text:
\section{Introduction}\label{sec:into}

Conformance testing is a practical and rigorous technique for verifying system correctness against a specification.
In the context of cyber-physical systems, several notions of conformance have been proposed for testing such systems against their hybrid systems models, of which  hybrid input-output conformance (\textit{hioco}) \cite{vanOsch2006} and $(\tau, \epsilon)$-hybrid conformance (\textit{hconf}) \cite{Abbas14,AbbasThesis15} are two notable examples.
As we noted earlier in  \cite{Mohaqeqi2014,Khakpour2015}, these two notions are substantially different in that hconf allows for approximate comparison of system dynamics, while hioco provides direct support for discrete actions and non-deterministic and partial specifications. Hence, as indicated in \cite{Mohaqeqi2014,Khakpour2015},  combining the advantages of these approaches will be beneficial and will lead to a flexible and  realistic definition of conformance with the above-mentioned characteristics.

In this paper, we define an approximate notion of conformance that allows for approximate comparison of both  (sampled) continuous signals and observable discrete actions. To this end, we take the notion of hconf \cite{Abbas14,AbbasThesis15}, and interpret discrete actions as special signals with discrete values ($0$ and $\infty$). As a first sanity check, we set out to prove that our notion of conformance is a pre-order (i.e., it is reflexive and transitive), making it suitable for step-wise conformance testing and refinement. Reflexivity immediately follows from the definition; transitivity, however, requires a twist in accumulating the conformance bounds (error margins).
We formulate the notion of semi-transitivity and prove that our proposed notion is indeed semi-transitive.

We illustrate the definitions using a simple example of a thermostat.  The thermostat has two input discrete actions ON and OFF and one output (continuous) variable $x$ denoting the current temperature. The dynamics of the temperature are different depending on the state of the thermostat, which is in turn determined by the discrete input actions. We provide a formalization of an ideal thermostat in hybrid (I/O) automata and analayze the conformance of its implementation with respect to the specification.

The rest of this paper is organized as follows.
In Section \ref{sec:rel}, we present an overview of the related work and position our approach with respect to the state of the art in the literature.
In Section \ref{sec:pre}, we introduce the preliminary definitions regarding hybrid I/O automata and their traces. In Section \ref{sec:hioco}, we review the notion of hioco.
In Section \ref{sec:ext}, we first review the notion of hconf and show how it can be extended to treat conformance for systems with discrete actions. Additionally, we state and prove it semi-transitivity property. In Section \ref{sec:conc}, we conclude the paper and present some directions for future research.

\section{Related Work}\label{sec:rel}

There is a substantial literature on approximate notions of behavioral equivalence and/or pre-order for hybrid systems. We have already mentioned two examples of such notions, namely
hybrid input-output conformance (\textit{hioco}) \cite{vanOsch2006} and $(\tau, \epsilon)$-hybrid conformance (\textit{hconf}) \cite{Abbas14,AbbasThesis15}. Additionally, there are a number of notions of approximate equivalence / pre-order based on Metrics such as \cite{Girard2011,Julius2009,Tabuada07}.
In \cite{Mohaqeqi2014,Khakpour2015}, we formally relate some of the basic notions of conformance for hybrid systems and their underlying semantic models.

We expect the extension of our notion to the non-deterministic case to be straightforward; in \cite{vanOsch2006} and  \cite{AbbasThesis15} already non-deterministic models are considered.
The extension to richer semantic domains, e.g., with probabilistics or stochastics, are far less trivial. We refer to \cite{Julius09,Zamani15} for related ideas in this direction.

\section{Preliminaries}\label{sec:pre}

This section introduces the basic definitions and notations used throughout the rest of the paper, including a formal definition of hybrid I/O automata and their associated traces.

\subsection{Notation and Basic Definitions}

We denote the set of real numbers by $\mathbb{R}$ and the set of non-negative integers by $\mathbb{N}$.
Given  a tuple $\alpha=(a_1,a_2,\ldots,a_n)$, the $i$th element of the tuple is denoted by $\gamma_i(\alpha)$, that is, $\gamma_i(\alpha)=a_i$.

For a function $f$, we denote its domain by $\mathit{dom}(f)$. For a set $S$, the restriction of $f$ to $S$, denoted by $f \lceil S$, is a function with the domain $\mathit{dom}(f) \cap S$, where $(f \lceil S)(c) = f(c), \forall c \in S$.

In order to describe the system dynamics, we define and use the following notions of valuation and trajectory.

\begin{definition}[Valuation]
Consider a set of real-valued variables $V$. A valuation of $V$ is a function of type $V \mapsto \mathbb{R}$, which assigns a real number to each variable $v \in V$. The set of all valuations of $V$ is denoted by $\mathit{Val}(V)$.
\end{definition}

\begin{definition}[Trajectory]
Let $D \subset \mathbb{R}$ be an interval. A trajectory $\sigma$ is a function $\sigma : D \rightarrow \mathit{Val}(V)$ that maps each element
in interval $D$ to a valuation. The set of all trajectories associated to $V$ is denoted by $\mathit{trajs}(V)$.
\end{definition}

Consider a set of variables $V$ and the corresponding set of valuations $\mathit{Val}(V)$. The \textit{restriction} of a valuation $\mathit{val} \in \mathit{Val}(V)$ to a set $V' \subseteq V$, denoted by $\mathit{val}\downarrow V'$, is a valuation $\mathit{val}' \in \mathit{Val}(V')$ where for all $v \in V'$, $\mathit{val}'(v)=\mathit{val}(v)$. Also, the restriction of a trajectory $\sigma : D \rightarrow \mathit{Val}(V)$ to a set $V' \subseteq V$, denoted by $\sigma \downarrow V'$, is a trajectory $D \rightarrow \mathit{Val}(V')$ where $(\sigma \downarrow V')(t)=\sigma(t) \downarrow V'$ for all $t \in D$.

For a trajectory $\sigma$, the first valuation of $\sigma$ (i.e., $\sigma(min(\mathit{dom}(\sigma)))$), if it exists, is denoted by $\sigma.\mathit{fval}$. Moreover, if $\mathit{dom}(\sigma)$ is right-closed, then  $\sigma.\mathit{lval}$ is defined as the last valuation of $\sigma$.

For an interval $D \subseteq \mathbb{R}$ and an arbitrary $ t \geq 0$, we define $D+t \triangleq \{t' + t | t' \in D\}$ as the time shift of $D$. Moreover, for a trajectory $\sigma$ with domain $D$, $\sigma+t$ is defined as a function with domain $D+t$ such that $\forall t' \in D: (\sigma+t)(t' + t)=\sigma(t')$.

For a trajectory $\sigma$, define $\tau \unrhd t \triangleq (\tau \lceil [t,\infty))-t$. Additionally, trajectory $\sigma$ is a prefix of trajectory $\sigma'$, denoted as $\sigma \leq \sigma'$, if and only if $\sigma=\sigma' \lceil \mathit{dom}(\sigma)$. Also, the concatenation of two trajectories  $\sigma$ and $\sigma'$ is defined by $\sigma ^\frown \sigma' \triangleq \sigma \cup \sigma'$.

\subsection{Hybrid Automata}

Hybrid I/O automata (HIOA) \cite{lynch2003hybrid} are defined as the extension of hybrid automata \cite{Alur1993} with an additional classification of external actions and variables as inputs and outputs. Hence, we first review the definition of hybrid automata and then introduce its extension to input and output actions and variables.

\begin{definition}[Hybrid Automata]
A hybrid automaton (HA) $\mathcal{A}=(W,X,Q,\Theta,E,H,D,\mathscr{T}$) consists of
\begin{itemize}
\item
A set $W$ of external variables and a set $X$ of internal variables, disjoint from each other. We write $V \equiv W \cup X$.

\item
A set $Q \subseteq \mathit{Val}(X)$ of states.

\item
A nonempty set $\Theta \subseteq Q$ of start states.

\item
A set $E$ of \textit{external actions} and a set $H$ of \textit{internal actions}, disjoint from each other. We write $A \equiv E \cup H$.

\item
A set $D \subseteq Q \times A \times Q$ of \textit{discrete transitions}. We use $\mathbf{x} \stackrel{a}{\rightarrow}_\mathcal{A} \mathbf{x'}$ as shorthand for $(\mathbf{x}, a, \mathbf{x'}) \in D$. We sometimes drop the subscript and  write $\mathbf{x} \stackrel{a}{\rightarrow} \mathbf{x'}$, when we think $\mathcal{A}$ should be clear from the context. We say that $a$ is enabled in $\mathbf{x}$ if there exists an $\mathbf{x'}$ such that $\mathbf{x} \stackrel{a}{\rightarrow} \mathbf{x'}$.

