Short Quiz 6 Example
type
rules
1) What is the purpose of type system? (correct errors of input / write program easily / both ) •2) IfWboth
operands
ofin the
operators of
hich system is included type arithmetic
system ? ( type checking system / type inference system / both ) addition, subtraction, and multiplication are of type
integer, then the result is of type integer (Pascal
3) What is the result of following C program ? (notice the ascii code for a is definition).
96) Rule for + (analogue rules for − and ∗):
Example
type
rules
#include <ctype.h> #define toupper(c) ( (islower(c)) ? _toupper(c) : (c) ) int a = 'm'; a = toupper(a-­‐-­‐); 1
•(If
both
of the arithmetic 2operators of
printf “%d”, a ) ;operands
addition, subtraction,1 and 2multiplication are of type
94 integer, then the result is of type integer (Pascal
definition).
4) If both operands of the arithmetic operators of addition, subtraction, and Rule for
forresult − and
multiplication are +
of t(analogue
ype integer, rules
then the is of ∗):
type integer (Pascal definition). Rule for + (analogue rules for − and * ): E � e : integer E � e : integer
E � (e + e ) : integer
where E is a type environment that maps constants
and variables to their types.
E � e2 :two
integer
E � ewith
1 : integer
In combination
the following
axioms in the
e2) and
: integer
type system {c : Eα}� �(e1c+: α
{v : α} � v : α
Eenvironment isinfer,
a typethat
environment
maps
constants
where Ewhere
is a now
type that constants nd variables to their we can
(2maps + 3)
isthat
of atype
integer:
types. Pand
lease finish the tto
ype inference procedure ) variables
their
types.
E � 2 : integer E � 3 : integer
the
In combination with the following two axioms in
E : �α}|-­‐ (2�(2
+
3)
:and
integer
: integer
type system {c E
c :+
α 3)
{v : α} � v : α
we can now infer, that (2 + 3) is of type integer:
where E
E == {2
{2:: integer,
integer, 33: :integer}.
where
integer}. E � deduction
2 : integer proofs
E � 3 work
: integer
In general, type
bottom up.
E � (2 + 3) : integer
CS 314
spring’13 lecture 27, page 4
where E = {2 : integer, 3 : integer}.
the result
objectofreferred
to by&the
operand.
the operand
• The
the unary
operator
is aIfpointer
to
is ofThe
type
“foo”,
the type of
result
is a
result
of thethen
unary
is athe
pointer
the •object
referred
to by& operator
the operand.
If to
the operand
“pointer
to foo”.
and
definition)
the object
referred(C
to by
theC++
operand.
If the operand
is of type
“foo”,
then
the
type
of
the
is a
is of type “foo”, then the type of the result result
is a
5) Please write the type inference procedure for following C/C++ program: tounary foo”.&(C
and
C++
definition)
“pointer
foo”.
and
C++
definition)
The r“pointer
esult oto
f the (C
operator(It means to get the pointer of current E�e:α
symbol/variable/function name) is a pointer to the object referred to by the operand. If the operand is of type “foo”, then what is the type of result? E�e:α
E � &e : pointer(α)
E � &e : pointer(α)
E�e:α
E�
: pointer(α)
E &e
|-­‐ &e
: pointer(α) • Two
expressions can only be compared if they have
• Two expressions can only be compared if they have
the
sametypes.
types. The
is ofistype
boolean.
the
same
Theresult
result
of type
boolean.
6) Please finish the following type inference of following C/C++ program: Two expressions can only be compared if they have the same types. The result is of type boolean. : α Ebe� compared
e2 : α
� e1only
• Two expressionsEcan
if they have
:
α
E
�
e
:
α
E
�
e
E � (e11= e2) : boolean 2
the same types.EThe
result
is
of
type
boolean.
� (e = e ) : boolean
1
E |-­‐ (e1 =
2
e 2)
: Boolean
E � e2 for
: αother
E �which
e1 :isαa placeholder
α is a type variable,
type expressions.E � (e = e ) : boolean
1
2
α is a type variable, which is a placeholder for other
type 7) expressions.
Please write down result of following ML program: CS 314
-­‐ 1 * 2 + 3 * 4 ; val it = 14 : int -­‐ "abc" ^ "def"; val it = "abcdef" : string CS 314
-­‐ if (1 < 2) then 3.0 else 4.0; val it = 3.0 : real -­‐ 1 < 2 orelse (1 div 0) = 0; val it = true : bool CS 314 spring’13 lecture 27, page 6
α is a type variable, which is a placeholder for other
spring’13 lecture 27, page 6
type expressions.
spring’13 lecture 27, page 6
8) What is wrong with following ML program: -­‐ 10 / 5; stdIn:5.1-­‐5.5 Error: operator and operand don't agree [literal] operator domain: real * real operand: int * int -­‐ #"a" = #"b" or 1 = 2; stdIn:1.1-­‐2.8 Error: operator and operand don't agree [literal] operator domain: bool * bool operand: bool * int -­‐ 1.0 = 1.0; stdIn:1.1-­‐1.10 Error: operator and operand don't agree [equality type required] operator domain: ''Z * ''Z operand: real * real -­‐ if (1<2) then 3 else 4.0; stdIn:1.1-­‐2.14 Error: types of if branches do not agree [literal] then branch: int else branch: real -­‐ if (1<2) then 3; no such construct 9) What is the result of the following ML program ? -­‐ square 7 + 6 ; 55 10 ) What if anything is wrong with each of the following expressions? trunc 5 stdIn:1.1-­‐1.8 Error: operator and operand don't agree [literal] trunc
(fn: real à int)
operator domain: real operand: int ord (fn: char à int)
ord "a" chr (fn: int à char)
stdIn:1.1-­‐1.8 Error: operator and operand don't agree [tycon mismatch] str (fn: char à string)
operator domain: char operand: string if 0 then 1 else 2 stdIn:1.1-­‐7.10 Error: test expression in if is not of type bool [literal] test expression: int in expression: if true then 1 else 2.0 stdIn:1.1-­‐7.15 Error: types of if branches do not agree [literal] then branch: int else branch: real chr(trunc(97.0)) val it = #"a" : char chr(trunc 97.0) val it = #"a" : char chr trunc 97.0 stdIn:9.1-­‐9.15 Error: operator and operand don't agree [tycon mismatch] operator domain: int operand: real -­‐> int