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Merged
conrad-watt merged 3 commits into from
Sep 4, 2023
Merged

WIP: Memory model aux defs #209

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23337da
Preliminary addition of auxiliary definitions in the relaxed memory m…
ioannad Sep 1, 2023
21839b0
Added macro for trace
ioannad Sep 1, 2023
749eceb
Found loose b.
ioannad Sep 1, 2023
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79 changes: 58 additions & 21 deletions document/core/exec/relaxed.rst
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Expand Up @@ -55,9 +55,9 @@ A trace is a coinductive set of :ref:`events <syntax-evt>`. A trace is considere

.. math::
\frac{
\config \stepto^{\evt} \config' \qquad \vdash \config' : \X{tr} \qquad \timeevt(\evt) \notin \timeevt^\ast(\X{tr})
\config \stepto^{\evt} \config' \qquad \vdash \config' : \trace \qquad \timeevt(\evt) \notin \timeevt^\ast(\trace)
}{
\vdash \config : \evt~\X{tr}
\vdash \config : \evt~\trace
}

When a WebAssembly program is executed, all behaviours observed during that execution must correspond to a single :ref:`consistent <relaxed-consistent>` pre-execution of that execution's starting :ref:`configuration <syntax-config>`.
Expand All @@ -70,57 +70,94 @@ Consistency

.. todo:: define auxiliary functions (either here or in Runtime Structure)

.. math::
\begin{array}{lcl}
\ordaux(\ARD_{\ord}~\loc~\byte^\ast~\NOTEARS^?) & = & \ord \\
\ordaux(\AWR_{\ord}~\loc~\byte^\ast~\NOTEARS^?) & = & \ord \\
\ordaux(\ARMW~\loc~{\byte_1}^\ast~{\byte_2}^\ast) & = & \SEQCST \\
&&\\
\locaux(\ARD_{\ord}~\loc~\byte^\ast~\NOTEARS^?) & = & \loc \\
\locaux(\AWR_{\ord}~\loc~\byte^\ast~\NOTEARS^?) & = & \loc \\
\locaux(\ARMW~\loc~{\byte_1}^\ast~{\byte_2}^\ast) & = & \loc \\
&&\\
\sizeaux(\ARD_{\ord}~\loc~\byte^n~\NOTEARS^?) & = & n \\
\sizeaux(\AWR_{\ord}~\loc~\byte^n~\NOTEARS^?) & = & n \\
\sizeaux(\ARMW~\loc~{\byte_1}^n~{\byte_2}^n) & = & n \\
&&\\
\readaux(\ARD_{\ord}~\loc~\byte^\ast~\NOTEARS^?) & = & \byte^\ast \\
\readaux(\AWR_{\ord}~\loc~\byte^\ast~\NOTEARS^?) & = & \epsilon \\
\readaux(\ARMW~\loc~{\byte_1}^\ast~{\byte_2}^\ast) & = & {\byte_1}^\ast \\
&&\\
\writeaux(\ARD_{\ord}~\loc~\byte^\ast~\NOTEARS^?) & = & \epsilon \\
\writeaux(\AWR_{\ord}~\loc~\byte^\ast~\NOTEARS^?) & = & \byte^\ast \\
\writeaux(\ARMW~\loc~{\byte_1}^\ast~{\byte_2}^\ast) & = & {\byte_2}^\ast \\
&&\\
\addraux(\act) & = & \addraux(\regionaux(\act) \\
\addraux(\loc) & = & \addraux(\regionaux(\reg) \\
\addraux(\addr.\fld) & = & \addr \\
&&\\
\regionaux(\act) & = & \regionaux(\locaux(\act) \\
\regionaux(\reg) & = & \reg \\
\regionaux(\reg[i]) & = & \reg \\
&&\\
\offsetaux(\act) & = & \offsetact(\locaux(\act)) \\
\offsetaux(\reg) & = & 0 \\
\offsetaux(\reg[i]) & = & i \\
\end{array}

