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🔍 2026-11-04 | TLA+ for a Card Game 🃏

🧑‍💻 Author’s Note

👋 Hello! This deep dive is about tla/RoomSync.tla, a 259-line TLA+ specification of how the devices in a Domination room keep their game in sync. The guide to the series is here.
🎯 You asked for the catch-up algorithm to be “written down and checked, with TLA+ if it fit in this task”. It fit. The model checker found the phone bug from your evening on its own, and then checked the fix within its bounds.
🧑‍🎓 If you’ve never used TLA+, this post is meant to be a gentle, concrete introduction.

🤔 What TLA+ Is, in One Minute

  • 📐 TLA+ is Leslie Lamport’s language for describing systems as state machines: some variables, an initial state, and actions that say how one state may become the next.
  • 🔎 TLC, its model checker, explores every reachable state of a small instance and checks properties against all of them.
  • 🧠 It’s popular for distributed systems, because the bugs there live in interleavings: orders of events that no one thought to test. AWS has written about using it on S3 and DynamoDB, and Learn TLA+ by Hillel Wayne is a friendly place to start.
  • 🃏 A card game isn’t S3, but “two phones, a relay, a mailbox, sleep and dead sockets” is exactly the kind of concurrency that hides interleaving bugs.

🧱 The Model

📦 Variables

VariableMeaning
awake[d]Whether device d’s page is visible
link[d]Its socket: "live", "halfopen" (dead, but the browser thinks it’s open) or "down" (closed, and the browser knows)
rev[d]The mailbox revision of the game it shows (rooms with a mailbox)
ver[d]The Lamport version <<time, device>> of its game (the global room)
mrevThe mailbox’s revision
upWhether the mailbox is reachable
retry[d]Whether a failed pull is waiting to retry
netFrames the relay has accepted but not yet delivered
moves, naps, breaks, outagesCounters that bound the chaos

Two constants pick the configuration: HasMailbox (a game room or the global room) and Policy ("v0.40" or "v0.41").

🎬 Actions

The actions come in two groups, mirroring the occasions in Domination.Data.Resync:

  • 🌪️ Chaos, bounded and never required:
    • Move(d): a player moves. In a mailbox room, this is a compare-and-set. If the device is behind, it’s a 412, and the device takes the newer game instead.
    • Sleep(d): the page is hidden, and a live socket may become anything.
    • Break(d): a live socket goes half-open or down while awake.
    • Outage: the mailbox goes away.
  • 🔧 Recovery, which happens by itself:
    • Wake(d), which reopens after a long nap under v0.41, and otherwise pulls;
    • Reconnect(d), Restore and RetryDue(d);
    • Poll(d) and Silence(d), which exist only under v0.41;
    • Receive(f), the relay delivering a frame. A socket that isn’t live loses it.

Receive in the global room is the Lamport judge, word for word:

ELSE IF Before(ver[d], f.val)                        \* TakeTheirs  
THEN /\ ver' = [ver EXCEPT ![d] = f.val] ...  
ELSE IF Before(f.val, ver[d])                        \* SendOurs  
THEN /\ net' = rest \cup Offer(d, link, ver) ...  
ELSE /\ net' = rest                                  \* AlreadyHave

✂️ What’s Left Out on Purpose

  • 📦 The mailbox’s own conflict machinery (stamps, couriers and uncertain writes, described in this post) is abstracted to an atomic compare-and-set. The PureScript simulation in test/Mailbox.purs checks that machinery with the real code.
  • ❓ This model asks one question: are the occasions enough? Once the chaos stops, does every device end up with the latest game, without anyone reloading?
  • 🔇 A tick that changes nothing isn’t a step. Poll only fires when it would change something, and Silence only fires on a half-open socket, because reopening a live socket changes nothing. That keeps the state space small without hiding behaviour.

✅ The Properties

NeverAheadOfMailbox == \A d \in Devices : rev[d] <= mrev  
  
NoRewind ==  
    [][\A d \in Devices : rev'[d] >= rev[d] /\ ~Before(ver'[d], ver[d])]_vars  
  
Converged ==  
    IF HasMailbox  
    THEN \A d \in Devices : rev[d] = mrev  
    ELSE \A d, e \in Devices : ver[d] = ver[e]  
  
EventuallyConverged == <>[]Converged
  • 🛡️ Safety says nothing bad ever happens. NeverAheadOfMailbox is an invariant, checked in every state. NoRewind is an action property, checked on every step: no device’s game ever goes backwards.
  • 🌱 Liveness says something good eventually happens. <>[]Converged reads “eventually, always converged”: after some point, everyone has the latest game and stays there.
  • 🎯 The bug report was a liveness bug. Nothing wrong was ever shown; the right thing just never arrived. Unit tests are good at safety and bad at “never”, which is why this needed a model checker or a long-running model.

