1. David Evans http://www.cs.virginia.edu/~evans CS200: Computer Science University of Virginia Computer Science Lecture 29: Typed Scheme MC Escher, Liberation

2. 3 April 2002CS 200 Spring 20022 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

3. 3 April 2002CS 200 Spring 20023 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-procedure-type inputs outputs) (list ‘procedure-type inputs outputs)

4. 3 April 2002CS 200 Spring 20024 Changing Frames global environment + : # double: x: 3 parameters: x body: (lambda (x) (+ x x)) global environment +: (-> (x Number Number) Number) # double: (-> Number Number) x: Number 3 parameters: x Number body: (lambda (x) (+ x x))

5. 3 April 2002CS 200 Spring 20025 Defining typeof

6. 3 April 2002CS 200 Spring 20026 typeof Examples > (typeof '(+ 3 4) the-global-environment) (primitive-type number) > (typeof '(+ 3 +) the-global-environment) Type mismatch. Application (+ 3 +) parameter types are (Number x Number) -> (Number) x Number, should be Number x Number. (error-type) > (typeof '(+ 3) the-global-environment) Type mismatch. Application (+ 3) parameter types are Number, should be Number x Number. (error-type)

7. 3 April 2002CS 200 Spring 20027 Start with meval (define (meval expr env) (cond ((self-evaluating? expr) expr) ((variable? expr) (environment-lookup-name expr env)) ((lambda? expr) (make-procedure (lambda-parameters expr) (lambda-body expr) env)) ((application? expr) (mapply (meval (application-operator expr) env) (map (lambda (subexpr) (meval subexpr env)) (application-operands expr)))) (else (error "Unknown expression: " exp))))

8. 3 April 2002CS 200 Spring 20028 typeof (define (typeof expr env) (cond ((self-evaluating? expr) (typeof-self-evaluating expr) ((variable? expr) (typeof-variable expr env)) ((lambda? expr) (typeof-procedure expr env)) ((application? expr) (typeof-application expr env)) (else (error "Unknown expression: " exp))))

9. 3 April 2002CS 200 Spring 20029 typeof-self-evaluating (define (typeof-self-evaluating expr) (cond ((number? expr) (make-number-type)) ((string? expr) (make-string-type)) ((primitive-procedure? expr) (error “Bad typeof-self-evaluating”))) (define (self-evaluating? expr) (or (number? expr) (string? expr) (primitive-procedure? expr)))

10. 3 April 2002CS 200 Spring 200210 Testing 1 2 3… > (typeof '3 the-global-environment) (primitive-type number) > (typeof '"test" the-global-environment) (primitive-type string) > (typeof '+ the-global-environment) (procedure-type (product-type (primitive-type number) (primitive-type number))

11. 3 April 2002CS 200 Spring 200211 typeof (define (typeof expr env) (cond ((self-evaluating? expr) (typeof-self-evaluating expr) ((variable? expr) (typeof-variable expr env)) ((lambda? expr) (typeof-procedure expr env)) ((application? expr) (typeof-application expr env)) ((definition? expr) (typeof-definition expr env)) (else (error "Unknown expression: " exp)))) This is why (typeof ‘+ the-global-environment) worked!

12. 3 April 2002CS 200 Spring 200212 typeof (define (typeof expr env) (cond ((self-evaluating? expr) (typeof-self-evaluating expr) ((variable? expr) (typeof-variable expr env)) ((lambda? expr) (typeof-procedure expr env)) ((application? expr) (typeof-application expr env)) (else (error "Unknown expression: " exp))))

13. 3 April 2002CS 200 Spring 200213 typeof-application (define (typeof-application expr env) (let ((operator (typeof (application-operator expr) env))) (if (procedure-type? operator) (let ((argument-types (typelist-to-product-type (map (lambda (operand) (typeof operand env)) (application-operands expr))))) (if (type-match argument-types (procedure-type-params operator)) (procedure-type-result operator) (begin (printf "Type mismatch… " ) (make-error-type))))))) The type of an application of an operator of type params  result is the result type of the operator

14. 3 April 2002CS 200 Spring 200214 typeof-application (define (typeof-application expr env) (let ((operator (typeof (application-operator expr) env))) (if (procedure-type? operator) (let ((argument-types (typelist-to-product-type (map (lambda (operand) (typeof operand env)) (application-operands expr))))) (if (type-match argument-types (procedure-type-params operator)) (procedure-type-result operator) (begin (printf "Type mismatch… " ) (make-error-type))))))) But, also check the parameter types match!

