Exercise Bank — Knowledge, Belief, and Common Ground
The printed Chapter 6 keeps Exercises 1–6; the bank continues from Exercise 7.
Exercise 7. An orchestrator (agent 1) has fixed the release gate and must coordinate it with a coder (agent 2) over a channel that silently drops messages. Let \varphi be “the gate closes at 14:00”, and let G = \{1, 2\}. It is common knowledge that agent 1 knows the gate time it set, that every message sent is truthful, and that both agents follow the protocol. The trace: m_1, from 1 to 2, “the gate closes at 14:00” — delivered; m_2, from 2 to 1, “acknowledged” — delivered; m_3, from 1 to 2, “your acknowledgement arrived” — delivered; m_4, from 2 to 1, “noted” — lost. (a) After each of the four messages, state the largest k for which E_G^{k}\varphi holds, justifying each step by naming the nested-knowledge formula — K_1 K_2 \varphi, K_2 K_1 K_2 \varphi, and so on — that the delivery makes true, and the one that still fails. (b) At the end of the trace, exhibit the specific conjunct of C_G\varphi that fails, and say whose doubt, about which message, breaks it. (c) Using the fixed-point characterisation of Section 6.2.1, argue that no continuation of this protocol, however long, attains C_G\varphi — the argument is asked for, not the citation. (d) The channel is repaired: every message now arrives, but after a delay with no fixed bound. Show that the conclusion of (c) survives, by identifying what the sender of the most recent message can never rule out at any finite time.
Exercise 8. At step 4 a coder begins a feature against schema v1. At step 9 the orchestrator migrates the staging database to v2 and records the change in the issue tracker. The coder has not read the tracker since step 3, and at step 12 it posts “ready to merge”. (a) Say whether the diff will assume v1 or v2, and what the reviewer should predict this from — the fact, or the coder’s belief. Then map the scenario element-for-element onto the false-belief task of Section 6.3: who plays Sally, and what play the marble, the basket, the box, the leaving of the room, and the question “where will Sally look?”. (b) Writing v for “schema v1 is live”, the orchestrator as agent 1 and the coder as agent 2, express with the chapter’s operators the epistemic state the orchestrator must represent in order to see that a notification is needed, and the state that grounding evidence must establish once the notification has been sent and acknowledged. State the order — first, second, or third — of each formula you write. (c) The scenario is to become an evaluation probe for machine theory of mind. In the spirit of Ullman’s perturbations (2023), design two minimal variants that leave the story’s surface almost untouched while changing the correct answer — one of them a true-belief control — and state the pattern of answers across all three versions that would distinguish robust belief-tracking from template matching.
Exercise 9. A team of four — orchestrator O, coders C_1 and C_2, and tester T — coordinates through a shared scratchpad: free-text entries, a single plan field overwritten in place, last writer wins, and a convention that each agent reads the pad once, when it picks up its task. The trace: at t_1, O sets the plan to “target branch main; schema v1”. At t_2, C_1 reads the pad and starts its diff. At t_3, T posts “smoke tests green” — the run was against main at v1, but the entry names no branch, no schema, and no time. At t_4, O rewrites the plan: “target branch release-2.4; schema v2”. At t_5, C_2 reads the pad and starts its diff. At t_6, C_1, finishing, writes the plan field back from its t_2 copy with its own part marked done — silently discarding O’s rewrite. At t_7, O returns, sees a plan naming main and v1 with C_1’s part done and smoke tests green above it, and approves the release. (a) Section 6.6 names three hazards that follow a shared store about, and Section 6.5 a fourth failure the store inherits. Identify all four in this trace, each with the specific events that constitute it. (b) For each hazard, prescribe the minimal piece of bookkeeping that removes it, and name the earliest trace event at which your fix would have changed what happened. (c) A colleague proposes a simpler cure: every agent re-reads the entire pad immediately before every write and every action. Which of the four hazards does this fail to remove, and what new cost does it introduce?
Exercise 10 (project). The chapter’s remedies keep converging on bookkeeping that no context window supplies: beliefs that carry their sources, justifications that let a retraction find its dependents, and a ledger of who was told what, so that a revision can propagate. Build the module. A BeliefStore holds belief records — claim, source, supporting belief identifiers, status, and the agents the belief has been told to — together with an entrenchment map from sources to weights. It must support: asserting a belief with its provenance; recording that a belief was messaged to an agent; retracting a belief, with the truth-maintenance cascade of Section 6.4 — every belief whose support is lost goes with it — returning the notifications now owed as pairs of belief and agent; and revising, where a newcomer that contradicts an incumbent is admitted only if its source outweighs the incumbent’s, and is otherwise turned away. Demonstrate the module on the scenario of Exercise 4: check that the cascade and the notifications match your hand answer to 6(b), and that re-asserting the configuration-file claim after the revision is refused.
Exercise 11 (lab). Take the probe suite you designed in Exercise 8(c) — the original scenario and your two perturbed variants — and administer it to a current language model, each version in at least three paraphrases that vary names and wording but not the epistemic facts, each probe in a fresh context. Score every response for whether it predicts behaviour from the protagonist agent’s belief or from the fact. Analyse the outcome as the chapter’s dispute in miniature: state in advance which pattern across versions and paraphrases would support Kosinski’s reading (2024) and which Ullman’s (2023), then report which you observed. Close by stating what your probe cannot settle, and why the design lesson of Section 6.3 does not wait on how the dispute comes out. Record the model identifier and the date beside your results.