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CSE 452: Distributed Systems

Lecture 9: Knowledge — Whiteboard Descriptions

These are text descriptions of the whiteboard PDF from the lecture on April 17, 2026. See also the whiteboard PDF and the notes.

These materials were drafted by AI based on the live whiteboard PDF and audio transcript from the corresponding lecture and then reviewed and edited by course staff. They may contain errors. Please let us know if you spot any.

Muddy Foreheads

  • n children
  • k have mud on their forehead
  • teacher: "Somebody has mud"

Diagram: Five stick-figure children arranged in a circle. Two of them have brown mud drawn on their foreheads; the other three have clean foreheads. The children can see each other's foreheads but not their own.

Kinds of Knowledge

  • K_i φ (i is a node) — means "node i knows φ"
    • K_i φ → φ (knowledge implies truth)
  • S_G φ (G is a set of nodes) — someone in G knows φ

Kinds of Knowledge (cont.)

  • D_G φ — "distributed knowledge of φ": if you combined everything all of G knew, then you'd know φ
  • E_G φ — everyone in G knows φ
  • E_G^k φ — ("everyone knows")^k of φ (i.e., E_G applied k times)

Common Knowledge

  • C_G φ — "φ is common knowledge"
  • Defined as the infinite conjunction: C_G φ ≡ ⋀_{k=1}^∞ E_G^k φ

Evolution of a Protocol

A hierarchy showing how knowledge strengthens as a protocol runs, indicated by wavy arrows:

  • K_i φ ⤳ E_G φ (individual knowledge progresses toward everyone knowing)
  • D_G φ ⤳ S_G φ (distributed knowledge progresses toward someone-in-group knowing)

The paper for today proves that Gaining common knowledge is impossible. If C_G φ is true at some point during an execution, then it was also true at the beginning of the execution.

Impossibility of Coordinated Attack

Diagram: Two hilltops with a valley between them. Three red stick-figure soldiers stand on the left hilltop, three more on the right hilltop. In the valley between them is a small house (the enemy camp) with several purple stick figures (enemy soldiers) around it. The two armies on the hills must coordinate an attack on the camp below, but can only communicate via unreliable messengers that cross the valley.

Diagram: Two vertical process lines labeled A and B (time flowing downward). An arrow goes from A down-right to B (a message delivered). Then an arrow from B down-left to A (acknowledgement delivered). Dashed squiggles in the middle suggest further exchanges. At the bottom, a final attempted message from A to B is crossed out in red (shown as a red "X" near B), indicating a message that fails to arrive — no matter how many rounds of acknowledgement occur, the last message is always uncertain.

Consensus

  • Get nodes to agree on something
  • Majority voting
  • Impossible