.. todo:: Add more auxiliary functions

.. todo:: add prose intuition

.. math::
\frac{
\forall \reg, \, \vdash_{\reg} \X{tr} \consistentwith
\forall \reg, \, \vdash_{\reg} \trace \consistentwith
}{
\vdash \X{tr} \consistent
\vdash \trace \consistent
}

.. math::
\frac{
\begin{array}[b]{@{}l@{}}
\forall \evt_R \in \readingact_{\reg}(\X{tr}), \exists \evt_W^\ast,
\X{tr} \vdash_{\reg} \evt_R \readseachfrom \evt_W^\ast \\
\forall \evt_I, \evt \in \X{tr}, \,
\ordact_{\reg}(\evt_I) = \INIT \wedge
\forall \evt_R \in \readingact_{\reg}(\trace), \exists \evt_W^\ast,
\trace \vdash_{\reg} \evt_R \readseachfrom \evt_W^\ast \\
\forall \evt_I, \evt \in \trace, \,
\F{ord}_r(\evt_I) = \INIT \wedge
\evt_I \neq \evt \wedge
\overlapact(\evt_I, \evt) \Rightarrow \evt_I \prechb \evt
\end{array}
}{
\vdash_{\reg} \X{tr} \consistentwith
\vdash_{\reg} \trace \consistentwith
}

.. math::
\frac{
\begin{array}[b]{@{}c@{}}
\left|\evt_W^\ast\right| = |\readact_{\reg}(\evt_R)| \\
\forall i < |\evt_W^\ast|,
\X{tr} \vdash_{\reg}^i \evt_R \readsfrom \left(\evt_W^\ast[i]\right)
\trace \vdash_{\reg}^i \evt_R \readsfrom \left(\evt_W^\ast[i]\right)
\\
\vdash_{\reg} \evt_R \notear \evt_W^\ast
\end{array}
}{
\X{tr} \vdash_{\reg} \evt_R \readseachfrom \evt_W^\ast
\trace \vdash_{\reg} \evt_R \readseachfrom \evt_W^\ast
}

.. math::
\frac{
\begin{array}[b]{@{}l@{}}
\evt_R \neq \evt_W \\
\evt_W \in \writingact_{\reg}(\X{tr})
\evt_W \in \writingact_{\reg}(\trace)
\end{array}
\qquad
\begin{array}[b]{@{}r@{}}
\X{tr} \vdash_{\reg}^{i,k} \evt_R \valueconsistent \evt_W \\
\X{tr} \vdash_{\reg}^k \evt_R \hbconsistent \evt_W
\trace \vdash_{\reg}^{i,k} \evt_R \valueconsistent \evt_W \\
\trace \vdash_{\reg}^k \evt_R \hbconsistent \evt_W
\end{array}
\qquad
\X{tr} \vdash_{\reg} \evt_R \sclastvisible \evt^\ast_W
\trace \vdash_{\reg} \evt_R \sclastvisible \evt^\ast_W
}{
\X{tr} \vdash_{\reg}^i \evt_R \readsfrom \evt_W
\trace \vdash_{\reg}^i \evt_R \readsfrom \evt_W
}

.. math::
Expand All @@ -130,7 +167,7 @@ Consistency
k = \offsetact_{\reg}(\evt_R) + i &=& \offsetact_{\reg}(\evt_W) + j
\end{array}
}{
\X{tr} \vdash_{\reg}^{i,k} \evt_R \valueconsistent \evt_W
\trace \vdash_{\reg}^{i,k} \evt_R \valueconsistent \evt_W
}

