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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 =====
-<code java>
-//{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}
-</code>
-===== 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}}