\item
A set $\mathscr{T}$ of trajectories for $V$ such that $\tau(t) \lceil X \in Q$ for every $\tau \in \mathscr{T}$ and $t \in \mathit{dom}(\tau)$. Given a trajectory $\tau \in \mathscr{T}$ we denote $\tau.\mathit{fval} \lceil X$ by $\tau.\mathit{fstate}$ and, if $\tau$ is closed, we denote $\tau.\mathit{lval} \lceil X$ by $\tau.\mathit{lstate}$.
We require that the following axioms hold:

\textbf{T1} (\textit{Prefix closure})

\quad For every $\tau \in \mathscr{T}$ and every $\tau' \leq \tau$, $\tau' \in \mathscr{T}$.

\textbf{T2} (Suffix closure)
\quad
For every $\tau \in \mathscr{T}$ and every $t \in \mathit{dom}(\tau)$, $\tau \unrhd t \in \mathscr{T}$

\textbf{T3} (Concatenation closure)
\quad
Let $\tau_0$, $\tau_1$, $\tau_2$, $\ldots$ be a sequence of trajectories in $\mathscr{T}$ such that, for each nonfinal index $i$, $\tau_i$ is closed and $\tau_i.\mathit{lstate} = \tau_{i+1}.\mathit{fstate}$. Then $\tau_0 ^\frown \tau_1 ^\frown \tau_2 \ldots \in \mathscr{T}$.
\end{itemize}
\end{definition}

\begin{example}
\label{ex:HA}
Figure \ref{subfig:HAofTHERM}, due to \cite{Aerts16}, shows the hybrid automaton of the thermostat described in Section \ref{sec:into}
 with the set of variables of $W=\{y\}$, $X=\{l, x\}$, where $l$ is a two-valued variable (mode\_ON and mode\_OFF) determining the \textit{location} of the automaton, $x$ is a continuous real-valued variable, and $y$ is always equal to $x$, i.e., $y(t)=x(t), \forall t \geq 0$. Further, the set of actions is specified as $E=\{\mathit{ON}\, \mathit{OFF}\}$, and $H=\{\}$. Also, we have
\begin{equation}
Q=\{(l,x) \mid l \in \{mode\_ON, mode\_OFF \}, \ 0 \leq x \leq 20  \} \nonumber
\end{equation}
and  $\Theta = \{(\mathit{mode\_ON},5)\}$.
Trajectories generated by this HIOA are specified according to the differential equations given for each location. Figure \ref{subfig:TRAJofTHERM} depicts a set of trajectories for variable $x$ (or equivalently external variable y).
\end{example}

\input{Figures/thrms}

\subsection{Hybrid I/O Automata}

As mentioned before, an HIOA is a hybrid automaton in which the set of variables and actions are classified as input and output, as specified below.

\begin{definition}[Hybrid I/O Automata \cite{lynch2003hybrid}]
A hybrid I/O automaton (HIOA) is a tuple ($\mathcal{H},W_I,W_O,E_I,E_O$) where
\begin{itemize}
\item
$\mathcal{H}=(W,X,Q,\Theta,E,H,D,\mathscr{T})$ is a hybrid automaton.

\item
$W_I$ and $W_O$ partition $W$ into \textit{input} and \textit{output} variables, respectively. Variables in $Z\equiv X \cup W_O$ are called locally controlled.

\item
$E_I$ and $E_O$ partition $E$ into \textit{input} and \textit{output} actions, respectively. Actions in $L\equiv H \cup E_O$ are called locally controlled.

\item
The following additional axioms are satisfied:
\begin{itemize}
\item
\textbf{E1} (Input action enabling)

For every  $\mathbf{x} \in Q$ and every $a \in E_I$, there exists $\mathbf{x'} \in Q$ such that $\mathbf{x} \stackrel{a}{\rightarrow} \mathbf{x'}$.

\item
\textbf{E2} (Input trajectory enabling)

For every  $\mathbf{x} \in Q$ and every $v \in \mathit{trajs}({W_I})$, there exists $\tau \in \mathscr{T}$ such that $\tau.\mathit{fstate} = \mathbf{x}$,
$\tau \downarrow {W_I} \leq v$ and either
\begin{enumerate}
\item
$\tau \downarrow {W_I} = v$, or

\item
$\tau$ is closed and some $l \in L$ is enabled in $\tau.\mathit{lstate}$.

\end{enumerate}

\end{itemize}
\end{itemize}
\end{definition}

\begin{example}
\label{ex:HIOA}
Consider the hybrid automaton described in Example \ref{ex:HA}. Classifying the set of variables into input and output variables $W_I=\{\}$, $W_O=\{y\}$ and the set of actions into input and output actions as $E_I=\{\mathit{ON}\}$, $E_O=\{\mathit{OFF}\}$ reveals the corresponding HIOA.
\end{example}

{For an HIOA, a state is said to be \textit{agile} if it affords some trajectory, i.e., there exists trajectory $\sigma \in \mathscr{T}$ for which $s \stackrel{\sigma}{\rightarrow} $. To show that a state is agile, we add a spacial action $\xi$ to the set of output actions $E_O$ (the usage of this action is explained in the next section).}

\subsection{Hybrid sequences, executions, traces and solution pairs}
To describe the behavior of a hybrid (I/O) automaton, we use the notions of execution, trace and solution pair. To this end, we first define a notation of hybrid sequence.

\begin{definition}[Hybrid Sequence \cite{lynch2003hybrid}]
Take a set of variables $V$ and a set of actions $A$. Then, an ($V,A$)-sequence is defined as a finite or infinite sequence $\alpha=\tau_0,a_1,\tau_1,a_2,\ldots$, in which
\begin{itemize}
\item
each $\tau_i$ denotes a trajectory in $\mathit{trajs(V)}$;
\item
each $\alpha_i$ denotes an action in $A$;
\item
if $\alpha$ is finite then it ends with a trajectory;
\item
if $\tau_i$ is not the last trajectory in the sequence then  $\mathit{dom}(\tau_i)$  is closed.
\end{itemize}

Any ($V,A$)-sequence is called a hybrid sequence.
\end{definition}

Next, we define the notions of execution fragment and execution for a hybrid I/O automaton.

\begin{definition}[Execution Fragment and Execution \cite{lynch2003hybrid}]
A hybrid sequence $\alpha=\tau_0,a_1,\tau_1,a_2,\ldots$ is an execution fragment for an HIOA $A$ if (i) each $\tau_i$ is a trajectory in $\mathscr{T}$, and (ii) if $\tau_i$ is not the last trajectory then $\tau_i.\mathit{lstate} \stackrel{a_{i+1}}{\rightarrow} \tau_{i+1}.\mathit{fstate}$.
For an execution fragment $\alpha$, we define $\alpha.\mathit{fstate}$ to be $\tau_{0}.\mathit{fstate}$.

An execution fragment $\alpha=\tau_0,a_1,\tau_1,a_2,\ldots$ is called an \emph{execution} of a hybrid automata $\mathcal{A}$  if
$\tau_0.\mathit{fstate}$ is a start state, i.e., $\tau_0.\mathit{fstate} \in \Theta$.

We write $\mathit{execs}_\mathcal{A}$ to denote the set of all executions of $\mathcal{A}$.
\end{definition}

To capture the \textit{external} behavior of a hybrid (I/O) automaton, we use the notion of \textit{trace}. Intuitively, a trace of an execution specifies the sequence of its external actions and the evolution of the external variables.
\begin{definition}[Trace \cite{lynch2003hybrid}]
\label{def:trace}
Consider an execution $\alpha$. The trace of $\alpha$, denoted by $\mathit{trace}(\alpha)$, is the ($E,W$)-restriction of $\alpha$ (It is recalled that $E$ and $W$ denote the set of external actions and external variables, respectively).
\end{definition}

\begin{example}
\label{ex:trace}
Figure \ref{subfig:TRAJofTHERM} depicts a set of trajectories for variable $x$ of the HIOA specified in Example \ref{ex:HIOA}. According to this sample behavior, $\alpha=\tau_0,OFF,\tau_1,ON,\tau_2$ specifies a trace of the HIOA.
\end{example}

\begin{definition}[Solution Pair \cite{AbbasThesis15}]
\label{def:sol}
Assume that $u$ and $y$ are two traces for a HIOA $\mathscr{H}$; ($u$, $y$) is a solution pair to $\mathscr{H}$ if
\begin{itemize}
\item
$\mathrm{dom}(u)=\mathrm{dom}(y)$, and
\item
there exists a trace $\alpha$ to $\mathscr{H}$ such that $\mathrm{dom}(\alpha)=\mathrm{dom}(u)$, $u=\alpha \downarrow W_I$, and $y=\alpha \downarrow W_O$.
\end{itemize}
\end{definition}

\section{I/O Conformance for Hybrid I/O Automata}\label{sec:hioco}
In this section, we review the notion of hioco \cite{vanOsch2006}, which is an exact notion of conformance, and adopt it for the HIOA formalism.
Subsequently, in the next section, we will adapt the ideas explored in this section to
come up with an approximate notion of conformance.
To this end, we first define the output and the trajectories associated with a state of an HIOA.

