Unformatted text preview: given a boolean, it returns an integer. $
¨$ $
¨$ The value of
has type
.
When given an integer, it returns an integer.
©$
¨$ ¡ ( &
0 ' %
" ¢ ¤¢
¥
) ¦
§¤ ¤
£©
£¡ %(
¨( ¡ 2 4'
¨( The value of
has no type in our
type system.
When given a boolean it might return either an integer or a boolean, and we have
no type that describes this behavior.
$$ ©
¨& %
" ¤¢
¥¡
) ¦
§¤ ¡ ¡ 4('
¨)$ ¡ 4'
¨( The value of
has type
.
When given a boolean, it returns a procedure from booleans to integers. §
¥§ ©$
¨$ %
"
¢ ¤
¥© ¤ ¢¥
¤
¤ ¥¢ ¡
) %(¨(
¦
£¡ ¦¤ ¤
¥© ¡ 2 ¦¡ ¢ ¤
1¡ 4'
¨( The value of
has type
.
When given a procedure from integers to booleans, it returns an integer.
¤ © % (
¨( ¦¡ 2 ¤
£© ¦¡ ¢ ¤¡
1¡ 4'
¨( The value of
has type
t
t for any type t.
When given a procedure of type
t , it returns a value of type t.
¦¡ ¦
£¡ ¢ ¤
£© ¤
1¡ ¡ ¤¥© 4'
¨( ¤
¥© ¢ ¤¡
1¡ ¦4'
£¡ ¨( The value of
has type t
t
t
t for any type t.
When given a procedure of type t
t , it returns another procedure which,
when given an argument of type t, returns a value of type t.
¤
¥© 5 ¡
¤
£© ¡ ¤
¥© ¤
¥© ¡
¡ Let’s write down a deﬁnition that captures these examples. It will be deﬁned by
induction on t. (See, we are following the Design Recipe!)
Deﬁnition 1 The property of an expressed value v being of type t is deﬁned by
induction on t:
%#!
)"© ¢ ¤ 5 An expressed value is of type
5 It is of type
5 It is of type t1 t2 iff it is a
with the property that if it is given
an argument of type t1 , then one of the following things happen:
%#!©%(
$( . . %#!©4('
$"¨) 2 iff it is a iff it is a %(
¨( 2
¡ 1. it returns a value of type t2
2. it fails to terminate
3. it fails with an error other than a type error.
We occasionally say “v has type t” instead of “v is of type t.”
Puzzle: in this system can a value val have more than one type?
Puzzle: For the language LETREC, is it decidable whether an expression e has a
value that is of type t ? 6 6.3 Assigning a type to an expression ¢ ¦ ¨)£
(©¡ Requirement: Write a procedure
which, given an expression (call it
exp) and a type environment (call it tenv) mapping each variable to a type, assigns
to exp a type t with the property that:
¦ )
(© ¢ ¡
£ Speciﬁcation of Whenever exp is evaluated in an environment in which each variable has the type
speciﬁed for it by tenv, one of the following happens:
5 the resulting value has type t,
5 the evaluation does not terminate, or
5 the evaluation fails on an error other than a type error. Another way of writing the permissible outcomes: The evaluation does not cause a type error, and if it terminates, its value is of type
t. Our analysis will be based on the principle that if we can predict the types of the
values of each of the subexpressions in an expression, we can predict the type of
the value of the expression. ¢ ¦ £
(© ¡ We’ll use this idea to write a speciﬁcation for
. We will write this speciﬁcation as a set of inference rules, as we have done elsewhere. Assume that tenv
is a type environment mapping each variable to its type. Then we should have: 7 $
%# tenv t2 t2 # #
H
t1 t1 $
9#
$
5# $%#
#
$
9#
$
9#
B¨¨@ $
¢¢¦A9# t tenv t
t $
5# # tenv t2 t2 $
5# D B¢¨¢¨¦@A$9#
('& $ tenv $
5# tenv var $
%# #
$
5# $
5# # tenv # tenv # 8 rator rand £!4¥©§©B©B( ©¢¦¤¢ 1 ¨§¥£¡
©©E¤¢ ¨§¥£¡
¨§¥£¡
©©E¤¢ rator tenv
rand tenv e1 e2 e3 £¤!"¥©§ GF ©¢¦¤¢ &
¨§¥£¡
"¢©E¤© ¨§¥£¡
"¢©E¤© ¨§¥£¡
¨§¥£¡
"¢©E¤© e1 tenv
e2 tenv
e3 tenv £¤!4¥¢§ " ©¢¦¤¢ ¥B ¨§¥£¡
©¢¦¤¢ ¨§¥£¡
¨§¥£¡
©¢¦¤¢ var e1 body C £¤!"¥©§©8¢337 ©¢¦¤¢ ¨2¥6 ¨§¥£¡
¨§¥£¡
©¢¦¤¢ e1 £!¥§2 )
¤4¢31 0 var £!¥ ¨
¤"©§ num
('& ( (
( ( (
(
( ( e1 tenv
t1
body var t1 tenv (
( ( e1 tenv ( ¨§¥£¡
©¢¦¤¢ ¨§¥£¡
©¢¦¤¢ (
( Simple typing rules What about procedures?
4('
¨) If
( x) e has type (t1
t2 ), then it is intended to be called on an argument
of type t1 . When its body e is evaluated, the variable x will be bound to a value of
type t1 . This suggests the following rule: tenv tvar H # C var body t2
t2 # t1 tenv $ var D £¤!"¥©©¢33£ ©©E¤¢ §¨2
¨§¥£¡
¨§¥£¡
"¢©¦© ( body 9#
$
$
5# ( 4('
¨)$ There’s only one proble...
View
Full Document
 Fall '09
 Logic, Expression, rator tenv

Click to edit the document details