Part of the challenge (and fun) of low-level systems code is in the optimizations they employ: developers might use manual memory management, they might use bit-packing and bit-twiddling optimizations, or they might use multi-threading to speed up their code. When dealing with such things for critical software, though, it can be difficult to ensure their correctness. This is why my research group is interested in the formal verification of systems software: ensuring through computer-checked mathematical proofs that software does what it is supposed to, and ideally not compromising on these optimizations.

For this purpose, we have been developing Verus , a verification tool for the Rust programming language . Rust is increasingly popular as a systems programming language today, but we didn’t (just) choose it because of its popularity. Rather, it turns out that the properties that make it attractive as a systems programming language—most notably, that it allows manual memory management while simultaneously guaranteeing memory-safety— also make it excellent in the setting of formal verification: in some ways straightforward, and in some ways surprising. In this blog post, I’ll explain what these ways are.

Verification, mutable memory, and Rust

First, we’re interested in proving code to be “correct”. What does that mean exactly? Let’s get our feet wet in verification with some simple examples and then talk about a challenge that Rust helps us solve.

Intro to Verus

The key idea behind Verus is to check additional properties of programs that Rust doesn’t check on its own. For example, consider the following valid Rust program, operating over an 8-bit integer.

fn double(i: u8) -> u8 {
    return i * 2;
}

Though it’s a valid program, it (potentially) has a problem: if the argument i is more than 127, then the multiplication will overflow the 8-bit integer. If you run Verus on it (which you can try yourself at the Verus playground ), Verus reports this error:

error: possible arithmetic underflow/overflow
 --> /playground/src/main.rs:6:16
  |
6 |         return i * 2;
  |                ^^^^^

To remedy this, the programmer can declare their intent : namely, that the double function should never be called with any argument greater than 127.

fn double(i: u8) -> u8
    requires i <= 127
{
    return i * 2;
}

The requires clause is not a standard Rust feature, but a feature of Verus: in general, Verus source code comprises both Rust code and extra directives for Verus, such as this requires clause, also known as a precondition . In any case, Verus now accepts the program ( playground link ) because with the new assumption, Verus can determine that this arithmetic operation never overflows.

Furthermore, any time the developer calls double from elsewhere in the program, Verus will check that the call satisfies the precondition. Keep in mind, also, that this is a check done statically, checked for all possible executions of the program, not a runtime check.

Specifications and program correctness

With Verus, we are actually interested in correctness criteria that go beyond simple arithmetic bounds checks. Usually, we are interested in proving that a program’s behavior meets some specification , as in this function that computes the maximum of two integers:

fn max(a: u64, b: u64) -> (result: u64)
    ensures
        result == a || result == b,
        result >= a,
        result >= b,
{
    if a > b {
        return a;
    } else {
        return b;
    }
}

Again, let’s highlight the Verus-specific parts: first, we have the ensures clause (also known as a postcondition ) serving as the function’s specification, along with the name result on the return type, which is referenced from said postcondition. Once again, the body of the max function is the Rust code we are verifying.

The ensures clause denotes a predicate that should hold true at the end of the call to max . This determines what it means for an implementation of max to be “correct”: it is correct if every execution of its code returns a result that meets its specification.

So, how does Verus actually check that this property holds? To do this, Verus (and similar tools) encode the correctness of max as logical formulae called verification conditions :

\[ a > b \implies result = a \implies (result = a \lor result = b) \land (result \ge a) \land (result \ge b) \]

\[ \lnot(a > b) \implies result = b \implies (result = a \lor result = b) \land (result \ge a) \land (result \ge b) \]

These conditions are simplified a bit for presentation, but they are close enough for intuition. The first of these would be read as, “if \( a < b \) (i.e., the first branch is taken), and if \( result \) is set to the return value \( a \), then the conditions of the ensures clause hold”. The second condition is similar, but for the else side of the branch.

If we prove the verification conditions are correct, this implies the correctness of the program according to its specification. To do so, Verus uses an automated theorem prover—in this case, Z3 —to prove the verifification conditions hold for all values of a , b , and result . This example is simple enough that Z3 can validate the conditions quickly, though for more complicated examples, the developer may need to write additional proofs to help it out. If Z3 is unable to prove the theorem, either because it is wrong, or because it needs additional help to prove, then Verus outputs an error message like the one from the previous section.

Specification-checking is extremely useful for situations where an implementation is optimized and handles low-level details, but we would like to provide a higher-level, mathematically precise specification. For example:

A program uses the bitwise operation (x & (x - 1)) == 0 to determine if x is a power-of-2, but uses a more mathematically precise specification, \( \exists b.~ 2^b = x \).
A data-structure implements a hash table or a red-black tree, but has a specification stating that its operations are equivalent to those of a mathematical set.
A replicated data structure with a sophisticated synchronization algorithm uses a specification that it acts indistinguishably from a single copy of the data structure.