\begin{definition}
Assume that  $\mathcal{A}=$ ($\mathcal{H},W_I,W_O,E_I,E_O$) is a hybrid IO automaton with the sets $Q$ and $\Theta$ of states and starting states, respectively. For a state $s \in \Theta$ we define
\begin{eqnarray}
\mathbf{out}(s)=\left\lbrace \begin{array}{ll}
                  \{o \in E_O \mid \{\xi\}\cup s\stackrel{o}{\rightarrow}\}
                  			& \mathrm{, if \ } s \mathrm{\ is \ agile} \\
                  \{o \in E_O \mid s \stackrel{o}{\rightarrow} \}
                  			& \mathrm{, otherwise}
                \end{array}
              \right.
\end{eqnarray}
Furthermore, for a set $C \subseteq Q$ we define:
\begin{equation}
\mathbf{out}(C) = \bigcup_{s \in C}  \mathbf{out}(s). \nonumber
\end{equation}
\end{definition}

\begin{definition}[$\mathbf{after}$ operator]
For an HIOA $\mathcal{A}$ and a trace $t$, $\mathbf{after}$ operator is defined as
\begin{equation}
\mathcal{A} \mathbf{after}\ t = \{ s \mid s_0 \stackrel{t}{\rightarrow}  s \}
\end{equation}
\end{definition}

\begin{definition}[Trajectories of a state]
For an HIOA $\mathcal{A}$ and a state $s$, $\mathbf{traj}(s)$ is defined as
\begin{equation}
\mathbf{traj} (s) = \{ \sigma \in \mathscr{T} \mid s \stackrel{\sigma}{\rightarrow}  \}
\end{equation}
\end{definition}

In order to define conformance, we need to focus on those  trajectories of the system under test for which a trajectory with the same behavior exists in the specification.
This is achieved by the following infilter operator.

\begin{definition}[$\mathbf{infilter}$]
Consider two sets of trajectories $\Sigma_I$ and $\Sigma_S$ on a set of variables $V$. Assume $V_I \subseteq V$ is the set of input variables; then
\begin{equation}
\mathbf{infilter} (\Sigma_I, \Sigma_S) = \{ \sigma \in \Sigma_I \mid \exists_{\sigma' \in \Sigma_S} : \sigma \downarrow V_I = \sigma' \downarrow V_I  \}
\end{equation}
\end{definition}

Finally, the notion of HIOCO is defined HIOA below.

\begin{definition}[extension of $\mathbf{hioco}$\label{def:hioco}]
An (input-enabled) HIOA $I$ is said to hybrid input-output conform to another HIOA  $S$, denoted by $I\ \mathbf{hioco}\ S$, if and only if for all traces $t \in \mathit{traces}(S)$:
\begin{eqnarray}
& & \mathbf{out}(I\ \mathbf{after}\ t) \subseteq \mathbf{out}(S\ \mathbf{after}\ t), \mathrm{and} \nonumber \\
& & \mathbf{infilter} \left(\mathbf{traj}(I\ \mathbf{after}\ t),\mathbf{traj}(S\ \mathbf{after}\ t)\right) \subseteq \mathbf{traj}(S\ \mathbf{after}\ t)\nonumber
\end{eqnarray}
\end{definition}

\begin{example}
\label{ex:hioco}
Assume that we refer to the HIOA specified in Example \ref{ex:HIOA} by $S$. As specified in Example \ref{ex:trace}, $\alpha=\tau_0,\textit{OFF},\tau_1,\textit{ON},\tau_2$ is a trace of $S$. Then, we have $\mathbf{out}(S\ \mathbf{after}\ \alpha) = \{\textit{OFF} \}$.
Now, assume an imaginary HIOA $I$ which generates a trace $\alpha'=\tau_0,\textit{OFF},\tau_1,\textit{ON},\tau_2,\textit{ON}$.
It is seen that $\textit{ON} \in \mathbf{out}(I\ \mathbf{after}\ \alpha)$. As a consequence, $I$  does not conform to $S$ since $\mathbf{out}(I\ \mathbf{after}\ \alpha) \not\subseteq \mathbf{out}(S\ \mathbf{after}\ \alpha)$.
\end{example}

\section{Approximate Conformance for Hybrid I/O Automata}\label{sec:ext}

As it can be noted in Definition \ref{def:hioco},
HIOCO considers an implementation to be conforming when it exactly follows (features a non-empty subset of the behavior of) the specification. However, due to (unavoidable) inaccuracies during the implementation and testing of a CPS,  this is not a feasible and practical approach. We wish to allow for \textit{limited} deviations of the implementation from the specification. In this regard, Abbas et al. \cite{Abbas14,AbbasThesis15} define a notion of closeness which is measures the degree of similarity between the observed behavior of two systems. However, the approach ignores the input and output actions and mainly focuses on the continuous behavior of the system. In the remainder of this section, we first review the hybrid conformance relation proposed by Abbas et al.\ \cite{Abbas14,AbbasThesis15} and then extend it, following the recipe of HIOCO, with discrete input and output actions.

\subsection{Conformance Relation based on the Notion of Closeness}

In the hybrid conformance approach \cite{Abbas14,AbbasThesis15}, the behavior of a system is specified using a notion of trace in which the types of observable discrete actions are not explicitly specified.  Instead, the number of discrete jumps are considered relevant in the definition of hybrid conformance. This kind of trace, which we call an \textit{action-insensitive trace} (or an \textit{a-trace}) hereafter, is defined based on the notion of hybrid time domain, defined below.

\begin{definition}[Hybrid Time Domain \cite{AbbasThesis15}]
A hybrid time domain $E$ is a subset of $\mathbb{R}_{+} \times \mathbb{N}$ defined as
\begin{equation}
E=\bigcup_{j=0}^{J-1}[t_j,t_{j+1}]\times\{j \}
\end{equation}
where $0=t_0 \leq t_1 \leq t_2 \leq ... \leq t_J$. We denote the set of all hybrid time domains by $\mathbb{T}$.
\end{definition}

The hybrid time domain is used to specify the evolution of a CPS regarding both continuous evolution (using time intervals $[t_j,t_{j+1}]$) and discrete jumps (using integer numbers $j$). Next, we define the notion of action-insensitive trace which is subsequently used to define the approximate notion of conformance.\footnote{In \cite{AbbasThesis15}, the term \textit{Hybrid Arc} is used to refer to this concept.}

\begin{definition}[Action-Insensitive Trace \cite{AbbasThesis15}]
Take a hybrid time domain $E$ and a set of variables $V$. An action-insensitive trace (a-trace) over $E$ is a function $\phi: E \rightarrow \mathit{Val}(V)$, where  for every $j$, $t \mapsto \phi(t,j)$ is absolutely continuous in $t$ over the interval $I_j=\{t | (t,j) \in E \}$. The set of all a-traces defined over the variable set $V$ is denoted by a-trace$(V)$.
\end{definition}

As it can be noted, the notion of a-trace is an abstraction of a trace, as defined in Definition \ref{def:trace}. In fact, an a-trace can be obtained from a given trace by removing its actions and keeping only the number of jumps.

\begin{definition}[($\tau$,$\epsilon$)-closeness \cite{AbbasThesis15}]
\label{def:closeness}
Consider a test duration $T \in \mathbb{R}_{+}$, a maximum number of jumps $J \in \mathbb{N}$, and $\tau,\epsilon > 0$; then two a-traces $y_1$ and $y_2$ are said to be  ($\tau$,$\epsilon$)-close, denoted by $y_1 \approx_{(\tau,\epsilon)} y_2$, if
\begin{enumerate}
	\item
	for all $(t,i) \in \mathrm{dom}(y_1)$ with $t\leq T, i \leq J$, there exists $(s,j) \in \mathrm{dom}(y_2)$  such that  $|t-s| \leq \tau$ and $\|y_1(t,i) -y_2(s,j)\| \leq \epsilon$, and

	\item
	for all $(t,i) \in \mathrm{dom}(y_2)$ with $t\leq T, i \leq J$, there exists $(s,j) \in \mathrm{dom}(y_1)$  such that  $|t-s| \leq \tau$ and $\|y_2(t, i) -y_1(s, j)\| \leq \epsilon$.
\end{enumerate}
\end{definition}

\begin{definition}[Conformance Relation \cite{AbbasThesis15}]
\label{def:conformance}
Consider two hybrid I/O automata $\mathscr{H}_1$ and $\mathscr{H}_2$. Given a test duration $T \in \mathbb{R}_{+}$, a maximum number of jumps $J \in \mathbb{N}$, and $\tau,\epsilon > 0$, \emph{$\mathscr{H}_2$ conforms to $\mathscr{H}_1$}, denoted by $\mathscr{H}_2 \approx_{(\tau,\epsilon)} \mathscr{H}_1$, if and only if for all solution pairs $(u,y_1)$ of $\mathscr{H}_1$,
 there exists a solution pair $(u, y_2)$ of $\mathscr{H}_2$ such that the corresponding output a-traces $y_1$ and $y_2$ are ($\tau$,$\epsilon$)-close.
\end{definition}

{Figure \ref{fig:hconf} shows two a-traces $y_1$ and $y_2$ where
$y_1 \approx_{(0.8, 1)} y_2$. As mentioned, the notion of hconf ($\approx_{(\tau, \epsilon)}$) is not sensitive to the existence/absence of actions. As a result, if the HIOA generating $y_1$ produces  output action \textit{OFF} at $t=2$ and the other HIOA produces no action, the two a-traces are still regarded as conforming according to hconf.}

\input{Figures/hconf.tex}

\subsection{Extension}

The goal of this section is to extend the notion of closeness and the corresponding conformance relation to the case of hybrid I/O automata. The main difference of our extended conformance with that of Abbas et al. \cite{Abbas14} is that
we include the input and output actions for the comparison of two behaviors
(i.e., the comparison of specification and implementation).

