# [tlaplus] Re: Consensus spec safety in Paxos example

Hi Pavel,

>> For what I see, both Inv and THEOREM can be removed from this file, and the spec will remain rigorous
>> what value does the Inv and THEOREM  bring to this spec, given that it only states a necessary but not sufficient invariant for consensus?

On Inv being removed:

TypeOK states that chosen is a finite subset of Value, so chosen can be a finite set of any size.
Inv states that TypeOK is true and that the cardinality of chosen is zero or one.
So TypeOK is weaker than Inv. Should we remove it too?
There's nothing wrong with stating that the spec satisfies “weak” properties.

On THEOREM being removed and rigor:

Some more advanced users might explain it better than I do, but here's my take on proofs and rigor:

A THEOREM statement in a TLA+ specification serves as documentation to state something you believe to be true.
TLC ignores THEOREM statements because theorems can't be proved by model checking or simulation.
TLAPS, on the other hand, does not ignore THEOREM statements. You can see in the Consensus.tla file that there's a proof. This proof must've been checked by TLAPS.

So, in that regard, theorem proofs are the golden standard for rigor, even if these theorems are obvious to us.
Specifications with more proofs are more rigorous than specifications with fewer proofs.

>> can we state a more rigorous property here, instead of Inv, such as: [...]

Sure! What you seem to be trying to express is a liveness property instead of an invariant.
Just be careful not to assert about values in the next state, such as chosen'.

The Consensus module already contains a liveness property: Success.
Success states that chosen eventually isn't empty.
You might say that it doesn't assert about chosen's cardinality, and conclude it's a “weak” property.
So let's define a stricter liveness property: “a single value is eventually chosen and stays chosen”.

StaysChosen == <>[](\E v \in Value: chosen = { v })

To check a liveness property, you must specify liveness properties for your specification.
If your spec doesn't have liveness at all, you can't guarantee something happens: something might happen in some behaviors, nothing at all might happen in others.
In this case, that means the formulas Spec => Success and Spec => StaysChosen are false because they don't have liveness.

We usually specify the liveness of a TLA+ spec by declaring fairness for sub-actions of Next or Next itself.
In this case, Next doesn't contain sub-actions, so we'd declare weak or strong fairness for Next itself.

You can see that Consensus already defines a LiveSpec formula conjoining Spec with weak fairness for Next.
By specifying liveness for Spec, both formulas LiveSpec => Success and LiveSpec => StaysChosen will be true.

Jones

On Friday 10 May 2024 at 09:49:27 UTC-3 Pavel Kalinnikov wrote:
correction to above invariant:
chosen \subseteq chosen' /\ Cardinality(chosen') \leq 1

On Friday, May 10, 2024 at 1:45:26 PM UTC+1 Pavel Kalinnikov wrote:
Hi Jones,

Yes, I agree that Init/Next on its own correctly specifies consensus, with the temporal aspect of it. Hence was my original question: what value does the Inv and THEOREM bring to this spec, given that it only states a necessary but not sufficient invariant for consensus? For what I see, both Inv and THEOREM can be removed from this file, and the spec will remain rigorous.

And the second part of my question was: can we state a more rigorous property here, instead of Inv, such as:
"chosen \in chosen' /\ Cardinality(chosen') \leq 1"
and prove a THEOREM that more convincingly confirms that Init/Next spec is the right thing?

Thank you,
Pavel
On Thursday, May 9, 2024 at 4:07:07 AM UTC+1 Jones Martins wrote:
Hi Pavel,

If I understood you correctly, you're saying the formula Inv doesn't represent a consensus safety property: this invariant is too weak, and it's satisfied by a sequence of assignments for chosen such as {} -> {1} -> {} -> {2} -> {2} -> etc.

That's true. In fact, there's an infinite number of possible behaviors in the universe such that Inv is True, but these behaviors are irrelevant. It depends on what specification described these behaviors in the first place, and that chosen has the same meaning in both specifications. Meaning is essential.

> To demonstrate that, we can change the Next action in such a way that the invariant will still hold. For example, if we can allow a transition from {v} back to {}, the invariant is still true, but the safety is broken.

Indeed, but that wouldn't be a consensus specification anymore.

> The safety invariant here has to have a temporal aspect: once the chosen set is non-empty, it stays so.

What is so elegant about TLA+ is that the “temporal aspect” you mentioned is the specification itself. In other words, Consensus' Spec expresses that “once the chosen set is non-empty, it stays so”:  action Next in Consensus selects one value, then stutters forever.

Best,
Jones

P.S. I'd expand on the true danger of false positives through bad refinement mappings, but it's late here, and I need to sleep haha

On Wednesday 8 May 2024 at 19:51:00 UTC-3 Pavel Kalinnikov wrote:
Hi Jones,

Your explanation matches my understanding, however I think the Inv does not correctly represent a consensus safety property. To demonstrate that, we can change the Next action in such a way that the invariant will still hold. For example, if we can allow a transition from {v} back to {}, the invariant is still true, but the safety is broken.

The safety invariant here has to have a temporal aspect: once the chosen set is non-empty, it stays so.

I.e. the comment "Safety: At most one value is chosen" seems misleading here. It does express some invariant, but this invariant does not on its own guarantee safety.

Thank you,
Pavel

On Wednesday, May 8, 2024 at 10:58:17 PM UTC+1 Jones Martins wrote:
Hi Pavel,

The Voting module instantiates the Consensus module to prove that Voting's Spec implies Consensus' Spec.

The consensus specification itself is pretty abstract.
Init expresses that chosen = {} is true in the first state of every behavior.
Next expresses that if chosen = {} is true, then chosen' = { <some value> }.
This means that, when some value has been chosen, it never changes again, so the original  Inv invariant is strong enough.

By the way, Inv can't be augmented as you suggested. Invariants are state predicates, and state predicates don't contain primed variables.

Best,
Jones
On Wednesday 8 May 2024 at 16:41:29 UTC-3 Pavel Kalinnikov wrote:
Should this line

/\ Cardinality(chosen) \leq 1

be augmented with something like:

/\ Cardinality(chosen') \leq 1
/\ chosen = {} \/ chosen' = chosen

On Wednesday, May 8, 2024 at 8:24:27 PM UTC+1 Pavel Kalinnikov wrote:
Hello,

I am looking at the Consensus spec in tlaplus examples repo, which specifies the single-value consensus problem.

The Init/Next spec in this file correctly specifies consensus, because it precludes the chosen set from changing after it got to contain exactly one value.

What value does the the safety invariant in lines 36-37, and the theorem that follows it, bring to this spec? If I am interpreting it correctly, the invariant does not fully specify the safety property: it allows a sequence in which a value is chosen, then another value replaces it (the size of the chosen set is still 1, so the invariant remains true). Or a value is chosen, then the set is emptied, then another value is chosen; etc.

Is the value of this invariant+theorem in just demonstrating an interesting property? Is it not strictly needed to be in this spec?

Is there a way to fix this invariant, so that it fully represents the consensus safety property (rather than partially, like it seems).

Thank you,
Pavel

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