Quantifying over infinite sets

One of the main advantages of Vehicle compared to a testing framework is that it can be used to state and prove specifications that describe the network’s behaviour over an infinite set of values.

Suppose you have the following network which produces two outputs:

f : Tensor Rat [10, 10] -> Vector Rat 2

and would like to specify that for any input the network’s first output is always positive. This can be achieved by using the forall quantifier as follows:

forall x . f x ! 0 > 0

which brings a new variable x of type Tensor Rat [10, 10] into scope. The variable x has no assigned value and therefore represents an arbitrary input.

Similarly, if trying to specify that there exists at least one input for which the network’s first output is positive, the exists quantifier can be used as follows:

exists x . f x ! 0 > 0

As with lambda functions, the quantified variables can be annotated with their types:

exists (x : Tensor Rat [10, 10]) . f x ! 0 > 0

and multiple variables can be quantified over at once:

exists x i . f x ! i > 0

In many cases you don’t want the property to hold over all the values in the set, but only a (still infinite) subset of them. For example, network inputs are frequently normalised to lie within the range [0, 1]. If the quantified variable’s domain is not also restricted to this range, then Vehicle will produce spurious counter-examples to the specification.

In general such restrictions can be achieved by combining a quantifier with an implication as follows:

forall x. (forall i j . 0 <= x ! i ! j <= 1) => f x ! 0 > 0

Quantifying over finite sets

While most specifications will quantify over at least one variable with an infinite domain, sometimes one might also want to quantify over a finite set of values. There are multiple ways of doing this:

Quantifying over an Index type

The first approach is to quantify over a variable with the Index type. For example:

pointwiseLess : Vector Rat 3 -> Vector Rat 3 -> Bool
pointwiseLess x y = forall (i : Index 3) . x ! i < y ! i

will get automatically expanded to:

pointwiseLess : Vector Rat 3 -> Vector Rat 3 -> Bool
pointwiseLess x y = x ! 0 < y ! 0 and x ! 1 < y ! 1 and x ! 2 < y ! 2

The type annotation Index 3 on the quantified variable i is included for clarity but are not need in practice as it can be inferred by the compiler.

The in keyword

Alternatively quantifiers can be modified with the in keyword to quantify over all the values contained within a List, Vector or Tensor:

myList : List Rat
myList = [0.4, 1.1, 0.2]

myListInRange : Bool
myListInRange = forall x in myList . 0 <= f x <= 1

During compilation Vehicle will automatically expand this out to a sequence of conjunctions as follows:

myListInRange : Bool
myListInRange = 0 <= f 0.4 <= 1 and 0 <= f 1.1 <= 1 and 0 <= f 0.2 <= 1


One hard constraint enforced by both training and verification tools is that you may not use both a forall and an exists that quantify over infinite domains within the same property. For example, the following is not allowed:

f : Vector Rat 2 -> Vector Rat 1

surjective : Bool
surjective = forall y . exists x. f x == y

This remains true even if you move one or more of the quantifiers to separate functions. For example, the following is not allowed either:

f : Vector Rat 2 -> Rat

hits : Vector Rat 2 -> Bool
hits y = exists x . f x == y

surjective : Bool
surjective = forall y . hits y

However, you can have both types of quantifiers within the same specification as long as they belong to different properties. For example, the following is allowed:

f : Vector Rat 2 -> Rat

prop1 : Bool
prop1 y = exists x . f x >= 2

prop2 : Bool
prop2 = forall x . 1 <= f x <= 3