The earlier lessons on pre- and post-conditions, assert and assume stastements all showed examples of expressions used in JML clauses. As expresions are a building block for nealry all other JML constructs, we include a couple lessons at this point to introduce JML expressions.

JML expressions look very much like Java expressions. Indeed, the JML expression syntax includes all of the Java expression syntax, with this exception: JML expressions are not allowed to have side-effects. So the ++, --, and op= (e.g., +=) operators are not allowed in JML expressions. The meaning of Java operators in JML is also unchanged, except for two matters:

Note that the non-short-circuiting boolean operators & and | are legal in both Java and JML, though the short-circuiting versions && and || are much more commonly used in Java because they can be more efficient. In static reasoning, however, execution efficiency does not matter. In static reasoning, & and | are simpler to reason about, but && and || are often needed because of well-definedness considerations.

But JML also adds to Java some new operators and expression syntax. The new operators are these:

• ==> (implication): this binary operator takes two boolean operands, e.g., p and q; p ==> q is read a “p implies q” and means the same as logical implication, that is, the same as “not p or q”. The implication operator is short-circuiting. That is, the value and well-definedness of the right-hand-side is immaterial, unless the left-hand-side is true. In other words, p ==> q is precisely eqiuvalent to !p || q. The implication operator has lower precedence than && and ||, so p && q ==> r || s means (p && q) ==> (r || s). ==> is right associative, so that p ==> q ==> r means p ==> (q ==> r), which is the same as (p && q) ==> r.
• <==> (equivalence): this is also a binary operation between boolean values. p <==> q means the same as p == q, except that <==> has lower precedence than && and ||, whereas == has higher precedence. Thus p && q == r || s means (p && (q == r)) || s, whereas p && q <==> r || s means (p && q) <==> (r || s). Also <==> is left associative (p <==> q <==>r is (p <==> q) <==> r), though (a) the meaning is the same as if it were right associative and (b) expressions that rely on associativity of <==> tend to be confusing and should be avoided.
• <=!=> (inequivalence): this operator is simply the negation of <==> and has the same precedence as <==>. <=!=> is left associative: p <=!=> q <==> r is (p <=!=> q) <==> r, though again, expressions relying on associativity of <=!=> and <==> tend to be confusing and should be avoided.

Another important addition to JML is the chaining of relational operators. Instead of writing i <= j && j < k, one can write i <= j < k. Similarly i > j > k means i > j & j > k. < and <= can be chained together and > and>= can be chained together, but the two groups cannot be mixed. Furthermore, == does not chain and in fact has a lower precedence than the relational operators: a < b == c < d means (a < b) == (c < d) in both Java and JML. These chained operations are particularly convenient for writing ranges of indices. For example, for an array a one might constrain an index variable i as 0 <= i < a.length.

In addition quantified expressions are described below and two other advanced operators are presented in Advanced topics lessons:

Finally, there are many keywords that designate either singleton values (e.g. \result) or function-like operations (e.g., \typeof(...). These will be explained as needed in future lessons, although one, \result, you have already seen. All JML keywords used within expressions begin with a backslash, so they cannot conflict with Java identifiers.

Of course, these JML operators and functions (and all other JML syntax) can only be used within JML annotations, not in Java.

TODO - need some examples of these operators in action

### Quantified expressions

A general point about all these quantified expressions is that any numeric subexpressions are evaluated in bigint-math mode so that there is no conern about overflow in evaluating the expression. Arithmetic mode operators are not allowed within the quantified expression. The result of the expression may be cast to the result type of the expression when the computation is complete. Examples are given below.

TODO - need worked examples of these

#### forall and exists

Quantified expressions are common in logic and are just as necessary in JML to express properties over collections of objects. The two most common expressions are universal and existential quantification. Here are some common examples

• \forall int i; 0 <= i < a.length; a[i] == 2*i
• \exists int i; 0 <= i < a.length; a[i] == 0

The first states the property that each of the elements of array a have a value equal to twice the index of the element. The second states that for some index within the array, the value at that index equals 0. Both of these expressions have the same general form: the keyword, a variable declaration (which is local to the expression), a boolean range predicate R, and a boolean value predicate V. It is permitted to join the range and value predicates like this:

• \forall ...;; R ==> V;
• \exists ...;; R && V;

However separating them makes it easier to have an efficient implementation for runtime-assertion-checking. Note that if the range predicate is false (for example if the array has length 0 above), then the forall expression is true, but the exists expression is false.

These expressions are very commonly used in reasoning about loops, arrays, sequences and sets.

TODO - say more about situations where the index is not an int, e.g. over a set of elements.

#### choose

The \choose predicate is quite similar to the \exists predicate. Whereas \exists ...\; R; V is true if there is an index for which R && V is true, \choose asks what that value is. For example, the value of \choose int i; 0 <= i < a.length; a[i] == 0 is an int for which the range and predicate are true, that is in this example, for which the array element is 0. The type of the expression is always the type of the declaration of the local variable.

If there is more than one such index, the result of the expression might be any one of them, but always the same one for a semantically identical expression. But the fact that the value could be any satisfying index means that any assertion that uses that index must hold no matter what the value is.

If R && V is always false, that is there is no such index, then the expression is not well-defined.

#### max and min

The \max and \min predicates have the same form except that the value term is now numeri, so that maximum and minimum are meaningful concepts. You might ask, for example, for the maximum and minimum values of an array:

• \max int i; 0 <= i < a.length; a[i]
• \min int i; 0 <= i < a.length; a[i]

A first point to note is that these are each equivalent to a pair of forall and exists expressions: x == \max ...; R; V is equivalent to (\forall ... ; R; x >= V) && (\exists ... ; R; x == V). That is, the value of the \max expression is a number that is at least as large as all the elements being considered and is equal to at least one of them.

A second point is that these expressions are not well-defined if the range is empty (range predicate is always false).

A third point is that the type of the expression is the same as the type of the value term. However the value term itself is evaluated in bigint-math mode and only when the max or min has been determined is the result cast back to the final type.

#### sum and product

The \sum and \product quantifiers add up or multiply up all the values of the value term for which the range term is true. For example the sum or product of all the elements in an array a would be expressed as

• \sum int i; 0 <= i < a.length; a[i]
• \product int i; 0 <= i < a.length; a[i]

The type of these operations is \bigint simply because overflow is a distinct possibility. The result can always be cast to a desired final type, at which point the final value is checked that it is actually in range for the desired type.

If the range predicate is empty (always false) then the sum is 0 and the product is 1.

The implementation of these two expressions in JML tools is in progress. Don’t count on them working yet.

#### num_of

A final quantified expression is \num_of which counts the number of times the boolean value term is true when the range term is also true. For example, \numof int i; 0 <= i < a.length; a[i] == 0 counts the number of elements of the array that are 0. Quite obviously \num_of ...; R; V is equivalent to \sum ... ; R && V; 1. The type of a \num_of expression is \bigint and it can be cast to a desired final type; if the range is empty the value of the expression is 0.

The implementation of this expression in JML tools is in progress. Don’t count on it working yet.