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 sav08:hoare_logic [2009/03/04 11:03]vkuncak sav08:hoare_logic [2015/04/21 17:30] Line 1: Line 1: - ====== Hoare Logic ====== - Hoare logic is a way of inserting annotations into code to make proofs about program behavior simpler. - - - ===== Example Proof ===== - - - //{0 <= y} - i = y; - //{0 <= y & i = y} - r = 0; - //{0 <= y & i = y & r = 0} - while //{r = (y-i)*x & 0 <= i} - (i > 0) ( - //{r = (y-i)*x & 0 < i} - r = r + x; - //{r = (y-i+1)*x & 0 < i} - i = i - 1 - //{r = (y-i)*x & 0 <= i} - ) - //{r = x * y} - ​ - - - ===== Hoare Triple for Sets and Relations ===== - - When $P, Q \subseteq S$ (sets of states) and $r \subseteq S\times S$ (relation on states, command semantics) then - Hoare triple - $- \{P \}\ r\ \{ Q \} -$ - means - $- \forall s,s' \in S. s \in P \land (s,s') \in r \rightarrow s' \in Q -$ - 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.) - - - - - - ===== Strongest Postcondition - sp ===== - - Definition: for $P \subseteq S$, $r \subseteq S\times S$, - $- ​sp(P,​r) = \{ s' \mid \exists s. s \in P \land (s,s') \in r \} -$ - - This is simply relation image of a set. (See [[Sets and relations#​Relation Image]].) - - {{sav08:​sp.png?​400x250|}} - - - ==== Lemma: Characterization of sp ==== - - $sp(P,r)$ is the the smallest set $Q$ such that $\{P\}r\{Q\}$,​ that is: - - $\{P\} r \{ sp(P,r) \}$ - - $\forall Q \subseteq S.\ \{P\} r \{Q\} \rightarrow sp(P,r) \subseteq Q$ - - - ===== Weakest Precondition - wp ===== - - Definition: for $Q \subseteq S$, $r \subseteq S \times S$, - $- ​wp(r,​Q) = \{ s \mid \forall s'. (s,s') \in r \rightarrow s' \in Q \} -$ - - Note that this is in general not the same as $sp(Q,​r^{-1})$ when relation is non-deterministic. - - {{sav08:​wp.png?​400x250|}} - - ==== Lemma: Characterization of wp ==== - - $wp(r,Q)$ is the largest set $P$ such that $\{P\}r\{Q\}$,​ that is: - - $\{wp(r,​Q)\} r \{Q \}$ - - $\forall P \subseteq S.\ \{P\} r \{Q\} \rightarrow P \subseteq wp(r,Q)$ - - ===== Some More Laws on Preconditions and Postconditions ===== - - We next list several more lemmas on properties of wp, sp, and Hoare triples. - - ==== Postcondition of inverse versus wp ==== - - 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: - $- S \setminus wp(r,Q) = sp(S \setminus Q,r^{-1}) -$ - - ==== Disjunctivity of sp ==== - - $- ​sp(P_1 \cup P_2,r) = sp(P_1,r) \cup sp(P_2,r) -$ - $- ​sp(P,​r_1 \cup r_2) = sp(P,r_1) \cup sp(P,r_2) -$ - - ==== Conjunctivity of wp ==== - - $- wp(r,Q_1 \cap Q_2) = wp(r,Q_1) \cap wp(r,Q_2) -$ - - $- wp(r_1 \cup r_2,Q) = wp(r_1,Q) \cap wp(r_2,Q) -$ - - ==== Pointwise wp ===== - - $- wp(r,Q) = \{ s \mid s \in S \land sp(\{s\},r) \subseteq Q \} -$ - - ==== Pointwise sp ===== - - $- ​sp(P,​r) = \bigcup_{s \in P} sp(\{s\},​r) ​ -$ - - ==== Three Forms of Hoare Triple ==== - - The following three conditions are equivalent: - * $\{P\} r \{Q\}$ - - * $P \subseteq wp(r,Q)$ - - * $sp(P,r) \subseteq Q$ - - - ===== Hoare Triples, Preconditions,​ Postconditions on Formulas and Commands ===== - - 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: - $- \{ P \} c \{ Q \} ​ -$ - means - $- \forall s_1, s_2.\ f_T(P)(s_1) \land (s_1,s_2) \in r_c(c) \rightarrow f_T(Q)(s_1) -$ - In words: if we start in a state satisfying $P$ and execute $c$, we obtain a state satisfying $Q$.  ​ - - We then similarly extend the notion of $sp(P,r)$ and $wp(r,Q)$ to work on formulas and commands. ​  We use the same notation and infer from the context whether we are dealing with sets and relations or formulas and commands. - - - - ===== Composing Hoare Triples ===== - - $- \frac{ \{P\} c_1 \{Q\}, \ \ \{Q\} c_2 \{R\} } - { \{P\} c_1 ; c_2 \{ R \} } -$ - - We can prove this from - * definition of Hoare triple ​ - * meaning of ';'​ as $\circ$ - - ===== Further reading ===== - - * {{sav08:​backwright98refinementcalculus.pdf|Refinement Calculus Book by Back, Wright}}