Question: Given a system where an input can be any int, how can you describe it with TLA+ in a way that TLC can check?

[Ari Sweedler]

I first asked this question on Reddit, https://www.reddit.com/r/tlaplus/comments/kc75n0/given_a_system_where_an_input_can_be_any_int_how/, but figured this was a better place to ask. I'll copy/paste my post & question here to save y'all from having to click that link:

Given a system where an input can be any int, how can you describe it with TLA+ in a way that TLC can check?

Btw, extreme noob here. Started Hillel Wayne's book. Saw this https://www.learntla.com/introduction/example/

    If we actually wanted to test all possible cases, we could replace money \in 1..20 with money \in Nat, where Nat is the set of natural numbers. This is perfectly valid TLA+. Unfortunately, it’s also something the model checker can’t handle. TLC can only check a subset of TLA+, and infinite sets aren’t part of that.

And this TLA+ started as a notation, and the model checker, TLC, only came out 5 years later. As it was never intended to be run, there’s some assumptions of it being a read document instead of being runnable code.

So I kinda get it, the mathematical proof of "All natural numbers" isn't computable. However, I am first a computer scientist and second,,, well not even second. And thirdly a (wannabe) mathematician. So the fact that Nat is not a finite number. To put it in the language of mathematics, this is the finite set of the values of all valid ints: GF(1)32. (On a 32-bit system, that is).

So should I write something like \in 0..4294967296 or \in -2147483648..2147483647 everywhere in my code? That's not reasonable. Unless TLA+ does something fancy with efficiency (something inductive, to prove that the remaining values follow from prior cases), then I am wrong to want to test all integers.

But then... what's the point of a spec that isn't exhaustive? Can someone help explain?

Thanks,

Ari

[Stephan Merz]

TLC exhaustively checks finite (and typically small) instances. The underlying assumption is that for actual systems, this is an effective way of finding bugs. If you need higher assurance, you can do theorem proving, which can be applied to arbitrary (including infinite-size) instances, at the cost of more human effort.

Stephan

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[Leslie Lamport]

Hi Ari,

That's a good question, and it deserves a good answer.  Let's see what

I can do.

Why do we have an infinite set of natural numbers?  Before there were

computers, no one was ever going to write a number with more that a

few hundred digits.  With a computer, no one is ever going to write a

number with more than a few billion digits.  Why don't we just declare

that there is some largest natural number so we don't have to deal

with an infinite set?

The answer is: for simplicity.  The laws of arithmetic would be much

more complicated with a finite number of integers.  For example, 

3*n - 2*n = n  would be false for must integers n.

Similarly, why did you want to write a spec of a system in which a

variable x can equal any natural number?  A system that implements the

spec will break in an execution described by a behavior of the spec in

which the value of x gets too large.  The answer is the same: for

simplicity.

A specification is an abstract model of a system.  As someone almost

said:

  No model is accurate; some models are useful.

You wrote a spec in which x can get arbitrarily large because either

(i) you believe that in practice the value of x will never get large

enough to worry about, or (ii) what to do if x gets too large is a

detail that is irrelevant to the purpose for which you wrote the

spec.  In case (ii), the implementation will have to deal with the

value of x getting too big, but you didn't want to worry about that 

in your spec.  You wrote the spec to model some other aspect of the

system.

As Stephan wrote, math has no trouble dealing with an infinite set of

integers, and using them will make the proof simpler.  But TLC can't

deal with infinite sets.  With most model checkers, you would have to

rewrite the spec to check it, making it allow only a finite set of

possible states.  The goal of TLC is to be able to check any spec

without modifying it.  You do that by letting TLC check a finite model

of the spec.  TLC models provide two features for doing that.  First,

you can tell it to replace the set Nat by a finite set of numbers.

Second, you can tell the model checker to ignore states in which the

value of x is greater than some value.  

Checking a model of the spec doesn't prove that the spec is correct.

But correctness of the spec doesn't prove that the system is correct.

Remember, that a spec is an abstract model.  It's not supposed to be

accurate; it's supposed to be useful.  Its use is to eliminate some

class of errors--in particular, what one would call design errors

rather than coding errors.  Checking a model of the spec is useful if

it can catch errors in the spec.

In practice, checking a large enough model is a good way of finding

errors.  In principle, a violation of a safety property can be always

be found by checking a large enough model.  That's not true of

liveness properties.  In practice, a finite model most often causes

TLC to find a liveness error that doesn't exist in the spec.  Ensuring

that something eventually happens in all executions can require

executions in which the value of x becomes arbitrarily large.  It's

also possible that a finite model hides the fact that a liveness

property isn't satisfied, but that seems to happen less often.

