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sav08:hoare_logic [2009/03/04 11:03]
vkuncak
sav08:hoare_logic [2015/04/21 17:30] (current)
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When $P, Q \subseteq S$ (sets of states) and $r \subseteq S\times S$ (relation on states, command semantics) then When $P, Q \subseteq S$ (sets of states) and $r \subseteq S\times S$ (relation on states, command semantics) then
Hoare triple Hoare triple
-$+\begin{equation*} \{P \}\ r\ \{ Q \} \{P \}\ r\ \{ Q \} -$+\end{equation*}
means means
-$+\begin{equation*} \forall s,s' \in S. s \in P \land (s,s') \in r \rightarrow s' \in Q \forall s,s' \in S. s \in P \land (s,s') \in r \rightarrow s' \in Q -$+\end{equation*}
We call $P$ precondition and $Q$ postcondition. We call $P$ precondition and $Q$ postcondition.

Note: weakest conditions (predicates) correspond to largest sets; strongest conditions (predicates) correspond to smallest sets that satisfy a given property (Graphically,​ a stronger condition $x > 0 \land y > 0$ denotes one quadrant in plane, whereas a weaker condition $x > 0$ denotes the entire half-plane.) Note: weakest conditions (predicates) correspond to largest sets; strongest conditions (predicates) correspond to smallest sets that satisfy a given property (Graphically,​ a stronger condition $x > 0 \land y > 0$ denotes one quadrant in plane, whereas a weaker condition $x > 0$ denotes the entire half-plane.)
+

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Definition: for $P \subseteq S$, $r \subseteq S\times S$, Definition: for $P \subseteq S$, $r \subseteq S\times S$,
-$+\begin{equation*} ​sp(P,​r) = \{ s' \mid \exists s. s \in P \land (s,s') \in r \} ​sp(P,​r) = \{ s' \mid \exists s. s \in P \land (s,s') \in r \} -$+\end{equation*}

-This is simply ​relation image of a set. (See [[Sets and relations#​Relation Image]].)+This is simply [[Sets and relations#​Relation Image]] ​of a set.

{{sav08:​sp.png?​400x250|}} {{sav08:​sp.png?​400x250|}}
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Definition: for $Q \subseteq S$, $r \subseteq S \times S$, Definition: for $Q \subseteq S$, $r \subseteq S \times S$,
-$+\begin{equation*} ​wp(r,​Q) = \{ s \mid \forall s'. (s,s') \in r \rightarrow s' \in Q \} ​wp(r,​Q) = \{ s \mid \forall s'. (s,s') \in r \rightarrow s' \in Q \} -$+\end{equation*}

Note that this is in general not the same as $sp(Q,​r^{-1})$ when relation is non-deterministic. Note that this is in general not the same as $sp(Q,​r^{-1})$ when relation is non-deterministic.
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If instead of good states we look at the completement set of "error states",​ then $wp$ corresponds to doing $sp$ backwards. ​ In other words, we have the following: If instead of good states we look at the completement set of "error states",​ then $wp$ corresponds to doing $sp$ backwards. ​ In other words, we have the following:
-$+\begin{equation*} S \setminus wp(r,Q) = sp(S \setminus Q,r^{-1}) S \setminus wp(r,Q) = sp(S \setminus Q,r^{-1}) -$+\end{equation*}

==== Disjunctivity of sp ==== ==== Disjunctivity of sp ====

-$+\begin{equation*} ​sp(P_1 \cup P_2,r) = sp(P_1,r) \cup sp(P_2,r) ​sp(P_1 \cup P_2,r) = sp(P_1,r) \cup sp(P_2,r) -$ +\end{equation*}
-$+\begin{equation*} ​sp(P,​r_1 \cup r_2) = sp(P,r_1) \cup sp(P,r_2) ​sp(P,​r_1 \cup r_2) = sp(P,r_1) \cup sp(P,r_2) -$+\end{equation*}

==== Conjunctivity of wp ==== ==== Conjunctivity of wp ====

-$+\begin{equation*} wp(r,Q_1 \cap Q_2) = wp(r,Q_1) \cap wp(r,Q_2) wp(r,Q_1 \cap Q_2) = wp(r,Q_1) \cap wp(r,Q_2) -$+\end{equation*}

-$+\begin{equation*} wp(r_1 \cup r_2,Q) = wp(r_1,Q) \cap wp(r_2,Q) wp(r_1 \cup r_2,Q) = wp(r_1,Q) \cap wp(r_2,Q) -$+\end{equation*}

==== Pointwise wp ===== ==== Pointwise wp =====

-$+\begin{equation*} wp(r,Q) = \{ s \mid s \in S \land sp(\{s\},r) \subseteq Q \} wp(r,Q) = \{ s \mid s \in S \land sp(\{s\},r) \subseteq Q \} -$+\end{equation*}

==== Pointwise sp ===== ==== Pointwise sp =====

-$+\begin{equation*} ​sp(P,​r) = \bigcup_{s \in P} sp(\{s\},​r) ​ ​sp(P,​r) = \bigcup_{s \in P} sp(\{s\},​r) ​ -$+\end{equation*}

==== Three Forms of Hoare Triple ==== ==== Three Forms of Hoare Triple ====
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Let $P$ and $Q$ be formulas in our language $F$ (see [[simple programming language]]). We define Hoare triples on these syntactic entities by taking their interpretation as sets and relations: Let $P$ and $Q$ be formulas in our language $F$ (see [[simple programming language]]). We define Hoare triples on these syntactic entities by taking their interpretation as sets and relations:
-$+\begin{equation*} \{ P \} c \{ Q \} ​ \{ P \} c \{ Q \} ​ -$+\end{equation*}
means means
-$+\begin{equation*} \forall s_1, s_2.\ f_T(P)(s_1) \land (s_1,s_2) \in r_c(c) \rightarrow f_T(Q)(s_1) \forall s_1, s_2.\ f_T(P)(s_1) \land (s_1,s_2) \in r_c(c) \rightarrow f_T(Q)(s_1) -$+\end{equation*}
In words: if we start in a state satisfying $P$ and execute $c$, we obtain a state satisfying $Q$.  ​ In words: if we start in a state satisfying $P$ and execute $c$, we obtain a state satisfying $Q$.  ​

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===== Composing Hoare Triples ===== ===== Composing Hoare Triples =====

-$+\begin{equation*} \frac{ \{P\} c_1 \{Q\}, \ \ \{Q\} c_2 \{R\} } \frac{ \{P\} c_1 \{Q\}, \ \ \{Q\} c_2 \{R\} } { \{P\} c_1 ; c_2 \{ R \} } { \{P\} c_1 ; c_2 \{ R \} } -$+\end{equation*}

We can prove this from  We can prove this from