Chapter 4, Metalinguistic Abstraction

Exercise 4.15


Let’s first see the code again:

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(define (run-forever) (run-forever))

(define (try p)
  (if (halts? p p)
      (run-forever)
      'halted))

(Knowing how to structure a proof, that I learned from book How to prove it helps)

Let’s assume that such a program halts? is possible. Thus it means that this procedure halts? can also tell whether (try try) halts or not!

Now, we evaluate (try try).

Assume that (halts? try try) returns true.

Thus (run-forever) executes and program goes into infinite loop. Thus (try try) does not terminate. But halts? says (try try) terminates by returning true! It follows that our assumption that (halts? try try) returned true is wrong.

So, let’s assume that (halts? try try) returns false.

Thus 'halted is printed and thus (try try) terminates. But halts? says (try try) does not terminate! Or our assumption is wrong again!

That means halts? can neither return true and nor it can return false for procedure try!

It follows that such program halts? that can tell weather any program halts or not is impossible.

One important thing to focus here:

This only proves that such procedure halts? is not possible for every program. There can be halts? procedure that can work for specific programs!


Since this marks the end of a sub-section and many new things have been added in the simulator as part of exercises. I think I shall put the complete code here till now:

(note that evaluator does not use data driven approach for application? procedure but apart from this every exercise is built on top of the previous changes in the evaluator as part of previous exercies).

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;;;;METACIRCULAR EVALUATOR FROM CHAPTER 4 (SECTIONS 4.1.1-4.1.4) of
;;;; STRUCTURE AND INTERPRETATION OF COMPUTER PROGRAMS

;;;;Matches code in ch4.scm

;;;;This file can be loaded into Scheme as a whole.
;;;;Then you can initialize and start the evaluator by evaluating
;;;; the two commented-out lines at the end of the file (setting up the
;;;; global environment and starting the driver loop).

;;;; resetting the environment so that we can reload the file
(ge (make-top-level-environment))

;;;;This warning here now not applicable in our case because of the statement above.
;;;;**WARNING: Don't load this file twice (or you'll lose the primitives
;;;;  interface, due to renamings of apply).

;;;from section 4.1.4 -- must precede def of metacircular apply
(define apply-in-underlying-scheme apply)

;;;SECTION 4.1.1

(define (eval exp env)
  (cond ((self-evaluating? exp) exp)
        ((variable? exp) (lookup-variable-value exp env))
        ((quoted? exp) (text-of-quotation exp))
        ((assignment? exp) (eval-assignment exp env))
        ((definition? exp) (eval-definition exp env))
        ((unbound!? exp) (eval-unbound! exp env))
        ((if? exp) (eval-if exp env))
        ((lambda? exp)
         (make-procedure (lambda-parameters exp)
                         (lambda-body exp)
                         env))
        ((begin? exp) 
         (eval-sequence (begin-actions exp) env))
        ((cond? exp) (eval-cond exp env))
		((logical? exp) (eval-logical exp env))
		((let? exp) (eval (let->combination exp) env))
		((let*? exp) (eval (let*->let exp) env))
		((for? exp) (eval (for->let exp) env))
        ((application? exp)
         (apply (eval (operator exp) env)
                (list-of-values (operands exp) env)))
        (else
         (error "Unknown expression type -- EVAL" exp))))

(define (apply procedure arguments)
  (cond ((primitive-procedure? procedure)
         (apply-primitive-procedure procedure arguments))
        ((compound-procedure? procedure)
         (eval-sequence
           (procedure-body procedure)
           (extend-environment
             (procedure-parameters procedure)
             arguments
             (procedure-environment procedure))))
        (else
         (error
          "Unknown procedure type -- APPLY" procedure))))

(define (list-of-values exps env)
  (if (no-operands? exps)
      '()
      (cons (eval (first-operand exps) env)
            (list-of-values (rest-operands exps) env))))

(define (eval-if exp env)
  (if (true? (eval (if-predicate exp) env))
      (eval (if-consequent exp) env)
      (eval (if-alternative exp) env)))