According to Definition \ref{def:closeness}, the notion of closeness is specified on the basis of comparing a-traces. However, as mentioned in the previous section, an a-trace is an abstraction of a trace of a HIOA, in which, the actions are not explicitly specified. As a result, a natural way for extending the notion of closeness to HIOA is to augment the closeness definition (Definition \ref{def:closeness}) with additional conditions that also compare the corresponding actions of the traces.

We extend the definition of a-trace such that it can reflect the occurrence of  actions as well as the change of continuous variables. For this purpose, we consider each action as a special continuous variables. This variable is assumed to get only two values: $0$ and $\infty$, where a value of $0$ denotes the absence of the corresponding action while the value of $\infty$ denotes an instantaneous occurrence of the action.

To this end, we first assume that the set of real numbers is augmented with a special number $\infty$ which is regarded to be larger than any number in $\mathbb{R}$, and hence larger than any arbitrary $\epsilon$. Further, for notational convenience,  we define $\infty-\infty=0$.
 Let $\mathbb{R}_\infty$ denote $\mathbb{R}$ extended with $\infty$.

We extend the notion of trace and solution pair (Definitions \ref{def:trace} and \ref{def:sol}) such that for any action in a given HIOA, there is a  fresh variable. Such variables get only two values, namely 0 and $\infty$. This naturally extends the notion of a-trace to our notion of trace which has the same type (apart from the fact that $Val(V)$ ranges over  $\mathbb{R}_\infty$); the only difference is that the set of variables also includes the freshly introduced action variables.

In order to extend the notion of closeness (and thus, hconf conformance relation), we need to define the meaning of a norm ($\| \|$) for the signals containing a mixture of normal real-valued variables and these extended type of variables.
Assume $y_1$ and $y_2$ are two vector of variables of the same length, i.e.,
\begin{eqnarray}
y_1=(y_1[1],\ldots, y_1[n])  \nonumber \\
y_2=(y_2[1],\ldots, y_2[n])  \nonumber
\end{eqnarray}
We define the distance of $y_1$ and $y_2$, denoted by $\| y_2 - y_1 \|_e$, as
\begin{eqnarray}
\label{eq:normExt}
\| y_2 - y_1 \|_e =
	\begin{cases}
		\infty,  & \exists i \in \{1, \ldots, n \} s{.}t{.} (y_1[i] = \infty \ and \ y_2[i] \neq \infty)\ or\ (y_1[i] \neq \infty \ and\ y_2[i] = \infty)\\
		\| y_2 - y_1 \|, & \text{otherwise}
	\end{cases}
\end{eqnarray}
where $\| y_2 - y_1 \|$ denotes the ordinary Euclidean distance.
Then, we can modify the definition of ($\tau$,$\epsilon$)-closeness based on this notion of distance.

{
\begin{definition}[Modified ($\tau$,$\epsilon$)-closeness]
\label{def:modifCloseness}
Consider a test duration $T \in \mathbb{R}_{+}$, a maximum number of jumps $J \in \mathbb{N}$, and $\tau,\epsilon > 0$; then two a-traces $y_1$ and $y_2$ are said to be  ($\tau$,$\epsilon$)-close, denoted by $y_1 \approx_{(\tau,\epsilon)}^e y_2$, if
\begin{enumerate}
	\item
	for all $(t,i) \in \mathrm{dom}(y_1)$ with $t\leq T, i \leq J$, there exists $(s,j) \in \mathrm{dom}(y_2)$  such that  $|t-s| \leq \tau$ and $\|y_1(t,i) -y_2(s,j)\|_e \leq \epsilon$, and
	\item
	for all $(t,i) \in \mathrm{dom}(y_2)$ with $t\leq T, i \leq J$, there exists $(s,j) \in \mathrm{dom}(y_1)$  such that  $|t-s| \leq \tau$ and $\|y_2(t, i) -y_1(s, j)\|_e \leq \epsilon$.
\end{enumerate}
\end{definition}
}

{
\begin{definition}[Conformance Relation]
\label{def:modifConformance}
Consider two hybrid I/O automata $\mathscr{H}_1$ and $\mathscr{H}_2$. Given a test duration $T \in \mathbb{R}_{+}$, a maximum number of jumps $J \in \mathbb{N}$, and $\tau,\epsilon > 0$, \emph{$\mathscr{H}_2$ conforms to $\mathscr{H}_1$}, denoted by $\mathscr{H}_2 \approx_{(\tau,\epsilon)}^e \mathscr{H}_1$, if and only if for all solution pairs $(u,y_1)$ of $\mathscr{H}_1$,
 there exists a solution pair $(u, y_2)$ of $\mathscr{H}_2$ such that the corresponding output a-traces $y_1$ and $y_2$ are ($\tau$,$\epsilon$)-close, considering the modified notion of closeness in Definition \ref{def:modifCloseness}.
\end{definition}
}

\begin{example}
{Consider two traces $\alpha=\tau_1,\text{OFF},\tau_2$ and $\alpha'=\tau'_1,\tau'_2$ where $\tau_1$, $\tau_2$, $\tau'_1$, and $\tau'_2$ are trajectories shown in Fig. \ref{fig:hconf}. Let $\phi_1$ and $\phi_2$ be the corresponding a-traces obtained according to the abovementioned method for encoding the actions as numeric variables. Then, according to the modified notion of closeness, we will have $\phi_1 \not\approx_{(\tau,\epsilon)}^e \phi_2$.}
\end{example}

\subsection{Step-wise Approximate Refinement}
In order to apply our notion of conformance in a step-wise refinement trajectory,
one needs to compose approximate conformance relations.
This can be achieved through the following notion of semi-transitivity.

\begin{definition}[Semi-Transitivity] \label{def:tran}
Let $\mathscr{H}_1$, $\mathscr{H}_2$, and $\mathscr{H}_3$ be three arbitrary hybrid I/O automata such that $\mathscr{H}_2 \approx_{(\tau_1,\epsilon_1)} \mathscr{H}_1$ and $\mathscr{H}_3 \approx_{(\tau_2,\epsilon_2)} \mathscr{H}_2$ for some $\tau_1,\tau_2,\epsilon_1,\epsilon_2 > 0$. Then, the conformance relation is \emph{semi-transitive} if  $\mathscr{H}_3 \approx_{(\tau_1+\tau_2,\epsilon_1+\epsilon_2)} \mathscr{H}_1$.
\end{definition}

\begin{theorem} The extended conformance relation is semi-transitive.
\end{theorem}

\begin{proof}
For the proof, we first formally specify the assumptions and the thesis of the theorem; the latter is further refined and proven in a number of steps:

\begin{enumerate}
\item
Assumption: $\mathscr{H}_1 \approx_{(\tau_1,\epsilon_1)}^e \mathscr{H}_2$ and $\mathscr{H}_2 \approx_{(\tau_2,\epsilon_2)}^e \mathscr{H}_3$.
\\
Claim: $\mathscr{H}_1 \approx_{(\tau_1+\tau_2,\epsilon_1+\epsilon_2)}^e \mathscr{H}_3$.

\item
Assumption Rewriting (according to Definition \ref{def:modifCloseness}):
\begin{eqnarray}
\forall (u,y_1) \in \mathscr{H}_1: \exists (u,y_2) \in \mathscr{H}_2 \text{ such  that } y_1 \approx_{(\tau_1,\epsilon_1)}^e y_2, \\
\forall (u,y_2) \in \mathscr{H}_2: \exists (u,y_3) \in \mathscr{H}_3 \text{ such that } y_2 \approx_{(\tau_2,\epsilon_2)}^e y_3.
\end{eqnarray}

Claim:
$\forall (u,y_1) \in \mathscr{H}_1: \exists (u,y_3) \in \mathscr{H}_3 \text{ such  that } y_1 \approx_{(\tau_1+\tau_2,\epsilon_1+\epsilon_2)}^e y_3$.

