1. David Evans http://www.cs.virginia.edu/~evans
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

2. Menu Types and Type Checking Typed Scheme 1 April 2002
CS 200 Spring 2002

3. Types Type is a (possibly infinite) set of values
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
1 April 2002
CS 200 Spring 2002

4. 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)
1 April 2002
CS 200 Spring 2002

5. 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.
1 April 2002
CS 200 Spring 2002

6. 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)
1 April 2002
CS 200 Spring 2002

7. Scheme  Java Scheme has Latent, Dynamic types
Java has Manifest, Static types
1 April 2002
CS 200 Spring 2002

8. Java Example class Test { int tester (String s) { int x; x = s;
return "okay"; }
javac types.java
types.java:5: Incompatible type for =. Can't convert java.lang.String to int.
x = s; ^
types.java:6: Incompatible type for return. Can't convert java.lang.String to int.
return "okay";
2 errors
The result
is an integer
x is an integer
1 April 2002
CS 200 Spring 2002

9. Java Example class Test { int tester (String s) { int x;
tester (“hello”);
x = s;
return "okay"; }
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
1 April 2002
CS 200 Spring 2002

10. What do we need to do change our Mini-Scheme evaluator to provide Java-like type checking?
1 April 2002
CS 200 Spring 2002

11. Types in Mini-Scheme Type ::= PrimitiveType Type ::= ProcedureType
Type ::= ProductType
ProcedureType ::= Type  Type
ProductType ::= Type x Type
PrimitiveType ::= Number | String
1 April 2002
CS 200 Spring 2002

12. Examples 3 Number + Number x Number  Number (+ 3 3)
Type ::= PrimitiveType | ProcedureType | ProductType
ProcedureType ::= Type  Type
ProductType ::= Type x Type
PrimitiveType ::= Number | String
Examples 3
Number +
Number x Number  Number
(+ 3 3)
(lambda ((x number) (y number)) (+ x y))
Changed lambda form:
Expression ::= (lambda (((Name Type))*) Expr*)
1 April 2002
CS 200 Spring 2002

13. 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
1 April 2002
CS 200 Spring 2002

14. 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)))
1 April 2002
CS 200 Spring 2002

15. Representing Types (define (make-procedure-type params result)
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 (procedure-type-params type)
(cadr type))
(define (assert pred)
(if (not pred)
(error
"Assertion failed!")))
1 April 2002
CS 200 Spring 2002

16. Type of + (make-procedure-type (make-product-type (make-number-type)
1 April 2002
CS 200 Spring 2002

17. Changing Frames parameters: x body: (lambda (x) (+ x x))
global
environment
global
environment
+: (-> (x Number Number)
Number)
#<primitive:+>
+ : #<primitive:+>
double:
x: 3
double:
(-> Number
Number)
x: Number 3
parameters: x body: (lambda (x) (+ x x))
parameters: x Number body: (lambda (x) (+ x x))
1 April 2002
CS 200 Spring 2002

18. 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))
1 April 2002
CS 200 Spring 2002

19. 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
(environment-lookup-value
name
(enclosing-environment env)))))
(define (typeof-variable name env)
(if (null? env)
(error "No binding for" name)
(if (frame-contains? name
(first-frame env))
(frame-lookup-type name
(typeof-variable
name
(enclosing-environment env)))))
1 April 2002
CS 200 Spring 2002

20. 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)
(cadr (car frame))
(frame-lookup-type name (cdr frame)))))
1 April 2002
CS 200 Spring 2002

21. Charge Wednesday: Finish TypedScheme Friday: PS7 Due
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
1 April 2002
CS 200 Spring 2002