Challenge: handling mutable memory

One such “low-level detail” we often have to reason about is mutable heap state . To see why this is challenging without Rust’s help, let us set aside Rust for a moment, and imagine we designed a programming language with general pointer types, like in C. Consider a simple function that takes two pointers and updates one of them:

// Imagined C-like verification language
void compute_boolean_not(bool* x, bool* x_not)
    ensures (*x_not) == !(*x)
{
    bool tmp = *x;
    *x_not = !tmp;
}

This program looks straightforward at first, but it actually has a problem: what if x and x_not point to the same memory? Then *x would be updated when we update *x_not . Therefore, a tool would never be able to prove this code matches its specification—it simply isn’t true.

Visual representation of the above example

Left: what the developer imagines happening. Right: what might actually happen.

One solution is to specify that the pointers do not alias with each other, i.e., that they don’t point to the same memory location:

// Imagined C-like verification language
void compute_boolean_not(bool* x, bool* x_not)
    requires x != x_not        // This line has been added
    ensures (*x_not) == !(*x)
{
    bool tmp = *x;
    *x_not = !tmp;
}

Recall the requires clause here indicates an assumption the function can make at the beginning of its execution. By making this assumption, Verus can now check that the specification holds, although now every call to compute_boolean_not will need to uphold this contract.

Unfortunately, adding these “non-aliasing conditions” gets unwieldy fast, as data structures increase both in breadth and depth. This was our experience when we wrote the first version of VeriBetrKV , a key-value store developed in Dafny , which has a similar aliasing situation to our C-like language. Not only were the conditions difficult to write manually, but getting them wrong often led to error messages that were difficult to diagnose.

Rust to the rescue

In Rust, it isn’t common to use general-purpose pointer types. Instead, Rust uses more restricted reference types . In Rust, the types &T and &mut T each denote a reference to a value of type T . In the case of &mut T , which is specifically a mutable reference, the user is able to modify the value behind the pointer. Thus, in Rust/Verus, our boolean-negation example would look like this, with the x_not parameter marked as a mutable reference.

fn compute_boolean_not(x: &bool, x_not: &mut bool)
    ensures (*x_not) == !(*x)
{
    let tmp: bool = *x;
    *x_not = !tmp;
}

At the machine code level, these references are just like pointers, but the Rust type system enforces additional properties: namely, a &mut reference to a piece of data can never coexist with another reference to that data. Rust enforces this property because it is crucial to Rust’s guarantees about memory safety.

However, this property is also a huge boon for software verification. Because the non-aliasing property is checked by Rust’s type system, the developer no longer has to write the non-aliasing conditions manually. Furthermore, Rust’s type system is fast and often presents high-quality error messages when the property is violated.

One can think of this as if these non-aliasing conditions are inserted automatically, so the developer doesn’t have to worry about it, but in fact, the situation is even better: the verification tool can simplify the verification conditions to not include any notion of pointer addresses in the first place! In fact, some of my colleagues have published a paper quantifying the gains from this kind of simplification.

Are reference types all we need?

The fact that Rust works as a language at all is evidence that reference types are sufficient most of the time. Unfortunately, most of the time isn’t good enough. The non-aliasing reference problem gets in the way for implementing any of the following:

Doubly-linked lists
Reference-counted pointers (e.g., Rust’s Rc , similar to C++’s shared_ptr )
Any manner of concurrent algorithm: locks, message-passing queues, memory allocators, systems with domain-specific logic for avoiding data races

The reason these examples give difficulty is because Rust’s type system enforces that any object have a unique “owner” (unless those owners are immutable references). However, these examples seemingly need to violate the restriction:

Visual representation of a doubly-linked list. Each node has two incoming pointers from its neighbors, and two outgoing pointers to its neighbors.

In a doubly-linked list, each node has two neighbors which point to it. Thus, these nodes do not have unique owners.

Visual representation of reference-counted smart pointer, Rc. The shared object has multiple reference objects pointing to it.

When working with reference-counted smart pointers, each object may have multiple reference objects. These objects need to coordinate via the reference count to drop the given object at the appropriate time. This counter does not have a unique owner.

Visual representation of message-passing queue. The producer thread and the consumer thread each have a pointer to a shared queue buffer.

In a message passing queue, the producer thread and the consumer thread have to share a queue buffer to store in-flight messages. This buffer does not have a unique owner.

So how can we tackle these kinds of problems?

For such things, Rust programmers need to use Rust’s notorious “unsafe code” which opts in to various Rust features that the type system is unable to validate are used safely. As such, the burden goes from the type-checker to the programer to ensure they are used correctly. Applications like the above are generally considered low-level, and they are often relegated to time-tested libraries. It’s these kinds of low-level systems, though, that we are especially interested in verifying! So what do we do?