\item

Assumption Rewriting:
\begin{multline}
\label{eq:firstAssum}
\forall (u,y_1) \in \mathscr{H}_1: \exists (u,y_2) \in \mathscr{H}_2 \text{ such  that } \\
 \forall(t,i) \in \mathrm{dom}(y_1): \exists(s,j) \in \mathrm{dom}(y_2) \text{ such that } |t-s| \leq \tau_1 \text{ and } \|y_1(t,i) -y_2(s,j)\|_e \leq \epsilon_1,
\end{multline}
and also
\begin{multline}
\label{eq:secondAssum}
\forall (u,y_2) \in \mathscr{H}_2: \exists (u,y_3) \in \mathscr{H}_3 \text{ such  that } \\
 \forall(s,j) \in \mathrm{dom}(y_2): \exists(v,k) \in \mathrm{dom}(y_3) \text{ such that } |v-s| \leq \tau_2 \text{ and } \|y_2(s,j) -y_3(v,k)\|_e \leq \epsilon_2,
\end{multline}

conditioned on that $t\leq T$ and $i \leq J$.

Claim:
\begin{multline}
\label{eq:claim1}
\forall (u,y_1) \in \mathscr{H}_1: \exists (u,y_3) \in \mathscr{H}_3 \text{ such  that }  \\
 \forall(t,i) \in \mathrm{dom}(y_1): \exists(v,k) \in \mathrm{dom}(y_3) \text{ such that } |v-t| \leq \tau_1+\tau_2 \text{ and } \|y_1(t,i) -y_3(v,k)\|_e \leq \epsilon_1+\epsilon_2.
\end{multline}
\end{enumerate}

Now, we proceed with  the proof.
Fix an arbitrary $(u,y_1) \in \mathscr{H}_1$; it follows from \eqref{eq:firstAssum} that  $\exists y_2$ such that $(u,y_2) \in \mathscr{H}_2$ for which $\forall(t,i) \in \mathrm{dom}(y_1)$  there exists $(s,j) \in \mathrm{dom}(y_2)$ such that
\begin{equation}
\label{eq:y12timeCloseness}
 |t-s| \leq \tau_1
\end{equation}
and
\begin{equation}
\label{eq:y12closeness}
\|y_1(t,i) - y_2(s,j)\|_e \leq \epsilon_1
\end{equation}
Similarly, from \eqref{eq:secondAssum}, it follows that considering $y_2$, $\exists y_3$ such that $(u,y_3) \in \mathscr{H}_3$ for which $\forall(s,j) \in \mathrm{dom}(y_2)$  there exists $(v,k) \in \mathrm{dom}(y_3)$ such that
\begin{equation}
\label{eq:y23timeCloseness}
 |s-v| \leq \tau_2
\end{equation}
and
\begin{equation}
\label{eq:y23closeness}
\|y_2(s,j) - y_3(v,k)\|_e \leq \epsilon_2.
\end{equation}

Now, fix an arbitrary $\forall(t,i) \in \mathrm{dom}(y_1)$ and assume $(s,j)$ and $(v,k)$ are the corresponding points as specified above.
Since $\epsilon_1$ has a bounded value (i.e., $\epsilon_1 < \infty$), we can conclude from \eqref{eq:y12closeness} that the value of those variables in $y_1(t,i)$ and $y_2(s',j')$ which are associated with the actions is exactly the same. This is because that otherwise $\|y_1(t,i) - y_2(s',j')\|_e = \infty > \epsilon_1$ (according to \eqref{eq:normExt}) which violates \eqref{eq:y12closeness}.
The same results hold for $y_2(s',j')$ with respect to $y_3(v,k)$.
As a consequence, the norm operator in \eqref{eq:y12closeness} and  \eqref{eq:y23closeness} is reduced to the normal Euclidean norm which yields the following relation (according to the triangular property of the Euclidean norm)
\begin{equation}
\|y_1(t,i) - y_3(v,k)\|_e \leq \epsilon_1 + \epsilon_2.
\end{equation}
Besides, from \eqref{eq:y12timeCloseness} and \eqref{eq:y23timeCloseness} it is concluded that
\begin{equation}
 |t-v| \leq \tau_1 + \tau_2
\end{equation}
Consequently, the claim declared in \eqref{eq:claim1} is achieved.
\end{proof}

\section{Conclusions and Discussion}\label{sec:conc}
In this paper, we proposed a notion of conformance parameterized by conformance bounds in time and value. In addition to approximate comparison of (sampled) continuous signals, our notion allows for approximate matching of discrete signals by allowing them to happen within specified time margins. In this sense, it consolidates two earlier notions of hybrid conformance by Abbas and Fainekos \cite{Abbas14}  and by van Osch \cite{vanOsch2006}.

In the remainder of this section, we present the directions of our ongoing research.
As an immediate next step, we would like to prove the conservative extension property of the extended conformance relation compared to the other conformance relations, namely hioco and hconf. This will generalize our earlier result in \cite{Mohaqeqi2014}.
Moreover, we envisage other important issues such as sound sampling rates \cite{Mohaqeqi2016}, test-case generation algorithms, coverage \cite{Dreossi2015,Fainekos2015}, links to temporal and modal logics \cite{Deshmukh15}, and compositionality \cite{Abbas2015,Benes2015}.

