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Merge pull request #43 from bkushigian/master

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Change overloaded term `S` in section 5.4
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Adam Chlipala 2020-04-24 09:29:23 -04:00 committed by GitHub
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frap_book.tex
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@ -1168,7 +1168,7 @@ Of course, bugs in this program, like forgetting to include the locking, could l
\encoding
To prove that we got the program right, let's formalize it as a transition system. First, our state set:
$$\begin{array}{rrcl}
\textrm{States} & S &::=& \mathsf{Lock} \mid \mathsf{Read} \mid \mathsf{Write}(n) \mid \mathsf{Unlock} \mid \mathsf{Done}
\textrm{States} & P &::=& \mathsf{Lock} \mid \mathsf{Read} \mid \mathsf{Write}(n) \mid \mathsf{Unlock} \mid \mathsf{Done}
\end{array}$$
Compared to the last example, here we see more clearly that kinds of states correspond to \emph{program counters}\index{program counters} in the imperative code.
@ -1178,8 +1178,8 @@ Only $\mathsf{Write}$ states carry extra information, in this case the value of
At every other program counter, we can prove that the value of variable \texttt{local} has no effect on further transitions, so we don't bother to store it.
We will account for the value of variable \texttt{global} separately, in a way to be described shortly.
In particular, we will define a transition system for a single thread as $\mathcal L = \angled{(\mathbb N \times \mathbb B) \times S, \mathcal L_0, \to_{\mathcal L}}$.
We define the state to include not only the thread-local state $S$ but also the value of \texttt{global} (in $\mathbb N$) and whether the lock is currently taken (in $\mathbb B$, the Booleans, with values $\top$ [true] and $\bot$ [false]).
In particular, we will define a transition system for a single thread as $\mathcal L = \angled{(\mathbb N \times \mathbb B) \times P, \mathcal L_0, \to_{\mathcal L}}$.
We define the state to include not only the thread-local state $P$ but also the value of \texttt{global} (in $\mathbb N$) and whether the lock is currently taken (in $\mathbb B$, the Booleans, with values $\top$ [true] and $\bot$ [false]).
There is one designated initial state.
$$\infer{((0, \bot), \mathsf{Lock}) \in \mathcal L_0}{}$$