, and we cannot present such a bound  in
polynomial space (explicitly); therefore our proof does not rely on any
explicit upper bound, and it yields a modification of countdown games
where one player first sets any initial counter value and only then the
original countdown game is played. This modified game is thus
EXPSPACE-complete. (EXPSPACE-membership follows from our reduction in
Section~\ref{sec:reachtosimul} and the known results for simulation
and bisimulation relations, but also a simpler direct proof can be
given.)

Finally we can note that our modified (EXPSPACE-complete) game does
not seem easily implementable by BPA processes, hence
the EXPTIME-hardness result
in~\cite{DBLP:journals/ipl/Kiefer13} has not been improved here.
(The known upper bound for bisimilarity on BPA is 2-EXPTIME.)

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