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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. | ||
| Line 86: | Line 87: | ||
| 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 ==== | ||
| Line 134: | Line 135: | ||
| 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 $Q$. | ||
| Line 149: | Line 150: | ||
| ===== 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 | ||