Tensions in Practice: Fewer messages in tension with deeper receipt knowledge¶
Two people exchanging a truthful one-bit report
A knows a fact p and sends it to B over a channel that may lose messages. In the displayed trace, B receives the report. Both now know p, but A cannot tell that delivery succeeded. In the second trace B sends a receipt and A receives it: A now knows B knows p. B still cannot tell whether that receipt arrived. The observer sees the entire trace; each participant sees only their own sent and received messages.
Keep the exchange short
Deliver the report without requesting a receipt.
Confirm receipt to the sender
Let A establish that B received the fact.
Why these aims pull against each other
A delivered receipt changes A’s knowledge at the cost of another message, while leaving uncertainty about that receipt’s own delivery at B. Finite confirmations do not create the entire infinite tower.
Choose an arrangement to see what changes and what remains difficult.
p is the same fact in every row. “A knows B knows p” means A can establish that B received it. No marks knowledge not established by the local message history.
What this choice protects
What it costs
When it fits
Compare the arrangements
Send the report
A’s report arrives at B. No receipt is requested.
| Established? | |
|---|---|
| A knows p | Yes |
| B knows p | Yes |
| A knows B knows p | No |
| B knows A knows B knows p | No |
- What it protects
- The actual trace conveys p using one message.
- What it costs
- A cannot distinguish successful delivery from a lost report.
- When it fits
- Fits a task that needs B to learn the fact but does not need A to establish B’s receipt.
Illustration note: The displayed delivery succeeds; the policy does not guarantee delivery on the lossy channel.
Return one receipt
After receiving the truthful report, B sends a receipt. In this trace A receives it.
| Established? | |
|---|---|
| A knows p | Yes |
| B knows p | Yes |
| A knows B knows p | Yes |
| B knows A knows B knows p | No |
- What it protects
- A can now distinguish delivery of the report from its loss.
- What it costs
- A second message is required; B cannot distinguish a delivered receipt from a lost receipt.
- When it fits
- Fits a task requiring this finite confirmation to A, without demanding exact common knowledge.
Illustration note: Each receipt is trusted and refers to this report. No clock, guaranteed delivery or other side channel reveals whether a message arrived.
What this illustration does—and does not—establish
The source supplies the structural tension. This bounded example makes a particular relation inspectable; the aims, conditions and residual costs are part of the comparison.
- p is one stipulated true fact. Agents reason correctly from the declared protocol and their own observations.
- No means not established from that participant’s local history, not that the embedded fact is false.
- The comparison stops at finite knowledge depth. It makes no coordinated-action or consensus guarantee.
Source entries
Common Knowledge
The canonical tension motivates this comparison. The setting and arrangements are declared editorial illustrations, not observed findings.
Exact common knowledge versus achievable approximation (boundary)
The failure mode is reasoning as if exact common knowledge had been achieved — designing a protocol or a commitment that assumes the top rung when only a few rungs were ever secured.
The source operation
Common knowledge is the *infinite-tower* limit of shared knowledge: ordinary "we both know" sits one or two rungs up the tower, and the full tower is a strictly stronger condition.