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The TLA+ Proof System (TLAPS) is a mechanical proof checker for checking a structured proof written in TLA+.

About Me

My name is Lorin Hochstein, and I'm a senior software engineer at Netflix.

Leftpad in PlusCal

Here's the leftpad algorithm in PlusCal syntax.

--algorithm LeftPad
    \* inputs
    c \in alphabet,
    n \in Nat,
    s \in Seq(alphabet),
    output = s,
    \* local vars
    pad = max(n - Len(s), 0),
    i = 0
a:  while i<pad do
        output := <<c>> \o output;
        i := i + 1;
    end while
end algorithm

I simply translated the implementation from the Dafny one into PlusCal. Too late did I discover that there's a PlusCal implementation of leftpad in Practical TLA+.

Proof by inductive invariance

Proof by induction: a refresher

We prove the correctness of leftpad by induction. Recall (from your days in high school or college math) that to prove a predicate P is true by induction:

  1. Prove that P is true in the base case (N=1).
  2. Prove that, if P is true for case N, it's true for case N+1.

To prove that a TLA+ state predicate, Inv, is true by induction, prove that:

  1. Inv is true for the initial state, Init
  2. If Inv is true for an arbitrary state s, and t is any state such that the action formula Next is true for step s→t, then Inv is true for state t.

In TLA+ syntax, those two statements are written as:

1. Init => Inv
2. Inv /\ Next => Inv'

Using induction to prove correctness

To use induction to prove correctness, we ned to strategically choose a state predicate called an inductive invariant (let's called it Inv) that has the following properties:

  1. We can prove Inv by induction.
  2. We can prove Inv implies the correctness property.

Written as a structured proof, proof of correctness by inductive invariant looks like this:

THEOREM Spec=>[]Correct
<1>1. Init => Inv
<1>2. Inv /\ [Next]_vars => Inv'
<1>3. Inv => Correct
<1>4. QED
    BY <1>1, <1>2, <1>3

where Inv is the inductive invariant and Correct is the property you want to prove. In this case, that property is the leftpad specification.

The leftpad inductive invariant

Here's the inductive invariant used in this proof:

Inv == /\ TypeOK
       /\ \E prefix \in Seq({c}) : output = prefix \o s
       /\ Len(output) = Len(s) \/ Len(output) <= n
       /\ Len(output) = Len(s) + i 
       /\ pad = max(n - Len(s), 0)
       /\ i>=0
       /\ Correct

where TypeOK asserts the correct type of the variables, and Correct is the correctness condition for leftpad.

The leftpad proof

If you're familiar with TLA+ syntax, you should be able to read the proof without too much trouble.

TLAPS is a declarative system. You just tell it "this statement can be proved by using these other statements", and then TLAPS tries to determine whether that's true. You often need to help TLAPS out by breaking statements into smaller steps that TLAPS can handle

For example, consider this part of the proof, where we're checking that the initial condition satisfies the inductive invariant:

<1>1. Init => Inv
  <2>2. \E prefix \in Seq({c}) : output = prefix \o s

Given that the initial condition, Init, contains the conjunct output = s, this should obviously be true, given that the definition of Seq includes zero-length sequences.

Unfortunately, TLAPS can't figure this out on its own, so we need to help it by breaking this down into sub-steps that TLAPS can check, like this:

<2>2. \E prefix \in Seq({c}) : output = prefix \o s
  <3>1. output = << >> \o s
  <3>2. << >> \in Seq({c})
    BY DEF Seq
  <3>3. QED BY <3>1, <3>2

The most interesting part of the proof is proving <1>2. Inv /\ [Next]_vars => Inv'. It's broken up into cases. At the top-level, we prove it for each possible value of the pc variable, and for each of those, we prove Inv' for each of the conjuncts of the inductive invariant.

Leftpad is simple enough that pc can only take on two values: {"a", "Done"}, so our case analysis is pretty simple. Thet trickier one is the case where pc="a", that's done in <2>1.

TLAPS needed the most help for proving <3>2. (\E prefix \in Seq({c}) : output = prefix \o s)', so that one is broken up into the most steps. The case where i<pad is where I needed to break it down into smaller steps:

<3>2. (\E prefix \in Seq({c}) : output = prefix \o s)'
  <4>2. CASE i<pad
    <5>1. \E prefix \in Seq({c}) : <<c>> \o output = <<c>> \o prefix \o s
    <5>2. \A p \in Seq({c}) : <<c>> \o p \in Seq({c})
    <5>3. QED

Checking the proof with TLAPS

To verify that the proof is correct:

  1. Install the TLA+ toolbox
  2. Install the TLA+ Proof system
  3. Open LeftPad.tla in the toolbox.
  4. Right click on the statement THEOREM Spec=>[]Correct and click "Prove Step or Module"

Many of the lines should turn green, including the line containing the THEOREM. TLAPS colors a statement green if it was able to verify the proof of that statement. When the top-level THEOREM statement is green, the entire theorem's proof has been checked.

Proof screenshot

Invariants vs inductive invariants

The correctness property is an invariant. However, in general, correctness conditions are not inductive invariants. That means that, while the correctness conditions are true in all possible states of the specification, they can't be proved by induction.

To understand why, let's consider an algorithm that's a little simpler than leftpad.

Here's an algorithm that just decrements a counter until it reaches 0, and then stops, written in PlusCal notation:

--algorithm Decrement
variable i \in {1, 2};

    while i>0 do
        i := i-1;
    end while
end algorithm

It should be obvious that i≥0 is an invariant of this specification. Let's assume that this is our invariant Inv. Now, consider the following program state:


Here, Inv is true, but Inv' is false, because on the next step of the behavior, i will be decremented to -1.

Venn diagram

The above diagram is a Venn diagram of states. A bubble represents a state in TLA+. A bubble is colored gray if it is an initial state in the specification (i.e. if the Init predicate is true of that state). An arrow represents a step, where the Next action is true for the pair of states associated with the arrow.

The inner region shows all of the states that are in all allowed behaviors of the specification.

The outer region represents an invariant: it contains all states where the invariant holds. The middle region represents an inductive invariant: it contains all states where an inductive invariant holds.

Note how there is an arrow from a state where the invariant holds to a state where the invariant doesn't hold. That's because invariants are not inductive in general.

In contrast, for states where the inductive invariant holds, all arrows that start in those states terminate in states where the inductive invariant holds.

Finding an inductive invariant

The hardest part of proof by inductive invariance is finding an invariant that's inductive. If the invariant you come up with isn't inductive, you wont't be able to write the proof.

You can use TLC to help find an inductive invariant. My strategy was to start with Inv = TypeOK /\ Correct and use TLC to check if Inv was, indeed, an inductive invariant. When a counterexample was found, I added another conjunction to tighten up my invariant.

To check if my invariant was inductive, I created the ISpec definition:

ISpec == Inv /\ [][Next]_vars

Then I ran the following TLC model:

Temporal formula: ISpec
What is the model?
    alphabet <- {"a", "b"}
Invariants: Correct
Definition Override:
    Nat <- 0..3
    Seq(S) <- UNION {[1..m -> S] : m \in Nat}

In my case, this model was small enough to run quickly and large enough to find all of the problems with my proposed inductive invariants. I know for certain that my invariant is an inductive one because I was able to write the proof.

In general, checking that an invariant is inductive with TLC is hard because the state space can be enormous. Lamport describes a strategy for using pseudo-random sampling of a larger state space in Using TLC to Check Inductive Invariance.