\bibliographystyle{eptcs}

redpajama_set_name: RedPajamaArXiv


text:
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The Hobby Center Foundation Announces Resignation of President & CEO Fran Macfarren
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The Hobby Center Foundation today announced that President & CEO Fran Macferran will leave the Hobby Center effective December 31. Macferran will re-engage the consulting business he established prior to accepting his position at the Hobby Center nearly 17 years ago. The Hobby Center Foundation will conduct a nationwide search for a new President & CEO in 2022.
Macfarren was appointed as President & CEO in 2005. During his tenure, Macfarren applied his accounting and business background, and his extensive experience in live entertainment to evolve the organization's business model. This included Macferran's renegotiation of the Hobby Center's strategic relationships and his creation of a VIP donor program.
Among other achievements, Macferran led the development of the Autism Theater Initiative which debuted in 2012. This program provides performance experiences specifically tailored for those with sensory sensitivities and their families in a welcoming and supportive environment. To date, the Hobby Center has offered sensory friendly presentation of The Lion King, The Radio City Christmas Spectacular and Disney's Aladdin, as well as other shows. Houston and the Hobby Center were the first to provide an autism friendly performance of a nationally touring production with the presentation of The Lion King.
"We are grateful for Fran's leadership of the Hobby Center Foundation and the Hobby Center for thePerforming Arts," says Stephen Ash, immediate past -Chairman. "Fran, with the support of the executive leadership team, leaves the organization in a strong financial position in-spite of COVID related challenges. Fran's passion and dedication to the Hobby Center is deeply appreciated, and we wish him the very best."
New Chairman of the Board of Directors, Rob Doty, adds, "Over the past several years, the Hobby Center has been fortunate to have assembled a senior executive leadership team which is second to none in the industry, and we are grateful to have such an extraordinary team in place to help shepherd the Hobby Center into its next phase of leadership."
"I feel I have contributed my best to the Hobby Center, and I support the leadership team and board as they look to the future," states Macferran. "I am looking forward to new challenges, including consulting with other arts organizations."
A search committee will be organized to recruit a new President & CEO. Inquiries regarding the search can be directed to Kimberly Cutchall at kimberly@ultdgroup.com.
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Wonderful story, Timi. My love for fantasy stories knows no bounds. That picture made me laugh and acted as a counterpoint to the mood of the story …you know, humorous picture, somber story.
I think what I take from this–what I am reminded of–is that finding and brewing love has no set formula. These days, I no longer read manuals, tutorials or whatever insights on the subject. I think that's wrong but I can't help feeling that only stories (real or fictional) have a right to educate one about love.
I hope our prince becomes a man again. Well done, Timi.
" . . . finding and brewing love has no set formula." Indeed what worked with Ngozi may fail with Christina 🙂 I am intrigued by the hunter and the hunted and how gender roles have evolved or not over time. I rarely do sequels, but have said that if I did one, this story will have a happy ending.
I'm really glad you enjoyed the story. You and Eno worked hard and the story finally saw the light of day, I appreciate you guys. Thanks for coming over to read the finished product.
I like your fantasy stories!
Yes, I agree with you – You once more brought with a great combination!
diamond – magical – common sense – brilliant!
Thanks Mihran. I liked the magic too, and the dialogue between princess Amera and prince Zonaltera.
If I were to write a sequel, it would end on a happy note. So, just complete the story in your head, any which way you like jare.
How nice, a love tale. I really enjoyed this. And wasn't expecting this.
Such a vivid description. I was watching a movie in my head. Not what I expected it to end with. i was thinking, princess, finds prince, they live happy ever after, tada the end. But then, we're reminded that Nothing really lasts forever.
@vivid description, thanks! It made the story longer, but we both agree it's better this way.
I really enjoyed the details in your story, Timi! I was intrigued by the Wise One's hut and his beard that almost caught on fire! I was dismayed that she chose to aim the arrow, even after he espoused his love for her. She made a 'fatal' mistake, not trusting him to come to her, after they professed their interest in each other. I think we all learn from our mistakes, but I feel bad that hers was rather devastating!
I'm glad you enjoyed the story Robin. I tend to be traditional myself.
I like the idea of throwing the frog back in the pond, but what if he is not meant to be gone from her sight? What if he were accidentally eaten! Horrors!
Aw, the right circumstances, how sweet. Many more happy years to you and Peggy!
As for Princess Amera, yes, she should drop the bow and arrow, and try your suggestion 😉 She went to great lengths to 'acquire' him after all!
Gripping tale of love, Timi. Ha! Once I sensed it had that element of romance, my pulse increased. Interesting to see that our princess got what she wanted. But in life nothing lasts forever. Her choice was fleeting – a lesson for every person.
Ps Sorry for the mix up in my former comment.
Choice is a rudder without hindsight. If she had waited for him to come to her, and assuming he came, would it have made a difference to the outcome?
@dialogue, I'm happy you did. It was difficult to write as I didn't want to sound too contemporary. I also wanted to write words that were pregnant with meaning, to convey gender issues with only a hint.
I like how this references different cultures and fairy tales. Women going after men they love will remain an interesting topic for a long time, not just because of our culture but also because of our religions in Africa. I don't even want to have an opinion about it, biko let the women do what they want jare.
Great story as always. I like how this made me read it twice.
To chase a guy or not to? I had interesting opinions and answers to my informal survey as I crafted this piece. This is the one that cracked me up the most: chase, but with style 🙂 Because we're not islands, these are conversations that we're going to have, and situations we'll likely face . . .
I like how your comment made me read the post again. Thanks Ife.
I like Khalil Gibran's quote because it appeals to something in me, but I am old enough to know better, at least in my experience. I like to think that there are different kinds of love, but for any to be enduring, it takes a decision to love and a commitment to follow through. His quote seems to find place in this story though . . .
You are welcome, Timi. Always a pleasure.
Hahahaha. Very intelligent piece Timi. It's funny because we women like to dream of the perfect man. And then 'perfection' comes. When I met my hubby, I made my decision for him purely on logic. We both had a calling for missions, we both were involved in the performing arts, he's a strong teacher and leader. His temperament is stronger than mine (I have a strong temperament). Anyhow, I didn't make any decision based on feelings. Because feelings come and go. Being in love is not love. Maybe that's why see so much divorce now-a-day. People think that 'being in love' is love. Marriage to me means commitment. Sticking it through and working hard at it, every day. Two fallen people living together for the rest of their earthly lives is very challenging.
It takes discipline to not make decisions based on feelings. It would seem that Princess Amera made logical decisions with regards finding love, choosing not to wait, but to instead go after what she wanted.
So enjoy everyday together in fidelity and respect.
Hi Jean, I'm glad you did. I struggled with the characters thoughts and dialogue, to say so much, with very little, to write words that hold a depth of meaning.
"So enjoy everyday together in fidelity and respect." Good advice in a world that moves way too fast.
"Choice is a rudder without hindsight. A thing to be desired and yet a thing to be feared." Yet life is all about choice! I love this It is just the truth. May God bless you dear.
Thanks George. Indeed our lives consist of the choices we make, the consequences or rather the unfolding of which, we cannot fully see. Even when we make good choices, sometimes the path from the seed of the good choice to the fruit of good choice, takes us by surprise because of bumps along the way.
So our Princess got what she wanted, but her happiness was short lived.
Perhaps in our fight for gender equality and empowerment, we need to remember that both genders are different–not any less important– and play varying cultural roles.
When we spend all our energy stuck on the goal, we sometimes forget that the journey is just as important and the possible repercussions of our actions are somewhat dimmed.
I like this fight for owning ourselves as women, but I also like the roles men play and wouldn't want to usurp a couple of things. We don't want to take the real magic away, do we?
Nice story with lots of depth Timi.
P.S How did you coin all those awesome names?
And thus began the preference some people have for collecting frog-related knick knacks.
Be careful what you pursue . . .
We can only live "happily ever after" on a moment by moment basis. Good write.
Thanks Nancy. I usually say happiness is a moving target, so moment by moment seems fitting.


text:
Articles about the work of Greg Myre
NYT Blunders on Jerusalem Building
) -->The Times correctly reports on the Arab population growth in Israel's capital, but misrepresents the truth about the massive Arab building—both legal and illegal—within Jerusalem.
Al Dura Reprise
Once again, Israel has been blamed for the killing of a 10-year-old Palestinian child, and the death is expected to fuel Arab hatred of Israel. But was Israel really responsible?
The Temple Mount's Jewish History: More Than a Matter of Faith
The Temple Mount is the site of the first and second Jewish Temples, destroyed in 586 BCE and 70 CE, respectively–a historic fact accepted even by Muslim authorities. Nevertheless, that fact has not stopped some journalists from reporting on the Temple Mount's significance in Jewish history cautiously, as if its status is a matter of Jewish faith, or "belief," and not archeologic evidence.
New York Times Omits Major Reason Christians are Leaving Bethlehem
A New York Times report on the exodus of Christians from Bethlehem omitted a major reason for this demographic change – official and unofficial Muslim discrimination against Arab Christians, amid a rising tide of Islamism in the Palestinian territories.
Books by Greg Myre:


text:
EU postpones extention of blacklist for Russia to avoid ill-considered moves — diplomat
The declared formal reason for this decision is that "not all the countries have finalized the necessary internal procedures"
MOSCOW, February 26. /TASS/. The European Union has postponed a decision to extend its blacklist targeting Russia, as its countries don't want to take ill-considered moves, Russia's EU Ambassador Vladimir Chizhov told Rossiya 24 television on Monday.
"I think the main reason for their not doing it today is that back at the previous stage half-a-year ago, some EU countries warned the others and the EU leadership that the issue is rather serious and the timing is rather sensitive for passing decisions automatically, instead of discussing this situation each time," the ambassador said. "The declared formal reason is that not all the countries have finalized the necessary internal procedures," Chizhov added.
EU may formally decide on extension of Russian 'black list' in March
He explained that the formal decision on extending the EU blacklist for Russia must be passed at the level of the Council of the European Union before March 15, the date when this package of sanctions expires. "In order to extend sanctions, they need a session of the Council of the EU, a ministerial level. It should not be necessarily the foreign ministers, but ministers of transport or agriculture can take part," he said.
A total of 150 individuals and 38 organizations are on the EU blacklist for Russia.
Package of sanctions
The European Union imposed several sanctions packages against Moscow: economic sanctions, individual restrictions and Crimea-related sanctions.
Sectoral sanctions were initially levied on July 31, 2014, and covered financial, energy and defense sectors, as well as dual-use goods. Economic sanctions include restricted access to EU's primary and secondary capital markets for five Russian financial institutions and their subsidiaries founded outside the European Union, in which the state holds a majority stake, as well as for three biggest Russian energy companies and three defense firms.
They also include embargoed weapons trade and ban on exports of dual-use goods for military purposes to Russia, as well as restricted access to certain strategic technologies and services that may be used for oil exploration and production, for Russia.
There are also two independent sanctions packages, related to Ukraine. First, those are individual restrictions, meaning bans on trips and frozen banking assets in Europe (if such exist or have been revealed) for 150 persons and 38 organizations. Finally, there is a package of targeted sanctions against Crimea, which European businesses are banned to hold any ties with.
Foreign policySanctions vs. Russia


