Definition:Flow Chart/Interpretation

Definition
Let $C = \struct {F, P, V, E}$ be a flow chart.

Let $X$ be a set.

For each $g \in F$, let $f_g : X \to X$ be a partial mapping on $X$.

For each $q \in P$, let $p_q \subseteq X$ be a subset of $X$.

Then, the interpretation of $C$ defines a partial mapping $f_C : V_I \times X \to V_O \times X$.

For any $i \in V_I$ and $v \in X$, $\map {f_C} {i, v}$ is defined to be the unique $\tuple {o, v'}$, if it exists, such that there is a finite sequence $\sequence {\tuple {b_n, v_n}}_{n \le N}$ satisfying the following:
 * $b_j \in V$ and $v_j \in X$, for every $j \le N$.
 * $b_0 = i$ and $b_N = o$.
 * $v_0 = v$ and $v_N = v'$.
 * For every $j < N$, there is an arc $b_j b_{j + 1}$ in $C$.
 * For every $j < N$, $v_{j + 1} = v_j$, unless $b_j \in V_F$.
 * If $b_j \in V_F$, then $v_{j + 1} = \map {f_{b_j}} {v_j}$.
 * For every $j < N$, if $b_j \in V_P$, then:
 * If $v_j \in p_{b_j}$, then $b_j b_{j + 1}$ is labeled $\top$.
 * If $v_j \notin p_{b_j}$, then $b_j b_{j + 1}$ is labeled $\bot$.

Such a sequence is called a control path in $C$.

$\map {f_C} {i, v}$ can fail to exist if either:
 * The control path generated from the input does not end after a finite number of steps.
 * At some point, $\map {f_{b_j}} {v_j}$ fails to exist.

Also see

 * Flow Chart Interpretation is Well-Defined