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sav08:hoare_logic [2009/03/04 11:03] vkuncak |
sav08:hoare_logic [2015/04/21 17:30] |
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- | ====== Hoare Logic ====== | ||
- | Hoare logic is a way of inserting annotations into code to make proofs about program behavior simpler. | ||
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- | ===== Example Proof ===== | ||
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- | <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> | ||
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- | ===== Hoare Triple for Sets and Relations ===== | ||
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- | 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. | ||
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- | 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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- | ===== Strongest Postcondition - sp ===== | ||
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- | Definition: for $P \subseteq S$, $r \subseteq S\times S$, | ||
- | \[ | ||
- | sp(P,r) = \{ s' \mid \exists s. s \in P \land (s,s') \in r \} | ||
- | \] | ||
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- | This is simply [[Sets and relations#Relation Image]] of a set. | ||
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- | {{sav08:sp.png?400x250|}} | ||
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- | ==== Lemma: Characterization of sp ==== | ||
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- | $sp(P,r)$ is the the smallest set $Q$ such that $\{P\}r\{Q\}$, that is: | ||
- | - $\{P\} r \{ sp(P,r) \}$ | ||
- | - $\forall Q \subseteq S.\ \{P\} r \{Q\} \rightarrow sp(P,r) \subseteq Q$ | ||
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- | ===== Weakest Precondition - wp ===== | ||
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- | 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. | ||
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- | {{sav08:wp.png?400x250|}} | ||
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- | ==== Lemma: Characterization of wp ==== | ||
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- | $wp(r,Q)$ is the largest set $P$ such that $\{P\}r\{Q\}$, that is: | ||
- | - $\{wp(r,Q)\} r \{Q \}$ | ||
- | - $\forall P \subseteq S.\ \{P\} r \{Q\} \rightarrow P \subseteq wp(r,Q)$ | ||
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- | ===== Some More Laws on Preconditions and Postconditions ===== | ||
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- | We next list several more lemmas on properties of wp, sp, and Hoare triples. | ||
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- | ==== Postcondition of inverse versus wp ==== | ||
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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: | ||
- | \[ | ||
- | S \setminus wp(r,Q) = sp(S \setminus Q,r^{-1}) | ||
- | \] | ||
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- | ==== Disjunctivity of sp ==== | ||
- | |||
- | \[ | ||
- | sp(P_1 \cup P_2,r) = sp(P_1,r) \cup sp(P_2,r) | ||
- | \] | ||
- | \[ | ||
- | sp(P,r_1 \cup r_2) = sp(P,r_1) \cup sp(P,r_2) | ||
- | \] | ||
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- | ==== Conjunctivity of wp ==== | ||
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- | \[ | ||
- | wp(r,Q_1 \cap Q_2) = wp(r,Q_1) \cap wp(r,Q_2) | ||
- | \] | ||
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- | \[ | ||
- | wp(r_1 \cup r_2,Q) = wp(r_1,Q) \cap wp(r_2,Q) | ||
- | \] | ||
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- | ==== Pointwise wp ===== | ||
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- | \[ | ||
- | wp(r,Q) = \{ s \mid s \in S \land sp(\{s\},r) \subseteq Q \} | ||
- | \] | ||
- | |||
- | ==== Pointwise sp ===== | ||
- | |||
- | \[ | ||
- | sp(P,r) = \bigcup_{s \in P} sp(\{s\},r) | ||
- | \] | ||
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- | ==== Three Forms of Hoare Triple ==== | ||
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- | The following three conditions are equivalent: | ||
- | * $\{P\} r \{Q\}$ | ||
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- | * $P \subseteq wp(r,Q)$ | ||
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- | * $sp(P,r) \subseteq Q$ | ||
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- | ===== Hoare Triples, Preconditions, Postconditions on Formulas and Commands ===== | ||
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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: | ||
- | \[ | ||
- | \{ P \} c \{ Q \} | ||
- | \] | ||
- | 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 $Q$. | ||
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- | 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. | ||
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- | ===== Composing Hoare Triples ===== | ||
- | |||
- | \[ | ||
- | \frac{ \{P\} c_1 \{Q\}, \ \ \{Q\} c_2 \{R\} } | ||
- | { \{P\} c_1 ; c_2 \{ R \} } | ||
- | \] | ||
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- | We can prove this from | ||
- | * definition of Hoare triple | ||
- | * meaning of ';' as $\circ$ | ||
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- | ===== Further reading ===== | ||
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- | * {{sav08:backwright98refinementcalculus.pdf|Refinement Calculus Book by Back, Wright}} |