Let $I$ denote the set of blocks with user-provided initial values in a block diagram. Assume when $I$ is propogated through the block diagram, every block is has a value at time $t=0$. Let $d(t, b)$ denote the set of blocks with user-provided initial values which determine the value of block $b$ at time $t$. For example, in the diagram below colored blocks have initial values and the value of $d$ when applied to to each block for time $t=0$ is displayed. image Blocks have a single output, and take as input the output of $n$ blocks denoted $b_{i,1}\dots b_{i,n}$. If $b\in I$, then the value of $b(0)$ is user-provided, so $d(0, b)$ is ${b}$. Otherwise, $b(0)$ has a value if all $b$'s input blocks have one, so if for some input $b_i$, $b'\in d(0,b_i)$, then $b'\in b(0,b)$. During execution of a block diagram, initial values are propogated as above. Then the value of each $b\in I$ at $t=1$ is computed with the values of $b$'s inputs. Then, those new outputs are propogated through the diagram setting each $b\not\in I$'s value at $t=1$. To be pedantic, if $b\in I$ has no inputs, then it never receives a value at $t=1$ (nor will any blocks reachable from $b$ as there is nothing to propogate) and we say $b$ has no value at $t=1$.

When is a block diagram executable? For example, consider the block diagram below which has an initial value on the blue block. I'd say this is not executable. image On the left, the blue block has an initial value. No further execution is possible, as shown on the right. After adding an initial value for the green block, the diagram loops forever. I'd say this is executable.

image A block $b$ is composed of $N$ ports $b_{p1}\dots b_{pN}$, one terminal $b_t$, and a $N$-ary function mapping values on the ports to a value on the terminal. The value of this function is denoted $b(b_{p1}\dots b_{pN})$. image A wire connects a terminal to a port. For a wire $w$, let $w_p$ and $w_t$ denote its port and terminal. Terminals and ports are nodes. Nodes are composed of a set of values and a value. Let $U_n$ denote the set associated with the node $n$. At time $t$, let $n(t)$ denote the value of $n$. The value of a node is either an element of its space or nothing. That is, $\forall t(n(t)\in U_n\lor n(t)\equiv\varnothing)$.

image Consider the block diagram above which descibes the evolution of a point $(x, y) \in \mathbb{N}\times\mathbb{N}$. Suppose $\Delta$ is drawn from a random distribution with values spanning $[0, 2\pi)$. What happens to the value of the radius (r) over time? First, I'll show the radius never increases (Lemma 1). Then, I'll show for all possible values of the radius, there exists $\Delta$ which results in the radius decreasing (Lemma 4). Thus, given sufficent time the radius goes to zero. Lemma 1: The radius never increases. $$ \begin{aligned}