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Computer Assistant lets you register on the site and post your problem. Local freelance computer experts bid on the job. The site advertises same day remote and on-site service.
FreelancerMap is a site where IT experts, and the people who need them, meet up. There is a monthly or annual subscription required that lets you post jobs and see the applicants for those jobs. Skilled freelancers can sign up at Freelancer Map for a small monthly fee. There is a limited free account, too.
If your small business needs a consultant to manage an important business project, then SkillBridge is the place to look. The freelancers for hire here are "elite." Some are former C-level executives.
Is it just knowledge you need, such as to learn accounting? Need to get your employees trained on Microsoft Excel? Go to Tutor.com. Freelancers with knowledge to share can apply to be a tutor, too.
Arise acts as a clearinghouse for customer service work and directs work to qualified freelance personnel ("agents"). Services offered include outbound calling, live chat support and technical support. Freelancers work from home, but will need to form a company and have an email address at that company rather than a free email, like with Gmail or Yahoo.
Need a question answered about a legal issue or computer hardware? JustAnswer has experts to answer your questions within minutes. All experts' qualifications and expertise are vetted. The service will cost you less than hiring a local professional. Qualified freelancers can work with Just Answer.com to answer questions in their fields of expertise.
If your legal practice requires extra staff, then Update Legal can provide the qualified staff you need on either a temporary or a permanent basis. Legal freelancers will be attracted by the wide range of benefits available, including medical insurance and a 401(k) program.
While not freelance marketplaces in the traditional sense, these driving services are ways for freelancers with clean driving records and no criminal history to use their own vehicles to earn extra income. Small businesses could reduce travel expenses by using Uber and Lyft rather than regular taxis. Sidecar gives you a convenient way to outsource time-crucial deliveries.
Whether you are looking for freelance or temporary help in your business, or are looking to hang out your freelancing shingle, these freelancing sites will give you plenty of places to start.
Great list. I have definitely bookmarked this cause I just have to look into some of the freelance sites that I am unfamiliar with. Will definitely look into these.
Freelancer.com is a REALLY bad site for workers? I think, from the point of view of workers, that the site is little more than a scam or a fraud. Recently worked there on a project, and the employer refused to pay. I knew that the work I had done was good, and that I would therefore win any arbitration, so I arbitrated, and yes, I won, and freelancer ensured that he paid me. But then, AFTER the arbitration, they let him put a 'revenge comment' in my feedback! What sort of crazy system is that? How can I ever arbitrate against a non-paying employer in future? I wouldn't dare to, because I know they can 'get back at me for making them pay' so cheaply! And if I can't, what protection do I have on this worthless scam or fraud site?
It is really a wonderful list. Undoubtedly, this list will help many persons like me who wants to work from home. Thanks a lot Anita.
I hope you could also include Staff.com in the list. It's a freelance site that offers full time online jobs for everyone. I actually got my first job from them.
Would love to see http://www.goLance.com added to this list. While we are not a freelance job site geared specifically towards writers, we do have a number of freelance writing jobs that are posted on our job site. We are a great resource for freelancers because we have created a platform that is free for freelancers, with fees only coming out of earned income. In other words, it is free to sign up and free to apply to as many jobs as you want.
If we are talking about a business that has a very specific job that needs to get done, and are on a strict budget, http://www.talentpackages.com is a great option.
They specialize in freelance services at fixed prices and have more serious sellers than say Fiverr.
ZebraLancer.com: I'm not a technical person and I needed a software developer for my app. They gave me really good advice and the fee is pretty simple: 6%. No membership like other Freelancer websites; the price is crystal clear. So my budget was low and I really like it. No problem with the Freelancer that I hired.
There are a lot of scam when we are talking about lot prices. So I think you can include ZebraLancer as a trusted freelancer website in your list.
The trend is upward for freelancers working from remote location or even full time employees working for their company off site.
Yes, the biggest challenge in hiring freelancers that are out of your time zone is the communication where there is a delay of 24 hours between asking question and receiving your answer.
This and the language barrier, if any, will cause major delays and setbacks to any project.
We are working on launching this new Canadian startup where employers are encouraged to work with local or semi local freelancers where the end result is much more favourable.
The site is in beta and we've decided on workibility.com for our main domain.
https://www.skooli.com is also a great service to earn extra money as a tutor any time and from anywhere. It allows to connect with students in the digital classroom. As a Skooli tutor, I am my own boss and report to only myself. I set my tutoring calendar around my schedule so students can easily connect with me in my available time.
Also https://2polyglot.com is great for linguistic freelancers. There are for types of jobs, so I recommend to check it out.
I agree with Lori, tutoring is a great way to earn extra cash and your own convenience.
Great list Anita. But you missed DesignCrowd, Dribbble & Demand Media. These sites are also good platform for hiring and finding jobs online.
A common room for all to meet the world's most passionate and talented freelancers.
Awesome, much appreciated! Thanks so much for including the article on your list!Great post! These are some really cool ideas. Your blog is awesome, thanks so much and keep up the great work!! i have recently find your blog and its just awesome.Got a lot of unique and informative stuff here .thanks a lot and keep sharing up.Good Luck!


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Description: On August 24, Union II Corps moved south along the Weldon Railroad, tearing up track, preceded by Gregg's cavalry division.
On August 25, Maj. Gen. Henry Heth attacked and overran the faulty Union position at Ream's Station, capturing 9 guns, 12 colors, and many prisoners. The old II Corps was shattered. Maj. Gen. Winfield Scott Hancock withdrew to the main Union line near the Jerusalem Plank Road, bemoaning the declining combat effectiveness of his troops.
The Second Battle of Ream's Station (also Reams or Reams's) was fought during the Siege of Petersburg in the American Civil War on August 25, 1864, in Dinwiddie County, Virginia. A Union force under Maj. Gen. Winfield S. Hancock began destroying part of the Weldon Railroad, which was a vital supply line for Gen. Robert E. Lee's Confederate army in Petersburg, Virginia. Lee sent a force under Lt. Gen. A.P. Hill to challenge Hancock and the Confederates were able to rout the Union troops from their fortifications at Reams Station. However, they lost a key portion of the railroad, causing further logistical difficulties for the remainder of the Richmond-Petersburg Campaign.


Gah! Count down to Canada begins! Attempting to finish up my strips to last throughout my strips and finish packing my bags. T_T; Flight leaves on Thursday evening~ Still gots much to do~!
Darn it, if I could afford to go, I totally would!


Chef José Perez, a born and raised Spaniard, is the creative force behind the exquisite tapas served in Tia Rosa. Growing up in the city of Malaga, José learned how to cook good Spanish dishes right at home. Now, with almost thirty years of experience in the kitchen, he still loves to create the most delicious tapas, always with his own modern twist. José enjoys working in an open kitchen, because it enables him to make contact with the guests in the restaurant.
Our chef loves to make tasty dishes on demand. On the basis of your preferences, he will create something scrumptious with beautiful fresh ingredients.
So do not be shy to ask for the chef's special!


Director Vijay Milton's 10 Endrathukulla is marred by overzealous performances, ordinary music, tacky CGI and poor execution, warns S Saraswathi.
It is hard to believe that Vikram's 10 Endrathukulla is directed by Vijay Milton.
The director's earlier film Goli Soda was an action-packed entertainer and one of biggest hits of 2014.
10 Endrathukulla is not in the same league.
The much-hyped road film jointly -- produced by AR Murugadoss and Fox Star Studios -- barely manages to hold your attention.
The first scene has Vikram's car taking a giant leap through a bridge, somersaulting in the air, landing atop a speeding train, and continues as one ludicrous scene after the other.
Vikram is a daredevil car driver, who is forced to take up jobs ferrying illegal goods to provide for his comatose sister. Interestingly, he has no name in the film, and calls himself James Bond, Mani Ratnam or Sunil Gavaskar.
He meets Shakeela (Samantha) at a driving institute, where he is a trainer. An orphan, Shakeela is determined to earn her driving license.
The director has the pretty Samantha play a dimwit, who loves to make a fool of herself. The jokes become stale after a while.
Meanwhile, we have a small-time goon Dass (Pasupathy), who hires Vikram to transport a vehicle from Chennai to Mussoorie in Uttarakhand. What he fails to mention is that a kidnapped Shakeela is lying unconscious in the car.
The first half meanders aimlessly with a number of unnecessary songs at regular intervals, while the lead pair are busy falling in love. But the second half takes on a more serious note.
The buildup is enormous, as we are introduced to terrifying villains, played by Rahul Dev and Abhimanyu Singh. There are violent caste clashes, and an unknown person is soon to be hanged for the crime.
The director connects the dots between this mysterious person and Shakeela's kidnapping at the very end. The road trip that was to be the crux of the film is almost nonexistent. The car chases are unremarkable and the over-the-top graphics utterly ridiculous.
There is barely any chemistry between Vikram and Samantha, who seem too enthusiastic and appear to be overplaying their parts.
Pasupathy, as the greedy goon with his straight face and casual dialogue delivery, provides plenty of laughter.
The quietly menacing Rahul Dev is impressive as the bad guy.
Though a decent plot, the director in his attempt to make it bigger and better than his previous film, has perhaps added too much melodrama.
Director Vijay Milton's 10 Endrathukulla is a tedious road film marred by overzealous performances, ordinary music, tacky CGI and poor execution.


