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*From*: Hillel Wayne <hwayne@xxxxxxxxx>*Date*: Fri, 28 Oct 2022 10:55:52 -0500*References*: <57ba9669-3f80-b725-a7d4-6e4daf3b6d33@kit.edu>*User-agent*: Mozilla/5.0 (Windows NT 10.0; Win64; x64; rv:102.0) Gecko/20100101 Thunderbird/102.4.0

Section 3.4 is about history variables, which track prior states
of the spec. In this case RandomH isn't actually using a history
variable.

I intuitively think of `Spec1 => \EE x: Spec2` as
saying that Spec1 refines an *aspect* of Spec2. We might use
it, for example, if Spec1 is a detailed server model and spec2 is
the high level interactions of a server and a client. then we'd
use this refinement to say that Spec1 refines the behavior of the
*server* in Spec2, if we hide all the behavior of the client.
In your case, you're showing that `Random!Spec` accurately
represents the part of `RandomH` related to a random
choice, if you hide all the behavior of the incrementing counter.

H

On 10/28/2022 8:44 AM, Grundmann,
Matthias (KASTEL) wrote:

Hello,--

I'm trying to understand what exactly it means if two specifications are "equivalent". As an example, I've created two specifications and, as far as I understand the method presented in [1], we can show that these specifications are equivalent although my intuition says that they are not. The first specification (CounterUp) models a simple counter that always increments a variable by 1 and the second specification (Random) sets a variable in each step to a random value.

The two specifications are defined as follows:

----------------------------- MODULE CounterUp -----------------------------

EXTENDS Integers

VARIABLE counter

CONSTANT max

Init == counter = 0

Next == counter' = IF counter < max THEN counter + 1 ELSE

Spec == Init /\ [][Next]_counter

=============================================================================

------------------------------- MODULE Random -------------------------------

EXTENDS Integers

VARIABLE choice

CONSTANT max

Init == choice \in 0..max

Next == choice' \in 0..max

Spec == Init /\ [][Next]_choice

=============================================================================

It is intuitive that CounterUp!Spec => Random!Spec as incrementing a value by 1 is a special case of choosing the next value arbitrarily.

To show this implication, we add "Random == INSTANCE Random WITH choice <- counter" to the module CounterUp. Now, we can show "THEOREM CounterUp!Spec => Random!Spec" by running TLC for a model for CounterUp with the temporal formula "Spec" checking the property "Random!Spec".

To show Random!Spec => CounterUp!Spec, we introduce a new specification RandomH defined as follows (along the lines of [1, Section 3.1]):

------------------------------ MODULE RandomH ------------------------------

EXTENDS Random

VARIABLE h

varsH == <<choice, h>>

InitH == Init /\ h = 0

NextH == Next /\ h' = IF h < max THEN h + 1 ELSE 0

SpecH == InitH /\ [][NextH]_varsH

CounterUp == INSTANCE CounterUp WITH counter <- h

THEOREM SpecH => CounterUp!Spec

=============================================================================

According to Theorem 1 of [1], Random!Spec is equivalent to \EE h : RandomH!SpecH (1).

Using the definition "CounterUp == INSTANCE CounterUp WITH counter <- h" (second last line of module RandomH), we can show that RandomH!SpecH => CounterUp!Spec (2).

According to [1, Section 3.4], it follows from (1) and (2) that Random!Spec => CounterUp!Spec (or maybe Random!Spec => \EE h : CounterUp!Spec ???).

Having shown the two implications, we have shown that Random!Spec is equivalent to CounterUp!Spec.

I conclude that a system that counts is equivalent to a system that chooses values arbitrarily. This result contradicts my intuition which says that counting is not equivalent to choosing values arbitrarily. -- It might be that my intuition is wrong. It might be that my conclusion is wrong as I might have misunderstood what "equivalence" means in this context. It might be that I have included a methodological flaw above (see the "???"). What are your thoughts? What does it mean for one specification to be equivalent to another specification?

Thanks!

Matthias

[1] Lamport, Leslie, and Stephan Merz. “Auxiliary Variables in TLA+”. https://lamport.azurewebsites.net/pubs/auxiliary.pdf

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**Follow-Ups**:**Re: [tlaplus] Meaning of "equivalence" of specifications***From:*Grundmann, Matthias (KASTEL)

**References**:**[tlaplus] Meaning of "equivalence" of specifications***From:*Grundmann, Matthias (KASTEL)

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