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sav08:review_of_fixpoints_in_semantics [2008/04/30 10:43] vkuncak |
sav08:review_of_fixpoints_in_semantics [2008/05/07 22:58] giuliano |
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| ====== Review of Fixpoints in Semantics ====== | ====== Review of Fixpoints in Semantics ====== | ||
| - | |||
| - | **Definition:** Given a set $A$ and a function $f : A \to A$ we say that $x \in A$ is a fixed point (fixpoint) of $f$ if $f(x)=x$. | ||
| - | |||
| - | **Definition:** Let $(A,\le)$ be a partial order, let $f : A \to A$ be a monotonic function on $(A,\le)$, and let the set of its fixpoints be $S = \{ x \mid f(x)=x \}$. If the least element of $S$ exists, it is called the **least fixpoint**, if the greatest element of $S$ exists, it is called the **greatest fixpoint**. | ||
| - | |||
| - | Note: | ||
| - | * a function can have various number of fixpoints. Take $f : {\cal Z} \to {\cal Z}$, | ||
| - | * $f(x)=x$ the set of fixedpoints is | ||
| - | * $f(x)=x^2$ the set of fixedpoints is | ||
| - | * $f(x)=3x-6$ has exactly one fixpoint | ||
| - | * a function can have at most one least and at most one greatest fixpoint | ||
| - | |||
| - | We can use fixpoints to give meaning to recursive and iterative definitions | ||
| - | * key to describing arbitrary long executions in programs | ||
| ===== Transitive Closure as Least Fixedpoint ===== | ===== Transitive Closure as Least Fixedpoint ===== | ||
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| Program points are CFG nodes. Statements are labels on CFG edges. | Program points are CFG nodes. Statements are labels on CFG edges. | ||
| - | We look at a particular way of representing and computing sets of reachable, splitting states by program counter (control-flow graph node): **collecting semantics**. | + | We look at a particular way of representing and computing sets of reachable states, splitting states by program counter (control-flow graph node): **collecting semantics**. |
| $PS$ - the set of values of program variables (not including program counter). | $PS$ - the set of values of program variables (not including program counter). | ||
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| Let $p_0$ be initial program counter and $I$ the set of values of program variables in $p_0$. | Let $p_0$ be initial program counter and $I$ the set of values of program variables in $p_0$. | ||
| - | The set of reachable states is defined as the least solution of: | + | The set of reachable states is defined as the least solution of constraints: |
| + | \[ | ||
| + | I \subseteq C(p_0) | ||
| + | \] | ||
| + | \[ | ||
| + | \bigwedge_{(p_1,p_2) \in E} sp(C(p),r(c(p_1,p_2)))) \subseteq C(p_2) | ||
| + | \] | ||
| + | where $c(p_1,p_2)$ is command associated with edge $(p_1,p_2)$, and $r(c(p_1,p_2))$ is the relation giving semantics for this command. | ||