text:
Cyril Ramaphosa stands firm on Reserve Bank issue
Politics / 16 June 2019, 10:50am / Siyabonga Mkwanazi
President Cyril Ramaphosa. File photo: Laurent Gillieron/Keystone via AP.
President Cyril Ramaphosa has backed the South African Reserve Bank as the debate rages on in the ruling party and outside about its mandate.
Ramaphosa said he would defend the Sarb and ensure it maintained its reputation in the international markets.
ANC leaders came out against each other this week after secretary-general Ace Magashule said the national executive committee lekgotla had agreed to extend the mandate of the bank to include growth and employment.
But he was contradicted by Finance Minister Tito Mboweni and ANC head of economic transformation Enoch Godongwana, who said there was no such decision.
Political parties in Parliament said yesterday Ramaphosa must use the State of the Nation Address on Thursday to clarify the role and mandate of the Sarb. They said it would not make sense for Ramaphosa to talk about other issues in the address, and not touch on such a sensitive and important matter.
The DA, IFP and African Christian Democratic Party said the president needed to deal with the issue of policy uncertainty and Parliament was the right platform to do this.
Ramaphosa's spokesperson Khusela Diko said the president met the Sarb board on Friday including its Governor Lesetja Kganyago at his guest house in Pretoria. The meeting was to discuss the vacancies at the bank.
Kganyago's term comes to an end in November, while that of his deputy Daniel Mmimele expires at the end of the month.
Another former deputy governor, Francois Groepe, resigned in January this year.
At the meeting, Ramaphosa backed the Sarb. "The president has reiterated his commitment to safeguarding the high reputation the Sarb enjoys in the international financial community and committed to ensuring directors of a high calibre befitting the importance our country attaches to the Sarb are appointed," said Diko.
Deputy minister of finance David Masondo met the board with Ramaphosa .
The issue of the Reserve Bank has also brought in the ANC's tripartite alliance partners Cosatu and the SACP, which said they support the decision to expand the mandate of the bank.
They said this was the resolution of the ANC and it must be implemented.
ANC head of presidency Zizi Kodwa said last week Ramaphosa has spoken on the matter and there was no need to open the debate.
This was after the ANC president issued a statement on the NEC decision regarding the bank, and he considered the matter closed.
The ANC Top Six had met after Magashule's statement days earlier.
Political Bureau
Ramaphosa weighs in on debate over Sarb mandate, independence
Reserve Bank debate will not feature at Monday's Top 6 meeting
Enoch Godongwana backs Ramaphosa's statement on Reserve Bank mandate
ANC's policy on Reserve Bank has not changed: Ramaphosa


(Reuters) – Time Warner Inc unit HBO is considering moving film production of the Emmy award-winning show "Veep" to California from Maryland for better tax incentives, a spokesman said on Wednesday.
"Veep," the comedy starring Julia Louis-Dreyfus as a U.S. vice president, is one of four shows selected to receive tax credits for relocating production to California, according to the California Film Commission.
The panel has earmarked $6.5 million in film tax credits to "Veep" if the show moves to California, the commission's executive director, Amy Lemisch, said in a statement on Tuesday.
"Veep" has been filmed in Maryland for four years, primarily in Columbia, Baltimore and Sykesville.
"Veep" received $13.9 million in tax credits in its first three seasons, said Karen Bell Hood of Maryland's Department of Business and Economic Development.
The state has not earmarked tax credits toward the next season, she said.
In November, Maryland lawmakers proposed ending the tax credits in 2016. In a report, they said the programme was too costly to taxpayers and filming production brought little economic benefits.
The report said the state had given $62.5 million in tax credits to film and television productions since 2012. Of that amount, $60.2 million had gone to "Veep" and to "House of Cards," the Washington political thriller starring Kevin Spacey and Robin Wright.
California last year tripled its tax incentive fund for television and film production to $330 million.
This entry was posted on June 5, 2015, 5:55 am and is filed under California/Los Angeles, Maryland, Tax Incentives. You can follow any responses to this entry through RSS 2.0. Both comments and pings are currently closed.


Activities and exhibits on this special day focus on the energy that drives Alaska's economy. Also, help out our local food banks and save on your admission! Between noon and 2 p.m. with a suggested donation of two shelf-stable food items per person you can get into the Fair for $2. (Must be through the gate by 2 p.m.) Come check out the Plaza to help AOGA celebrate "Energy Day" – and don't forget to get your photo taken in the free photo booth!


The manufacturing brand of underwear of the big Bravissimo sizes has presented an unusual swimming mattress. Mirror reports about it. Under a pillow have made special space that with a big breast it was convenient to girls to lie on a stomach during swimming on a mattress. According to representatives of Bravissimo, they have posted the publication with allegedly new product in social networks for the sake of a joke. But users have accepted a novelty with such delight that the company has decided to start mass production of a beach accessory. At the moment the bright pink swimming mattress of CUP holders, Bravissimo style is on sale on the official site of a brand for 33 dollars. Materials on teme00:02 — Sea котикиКак spent on May 30 summer of supermodel on a paradise ostrovev June the Brazilian model Sofia Resing together with the Argentina designer Mary Rakauchi has released a collection of Mery Playa by Sofia Resing beachwear. According to Resing, she all life dreamed of the unique line of bathing suits therefore she when developing a collection has thought over each detail. "We have made an elastic insert in the lower part of a bathing suit. She will raise buttocks, but not to flatten them" — the 26-year model has told. It is more than important news in Film of Day Telegram-channel. Subscribe!


text:
Parador de Tui
Galician tradition on the border with Portugal
Avda. Portugal, s/n 36700 Tui
Register number: H-PO-001282
Stay with your pet in this Parador
Check conditions before booking
From 02/01/2022 until 25/02/2022
THE DATES YOU'VE SELECTED AREN'T AVAILABLE AT THIS RATE.
On the border between Galicia and Portugal, in the centre of the city of Tui and next to the banks of the Miño river, is the Parador de Tui. The building faithfully reproduces the architecture of a traditional Galician pazo and is surrounded by a mountainous and river environment. It is the perfect destination for nature lovers who want to combine their activity with river sports such as canoeing. The hotel has elegant lounges, cosy rooms, a large garden with a swimming pool and a tennis court.
Tui is at the head of the Baixo Miño region, a region bathed by the last kilometres of the river Miño before it reaches the Atlantic. The city has river beaches and natural landscapes of great beauty. Its historic centre is also of singular beauty, where a succession of cobbled streets reveals emblazoned houses and monuments at every turn, including the Santa María cathedral, the only one in the province of Pontevedra.
Tui's surroundings include the Monet Aloia Natural Park, the beautiful fishing village of A Guarda and the impressive Cistercian monastery of Santa María, in the small village of Oia. And in the landscape, the exceptional vineyards where the prestigious wines with D.O. Rías Baixas are grown.
Handicapped facilities
Historic hotels in Spain with a 20% discount
20% discount if booked from 1 September to 31 December 2022 (for travel from 1 October 2022 to 31 March 2023).
10 % discount for people over 55 years old
What to do in Parador de Tui
Tui has a beautiful old town centre with an interesting collection of historical-artistic buildings. Tui was a major town during the Middle Ages, being the focal point of trade and defence against Portugal, which it borders. The best way to discover its history is to walk around it. Stroll through its cobbled streets, discovering the emblazoned houses and the historic buildings. The most outstanding is the cathedral of Santa María, the only cathedral in the province of Pontevedra. Also of great interest are the churches of San Bartolomé, San Francisco and San Telmo.
Tui's natural landscape is one of its main attractions. Its setting next to the river Miño means it has river beaches that are suitable for bathing and river sports such as canoeing. It is also ideal for hiking. Near Tui is the Monte Aloia Natural Park, at an altitude of over 600 metres, where you can see the remains of a Cyclopean wall, the Cama de San Xiao rock, the Mirador de la Cruz and the chapel of San Xiao.
Tui is an ideal starting point to explore the Ribeira Sacra, an area comprising the banks of the rivers Sil and Miño, where you can discover the monasteries and temples along its slopes. And, on the other side of the Miño, you can visit nearby Valença do Minho, a picturesque Portuguese walled town.
Salones del Parador de Tui
10 x 9 x 3,5 m
tv monitors
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Elegance, comfort and landscape
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I put in a query to Pick n Pay Lemon Tree and still have not had a response from them. If this matter is not addressed, I will take it further. Below is the response from them and to date, nothing further.
Please can somebody assist me with this product.
I purchased the product last night for consumption and it had a weird taste to it.
I just want to make sure it doesn't contain something that I don't eat.
It is important to note that Freddy Hirsch only supplies the dry ingredients and possibly the casings to the butchers to make up this sausage. The butcher/ retailer then makes the sausage you show in the pictures, over which we have no control.
You have to take the product to the butcher/ retail outlet where you purchased the product and complain, showing them the pictures you took.
They will then investigate their side, and contact us with the batch details should they feel that the spice or our casings were at fault.
We, unfortunately, cannot investigate any fault our side without batch and product code details used in the make up of this particular product as we produce tons of this product per month, sold to thousands of different stores.
Please note that I did not supply pictures of the product. I merely want to determine what meat was used for this product please.
I have also bought meat from Pick n Pay, more than I can count, that was off, and I had to chuck it right after opening it. Sometimes the smell is unbearable! Looked at the expiry date but gives no indication. Not to mention, that the meat is quite expensive! Someone needs to look into this. It is very dangerous to sell meat/dairy that is off. Other than that, the bread quality is horrendous! Sometimes it literally falls apart as you cut into it. Same with some of the cheeses.