XLang

To complement the core Pure Scala language, the XLang module of Leon brings a few extensions to that core language.

These extensions are kept as an optional feature, that needs to be explicitly enabled from the command line with the option --xlang. If you do not specify the option, Leon will reject any program making use of an XLang feature.

On the technical side, these extensions do not have specific treatment in the back-end of Leon. Instead, they are desugared into Pure Scala constructs during a preprocessing phase in the Leon front-end.

Imperative Code

XLang lets you introduce local variables in functions, and use Scala assignments syntax.

def foo(x: Int): Int = {
  var a = x
  var b = 42
  a = a + b
  b = a
  b
}

The above example illustrates three new features introduced by XLang:

1. Declaring a variable in a local scope
2. Blocks of expressions
3. Assignments

You can use Scala variables with a few restrictions. The variables can only be declared and used locally, no variable declaration outside of a function body. There is also support for variables in case classes constructors. XLang introduces the possibility to use sequences of expressions (blocks) – a feature not available in Pure Scala, where you’re only option is a sequence of val which essentially introduce nested let declarations.

While loops

You can use the while keyword. While loops usually combine the ability to declare variables and make a sequence of assignments in order to compute something useful:

def foo(x: Int): Int = {
  var res = 0
  var i = 0
  while(i < 10) {
    res = res + i
    i = i + 1
  }
  res
}

Leon will automatically generate a postcondition to the while loop, using the negation of the loop condition. It will automatically prove that verification condition and you should see an invariant postcondition marked as valid.

Leon internally handle loops as a function with a postcondition. For the end-user it means that Leon is only going to rely on the postcondition of the loop to prove properties of code relying on loops. Usually that invariant is too weak to prove anything remotely useful and you will need to annotate the loop with a stronger invariant.

You can annotate a loop with an invariant as follows:

var res = 0
var i = 0
(while(i < 10) {
  res = res + i
  i = i + 1
}) invariant(i >= 0 && res >= i)

The strange syntax comes from some Scala magic in order to make the keyword invariant a valid keyword. Leon is defining an implicit conversion from Unit to an InvariantFunction object that provides an invariant method. The invariant method takes a boolean expression as a parameter and its semantics is to hold at the following points during the execution of the loop:

1. When first entering the loop: initialization.
2. After each complete execution of the body.
3. On exiting the loop.

Leon will generate verification conditions invariant inductive and invariant postcondition to verify points (2) and (3) above. It will also generate a precondition corresponding to the line of the while loop. This verification condition is used to prove the invariant on initialization of the loop.

Arrays

PureScala supports functional arrays, that is, the operations apply and updated which do not modify an array but only returns some result. In particular, updated returns a new copy of the array.

def f(a: Array[Int]): Array[Int] = {
  a(0).updated(0, a(1))
}

However, using functional arrays is not the most natural way to work with arrays, and arrays are often used in imperative implementations of algorithms. XLang adds the usual update operation on arrays:

val a = Array(1,2,3,4)
a(1) //2
a(1) = 10
a(1) //10

Leon simply rewrite arrays using update operation as assignment of function arrays using updated. This leverages the built-in algorithm for functional array and rely on the elimination procedure for assignments. Concretely, it would transform the above on the following equivalent implementation:

var a = Array(1,2,3,4)
a(1) //2
a = a.updated(1, 10)
a(1) //10

Then Leon would apply the same process as for any other XLang program.

Mutable Objects

A restricted form of mutable classes is supported via case classes with var arguments:

case class A(var x: Int)
def f(): Int = {
  val a = new A(10)
  a.x = 13
  a.x
}

Mutable case classes are behaving similarly to Array, and are handled with a rewriting, where each field updates becomes essentially a copy of the object with the modified parameter changed.

Aliasing

With mutable data structures comes the problem of aliasing. In Leon, we maintain the invariant that in any scope, there is at most one pointer to some mutable structure. Leon will issue an error if you try to create an alias to some mutable structure in the same scope:

val a = Array(1,2,3,4)
val b = a //error: illegal aliasing
b(0) = 10
assert(a(0) == 10)

However, Leon correctly supports aliasing mutable structures when passing it as a parameter to a function (assuming its scope is not shared with the call site, i.e. not a nested function). Essentially you can do:

case class A(var x: Int)
def updateA(a: A): Unit = {
  a.x = 14
}
def f(): Unit = {
  val a = A(10)
  updateA(a)
  assert(a.x == 14)
}

The function updateA will have the side effect of updating its argument a and this will be visible at the call site.

Annotations for Imperative Programming

XLang introduces the special function old that can be used in postconditions to talk about the value of a variable before the execution of the block. When you refer to a variable or mutable structure in a post-condition, leon will always consider the current value of the object, so that in the case of a post-condition this would refer to the final value of the object. Using old, you can refer to the original value of the variable and check some properties:

case class A(var x: Int)
def inc(a: A): Unit = {
  a.x = a.x + 1
} ensuring(_ => a.x == old(a).x + 1)

old can be wrapped around any identifier that is affected by the body. You can also use old for variables in scope, in case of nested functions:

def f(): Int = {
  var x = 0
  def inc(): Unit = {
    x = x + 1
  } ensuring(_ => x == old(x) + 1)

  inc(); inc();
  assert(x == 2)
}

Epsilon

XLang introduces the epsilon keyword to express non-determinism. The concept is inspired from Hilbert’s epsilon calculus. An epsilon expression takes a predicate as parameter and is defined to return a value that satisfies the predicate:

def getInRange(from: Int, to: Int): Int = {
  epsilon(n => from <= n && n <= to)
}

You can use epsilon when you only know the interface of some function but cannot provide a concrete implementation.