The specification is a lot smaller than the code, and so it's easier to read and manually verify that it's correct.
specification is a lot
smaller than the code
How can the (full) specification of a program be smaller than the program? This would mean we can always compress a program into a smaller program (after all a specification can be seen as another program -- alternatively a program can be seen as its own specification).
A real-world program is usually more complex and intricate than a specification, often a lot more complex. This is usually done in the interest of optimization / performance, so that you program runs at least somewhat efficiently.
A specification is usually a lot simpler, because typically you only need to express the simplest possible way to do or to describe something. Even if the specification ends up looking like a very simple, naive and horribly inefficient program to a typical programmer, it doesn't matter: it's not going to be executed, it's only going to be used to prove that your program behaves in the same way, even though your program can be arbitrarily more complex.
Also, in your specification, sometimes you don't even need to express how to do something - you only need to express what the result looks like.
For example, let's say you want to prove that a sorting algorithm is implemented correctly. You literally only have to specify "after running the function sort() on an array, the resulting array is sorted. A sorted array means that for any X and Y elements of the array, if the index of X is less than the index of Y, then the value of X is less than or equal to the value of Y".
It's easy to see how that specifies correct sorting behavior, even though it doesn't even say how the sorting is supposed to be done. In contrast, the actual implementation of the sorting algorithm can be way more complex or difficult to see that is correct than that, especially if your program is written in a programming language prone to errors, with pointer manipulation and potential undefined behavior (such as C, for instance).
A specification is usually
a lot simpler, because ...
- Most verification does not tackle the full specification, but rather some aspect.
- Verification needs many intermediate specifications (e.g. induction hypotheses in proofs), which need to be accounted for.
I'm happy to be convinced otherwise.
prove that a sorting
algorithm is
In the context of the original poster, the size of the specification that matters with respect to being a potential source of errors is the specification of the final (initial?) theorems that you want to prove, not the actual proof, nor the intermediate lemmas/theorems, induction hypothesis, etc. The latter ones can be all incorrect for all you care, but it doesn't matter because the theorem prover will always tell you that your implementation does not implement the specification unless the proof and the intermediate theorems are also correct.
In the sorting example, what matters is that you correctly specify what is a sorted array and that the result of the sort() function should be a sorted array with the same elements as the input. If you decide to prove that your implementation of Quicksort implements that specification, you can use whatever implementation code you want and whatever intermediate lemmas or proofs you want, but again, if you make any mistake in implementing the algorithm or specifying the intermediate lemmas and proofs or even the final proofs, the theorem prover will complain loudly.
If you simply mean that the total size of all the specification, including proofs and intermediate theorems and proofs can be as large or larger than the actual implementation, I agree with you, that can be the case and that is definitely a problem, because that is one of the reasons why most programmers and the industry in general won't find formal verification to be worth it.
That said, SMT solvers (if you are so inclined) and proof automation in general can really help to reduce the size of the complete specification, including intermediate results and proofs (this is the main reason why I prefer Isabelle/HOL to Coq).
(As an aside, I was pretty sure I knew of an example where a real-world system was verified with an SMT solver and where the complete specification and proofs ended up being around 10% of the code size of the implementation, but for the life of me I cannot find that example... perhaps I am misremembering).
If by full verification you mean verifying all the way down to the machine code, I would argue that this kind of verification probably belongs in the compiler (e.g. CompCert or CakeML) rather than being something that the typical programmer should do.
But perhaps, and most likely, I have not understood you completely :)
The latter ones can
be all incorrect for
all you care
In my experience the TSP is subtle and contains most of the intermediate definitions too. A few years back I had a very large Isabelle/HOL proof invalidated because of some small mistake in an obscure part of the specification that I never though could be important. It required a redesign from scratch of almost all of the proof.
only have to specify "after
running the function sort()
on an array,
all the way down
to the machine code
I'm afraid that's not enough in practise: you will also have to specify (and verify) things like: the implementation does not touch memory except the array to be sorted (and some auxiliary variables). If you don't, it will typically be difficult to use the proof of the sorting function in the context of the verification of a larger program.
See [1] for an example of a formally proved implementation of an array search algorithm (implemented in the Dafny language, in this case. You can press the play button to see if the theorem prover approves of this implementation).
As you can see, in Dafny there's no need to prove that no other memory is touched, as it's implicit in the type of the method. However, even this example is not as good as it could be because:
- Ideally, you wouldn't need to specify that function arguments aren't null. This would have been implicit had the implementation language (Dafny) not had the concept of null (like Rust doesn't, for instance).
- Also ideally, you would refactor the pre- and post-conditions into a separate predicate, like it was done for sorted(), so that you could prove any array search function to be correct using the same short specification (and if a bug is found in the specification, the fix would be applied to all implementations of array search functions automatically).
Regarding having a large Isabelle/HOL proof invalidated, that sounds very interesting and would love to read more about it! I can definitely admit I have never done any large proofs yet, so I am interested in hearing about your experience.
Ideally there should be some way of isolating parts of a large specification such that a bug in one part could not affect the other parts (similar to how functional languages guarantee that a function can never read anything but its input, and can never write to anything but its output), but I have absolutely no idea if this is even logically possible, as I am still just a beginner in this area.
Thanks for the interesting discussion!
Depending on the programming
language being proved
not had the concept of null
would love to read more
about it
For example I recently verified a JIT compiler, and the compiler spec was functional correctness but we really wanted to ensure it was correct AND only used the memory we allocated for it. In this case we had to grow the specification to be larger, its not longer just being correct, but isolated in some way. This of course has a trickle effect where we need intermediate facts, and proofs in order to be able to prove the top level theorem.
As some posters remarked above you should still be able to give a concise top level specification that is much smaller then the implementation. You really just want to say (where m is pre-sort, and m' is post-sort):
(forall addr m m',
addr < addr_of(array) \/ addr_of(array) + length < addr -> read addr m = read addr m')
You can also make proofs obligations like this easier by using regions, a framing rule, or some programming model that gives some amount of memory isolation.
calling context
spec very similar to this
I recently verified a JIT compiler,
With JIT compilers the issue of self-modification becomes pertinent. I know that Magnus Myreen has verified a small JIT compiler a few years back but I don't remember exactly what theorem he proves. IIRC he was using a tailor-made Hoare logic for x86 code that basically moved some of the complexity of the spec into the logic.
But in practise, you need to specify what sortedness means, and in a generic sorting algorithm that means axiomatising the comparison predicate, including its side-effects. Note for example the C standard library implementation of Quicksort does take an arbitrary predicate, which may not be transitive, may have arbitrary side effects etc. The behaviour of sorting becomes pretty complicated in this scenario.
>> How is that the case in this specific example? It looks a lot harder to check for correctness.
Author here.
Here is the specification for the stochastic backpropagation algorithm:
https://github.com/dselsam/certigrad/blob/master/src/certigr...
The preconditions do not need to be inspected carefully because we prove that the models of interest satisfy them (example: https://github.com/dselsam/certigrad/blob/master/src/certigr...)
Here is the part of the specification that needs to be inspected carefully:
https://github.com/dselsam/certigrad/blob/master/src/certigr...
The syntax may seem strange to those unfamiliar with Lean, and a few of the functions involved may not be self-explanatory without reading their definitions, but the statement is conceptually simple: the stochastic backpropagation algorithm correctly computes unbiased estimates of the gradient of the expected loss.
It is subjective, but I personally think that this specification is vastly easier to understand and to check for correctness than the actual implementation of the stochastic backpropagation algorithm.