\newglossaryentry{state and action space}
{
name={\ensuremath{\pspace}},
sort={LR},
description={A state and action space, i.e. a directed graph, $\pactions \subset \pstates \times \pstates$.}
}

\longnewglossaryentry{finite paths}
{
  name={\ensuremath{\pstates^\ast}},
  sort={L*}
}
{%
A set of finite paths in a state and action space $\pspace$, i.e.
\[
\pstates^\ast = \{ p = (p_1, p_2, \ldots, p_k) \colon p_i \in \pstates, (p_i, p_{i+1}) \in \pactions \}.
\]
We allow an empty path, for $k=0$, denoted $\epsilon$.
}

\longnewglossaryentry{concatenation}
{%
  name={\ensuremath{p \cdot q}},
  sort={pq}
}
{
If $p, q \in \pstates^\ast$, then $p \cdot q$ is concatenation of paths, i.e.
\[
p \cdot q = (p_1, p_2, \ldots, p_k, q_1, q_2, \ldots, q_l),
\]
where $p = (p_1, p_2, \ldots, p_k)$ and $q = (q_1, q_2, \ldots, q_l)$.
If we write $p \cdot q$, then we implicitly assume that $p \cdot q \in \pstates^\ast$, i.e. $(p_k, q_1) \in \pactions$.
}

\newglossaryentry{length of path}
{
name={\ensuremath{|p|}},
sort={p},
description={For $p = (p_1, p_2, \ldots, p_k) \in \pstates^\ast$, we let $|p| = k$ denote the length of $p$.}
}

\newglossaryentry{subshift}
{
name={\ensuremath{\psubshift}},
sort={L},
description={A set of infinite paths in $\pspace$, i.e.
\[
\psubshift = \{ (s_1, s_2, s_3, \ldots) \colon s_i \in \pstates, (s_i, s_{i+1}) \in \pactions \}.
\]}
}

\newglossaryentry{shift}{
name={\ensuremath{\shift}},
sort={shift},
description={A shift operator $\shift \colon \psubshift \to \psubshift$, i.e.
\[
\shift(s_1, s_2, s_3, \ldots) = (s_2, s_3, \ldots).
\]}
}

\newglossaryentry{cone}{
name={\ensuremath{\cone{p}}},
sort={cone},
description={
A cone in $\psubshift$ over $p \in L^\ast$, i.e.
\[
\cone{p} = \{ q \in \psubshift \colon p_i = q_i \text{ for } i \leq |p| \}.
\]
}
}

\newglossaryentry{optimal strategy}{
name={\ensuremath{\strategy^\ast}},
sort={mu*},
description={An optimal defender strategy, i.e. 
\[
\strategy^\ast \in \argmax_{P} \gamevalue(\strategy).
\]
}}

\newglossaryentry{optimal strategy against}{
name={\ensuremath{\strategy^\ast_\actionable}},
sort={mu*I},
description={An optimal defender strategy against $\actionable$, i.e. 
\[
\strategy^\ast_\actionable \in \argmax_{\strategy} \gamevalue_\actionable(\strategy).
\]
}}

\newglossaryentry{attack plans}{
name={\ensuremath{\targets}},
sort={T},
description={The set of attack plans of the attacker.
}}

\newglossaryentry{attack duration}{
name={\ensuremath{\tau_j}},
sort={tau},
description={The attack duration of attack plan $j \in \targets$.
}}

\newglossaryentry{payoff function}{
name={\ensuremath{\payoff}},
sort={G},
description={The payoff function of the defender $\payoff \colon \psubshift \times \targets \to \mathbb{R}$, where $G(p,j)$ is the payoff of the defender if the attacker executes an attack plan $j$ at time~$0$ against the patrol schedule $p$.
}}

\newglossaryentry{payoff function from attack plan}{
name={\ensuremath{\payoff_j}},
sort={Gj},
description={The payoff function of the defender when attacker executes attack plan $j \in \targets$, i.e., $\payoff_j = \payoff(\cdot, j)$.
}}

\newglossaryentry{shifted payoff function}{
name={\ensuremath{\payoff^n_j}},
sort={Gnj},
description={
The defender payoff if the attacker executes attack plan $j \in \targets$ at time $n \in \mathbb{N}$ against the patrol schedule $p \in \psubshift$, i.e.
\[
\payoff^n_j = \payoff_j \circ \shift^n.
\]
}}

\newglossaryentry{attacker strategy}{
name={\ensuremath{(i,j)}},
sort={ij},
description={
An attacker strategy, i.e., a pair $(i, j) \in \pstates^\ast \times \targets$ of a moment of the attack $i \in \mathbb{N}$ and an attack plan $j \in T$.
}}

\newglossaryentry{actionable observations}{
name={\ensuremath{\actionable}},
sort={I},
description={
Actionable observations of the attacker, i.e. a set $\actionable \subset \pstates^\ast$, i.e., upon observing $i \in \actionable$, the attacker can decide to take action, and otherwise they definitely does not make motion to perform an attack plan.
}}

