sml

18.1 Signatures 145
signature QUEUE WITH EMPTY =
sig
include QUEUE
val is empty : ’a queue -> bool
end
As the notation suggests, the signature QUEUE is included into the body of
the signature QUEUE WITH EMPTY, and an additional component is added.
It is not strictly necessary to use include to define this signature. Indeed,
we may define it directly using the following signature binding:
signature QUEUE WITH EMPTY =
sig
type ’a queue
exception Empty
val empty : ’a queue
val insert : ’a * ’a queue -> ’a queue
val remove : ’a queue -> ’a * ’a queue
val is empty : ’a queue -> bool
end
There is no semantic difference between the two definitions of QUEUE WITH EMPTY.
Signature inclusion is a convenience that documents the “history” of how
the more refined signature was created.
Signature specialization is used to augment an existing signature with
additional type definitions. For example, if we wish to refine the signature
QUEUE to specify that the type constructor ’a queue must be defined as a
pair of lists, we may proceed as follows:
signature QUEUE AS LISTS =
QUEUE where type ’a queue = ’a list * ’a list
The where type clause “patches” the signature QUEUE by adding a definition
for the type constructor ’a queue.
The signature QUEUE AS LISTS may also be defined directly as follows:
REVISED 11.02.11 DRAFT VERSION 1.2