2003:27 - Karlstads universitet

B.2 Dharas definition av en subtyp
Dharas definition [3]:
Definition of Pre-Behavioral Subtyping (PBS): S is a pre-behavioral subtype of T with
respect to a binary relation ≤ on types if an only if the following properties are satisfied:
hold:
Syntactic: For every method m in T there exists a method m in S such that the following
Contravariance of arguments: If the types of the additional arguments of m in S and T are
U and V respectively, then U V ∧ ( ∀ i V ≤ U )
= . .
Covariance of result: If the result types of m in S and T are U r and V r respectively, then
U r ≤ V r .
Exception rule: For each exception e: E S of m in S, m in T has an exception e: E T , and
ES ≤ E T .
Semantic: There exists a family of simulation functions, c X → Y : X ≤ Y , such that the
following hold:
Invariant rule: For all values S
v of S, ( v ) I ( c ( v ) )
36
i
I S S T S → T
Method’s rule: For all common methods m, if the types of the additional arguments types
of m in S and T are U and V respectively, and the result types of m in S and T are U r and
V r respectively, then for all objects self: S, y :V , and result: U r
- Precondition rule.
pre
m
T
- Postcondition rule.
m
( cS
→ T ( self ^) , y^)
preS
self ^,
c ( y^)
( )
i
V →U
m ( preS
( self ^ , y^)
m
post S ( self ^,
self ',
y^
, y',
result'
)
( m
T ( cS
→T
( self ^) , y^)
m
postT
( cS
→T
( self ^)
, cS
→T
( self ')
, y^
, y',
cU
→V
( result')
)
pre r r
S