Why Specifications Don't Compose
Why Specifications Don't Compose

    

    
    

    
  

Recently my friend Lars Hupel and I had a discussion on why formal methods don’t compose well. You can read the conversation here. We focused mostly on composing formally-verified code. I want to talk a little more about the difficulties in composing specifications. This is half because it’s difficult for interesting reasons and half because it’s a common question from beginners.

Beginners to formal specification expect specifications should be organized like programs: multiple independent modules that interact through public interfaces, where the modules don’t know about each others’ private implementations. This is more legible and reusable than a single giant program. We can test the individual modules and then only need to worry about how they combine. By contrast, most specifications are written as single global specs, with subcomponents rewritten from scratch per spec. Why can’t we make our specs as composable as programs?

Setup

Specs are Math

Specifications are mathematical expressions, usually a branch of logic. We use math to specify because it is precise, unambiguous, and expressive. The specification represents a system and its possible behaviors. Specifications have properties, statements that are true of all behaviors of the system. In many cases, we start out  knowing the properties we want our system to have, and then write the specification to satisfy them.

I’ll define composing two specifications as joining them in a way that preserves all properties of both specs, without requiring us to “dig in” to either spec. A good intuition here is function composition: if f(x) has type signature a -> b and g(x) has type signature b -> c, then g . f = g(f(x)) has type signature a -> c. We know that g . f preserves type safety without knowing anything about the internals of either function, just the top-level type signatures. Preferably, composing specs should work the same way: we don’t need to change the internals of either spec to combine the two.

Even without “perfect” composition, we can have a “spectrum” of composability. Let’s say a spec consists of three “components”. If we only need to change one component to combine it with another spec, it’s “more” composable than if we needed to change two components. This is more in line with how we think about composing programs. We might need to modify the interface or write a wrapper around the imports, but the internal implementation of each module stays the same.

Linear Temporal Logic

Let’s write our specifications in Linear Temporal Logic. In “normal” logic, the statement x means that x is true. In temporal logic, x means that x is true in the initial state. In future states, x doesn’t need to be true. To describe how the system evolves, we add three temporal operators:


NeXt (◯): ◯x means that x is true in the next state.
Final (◇): ◇x means that x is true in some future state.
Global (□): □x means that x is true in all future states.1






  
    Technical Stuff
  
  “Future states” includes the current state. ◇x is true if x is true in this state or a later state.







In all cases, I’ll write “x is true” as x and “x is false” as !x. All variables are booleans.

Let’s give an example. x && ◯!x means that x is true in the initial state and false in the next state. x && ◯(!x && ◯□x) means that x is initially true, then in the next state x is false, and permanently true in the state after.





  
    Breakdown
  
  
In the initial state, x is true
In the next state:


x is false
in the next state:


in all future states x is true.










We could also have written x && ◯!x && ◯◯□x, which would mean the same thing. ◯ distributes over both && and ||, while □ only distributes over && and ◇ over ||.

Our Example Spec

For the purposes of this discussion, we’ll use a “blinker”:


x starts as true or false
In the next state, x flips from true to false or false to true
This repeats infinitely.


Here’s the spec:

InitX = x || !x
BlinkX = (x => ◯!x) && (!x => ◯x)

SpecX = InitX && □BlinkX






  
    Technical Stuff
  
  
P => Q means that if P is true, then Q is true. So (x => ◯!x) means that if x is true now, then in the next state, x is not true.
SpecX is also a boolean formula. This lets us pick any arbitrary program behavior and ask if SpecX is true for it, ie SpecX is an accurate description of its behavior. It also means we can manipulate it like any other logical statement.








This has exactly two valid behaviors:

x; !x; x; !x; ...
!x; x; !x; x; ...


One of the properties of this spec is that x is always either true or false, never any other value.2

TypeInvariantX = □(x || !x) 
SpecX => TypeInvariantX


SpecX => TypeInvariantX means that if SpecX is true of a system– that is, SpecX is an accurate description of its behavior– then TypeInvariantX is also true of the system. In other words, any system satisfying SpecX also has property TypeInvariantX.

Composing SpecX

We’ll define an analogous specification SpecY, which is identical except it describes a different variable.3 We want to now model a system with two blinkers. The “obvious” way to do that is to compose SpecX and SpecY, rather than start from scratch. How bad could it be?

Let’s start by saying both are true:

Spec = SpecX && SpecY


This will guarantee that all properties of both specs are satisfied. There are exactly four valid behaviors:

x, y   ; !x, !y ; x, y   ...
x, !y  ; !x, y  ; x, !y  ...
!x, y  ; x, !y  ; !x, y  ...
!x, !y ; x, y   ; !x, !y ...


