VDM-10 Language Manual

Chapter 6. Expressions
all expression = ‘forall’, bind list, ‘&’, expression ;
exists expression = ‘exists’, bind list, ‘&’, expression ;
bind list = multiple bind, { ‘,’, multiple bind } ;
exists unique expression = ‘exists1’, bind, ‘&’, expression ;
Semantics: There are three forms of quantified expressions: universal (written as forall), existential
(written as exists), and unique existential (written as exists1). Each yields a
boolean value true or false, as explained in the following.
The universal quantification has the form:
✞
✡✝
forall mbd1, mbd2, ..., mbdn & e
where each mbdi is a multiple bind pi in set s (or if it is a type bind pi : type),
and e is a boolean expression involving the pattern identifiers of the mbdi’s. It has the value
true if e is true when evaluated in the context of every choice of bindings from mbd1,
mbd2, ..., mbdn and false otherwise.
The existential quantification has the form:
✞
✡✝
exists mbd1, mbd2, ..., mbdn & e
where the mbdi’s and the e are as for a universal quantification. It has the value true if
e is true when evaluated in the context of at least one choice of bindings from mbd1,
mbd2, ..., mbdn, and false otherwise.
The unique existential quantification has the form:
✞
✡✝
exists1 bd & e
where bd is either a set bind or a type bind and e is a boolean expression involving the
pattern identifiers of bd. It has the value true if e is true when evaluated in the context
of exactly one choice of bindings, and false otherwise.
All quantified expressions have the lowest possible precedence. This means that the longest
possible constituent expression is taken. The expression is continued to the right as far as it
is syntactically possible.
Examples: An example of an existential quantification is given in the function shown below,
QualificationOk. This function, taken from the specification of a nuclear tracking
system in [Fitzgerald&98], checks whether a set of experts has a required qualification.
49
✆
✆
✆