15. 3 April 2002CS 200 Spring 200215 typelist-to-product-type (define (typelist-to-product-type typelist) (if (null? typelist) (make-empty-type) (if (eq? (length typelist) 1) (car typelist) (make-product-type (car typelist) (typelist-to-product-type (cdr typelist))))))

16. 3 April 2002CS 200 Spring 200216 type-match (define (type-match t1 t2) (cond ((or (error-type? t1) (error-type? t2)) #t) ;; error types match anything ((number-type? t1) (number-type? t2)) ((string-type? t1) (string-type? t2)) ((procedure-type? t1) (and (procedure-type? t2) (type-match (procedure-type-params t1) (procedure-type-params t2)) (type-match (procedure-type-result t1) (procedure-type-result t2)))) ((product-type? t1) (and (product-type? t2) (type-match (product-type-first t1) (product-type-first t2)) (type-match (product-type-second t1) (product-type-second t2)))) (else (error "Bad type: " t1))))

17. 3 April 2002CS 200 Spring 200217 Testing 1 2 3… > (typeof '(+ 3 4) the-global-environment) (primitive-type number) > (typeof '(+ 3 +) the-global-environment) Type mismatch. Application (+ 3 +) parameter types are (Number x Number) -> (Number) x Number, should be Number x Number. (error-type) > (typeof '(+ 3) the-global-environment) Type mismatch. Application (+ 3) parameter types are Number, should be Number x Number. (error-type)

18. 3 April 2002CS 200 Spring 200218 typeof (define (typeof expr env) (cond ((self-evaluating? expr) (typeof-self-evaluating expr) ((variable? expr) (typeof-variable expr env)) ((lambda? expr) (typeof-procedure expr env)) ((application? expr) (typeof-application expr env)) (else (error "Unknown expression: " exp))))

19. 3 April 2002CS 200 Spring 200219 Examples (define (check-type expr) (display-type (typeof expr the-global-environment))) > (check-type '(lambda ((x number) (y number)) (+ x y))) "(Number x Number) -> (Number)" > (check-type '(lambda ((x number) (y number)) (+ x))) Type mismatch. Application (+ x) parameter types are Number, should be Number x Number. "(Number x Number) -> (Error)" > (check-type '(lambda ((x number) (y number)) +)) "(Number x Number) -> ((Number x Number) -> (Number))"

20. 3 April 2002CS 200 Spring 200220 (define (typeof-procedure expr env) (let* ((params (lambda-parameters expr)) (body (lambda-body expr)) (param-types (map (lambda (param) (parse-type (cadr param))) params)) (restype (typeof-sequence body (extend-environment (map (lambda (param) (car param)) params) ;; names param-types (map (lambda (param) 'unknown) params) ;; values env)))) (make-procedure-type (typelist-to-product-type param-types) restype)))) Figure out the type of the body

21. 3 April 2002CS 200 Spring 200221 (define (typeof-procedure expr env) (let* ((params (lambda-parameters expr)) (body (lambda-body expr)) (param-types (map (lambda (param) (parse-type (cadr param))) params)) (restype (typeof-sequence body (extend-environment (map (lambda (param) (car param)) params) ;; names param-types (map (lambda (param) 'unknown) params) ;; values env)))) (make-procedure-type (typelist-to-product-type param-types) restype))))

22. 3 April 2002CS 200 Spring 200222 typeof Examples > (check-type '(lambda ((x number)) (lambda ((y number)) (+ x y)))) "(Number) -> ((Number) -> (Number))" > (check-type '((lambda ((x number)) (lambda ((y number)) (+ x y))) 1)) "(Number) -> (Number)" > (check-type '(((lambda ((x number)) (lambda ((y number)) (+ x y))) 1) 2)) "Number" > (check-type '(((lambda ((x number)) (lambda ((y number)) (+ x y))) 1) "test")) Type mismatch. Application (((lambda ((x number)) (lambda ((y number)) (+ x y))) 1) test) parameter types are String, should be Number. "Error"

23. 3 April 2002CS 200 Spring 200223 What’s in the-global-environment ? (define the-global-environment (make-new-environment (list (list '+ (make-procedure-type (make-product-type (make-number-type) (make-number-type)) (make-primitive-procedure +)) the-empty-environment))

24. 3 April 2002CS 200 Spring 200224 Charge Friday: PS7 Due Friday: Exam 2 Out (due Wednesday) Mutation and Environmental Model of Evaluation –Classifying Problems –Evaluators (final question will involve modifying typed Scheme, so make sure you understand everything in today’s code) I will answer questions about it Friday, but not after handing out the exam!