“application: not a procedure” in binary arithmeti

问题:

I have a simple Racket definition for multiplying binary numbers together. It uses a well-tested "addWithCarry" definition that takes three parameters: two lists and a carry digit and returns the binary sum. The binary numbers are represented as lists in reverse order.

I stepped through the test line with the debugger, and it goes through the recursion properly. It performs the multBins each time shrinking the y list as appropriate, then conducts the addWithCarry functions as expected. As it rises back up the stack, it suddenly throws an exception "application: not a procedure, expected a procedure that can be applied to arguments" with the parameter '(0 0 0 1 0 1 1) which is the value of the highest "x" added to the total. I know this error can occur when you are attempting to apply the result of a function as a function with a parameter, but I don't see this here. Watching the debugger, everything seems to be working perfectly until the very end. Any ideas?

(define (multBins x y)
  (cond
    ((null? y)       '() )
    ((= (first y) 0) ((multBins (cons 0 x) (rest y))))
    (#t              ((addWithCarry x (multBins (cons 0 x) (rest y)) 0)))))  
(test (multBins '(1 0 1 1)'(1 1 0 1))'(1 1 1 1 0 0 0 1))

Here is the addWithCarry definition:

(define (addWithCarry x y carry)
  (cond
    ((and (null? x)(null? y)) (if (= carry 0) '() '(1)))
    ((null? x) (addWithCarry '(0) y carry))
    ((null? y) (addWithCarry x '(0) carry))
    ( #t  (let ((bit1 (first x))
            (bit2 (first y)))
               (cond
                 ((= (+ bit1 bit2 carry) 0) (cons 0 (addWithCarry (rest x) (rest y) 0)))
                 ((= (+ bit1 bit2 carry) 1) (cons 1 (addWithCarry (rest x) (rest y) 0)))
                 ((= (+ bit1 bit2 carry) 2) (cons 0 (addWithCarry (rest x) (rest y) 1)))
                 (   #t                     (cons 1 (addWithCarry (rest x) (rest y) 1))))))))

回答1:

In this line, you're calling multBins with (cons 0 x) and (rest y), and getting some result r, and then trying to call r:

((= (first y) 0) ((multBins (cons 0 x) (rest y))))
;                ^                              ^
;                +--- function application -----+

The same kind of thing is happening in the next line, where you're calling addWithCarry with some arguments, getting a result r, and trying to call r:

(#t              ((addWithCarry x (multBins (cons 0 x) (rest y)) 0)))))
;                ^                                                 ^
;                +-------------- function application -------------+

Presumable the unapplicable value '(0 0 0 1 0 1 1) is being returned by one of these.

In a very simplified case, consider this transcript from the DrRacket REPL:

> (define (value)        ; a function that returns the 
    '(0 0 0 1 0 1 1))    ; same value that yours should

> (value)                ; calling it produces the value 
(0 0 0 1 0 1 1)

> ((value))              ; calling it and then calling
                         ; return value causes the same
                         ; error that you're seeing
; application: not a procedure;
; expected a procedure that can be applied to arguments
;  given: (0 0 0 1 0 1 1)
;  arguments...: [none]

You didn't mention what editor/IDE/debugger you're using, but some should have made this a bit easier to spot. For instance, when I load your code (minus the call to test, whose definition I don't have, and with definitions of first and rest), DrRacket highlights the location of the offending call:

While both of the problematic calls that I've pointed out need to be fixed, the error that you're seeing right now is occurring in the second of the two.