Unsafe code in Verus, or: “condititionally safe code”

With Verus, we can recover the ability to implement such things while having a computer-checked guarantee of memory safety. A Rust feature being “unsafe” really just means that the developer has to uphold a certain contract to use it safely, which Rust cannot check. It is for this reason that I like to call unsafe code conditionally safe —i.e., it is safe subject to meeting certain conditions. Rust cannot check these conditions, but Verus can .

Here is a simple example: Rust’s common vector indexing operation performs a bounds-check to ensure there is no memory corruption from an out-of-bounds access. Therefore, this function is unconditionally “safe” to call, no matter what index the caller provides: even if the caller provides something out-of-bounds, the program might panic and exit, but it will never corrupt memory. However, there is a lesser-used get_unchecked operation which performs no such bounds check. Thus, get_unchecked is only safe to call if the index is already known to be in-bounds , thus making it unsafe (conditionally safe). This condition can be codified as a Verus requires clause:

unsafe fn get_unchecked<T>(vec: &Vec<T>, idx: usize) -> &T
    requires idx < vec.len()
    ...

Now, Verus will check that the index is in-bounds whenever get_unchecked is called. Thus, we can regain assurance in code that uses this function, provided that Verus is able to validate the code.

Handling unsafe ownership

Bounds-checking makes for an easy example, but when we consider programs like the ones diagrammed above, the situation gets a little more complicated. Recall that what characterizes these systems is that the objects may be pointed to from multiple owners, which have to coordinate their access somehow.

As a result, the “conditions” of the conditionally safe operations become substantially more involved. For example, accessing data through a pointer is only safe if there is no data race , i.e., another thread trying to access it at the same time. Such a condition seems inherently “non-local” as it involved talking about all threads at once, and therefore is much harder to check than that of a simple index being in bounds.

However, we have already discussed that Rust’s type system allows us to ensure the unique ownership of data, which then rules out illegal operations such as data races. Therefore, the kind of “condition” we need to check is already the exact kind of condition that Rust’s type system is designed to ensure. The problem here is just that these particular data structures do not use the specific types that are designed to ensure this. So how can we apply Rust’s philosophy anyway?

Since the data structures we want to verify use objects that don’t obey Rust’s unique ownership, our trick is to add new objects that do . However, we don’t want to bog down the program with extra data—that would defeat the point of writing optimized code—so these new objects are merely “conceptual proof objects.” In verification languages, such objects are often called ghost objects, not because they are spooky, but because they have no influence on the “physical world.” The real data structures in the compiled binary would be the ones diagramed above, but Verus treats the program as if the ghost objects were really there when generating its verification conditions.

For example, for a program that uses pointers, Verus programs can use a ghost object that represents “the right to read or write memory from the given location.” Just like for ordinary (“real”) data, Rust’s type system ensures that ownership this object is unique. Verus in turn ensures that such an object is present when the program accesses the data behind the pointer. Combining both results, we can be confident that such an access really is data-race-free. Even while multiple owners might point to the same piece of data, in the sense of physically having a pointer to it, only one owner at a time can have the right to manipulate that data.

To verify a doubly-linked list, then, we would arrange nodes with pointers in the usual way, but in addition to the “real” nodes, we would have an additional collection of ghost objects that represent the right to access those nodes. By writing additional Verus annotations, we can explain, mathematically, how these ghost objects relate to the structure of the linked list, and as a result we can use the ghost objects to traverse the list. For more details, you can see our paper , where we present the doubly-linked list in detail.

Conclusion

Rust’s type system, and similar type systems that enforce unique ownership over data, are enormously helpful in designing a verification language for low-level code. Just as Rust guarantees memory safety, thus taking the burden off the developer in the common case, Verus takes advantage of the same to remove the burden of complex aliasing conditions for verification developers. More surprisingly, though, we can apply Rust’s type system even for code that initially seems very un-Rust-like, which is common in highly-optimized systems code. Specifically, by utilizing ghost objects we recover the ability to use Rust’s ownership system (together with Verus to check conditionally safe code) to check code where the type system would not help in ordinary Rust.

Contents

Verus: A tool for verified systems code in Rust

Verification, mutable memory, and Rust

Intro to Verus

Specifications and program correctness

Challenge: handling mutable memory

Rust to the rescue

Are reference types all we need?

Unsafe code in Verus, or: “condititionally safe code”

Handling unsafe ownership

Further reading

Conclusion

Contents

Verification, mutable memory, and Rust

Intro to Verus

Specifications and program correctness

Challenge: handling mutable memory

Rust to the rescue

Are reference types all we need?

Unsafe code in Verus, or: “condititionally safe code”

Handling unsafe ownership

Further reading

Related work

Conclusion