What constitutes a "large enough" model?  The larger the model, the

more chance that it will catch all errors in the spec and the more

confidence it provides.  In practice, it's a matter of engineering

judgement whether the models that TLC is able to check give you

sufficient confidence in the correctness of the spec.  This seems to

work quite well for catching safety errors.  I don't have enough data

to know how well it works in industry for catching liveness errors.

My own personal experience has been that safety errors tend to be

more subtle and easier to make than liveness errors.

Leslie

[recepient recepient]

While I know far less than Mr. Lamport or Mr. Merz, I'd offer this input:

- formal methods are about naming and bounding risk through specification.

Therefore in practical software development where delivering code is the bottom line

formal methods is another tool (and powerful) in the toolbox. It exists in a continuum 

which includes unit test. So if you have a type that's supposed to be a variable length

integer (yet bounded in size for the reasons Lamport argues) testing that it

honors expected operations ala Hoare logic is better done as unit testing. Both

modes of testing are indicated. 

- Second, correctness often not always does not depend on the cardinality of domain

of the type. Key value hash correctness for example on integers esp in a concurrent

setting is more about other factors than the actual values in hand.

- Third plenty of violations can be found in bounded ranges precisely because of the

 previous point because they demonstrate a flaw in ordering, concurrency. locking,

bucket overflow etc. all issues not per se about the integer in hand.

- Fourth: Although above my pay grade I hazard a guess that model checking can

be bested in some cases through verification based on proofs which includes

induction. I think TLA has a proof mode but I know nothing about it. Induction

is about countable infinity ad we know. Still it's not immediately clear that having

induction helps in general. In the KV example assuming no other issues, the final

update of the hash amounts to assignment whose correctness doesn't really 

really depend on cardinality.

- Last we should not confuse domain (range) size with choice. If there is a bug

is it best explicated by working over the domain (range) or would it be demonstrated

earlier by allowing TLA to choose arbitrary values on a bounded range? If it's the

second case the issue it is much more likely the bug is about ordering of operations using 

integers not the integers itself. Here choose, either TLA commands are your 

friends. 

Computers in the end amounts to constructive proofs (which is what code

Is when bug free) on types whose storage is finite.

[Ari Sweedler]

Fantastically interesting responses; thank you both. For me to understand either
required me to fundamentally change how I think, which is a jarring and
uncomfortable feeling. Although that exact feeling was what attracted me to
Hillel's book & explanation of property-based tests in the first place.

***

To respond to Mr. Nets, (sorry, I don't know what else to call you :) ), points
2 and 3 that you, made give rise to a visualization for me that I'd like to
share - for posterity or just to explain my understanding so you don't feel like
you're shouting into a void. The visualization pertains to the inputspace of a
function with 1 integer argument. It can be an array, or a 2D board, or a 3D
rectangular prism, it doesn't matter.

Imagine coloring in every square red if TLC would find an error given that
square as a starting state.

I see that many classes of errors for many representations could potentially
color in a contiguous region, but when unraveling the inputspace to an array -
as one effectively does with TLC - we find that the array is either entirely
colored in, or periodically splotched. Like a bug leading to computational
chicken pox.

But - and this point you made I find quite compelling - we will rarely see a bug
concentrated in a small range! Regardless if that small range is far from 0 or
not. Thus, to check any large enough subset is to have a good chance of finding
any error, especially when <insert your 3rd point>.

***

To respond to you, Mr. Lamport,

> Why do we have an infinite set of natural numbers? ... For simplicity

I never thought about it like that. Infinite numbers to a mathematician are for
simplicity. Taking that as the reason instead of "because that's the way it is"
really helps me clearly see: this is a tool to catch a class of mistakes, it's
not a catch-all as true proofs are. And I heavily conflated the two in my head,
as I wasn't really sure what TLA+ is... I have since learned about TLAPS, and
the fact that such a system was invented answers a big question in my mind "What
did other people do when *they* wanted a computer to do this?" Because I *know*
that question's been asked before - it's just that TLA+ wasn't the answer!

> the implementation will have to deal with the value of x getting too big, but
> you didn't want to worry about that in your spec. You wrote the spec to model
> some other aspect of the system.

I see how this is both useful and powerful, now. Previously, I thought the
ability to replace Nat with a finite set was quite strange, but I understand it
now. The concept of telling the model checker to ignore states according to
conditions also seems quite powerful.

> Remember, that a spec is an abstract model. Its use is to eliminate some class

> of errors--in particular, what one would call design errors rather than coding
> errors.

If I had the knowledge I had now, this one sentence would have answered my
question. However, I didn't! And so I needed all the rest of the support from
the remaining 90% of the answers y'all provided. But I would just like to
especially highlight this in my response.

***

My model of formal specifications remains inaccurate, yet it has very much so
gotten more useful :).

Thank you both for your fantastic responses,
Ari