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*To*: willc%indiana.csnet@CSNET-RELAY.ARPA*Subject*: Re: assertions are better than types*From*: Gerald Jay Sussman <GJS%MIT-OZ@MIT-MC.ARPA>*Date*: Mon 18 Mar 85 14:01:07-EST*cc*: scheme@MIT-MC.ARPA*In-Reply-To*: Message from "Will Clinger <willc%indiana.csnet@csnet-relay.arpa>" of Sun 17 Mar 85 23:42:56-EST

Hi. Before you jump, I just want to clarify a few points about what I meant by type-restricted operators -- I know I didn't say this in the draft I sent you, but I didn't realize the scope of possible confusion. I rspond pointwise: Willc: Speaking for myself, I would not want to clutter my code with all these type-restricted operators as I write the code, because it would make the code less readable. (I guess readability is more important to my debugging processes than are type checks.) I would be willing to add type assertions when the time came to make the code run fast. Hence for me at least, the type restrictions would be more important as advice to the compiler than as debugging aids. GJS: I agree, usually we would not write the ugly kind of code which is illustrated below: (define (tak x y z) (if (not (<? y x)) ; notice that you can't say (integer <?) z (tak (tak ((integer -1+) x) y z) (tak ((integer -1+) y) z x) (tak ((integer -1+) z) x y)))) Instead we would write something like this: (define tak (let ((-1+ (integer -1+)) (<? (integer <?))) ;Note that we can say this, see below! (lambda (x y z) (if (not (<? y x)) z (tak (tak (-1+ x) y z) (tak (-1+ y) z x) (tak (-1+ z) x y)))))) Presumably, with a good compiler it would be just as efficient to write: (define (tak x y z) (let ((-1+ (integer -1+)) (<? (integer <?))) ;Note that we can say this, see below! (if (not (<? y x)) z (tak (tak (-1+ x) y z) (tak (-1+ y) z x) (tak (-1+ z) x y))))) Willc: The type restrictors require that all operands and the result be of the same type. This works reasonably well for numeric operators but is going to be awkward when the time comes to generalize it to other operators. How am I going to tell the compiler that the first argument to vector-ref is a vector of strings? Willc: These procedures take a numeric operator op and return a procedure that acts like op except that it is restricted to operate on numbers of the specified type and to return numbers of the specified type. GJS: The type restrictions are not intended to be simply error-checking on the type of the inputs and outputs, but rather to be an entry to a more coherent system built on the numerical tower. The idea is that for each kind of number there is a set of appropriate kinds of operations that make sense for it. For example, the INTEGERS are a ring, hence they do not support arbitrary division. This does not stop us from defining "/" for the INTEGERS to be anything we like, such as an entry to higher types on the tower. Additionally, I am not even attempting to clarify the types for things like strings or vectors of strings, since they do not have the mathematical structure of numbers... This is trying to be a coherent theory about numbers, rather than an incoherent theory of everything. For example, Hal and I have been hacking a new version of generic arithmetic based on these principles -- I include a fragment for your amusement. Basically, there is an operation table, and "restrictions" like INTEGER are things that transform the generic operator, like +, into the actual operation from the table. The following code creates a "rational package" for a given RING, the new objects have type FIELD. ;;; Rational operations generator (define (make-rational-operations! ring field) (let ((+r (ring +)) (-r (ring -)) (*r (ring *)) (negate-r (ring negate)) (=r (ring =)) (gcd (ring gcd)) (quotient (ring quotient)) (sign-r (ring sign)) (one-r (ring 'one)) (zero-r (ring 'zero))) (define (make-rational n d) (let ((s (sign-r d))) (cond ((=r one-r s) (cons '*numeric-type* (cons field (cons n d)))) ((=r zero-r s) (error "Bad rational construction" field (list n d))) (else (cons '*numeric-type* (cons field (cons (quotient n s) (quotient d s)))))))) (define numer caddr) (define denom cdddr) (define (+_rational x y) (let ((d1 (gcd (denom x) (denom y)))) (if (=r one-r d1) ;GIGO (make-rational (+r (*r (numer x) (denom y)) (*r (denom x) (numer y))) (*r (denom x) (denom y))) (let ((dx/d1 (quotient (denom x) d1))) (let ((tem (+r (*r (quotient (denom y) d1) (numer x)) (*r dx/d1 (numer y))))) (let ((d2 (gcd tem d1))) (make-rational (quotient tem d2) (*r dx/d1 (quotient (denom y) d2))))))))) (define (-_rational x y) (+_rational x (make-rational (negate-r (numer y)) (denom y)))) (define (*_rational x y) (let ((d1 (gcd (numer x) (denom y))) (d2 (gcd (numer y) (denom x)))) (make-rational (*r (quotient (numer x) d1) (quotient (numer y) d2)) (*r (quotient (denom x) d2) (quotient (denom y) d1))))) (define (/_rational x y) (*_rational x (make-rational (denom y) (numer y)))) (define (=_rational x y) (and (=r (numer x) (numer y)) (=r (denom x) (denom y)))) (define (negate-rational x) (make-rational (negate-r (numer x)) (denom x))) (define (recip-rational x) (make-rational (denom x) (numer x))) (define (raise i) (make-rational i one-r)) (put-coercion! raise ring field) (put-op! (raise zero-r) 'zero field) (put-op! (raise one-r) 'one field) (put-op! +_rational + field) (put-op! -_rational - field) (put-op! *_rational * field) (put-op! /_rational / field) (put-op! =_rational = field) (put-op! negate-rational negate field) (put-op! recip-rational recip field) (put-op! (lambda (x) (sign-r (numer x))) sign field) (put-op! numer numerator field) (put-op! denom denominator field) )) ;;;Now we can use this to define the rationals over integers (define (rational op) (get-op op rational)) (define (rational? x) (or (integer? x) (eq? (numeric-type x) rational))) (make-rational-operations! integer rational 'rational) ;;;For these particular rationals, we may also want some additional ;;;operations that are not sensible for arbitrary rationals, e.g., ;;;based upon polynomial rings. For example: (define (raise-rational-unary-operation op) (lambda (x) (op (rational->real x)))) (put-op! (raise-rational-unary-operation sqrt) sqrt rational) (put-op! (raise-rational-unary-operation sin) sin rational) (put-op! (raise-rational-unary-operation cos) cos rational) (put-op! (raise-rational-unary-operation tan) tan rational) (put-op! (raise-rational-unary-operation real-part) real-part rational) (put-op! (raise-rational-unary-operation imag-part) imag-part rational) (put-op! (raise-rational-unary-operation magnitude) magnitude rational) (put-op! (raise-rational-unary-operation angle) angle rational) -------

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