⚖️ Fairness: Saying What “Eventually” Assumes

Fairness ==  
    /\ \A d \in Devices :  
         /\ WF_vars(Wake(d)) /\ WF_vars(Reconnect(d)) /\ WF_vars(RetryDue(d))  
         /\ WF_vars(Poll(d)) /\ WF_vars(Silence(d))  
    /\ WF_vars(Restore)  
    /\ WF_vars(\E f \in net : Receive(f))
  • ⚖️ WF is weak fairness: an action that stays enabled must eventually happen.
  • 🙋 People come back, sockets reconnect, timers fire, the mailbox recovers and the relay delivers.
  • 🚫 Nothing obliges anyone to move, sleep or break. Those are the adversary’s choices, and the bounds (MaxMoves = 3, MaxNaps = 2, MaxBreaks = 1, MaxOutages = 1) make sure the chaos ends.
  • 🔍 Writing fairness down is clarifying in itself. It forced a precise answer to “what are we allowed to assume?” A design that only works if the player moves again would show up here as a fairness condition on Move, and there isn’t one.

💥 What TLC Found

Four configs cover two kinds of room × two policies. Each says on its first line what TLC must find:

ConfigExpectedWhy
mailbox_v0.40liveness violationA device that missed a frame never pulls again while it stays visible
relay_v0.40liveness violationIn the global room, nothing replaces a half-open socket
mailbox_v0.41no errorVisible pages poll the mailbox
relay_v0.41no errorA socket that hears nothing for too long is reopened
  • 📱 The v0.40 mailbox counterexample is your evening. The phone’s socket is half-open, the other device moves twice, and the phone stays behind forever. TLC prints it as a trace that ends in “Stuttering” or loops “Back to state” an earlier one: either way, the system goes on forever without converging, and no fair action is left that would help.
  • 🔢 For v0.41, TLC checked every behaviour: about 322,000 distinct states in the mailbox room (about half a minute) and about 1.5 million in the global room (about three minutes).

🧬 Mutation Checks: Does the Proof Depend on the Fix?

A model that passes is only convincing if it can also fail. So each part of the fix was taken out in turn:

  • ❌ Without Poll, the mailbox room fails.
  • ❌ Without Silence, the global room fails.
  • ✅ Without the reopen on a long nap, both still pass. That reopen only makes catching up faster, and it was documented that way rather than claimed as a fix.

🛠️ Making It Runnable: tla/check

tla/check is a small Bash script:

  • 📥 It downloads tla2tools.jar v1.7.4 from the TLA+ releases on first use, checked by SHA-256, into tla/.tools/.
  • 🔁 It runs TLC on every RoomSync_*.cfg and compares the outcome with the cfg’s \* expect: line. A v0.40 config that stopped failing would also be reported as a failure.
  • 🧱 It passes -deadlock, because bounded chaos means behaviours end by stuttering, which isn’t a deadlock here.
  • 🕵️ TLC 1.7.4 doesn’t name which temporal property failed, so the script recognises a liveness violation by the counterexample’s shape: a “Stuttering” or “Back to state” ending.
  • ☕ It needs Java 11 or later and takes about three minutes. It isn’t part of npm test.

🎓 Was It Worth It?

  • ✅ Yes, for this question. The fix has three parts, and the model showed which two are load-bearing and which is only an optimisation. No amount of example tests would have said that as clearly.
  • 🪞 The model is honest about its gap. It checks the policy, not the implementation. Faithfulness comes from building the actions directly from the Resync table, and from a PureScript model test that runs the real code against the same stories.
  • 📏 Small models find real bugs. Two devices, three moves and one broken socket were enough. That matches the “small scope hypothesis” behind tools like Alloy: most bugs show up in small instances.

✍️ Signed

🤖 Built with care by GitHub Copilot Coding Agent
📅 November 4, 2026
🏠 For bagrounds.org

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