.. math::
Expand All @@ -141,23 +178,23 @@ Consistency
\end{array}
\qquad
\begin{array}[b]{@{}l@{}}
\forall \evt'_W \in \writingact_{\reg}(\X{tr}),\\
\forall \evt'_W \in \writingact_{\reg}(\trace),\\
\quad \evt_W \prechb \evt'_W \prechb \evt_R \Rightarrow k \notin \rangeact_{\reg}(\evt'_W)
\end{array}
}{
\X{tr} \vdash_{\reg}^k \evt_R \hbconsistent \evt_W
\trace \vdash_{\reg}^k \evt_R \hbconsistent \evt_W
}

.. math::
\frac{
\begin{array}[b]{@{}l@{\qquad}l@{}}
\forall \evt'_W \in \writingact_{\reg}(\X{tr}), \evt_W \prechb \evt_R \Rightarrow \\
\forall \evt'_W \in \writingact_{\reg}(\trace), \evt_W \prechb \evt_R \Rightarrow \\
\quad \evt_W \prectot \evt'_W \prectot \evt_R \wedge \syncact_{\reg}(\evt_W, \evt_R) \Rightarrow \neg \syncact_{\reg}(\evt'_W, \evt_R) \\
\quad \evt_W \prechb \evt'_W \prectot \evt_R \Rightarrow \neg\syncact_{\reg}(\evt'_W, \evt_R) \\
\quad \evt_W \prectot \evt'_W \prechb \evt_R \Rightarrow \neg\syncact_{\reg}(\evt_W, \evt'_W)
\end{array}
}{
\X{tr} \vdash_{\reg} \evt_R \sclastvisible \evt_W
\trace \vdash_{\reg} \evt_R \sclastvisible \evt_W
}

.. math::
Expand Down
2 changes: 1 addition & 1 deletion document/core/exec/runtime.rst
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Original file line number Diff line number Diff line change
Expand Up @@ -706,7 +706,7 @@ Each event is annotated with a :ref:`time stamp <syntax-time>` that uniquely ide
\LDATA \\
\production{(store value)} & \storeval &::=&
\val ~|~
b^\ast \\
\byte^\ast \\
\end{array}

.. todo:: ensure identity of wait + wake operations is preserved
Expand Down
16 changes: 16 additions & 0 deletions document/core/util/macros.def
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Expand Up @@ -1113,6 +1113,18 @@
.. |storeval| mathdef:: \xref{exec/runtime}{syntax-storeval}{\X{storeval}}
.. |notears| mathdef:: \xref{exec/runtime}{syntax-ord}{\X{notears}}

.. Events, auxiliary functions

.. |ordaux| mathdef:: \xref{exec/relaxed}{exec-ordaux}{\F{ord}}
.. |locaux| mathdef:: \xref{exec/relaxed}{exec-locaux}{\F{loc}}
.. |sizeaux| mathdef:: \xref{exec/relaxed}{exec-sizeaux}{\F{size}}
.. |readaux| mathdef:: \xref{exec/relaxed}{exec-readaux}{\F{read}}
.. |writeaux| mathdef:: \xref{exec/relaxed}{exec-writeaux}{\F{write}}
.. |addraux| mathdef:: \xref{exec/relaxed}{exec-addraux}{\F{addr}}
.. |regionaux| mathdef:: \xref{exec/relaxed}{exec-regionaux}{\F{region}}
.. |offsetaux| mathdef:: \xref{exec/relaxed}{exec-offsetaux}{\F{offset}}


.. Events, meta functions
.. |tearing| mathdef:: \xref{exec/runtime}{syntax-ord}{\F{tearing}}

Expand Down Expand Up @@ -1150,6 +1162,10 @@
.. |prechb| mathdef:: \xref{exec/runtime}{relaxed-prechb}{\mathrel{\prec_{\K{hb}}}}
.. |prectot| mathdef:: \xref{exec/runtime}{relaxed-prectot}{\mathrel{\prec_{\K{tot}}}}

.. Relaxed Memory Model, non-terminals

.. |trace| mathdef:: \xref{exec/relaxed}{relaxed-trace}{\X{tr}}


.. Stack, terminals

Expand Down