Transcript slides
Lecture 28:
Types of Types
“It would appear that we have reached the limits
of what it is possible to achieve with computer
technology, although one should be careful with
such statements, as they tend to sound pretty
silly in five years.”
John Von Neumann, 1949
CS200: Computer Science
University of Virginia
Computer Science
David Evans
http://www.cs.virginia.edu/~evans
Menu
• Types and Type Checking
• Typed Scheme
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Types
Strings
Numbers
programs that halt
Colors
Beatle’s Songs that don’t end on the Tonic
lists of lists of lists of anything
• Type is a (possibly infinite) set of values
• You can do some things with some types,
but not others
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Why have types?
• Detecting program errors
– Better to notice error than report incorrect result
– Better to find error in development then in
execution
• Make programs easier to read, understand
and maintain
– Better than comments if they are checked and
can be trusted
• Security
– Can use types to constrain the behavior of
programs (not in CS200)
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Types of Types
Does regular Scheme have types?
> (car 3)
car: expects argument of type <pair>; given 3
> (+ (cons 1 2))
+: expects argument of type <number>; given (1 . 2)
Yes, without types (car 3) would produce some silly result.
Because of types, it produces a type error.
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Type Taxonomy
• Latent vs. Manifest
– Are types visible in the program text?
• Checked statically vs. checked
dynamically
– Do you have to run the program to know if it
has type errors?
• Checked weakly vs. strongly
– How strict are the rules for using types?
– Meaningless (just matter of degree)
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Scheme Java
• Scheme has Latent, Dynamic types
• Java has Manifest, Static types
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Java Example
javac types.java
class Test {
The result
types.java:5: Incompatible
int tester
is an integer
type for =. Can't convert
java.lang.String to int.
{
x = s;
int x;
^
x is an integer
types.java:6: Incompatible
x = s;
type for return. Can't convert
java.lang.String to int.
return
return "okay";
^
2 errors
}
}
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(String s)
"okay";
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Java Example
javac types.java
types.java:6: Incompatible
type for =. Can't convert
java.lang.String to int.
x = s;
^
types.java:7: Incompatible
type for return. Can't convert
java.lang.String to int.
return "okay";
^
2 errors
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class Test {
int tester (String s)
{
int x;
tester (“hello”);
x = s;
return "okay";
}
}
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What do we need to do change
our Mini-Scheme evaluator to
provide Java-like type checking?
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Types in Mini-Scheme
Type ::= PrimitiveType
Type ::= ProcedureType
Type ::= ProductType
ProcedureType ::= Type Type
ProductType ::= Type x Type
PrimitiveType ::= Number | String
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Examples
Type ::= PrimitiveType | ProcedureType | ProductType
ProcedureType ::= Type Type
ProductType ::= Type x Type
PrimitiveType ::= Number | String
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Number
+
Number x Number Number
(+ 3 3)
Changed lambda form:
Expression ::= (lambda (((Name Type))*) Expr*)
Number
(lambda ((x number) (y number)) (+ x y))
Number x Number Number
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Changing Evaluator
• Divide evaluation into two steps:
– Checking types
– Evaluating (essentially as before)
• How do we implement check-type?
– Represent types
– Put types in frame
– Change meval into typeof
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Representing
Types
Type ::= PrimitiveType | ProcedureType | ProductType
ProcedureType ::= Type Type
ProductType ::= Type x Type
PrimitiveType ::= Number | String
(define (make-primitive-type type) (list 'primitive-type type))
(define (primitive-type? type) (tagged-list? type 'primitive-type))
(define (make-number-type) (make-primitive-type 'number))
(define (number-type? type)
(and (primitive-type? type) (eq? (cadr type) 'number)))
(define (make-boolean-type) (make-primitive-type 'boolean))
(define (boolean-type? type)
(and (primitive-type? type) (eq? (cadr type) 'boolean)))
(define (make-string-type) (make-primitive-type 'string))
(define (string-type? type)
(and (primitive-type? type) (eq? (cadr type) 'string)))
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Representing
Types
Type ::= PrimitiveType | ProcedureType | ProductType
ProcedureType ::= Type Type
ProductType ::= Type x Type
PrimitiveType ::= Number | String
(define (make-procedure-type params result)
(list 'procedure-type params result))
(define (procedure-type? type)
(tagged-list? type 'procedure-type))
(define (procedure-type-result type)
(assert (procedure-type? type))
(caddr type))
(define (assert pred)
(if (not pred)
(error
"Assertion failed!")))
(define (procedure-type-params type)
(assert (procedure-type? type))
(cadr type))
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Type of +
(make-procedure-type
(make-product-type (make-number-type)
(make-number-type))
(make-number-type))
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global
environment
Changing global
Frames
environment
+: (-> (x Number Number)
Number)
#<primitive:+>
+ : #<primitive:+>
x: 3
double:
x: Number 3
double:
(-> Number
Number)
parameters: x
body: (lambda (x) (+ x x))
parameters: x Number
body: (lambda (x) (+ x x))
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Changing Frames
(define (extend-environment names values env)
(make-new-environment
(map (lambda (name value)
(cons name value))
names values)
env))
(define (extend-environment names types values env)
(make-new-environment
(map (lambda (name type value)
(list name type value))
names types values)
env))
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Looking Up Variables
(define (environment-lookup-name name env)
(if (null? env) (error "No binding for" name)
(if (frame-contains? name (first-frame env))
(frame-lookup-name name (first-frame env))
(environment-lookup-name name (enclosing-environment env)))))
(define (environment-lookup-value
name env)
(if (null? env)
(error "No binding for" name)
(if (frame-contains? name
(first-frame env))
(frame-lookup-value name
(first-frame env))
(environment-lookup-value
name
(enclosing-environment env)))))
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(define (typeof-variable name env)
(if (null? env)
(error "No binding for" name)
(if (frame-contains? name
(first-frame env))
(frame-lookup-type name
(first-frame env))
(typeof-variable
name
(enclosing-environment env)))))
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Frame Lookups
(define (frame-lookup-value name frame)
(if (null? frame) (error "Name not found in frame:" name)
(if (eq? (car (car frame)) name)
(caddr (car frame))
(frame-lookup-value name (cdr frame)))))
(define (frame-lookup-type name frame)
(if (null? frame) (error "Name not found in frame:" name)
(if (eq? (car (car frame)) name)
(cadr (car frame))
(frame-lookup-type name (cdr frame)))))
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Charge
• Wednesday: Finish TypedScheme
– Handling lambda
• Friday: PS7 Due
• Exam 2 out Friday or Monday (class
choice), due Weds next week
– Classify problems (undecidable, NP, P, etc.)
– Modify TypedScheme
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