(define (eval-cond exp env)
  (define (iter clauses)
	(if (null? clauses)
		false
		(let ((first (car clauses))
			  (rest (cdr clauses)))
		  (cond ((cond-else-clause? first)
				 (if (null? rest)
					 (eval (sequence->exp (cond-actions first)) env)
					 (error "Else not last clause"
							(cond-clauses exp))))
				((cond-=>-clause? first)
				 (let ((pred (eval (cond-=>-predicate first) env)))
				   (if (true? pred)
					   (apply (eval (cond-=>-recipient first) env) (list pred))
					   (iter rest))))
				(else (if (true? (eval (cond-predicate first) env))
						  (eval (sequence->exp (cond-actions first)) env)
						  (iter rest)))))))
  (iter (cond-clauses exp)))
	 
(define (eval-sequence exps env)
  (cond ((last-exp? exps) (eval (first-exp exps) env))
        (else (eval (first-exp exps) env)
              (eval-sequence (rest-exps exps) env))))

(define (eval-assignment exp env)
  (set-variable-value! (assignment-variable exp)
                       (eval (assignment-value exp) env)
                       env)
  'ok)

(define (eval-definition exp env)
  (define-variable! (definition-variable exp)
                    (eval (definition-value exp) env)
                    env)
  'ok)

(define (boolean-impl val)
  (if (true? val) val false))

(define (eval-logical exp env)
  (define (iter predicates test?)
	(let ((first (eval (first-exp predicates) env)))
	  (if (test? first)
		  (if (last-exp? predicates)
			  (boolean-impl first)
			  (iter (rest-exps predicates) test?))
		  (boolean-impl first))))
  
  (let ((test? (if (and? exp) true? false?))
		(predicates (logical-predicates exp)))
	(if (null? predicates)
		(boolean-impl (test? '()))
		(iter predicates test?))))
	
;;;SECTION 4.1.2

(define (self-evaluating? exp)
  (cond ((number? exp) true)
        ((string? exp) true)
        (else false)))

(define (quoted? exp)
  (tagged-list? exp 'quote))

(define (text-of-quotation exp) (cadr exp))

(define (tagged-list? exp tag)
  (if (pair? exp)
      (eq? (car exp) tag)
      false))

(define (variable? exp) (symbol? exp))

(define (assignment? exp)
  (tagged-list? exp 'set!))

(define (assignment-variable exp) (cadr exp))

(define (assignment-value exp) (caddr exp))

(define (definition? exp)
  (tagged-list? exp 'define))

(define (definition-variable exp)
  (if (symbol? (cadr exp))
      (cadr exp)
      (caadr exp)))

(define (definition-value exp)
  (if (symbol? (cadr exp))
      (caddr exp)
      (make-lambda (cdadr exp)
                   (cddr exp))))

(define (unbound!? exp)
  (tagged-list? exp 'unbound!))

(define (unbound!-variable-name exp)
  (if (symbol? (cadr exp))
	  (cadr exp)
	  (error "Unbound variable name is not a symbol" exp)))

(define (eval-unbound! exp env)
  (unbound-variable! (unbound!-variable-name exp) env))

(define (lambda? exp) (tagged-list? exp 'lambda))

(define (lambda-parameters exp) (cadr exp))
(define (lambda-body exp) (cddr exp))

(define (make-lambda parameters body)
  (cons 'lambda (cons parameters body)))

(define (if? exp) (tagged-list? exp 'if))

(define (if-predicate exp) (cadr exp))

(define (if-consequent exp) (caddr exp))

(define (if-alternative exp)
  (if (not (null? (cdddr exp)))
      (cadddr exp)
      'false))

(define (make-if predicate consequent alternative)
  (list 'if predicate consequent alternative))


(define (begin? exp) (tagged-list? exp 'begin))

(define (begin-actions exp) (cdr exp))