\newglossaryentry{game value against actionable}{
name={\ensuremath{\gamevalue_\actionable}},
sort={VI},
description={
Game value against the set of actionable observations $\actionable$, i.e., the value that is most beneficial for the attacker amongst the choice of an attack plan and an actionable observation.
}}

\newglossaryentry{game value against strategy}{
name={\ensuremath{\gamevalue(\strategy)}},
sort={V},
description={
Game value against the defender strategy $\strategy$, i.e.
\[
\gamevalue(\strategy) = \inf_{I \subset \pstates^\ast} \gamevalue_\actionable(\strategy).
\]
}}

\newglossaryentry{behavioral strategy}{
name={\ensuremath{\behavioral}},
sort={P},
description={
Behavioral strategy of the defender, i.e., $\behavioral\giventhat*{s}{p}$ is the probability that the defender enters location $s \in \pstates$, given that so far they have been following the patrolling schedule $p$.
}}

\newglossaryentry{strategy and action space}{
name={\ensuremath{\sspace}},
sort={SA},
description={
Strategy state and action space used by the defender to control their defensive resources.
}}

\newglossaryentry{projection}{
name={\ensuremath{\projection}},
sort={X},
description={
Projection, i.e., a graph homomorphism $\projection \colon \sstates \to \pstates$ that translates defender's internal scheduling to actions in the physical space.
}}


\newglossaryentry{Markov chain}{
name={\ensuremath{\mc}},
sort={M},
description={
A Markov chain on $\sspace$ with a transition matrix $\mcmatrix$ and a stationary distribution $\mcstationary$.
}}

\newglossaryentry{transition matrix}{
name={\ensuremath{\mcmatrix}},
sort={N},
description={
Transition matrix of the Markov chain.
}}

\newglossaryentry{stationary distribution}{
name={\ensuremath{\mcstationary}},
sort={sigma},
description={
Stationary distribution of the Markov chain.
}}

\newglossaryentry{Markov paths}{
name={\ensuremath{\ssubshift}},
sort={S},
description={
A set of infinite paths in $\sspace$.
}}

\newglossaryentry{Markov measure}{
name={\ensuremath{\mcmeasure}},
sort={nu},
description={
Markov measure induced by $\mc$ on $\ssubshift$.
}}

\newglossaryentry{defender strategy}{
name={\ensuremath{\strategy}},
sort={mu},
description={
The defender strategy $\strategy = \nu \circ X^{-1}$ with a hidden Markov model $\nu$.
}}

\newglossaryentry{patrolling units}{
name={\ensuremath{\units}},
sort={U},
description={
The set of patrolling units.
}}

\newglossaryentry{patrolling unit topology}{
name={\ensuremath{(\locations_u, \routes_u)}},
sort={LuRu},
description={
The set of locations $\locations_u$ patrolled by a particular unit $u \in \units$, and the set of routes $\routes_u$ that it traverses.
}}

\newglossaryentry{coverage function}{
name={\ensuremath{\coverage_u}},
sort={gamma},
description={
Coverage function $\coverage_u \colon \locations_u \times \targets \to [0, 1]$ defined for each patrolling unit $u \in \units$, where $\coverage_u(l, t)$ is the probability that the patrolling unit $u$ stationed at location $l$ will catch an intruder within a single unit of time while he attacks target $t$.
}}

\newglossaryentry{values}{
name={\ensuremath{\values}},
sort={V},
description={
Defender's values of targets $\values \colon \targets \to \mathbb{R}$.
}}

\newglossaryentry{attacker's observation}{
name={\ensuremath{\observation}},
sort={Y},
description={
Attacker's observation function $\observation \colon \sstates \to \actionable$.
}}

\newglossaryentry{history}{
name={\ensuremath{\observation_h}},
sort={Yh},
description={
History of length $h$, i.e., $\observation_h(p) = \shift^{|p| - h}(p)$.
}}

\newglossaryentry{observation lift}{
name={\ensuremath{\lift_h}},
sort={Yh*},
description={
Lift of the attacker's observation function, i.e.,  $\lift_h(s) = (M_X(s, h-1), M_X(s, h-2), \ldots, M_X(s, 0))$, where $M_X(s,i)$ maps $(s,i)$ to the past value of $X$ from $i$ steps before the defender reached the state $s$.
}}

\newglossaryentry{reward lift}{
name={\ensuremath{\widetilde\payoff_j}},
sort={Gj*},
description={
Lift of the payoff function, i.e.,  $\widetilde\payoff_j(s) = G_j((M_X(s, \tau_j-1), M_X(s, \tau_j-2), \ldots, M_X(s, 0))$, where $M_X(s,i)$ maps $(s,i)$ to the past value of $X$ from $i$ steps before the defender reached the state $s$.
}}