}{}\pgfsys@moveto{-8.533pt}{-3.533pt}\pgfsys@lineto{-8.533pt}{-1.533pt}\pgfsys@lineto{-6.533pt}{-1.533pt}\pgfsys@lineto{-6.533pt}{0.467pt}\pgfsys@lineto{-4.533pt}{0.467pt}\pgfsys@lineto{-4.533pt}{2.467pt}\pgfsys@lineto{-2.533pt}{2.467pt}\pgfsys@lineto{-2.533pt}{4.467pt}\pgfsys@lineto{-0.533pt}{4.467pt}\pgfsys@lineto{-0.533pt}{-3.53302pt}\pgfsys@closepath\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$ (a) agent can move
in any of the cardinal directions. Empty cells are implicitly labelled by
$\emptyset\in 2^{\mathcal{AP}}$. Moving against a wall keeps the agent in the
same state.

$q_{0}$start$q_{1}$$q_{2}$$\neg$ $\neg\mathbin{\leavevmode\hbox
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{}{}{{}}{}
{{{}{}}{}}{} {{{}{}}{}}{} {{{}{}}{}}{} {{{}{}}{}}{} {{{}{}}{}}{} {{{}{}}{}}{}
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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{{{}{}}{}}{} {{{}{}}{}}{} {{{}{}}{}}{} {{{}{}}{}}{} {{{}{}}{}}{} {{{}{}}{}}{}
{{{}{}}{}}{} {{{}{}}{}}{} {{{}{}}{}}{} {}\pgfsys@beginscope\pgfsys@invoke{
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}\definecolor{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-8.533pt}{-3.533pt}\pgfsys@lineto{-8.533pt}{-1.533pt}\pgfsys@lineto{-6.533pt}{-1.533pt}\pgfsys@lineto{-6.533pt}{0.467pt}\pgfsys@lineto{-4.533pt}{0.467pt}\pgfsys@lineto{-4.533pt}{2.467pt}\pgfsys@lineto{-2.533pt}{2.467pt}\pgfsys@lineto{-2.533pt}{4.467pt}\pgfsys@lineto{-0.533pt}{4.467pt}\pgfsys@lineto{-0.533pt}{-3.53302pt}\pgfsys@closepath\pgfsys@fillstroke\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
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specification is to grab coffee and then go upstairs.

Figure 7: A simple MDP (a) and TA (b) to show the Cone Lumping method.
$s_{00},q_{0}$start$s_{10},q_{0}$$s_{11},q_{0}$$s_{01},q_{0}$$s_{20},q_{0}$$s_{21},q_{0}$$s_{02},q_{1}$$s_{22},q_{0}$$s_{00},q_{1}$$s_{10},q_{1}$$s_{11},q_{1}$$s_{01},q_{1}$$s_{20},q_{1}$$s_{21},q_{2}$$s_{22},q_{1}$$s_{12},q_{1}$$\emptyset$$\emptyset$$\emptyset$$\emptyset$$\emptyset$$\mathbin{\leavevmode\hbox
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{}{}{{}}{}
{{{}{}}{}}{} {{{}{}}{}}{} {{{}{}}{}}{} {{{}{}}{}}{} {{{}{}}{}}{} {{{}{}}{}}{}
{{{}{}}{}}{} {{{}{}}{}}{} {{{}{}}{}}{} {}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@roundjoin\pgfsys@invoke{ }\pgfsys@roundcap\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-8.533pt}{0.0pt}\pgfsys@lineto{-8.533pt}{2.0pt}\pgfsys@lineto{-6.533pt}{2.0pt}\pgfsys@lineto{-6.533pt}{4.0pt}\pgfsys@lineto{-4.533pt}{4.0pt}\pgfsys@lineto{-4.533pt}{6.0pt}\pgfsys@lineto{-2.533pt}{6.0pt}\pgfsys@lineto{-2.533pt}{8.0pt}\pgfsys@lineto{-0.533pt}{8.0pt}\pgfsys@lineto{-0.533pt}{-0.00002pt}\pgfsys@closepath\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$$\emptyset$$\emptyset$$\emptyset$
Figure 8: NFA underlying the product MC constructed from MDP and DFA in Figure
7, under a policy which takes actions uniformly at random. MDP state $s_{00}$
represents the bottom left cell of the grid in Figure 7(a), while state
$s_{22}$ represents the top right, corresponding to the ordering $0,1,2$ of
cells for both the horizontal ($x$) and vertical ($y$) components of states
$s_{yx}$.