TLA+ is a language for specifying concurrent and distributed systems and for checking that these systems do what you expect. It has been used in industry:

• See "Formal Methods for Real-World Systems" on Page 3 of this CACM article.

and in academia:

• Raft: https://ramcloud.stanford.edu/~ongaro/thesis.pdf
• EPaxos: http://www.cs.cmu.edu/~imoraru/epaxos/tr.pdf
• TAPIR: https://syslab.cs.washington.edu/papers/tapir-tr-v2.pdf

Note that this tutorial is stolen from Leslie Lamport's excellent video course: https://lamport.azurewebsites.net/video/videos.html. If you want to learn TLA+, I recommend watching this series.

• DieHard
  • We model the execution of a system as a series of states.
  • A specification of a system describes the set of legal executions.
  • A specification does NOT provide instructions on how to transition states, just which state transitions are legal.
  • Play clip: https://youtu.be/6cAbgAaEOVE.
  • EXTENDS is like import.
  • We model the state of the system with two variables, big and small.
  • Note that TLA+ is untyped.
  • Init says the initial state of the system.
  • Next, we can specify formulas that describe legal state transitions.
  • A primed variable like small' represents the value of variable small in the next state.
  • Note that we have to constrain every variable. For example, if we did not constraint big' in FillSmall, then the value of big' could be anything.
  • /\ is conjunction, / is disjuntion.
  • Next is the disjunction of all our formula.
  • We can specify invariants for our executions. For example, small should be in the set {0, 1, 2, 3} and big should be in the range {0, 1, 2, 3, 4, 5}.
  • TLC is a model checker. It checks that all executions of our system satisfy the invariants we claim.
  • Show cfg file.
  • Run TLC with TypeOk only.
  • Flipping invariant checking on our head, we can also find particular executions, like NotFourGallons.
  • Pause for questions.
• AtomicCommit
  • Atomic commit is the problem where a set of nodes are all initially assigned a value 0 or 1. They must all agree to abort or commit. The nodes can only agree to abort if all initial values are 1.
  • Explain Constants and assumptions.
  • Explain functions.
  • Explain \A and \E.
  • Explain EXCEPT.
  • Explain /=.
  • Explain specs.
  • Explain theorem.
  • Note that we haven't really specified a system. We've specified the legal behaviors allowed by a particular problem, namely atomic commit.
• TwoPhaseCommit
  • Now, we'll look at a particular implementation of atomic commit, namely two-phase commit.
  • Explain tuples.
  • Explain records and messages. Messages never dropped.
  • I said two-phase commit implements atomic commit, but what do I mean by that? Can we formalize that?
  • Draw execution of two-phase commit, erase everything that is not state. The theorem says that anything satisfying the two-phase commit spec also satisfies the atomic commitment spec.

See #1 for clarifications!

• Model checking is not theorem proving.
  • Model checkers exhaustively enumerate executions looking for errors, but they're not proving anything about the correctness of your specification.
  • If there are infinite executions, you cannot check all of them.
  • Even if TLC does terminate, it often depends on a choice of constants.
  • Anecdote: Generalized Paxos has a TLA+ specification, but a later paper shows that this definition is incorrect (see 6.3)!
  • There is no substitute for a simple proof.
• TLA+ is for specifications, not for implementations.
  • You can't run TLA+ specs. Even if your specification is correct, you may implement it incorrectly.
  • Need to resort to automated theorem proving for really being confident that code is bug free.
• TLC can be slow, really slow
  • I ran TLC on my BPaxos stuff on 64 cores for hours and I exhausted 16 GB of disk before it could finish. Could take days to simulate.
  • See https://lamport.azurewebsites.net/video/video7-script.pdf slide 200.
• Sometimes hard to check if your system is doing what you expect.
  • Your specification might accidentally be disallowing certain actions that you think should be legal.
  • Your invariants are satisfied, so it's hard to tell when this is happening.
  • Liveness constraints can help discover this.
• Levels of testing:
  • unit tests
  • quickcheck
  • model checking
  • automated theorem proving

• Constant expression vs state expression vs action expression vs temporal formula
• Liveness
• More advanced forms of refinement
• PlusCal
• TLAPS

• Specifying Systems
• Introduction to TLA+ Model Checking in the Command Line
• TLA+ in Practice and Theory
• Practical TLA+
• Standard Library