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phd-dissertation
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Daniel Hillerström 5 years ago
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  1. 117
      thesis.tex

117
thesis.tex
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@ -10957,7 +10957,7 @@ Section~\ref{subsec:machine-correctness} easier to state.
\[
\bl
%\multicolumn{1}{c}{\stepsto \subseteq \MConfCat \times \MConfCat}\\[1ex]
\ba{@{}l@{\quad}r@{~}c@{~}l@{\quad}l@{}}
\ba{@{}l@{\quad}r@{~}c@{~}l@{~~}l@{}}
% \mlab{Init} & \multicolumn{3}{@{}c@{}}{M \stepsto \cek{M \mid \emptyset \mid [(\nil, (\emptyset, \{\Return\;x \mapsto \Return\;x\}))]}} \\[1ex]
% App
&&\multicolumn{2}{@{}l}{\stepsto \subseteq \MConfCat \times \MConfCat}\\
@ -10988,7 +10988,7 @@ Section~\ref{subsec:machine-correctness} easier to state.
% Deep resumption application
\mlab{Resume^\param} & \cek{ V\,\Record{W;W'} \mid \env \mid \shk}
&\stepsto& \cek{ \Return \; W \mid \env \mid (\sigma,(\env'[q \mapsto \val{W'}\env],q.\,H)) \cons \shk' \concat \shk},
&\text{if }\val{V}{\env} = (\sigma,(\env',q.\,H)) \cons \shk')^A \\
&\text{if }\val{V}{\env} = \shk' \concat [(\sigma,(\env',q.\,H))])^A \\
%
\mlab{Split} & \cek{ \Let \; \Record{\ell = x;y} = V \; \In \; N \mid \env \mid \shk}
&\stepsto& \cek{ N \mid \env[x \mapsto v, y \mapsto w] \mid \shk},
@ -11200,7 +11200,7 @@ provides a way to extract the final value of some computation from a
configuration.
\subsection{Putting the machine into action}
\newcommand{\Chiid}{\ensuremath{\Chi_{\text{id}}}}
\newcommand{\chiid}{\ensuremath{\chi_{\text{id}}}}
\newcommand{\kappaid}{\ensuremath{\kappa_{\text{id}}}}
\newcommand{\incr}{\dec{incr}}
\newcommand{\Incr}{\dec{Incr}}
@ -11222,7 +11222,12 @@ a deep, parameterised, and shallow handler.
\Handle\;m\,\Unit\;\With\\
~\ba{@{~}l@{~}c@{~}l}
\Return\;x &\mapsto& [x]\\
\OpCase{\Fork}{\Unit}{resume} &\mapsto& resume~\True \concat resume~\False
\OpCase{\Fork}{\Unit}{resume} &\mapsto&
\bl
\Let\;t \revto resume~\True\;\In\\
\Let\;f \revto resume~\False\;\In\\
t \concat f
\el
\ea
\el
\el
@ -11274,25 +11279,107 @@ a deep, parameterised, and shallow handler.