The two blinkers are always synchronous. One can’t run faster than the other, or consistently blink slightly earlier than the other, etc. This is rarely what we want when we compose specs; consider composing a heartbeat protocol with a general worker process. One should run more often than the other.





  
    Technical Stuff
  
  We can formally express this with a reachability property: that it’s always possible to reach a given state from any other state. Since there’s no way to reach x && !y from the starting state x && y, our system doesn’t satisfy the reachability property R(x && !y).

LTL cannot formally express reachability properties. This is a price it pays to make other (arguably more useful) properties expressible.







We want to instead say that one or the other can run. We could try writing this:

Spec = SpecX || SpecY


But this isn’t actually what we want either. This would also make one of the initial predicates optional: we want both of them to be true. We only want to || the Blink predicates. We’ve already lost simple composition, since we need to break the Spec predicates down into their subcomponents. That way we can write

Spec = (InitX && InitY) && □(BlinkX || BlinkY)


This still looks like “partial” composition, as long as our specs are of the form Init && □P. But to actually be composition, it should also preserve the properties of the local specs.

Breaking Properties

One of the properties of SpecX is TypeInvariantX, which is that □(x || !x). Does Spec also satisfy TypeInvariantX?

Nope. The following is a valid behavior of Spec:

x, y; x, !y; x="paprika", y; ...


The step between x to x="paprika" isn’t a valid under BlinkX, so wasn’t a valid step for SpecX. But Spec instead has □(BlinkX || BlinkY), and BlinkX can be false if BlinkY is true. This is called the frame problem. Since BlinkY doesn’t have x in its “frame”, it doesn’t restrict x to sensible values.

What we need to say is that if BlinkY happens, x is unchanged:

SameX = (x => ◯x) && (!x => ◯!x)

Spec = InitX
Spec = (InitX && InitY) 
       && □((BlinkX && SameY)
          ||(BlinkY && SameX))


We’ve now excluded both BlinkX and BlinkY from happening; the two blinkers cannot synchronize. Let’s leave that aside for now to focus on a different problem. In our original spec, we knew that there was at least one point where x was true and at least one point where x was false. These kinds of properties, that the system is guaranteed to eventually do something we want, are called liveness properties.

LivenessX = ◇x && ◇!x


This is satisfied by SpecX. But in Spec, the following is a valid behavior:

x, y; x, !y; x, y; x, !y; ...


The problem is that we could only do “one thing” in SpecX, while in Spec we can do one of two things. If we keep choosing BlinkY ad infinitum, BlinkX never happens and x never changes. We say here that BlinkX is unfair: even if it can happen, it isn’t guaranteed to happen. I cover fairness in much more detail here.

We can work around this by saying “BlinkX happens an infinite number of times”, aka “BlinkX always happens at least one point in the future:”

Spec = (InitX && InitY) 
       && □((BlinkX && SameY)
          ||(BlinkY && SameX))
       && □◇BlinkX && □◇BlinkY






  
    Technical Stuff
  
  To explain in a bit more detail: □◇BlinkX says that ◇BlinkX is always true. So once BlinkX happens, ◇BlinkX is still always true, meaning BlinkX must happen in the future again. So we have BlinkX an infinite number of times.

This isn’t an accurate representation of fairness, because it lacks conditionality. If something is preventing BlinkX from happening, we shouldn’t require it to happen infinitely often. I cover this in more detail in the fairness post.







Do we now have property composability? Does Spec guarantee everything SpecX does? Well, no. Here’s a property of SpecX:

BlinksTF = x => ◯!x


If x is true, it will be false in the next state. Since Spec is asynchronous, we can have two steps in a row where x is true.

Bummer.

Stuttering

We hit a wall. SpecX requires x to change every step, while Spec only requires it to change for some steps. Properties that use ◯ will break. There is no way around this. It is a fundamental problem with allowing composed specifications to run at different rates.

Would composition be easier if we forbid ◯ properties? We can do this by introducing stuttering: steps where nothing changes. We say a spec is stutter-invariant if we can insert a stutter step anywhere in the behavior without breaking the spec. Rewriting SpecX to be stutter-invariant:

SpecX = InitX && □(BlinkX || SameX)


As a simplification, I’ll borrow TLA+’s box notation: [P]_x is equivalent to “P or stutter”, aka P || UNCHANGED x.

SpecX = InitX && □[BlinkX]_x


We now can’t have any ◯ properties. Any property about how the values change can be immediately foiled by stuttering, aka not changing the values. Further, specs are now unfair by default. BlinkX isn’t guaranteed to happen because we could just stutter an infinite number of times. We’d have to add that to the spec, too:

SpecX = InitX && □[BlinkX]_x && □◇BlinkX


This pushes questions of fairness from the composed specification to the individual specification. This gives us some compositionality back. Writing SpecX && SpecY carries each local spec’s fairness properties with it; if the individual spec is fair then so is the composed spec.

