Tensions in Practice: Compact recurring rules in tension with explicit evaluation order¶
State updates · variables versus time instances
A counter starts at zero and adds one every tick. Its compact diagram loops: the output becomes the next input. For a three-tick calculation, we can instead draw a separate instance of the counter at each time. Every feedback update remains, but each arrow now points to a later instance. The acyclic picture is bought by expanding the time horizon, not by deleting the return dependency.
Describe an ongoing rule compactly
Use one state variable and a recurring update independent of how many ticks will run.
Expose a finite evaluation order
Make each instance and its dependency available for forward evaluation.
Why these aims pull against each other
A cycle among reusable variable names and an acyclic chain of time-indexed instances describe different levels of the same process. The latter needs a chosen horizon and grows as that horizon grows.
Choose an arrangement to see what changes and what remains difficult.
Qualitative paths and conditions, not measured costs, timings or performance guarantees.
What this choice protects
What it costs
When it fits
Compare the arrangements
Recurring rule
Keep the state and its next-step rule in a compact feedback representation.
- What it protects
- The rule does not need another drawn node for each future tick.
- What it costs
- The diagram alone is not a finite topological schedule; execution also needs the initial state, tick semantics and a stopping horizon if termination is required.
- When it fits
- The ongoing dynamics or a reusable simulation rule is the main object.
Illustration note: This editorial recurrence is x at the next tick = current x + 1. The return arrow includes a tick delay; no algebraic-loop solution is claimed.
Time instances
Replace each occurrence of the reusable state by a distinct time-indexed node, retaining every update.
- What it protects
- The three updates have an explicit forward evaluation order and reproduce values 1, 2 and 3.
- What it costs
- Longer horizons require more instances; the finite graph says nothing about ticks beyond its endpoint.
- When it fits
- A bounded computation, dependency audit or finite trace is required and the horizon is manageable.
Illustration note: Time unrolling is an editorial representation of the same stipulated recurrence. It does not make the underlying ongoing process cease to feed its output into its next input.
What this illustration does—and does not—establish
DAG Directed Acyclic Graph: Acyclic Order versus Genuine Feedback (sign/direction) supplies the warning against deleting real feedback. The explicit time-indexed recurrence demonstrates a scoped alternative representation.
- The example has a discrete delay and known initial value; instantaneous cyclic equations need different analysis.
- A finite unrolling does not prove a continuing process terminates or stabilizes.
- These arrows describe update dependencies, not a general license for causal inference.
Source entries
Directed Acyclic Graph
Directed Acyclic Graph: Acyclic Order versus Genuine Feedback (sign/direction) supplies the conflict examined here.
Acyclic Order versus Genuine Feedback (sign/direction)
The DAG buys ordering, termination, and bottom-up evaluation by forbidding cycles, but many real systems contain genuine feedback that is not an error to be removed.