%
\[
\bl
\dec{prod}\,\Unit \defas \Do\;\Yield\,(\Do\;\Incr\,\Unit); \dec{prod}\,\Unit\\
\dec{cons}\,\Unit \defas \Let\;x \revto \Do\;\Await\,\Unit\;\In\;\Let\;y \revto \Do\;\Await\,\Unit\;\In\;x+y
\dec{prod} : \UnitType \to \alpha \eff \{\Incr : \UnitType \opto \Int; \Yield : \Int \opto \UnitType\}\\
\dec{prod}\,\Unit \defas
\bl
\Let\;i \revto \Do\;\Incr\,\Unit\;\In\\
\Let\;x \revto \Do\;\Yield~i\\
\In\;\dec{prod}\,\Unit
\el\smallskip\\
\dec{cons} : \UnitType \to \Int \eff \{\Fork : \UnitType \opto \Bool; \Await : \UnitType \opto \Int\}\\
\dec{cons}\,\Unit \defas
\bl
\Let\;b \revto \Do\;\Fork\,\Unit\;\In\\
\Let\;x \revto \Do\;\Await\,\Unit\;\In\\
% \Let\;y \revto \Do\;\Await\,\Unit\;\In\\
\If\;b\;\Then\;x+2\;\Else\;x*2
\el
\el
\]
%
\begin{derivation}
&\nondet\,(\lambda\Unit.\incr\,\Record{0;\lambda\Unit.\Record{\Do\;\Fork\,\Unit;\Pipe\,\Record{\dec{prod};\dec{cons}}}})\\
&\nondet\,(\lambda\Unit.\incr\,\Record{0;\lambda\Unit.\Pipe\,\Record{\dec{prod};\dec{cons}}})\\
\stepsto& \reason{\mlab{Init} with $\env_0$}\\
&\cek{\nondet\,(\lambda\Unit.\incr\,\Record{0;\lambda\Unit.\Record{\Do\;\Fork\,\Unit;\Pipe\,\Record{\dec{prod};\dec{cons}}}}) \mid \env_0 \mid \sks_0}\\
\stepsto^+& \reason{\mlab{App}, \mlab{Handle}, \mlab{App}, \mlab{Handle^\param}, \mlab{App}, \mlab{Let}}\\
&\cek{\nondet\,(\lambda\Unit.\incr\,\Record{0;\lambda\Unit.\Pipe\,\Record{\dec{prod};\dec{cons}}}) \mid \env_0 \mid \sks_0}\\
\stepsto^+& \reason{$3\times$(\mlab{App}, \mlab{Handle^\delta})}\\
&\bl
\cek{c\,\Unit \mid \env_\Pipe \mid (\nil,\chi^\dagger_\Pipe) \cons (\nil, \chi^\param_\incr) \cons (\nil, \chi_\nondet) \cons \kappa_0}\\
\text{where }
\bl
\env_\Pipe = \env_0[c \mapsto (\env_0, \dec{cons}), p \mapsto (\env_0, \dec{prod})]\\
\chi^\dagger_\Pipe = (\env_\Pipe, H^\dagger_\Pipe)\\
\env_\incr = \env_0[m \mapsto (\env_0, \lambda\Unit.\Pipe\cdots),i \mapsto 0]\\
\chi^\param_\incr = (\env_\incr, H^\param_\Pipe)\\
\env_\nondet = \env_0[m \mapsto (\env_0, \lambda\Unit.\incr \cdots)]\\
\chi_\nondet = (\env_\nondet, H_\nondet)
\el
\el\\
\stepsto^+& \reason{\mlab{App}, \mlab{Let}}\\
&\bl
\cek{\Do\;\Fork\,\Unit \mid \env_0 \mid ([(\env_0,b,\Let\;x \revto \cdots)],(\env_\Pipe,H^\dagger_\Pipe)) \cons \kappa_1}\\
\text{where } \kappa_1 = (\nil, \chi^\param_\incr) \cons (\nil, \chi_\nondet) \cons \kappa_0
\el\\
\stepsto^+& \reason{\mlab{Forward}, \mlab{Forward}}\\
&\bl
\cek{\Do\;\Fork\,\Unit \mid \env_0 \mid ([(\env_0,x,\Record{x;\Pipe\,\Record{\dec{prod};\dec{cons}}})],\chi_\incr) \cons (\nil, \chi_\nondet) \cons \sks_0}\\
~\text{where } \chi_\incr \defas (\env_0[i \mapsto i],H_\incr)
\cek{\Do\;\Fork\,\Unit \mid \env_0 \mid [(\nil, \chi_\nondet),(\nil,\chiid)] \circ \kappa'}\\
\text{where } \kappa' = [([(\env_0,b,\Let\;x \revto \cdots)],(\env_\Pipe,H^\dagger_\Pipe)),(\nil, \chi^\param_\incr)]
\el\\
\stepsto^+& \reason{\mlab{Do}, \mlab{Let}}\\
&\bl