This also allows each spec to have a “flow of time”. Here’s some new behaviors of SpecX with stutter-steps:

x;  ; !x;  ; x;  ; !x
x;  ;  ; !x;  ;  ; x

x; !x;  ;  ;  ;  ; x


With the stutter steps, we can say the first two behaviors represent the blinker running at different rates and the third behavior represents a blinker that lags out a bit. That naturally gives us different specifications with different time flows. The slower spec just stutters more.

Once we make both SpecX and SpecY stutter-invariant, here’s how we can write Spec:

Spec = SpecX && SpecY






  
    Derivation Notes
  
  Is this equivalent to our old Spec? Yes, given two things:


By our definition of stutter-invariance, BlinkX <=> !StutterX
By the rules of modal logic, □(A && B) = □A && □B


Proving the two are equivalent is left as an exercise to the reader.







We’ve gotten composability back! This even allows both sync and async blinkers. We just had to lose boundedness properties and significantly restrict our expressiveness. Hooray!

Dependent Specs

As a rule, we can compose any number of stutter-invariant specs together and preserve all their individual properties… as long as they’re independent specs. The blinkers didn’t share any variables. Most of the time we want to compose specs that share some state, like a reader spec and writer spec that share a queue. This makes composition significantly more difficult.

Let’s take our blinker specs and add a flag. We’ll also have the two specs interact with it in different ways. SpecX will flip it on when it’s off, while SpecY will flip it off when it’s on.

On(f) = !f && ◯f
Off(f) = f && ◯!f

InitX(f) = (x || !x) && (f || !f)
InitY(f) = (y || !y) && (f || !f)

NextX(f) = BlinkX && (On(f) || (f && ◯f))
NextY(f) = BlinkY && (Off(f) || (!f && ◯!f))

SpecX(f) = InitX(f) && □[NextX(f)]_fx
SpecY(f) = InitY(f) && □[NextY(f)]_fy


Some important nuances:


We’ve parameterized SpecX on f: f is now a variable we can pass into the behavior. If we pass f into both specs, they’ll share the variable.4 Think of it as pass by reference, not pass by parameter.
We have to make sure that f is fully defined on every step of NextX, otherwise we can set f="paprika". When writing specs we need to define the next state of every variable for every step.
The stutter-step includes both variables. Either NextX(f) is true or neither change.


Now we compose them together and pass in the same variable to each.

Spec == SpecX(z) && SpecY(z)


Let’s say our initial state is !x, !y, !z. What happens when we blink x?


Both [NextX(z)]_zx and [NextY(z)]_zy need to be true. To blink x, NextX(z) is true (we don’t stutter).
Since z started out false and NextX(z) is true, On(z) is true and we have ◯z.
For NextY(z) to be true, we must have ◯!z. But by (2) we have ◯z, so NextY(z) is false.
This means that for [NextY(z)]_zy to be true, we need to stutter: y and z are unchanged.
But (2) changes z. If NextX(z) is true, then z must change and z must not change.
NextX(z) can never happen. QED


By sharing a flag between the two specs, we have permanently disabled one of them. The problem is that each spec needs to totally describe the behavior of all its component variables, meaning the composed spec can only make changes that are compatible with all of them.

This is even a well-behaved dependent spec, since the trouble is localized to the shared variable. You can imagine a SpecY where the value of y is what “blocks” NextY and forces stuttering. Then even though SpecX would change z in a way that’s compatible with NextY, SpecY stutters and forces z to stay unchanged. We’d have a deadlock.

Ultimately we have to give up on automatic composition and go back to manually stitching the specs together.

Next = (NextX(z) && Same(y)) || (NextY(z) && Same(x))
Spec = InitX && InitY && □[Next]_xyz


This also forces the specs always to be async and requires us to manually write out fairness requirements again, but at least it works.

Discussion

Composing specs is hard because specifications need to totally describe how their variables change, and component specs don’t know about each other’s variables. If they do, they likely want to change it in incompatible ways. We can work around this by manually integrating the specs together, but then we’re not composing the specs, since we need to examine the internals of each spec to do so. This doesn’t mean combining specs needs to be intrinsically hard: we can come up with new techniques to help stitch them together. But it means that there’s a predilection in the communities to write large global specs, not combine smaller ones.

Thanks to Andrew Helwer, Jay Parlar, and Liz Denhup for feedback.




    



  