\cek{resume~\True \mid \env_\nondet' \mid \kappa_0'}\\
\text{where }
\ba[t]{@{~}r@{~}c@{~}l}
\env_\nondet' &=& \env_\nondet[resume \mapsto \kappa' \concat [(\nil, \chi_\nondet)]]\\
\kappa_0' &=& [([(\env_\nondet',x,\Let\;y\revto\cdots)],\chiid)]
\el
\el\\
\stepsto^+& \reason{\mlab{Forward}, \mlab{Do^\param}}\\
\stepsto^+& \reason{\mlab{Resume}, \mlab{PureCont}, \mlab{Let}}\\
&\bl
\cek{resume~\True \concat resume~\False \mid \env_\nondet \mid \sks_0}\\
~\text{where } \env_\nondet \defas \env_0[resume \mapsto [([(\env_0,x,\Record{x;\Pipe\,\Record{\dec{prod};\dec{cons}}})],\chi_\incr), (\nil, \chi_\nondet)]
\el
\cek{\Do\;\Await\,\Unit \mid \env_0' \mid \kappa'}\\
\text{where }
\ba[t]{@{~}r@{~}c@{~}l}
\env_0' &=& \env_0[b \mapsto \True]\\
\kappa' &=& [([(\env_0',x,\If\;\cdots)], \chi^\dagger_\Pipe),(\nil,\chi^\param_\incr),(\nil, \chi_\nondet)] \concat \kappa_0'
\ea
\el\\
\stepsto^+& \reason{\mlab{Forward}, \mlab{Do^\dagger}}\\
&\bl
\cek{\Copipe\,\Record{resume;p} \mid \env_\Pipe' \mid \kappa'}\\
\text{where }
\ba[t]{@{~}r@{~}c@{~}l}
\env_\Pipe' &=& \env_\Pipe[resume \mapsto (\nil, [(\env_0',x,\If\;\cdots)])]\\
\kappa' &=& [(\nil,\chi^\param_\incr),(\nil, \chi_\nondet)] \concat \kappa_0'
\ea
\el\\
\stepsto^+& \reason{\mlab{App}, \mlab{Handle^\dagger}}\\
&\bl
\cek{p\,\Unit \mid \env_\Copipe \mid (\nil, \chi^\dagger_\Copipe) \cons \kappa'}\\
\text{where }
\ba[t]{@{~}r@{~}c@{~}l}
\env_\Copipe &=& \env_0[c \mapsto (\nil, [(\env_0',x,\If\;\cdots)]), p \mapsto (\env_0,\dec{prod})]\\
\chi_\Copipe &=& (\env_\Copipe,H^\dagger_\Copipe)\\
\ea
\el\\
\stepsto^+& \reason{\mlab{AppRec}, \mlab{Let}}\\
&\cek{\Do\;\Incr\,\Unit \mid \env_0 \mid ([(\env_0,i,\Let\;x\revto\cdots)],\chi^\dagger_\Copipe) \cons \kappa'}\\
\stepsto^+& \reason{\mlab{Forward}, \mlab{Do^\param}, \mlab{Let}, \mlab{App}, \mlab{PureCont}}\\
&\bl
\cek{resume\,\Record{1;0} \mid \env_\incr' \mid \kappa'}\\
\text{where }
\ba[t]{@{~}r@{~}c@{~}l}
\env_\incr' &=& \env_\incr[i' \mapsto 1, resume \mapsto [([(\env_0,i,\Let\;x\revto\cdots)],\chi^\dagger_\Copipe),(\nil,\chi^\param_\incr)]]\\
\kappa' &=& (\nil, \chi_\nondet) \cons \kappa_0'
\ea
\el
% &\bl
% \cek{\Do\;\Fork\,\Unit \mid \env_0 \mid ([(\env_0,x,\Record{x;\Pipe\,\Record{\dec{prod};\dec{cons}}})],\chi_\incr) \cons (\nil, \chi_\nondet) \cons \sks_0}\\
% ~\text{where } \chi_\incr \defas (\env_0[i \mapsto i],H_\incr)
% \el\\
% \stepsto^+& \reason{\mlab{Forward}, \mlab{Do^\param}}\\
% &\bl
% \cek{resume~\True \concat resume~\False \mid \env_\nondet \mid \sks_0}\\
% ~\text{where } \env_\nondet \defas \env_0[resume \mapsto [([(\env_0,x,\Record{x;\Pipe\,\Record{\dec{prod};\dec{cons}}})],\chi_\incr), (\nil, \chi_\nondet)]
% \el
% &\bl
% \cek{c\,\Unit \mid \env_{\Pipe} \mid (\nil,\chi_\Pipe) \cons (\nil,\chi_\incr) \cons (\nil, \chi_\nondet) \cons \sks_0}\\
% \ba[t]{@{~}r@{~}c@{~}l}

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