(define (last-exp? seq) (null? (cdr seq)))
(define (first-exp seq) (car seq))
(define (rest-exps seq) (cdr seq))

(define (sequence->exp seq)
  (cond ((null? seq) seq)
        ((last-exp? seq) (first-exp seq))
        (else (make-begin seq))))

(define (make-begin seq) (cons 'begin seq))


(define (application? exp) (pair? exp))
(define (operator exp) (car exp))
(define (operands exp) (cdr exp))

(define (no-operands? ops) (null? ops))
(define (first-operand ops) (car ops))
(define (rest-operands ops) (cdr ops))


(define (cond? exp) (tagged-list? exp 'cond))

(define (cond-clauses exp) (cdr exp))

(define (cond-else-clause? clause)
  (eq? (cond-predicate clause) 'else))

(define (cond-predicate clause) (car clause))

(define (cond-actions clause) (cdr clause))

(define (cond-=>-clause? clause)
  (eq? '=> (cadr clause)))

(define (cond-=>-predicate clause) (car clause))

(define (cond-=>-recipient clause)
  (let ((recp (cddr clause)))
	(if (last-exp? recp)
		(first-exp recp)
		(error "recipient can not have multiple expressions" clause))))

(define (cond->if exp)
  (expand-clauses (cond-clauses exp)))

(define (expand-clauses clauses)
  (if (null? clauses)
      'false                          ; no else clause
      (let ((first (car clauses))
            (rest (cdr clauses)))
        (if (cond-else-clause? first)
            (if (null? rest)
                (sequence->exp (cond-actions first))
                (error "ELSE clause isn't last -- COND->IF"
                       clauses))
            (make-if (cond-predicate first)
                     (sequence->exp (cond-actions first))
                     (expand-clauses rest))))))

(define (and? exp) (tagged-list? exp 'and))

(define (or? exp) (tagged-list? exp 'or))

(define (logical? exp) (or (and? exp) (or? exp)))

(define (logical-predicates exp) (cdr exp))

(define (logical->if exp)
  (expand-predicates (and? exp) (logical-predicates exp)))

(define (expand-predicates and-type preds)
  (if (null? preds)
	  (if and-type 'true 'false)
	  (let ((first (car preds))
			(rest (cdr preds)))
		(let ((rest-exp (if (null? rest)
							first
							(expand-predicates and-type rest))))
		  (if and-type
			  (make-if first rest-exp 'false)
			  (make-if first first rest-exp))))))

(define (let? exp) (tagged-list? exp 'let))

(define (let-has-name? exp) (symbol? (cadr exp)))

(define (let-name exp) (cadr exp))

(define (let-vardefs exp)
  (if (let-has-name? exp)
	  (caddr exp)
	  (cadr exp)))

(define (let-body exp)
  (if (let-has-name? exp)
	  (cdddr exp)
	  (cddr exp)))

(define (let->combination exp)
  (let ((res (fold-right
			  (lambda (new rem)
				(cons (cons (car new) (car rem))
					  (cons (cadr new) (cdr rem))))
			  (cons '() '())
			  (let-vardefs exp))))
	(let ((vars (car res))
		  (vexps (cdr res)))
	  (define proc (make-lambda vars (let-body exp)))
	  (if (let-has-name? exp)
		  (cons
		   (make-lambda '()
						(list (list 'define (let-name exp) proc)
							  (cons (let-name exp) vexps)
							  )) '())
		  (cons proc vexps)
	  ))))

;; var-defs and body should be list
(define (make-let var-defs body)
  (cons 'let (cons var-defs body)))

;; var-defs and body should be list
(define (make-named-let name var-defs body)
  (cons 'let (cons name (cons var-defs body))))

(define (let*? exp) (tagged-list? exp 'let*))

(define (let*->let exp)
  (car
   (fold-right (lambda(new rem)
				 (list (make-let (list new) rem)))
			   (let-body exp)
			   (let-vardefs exp))))

(define (for? exp) (tagged-list? exp 'for))

(define (for-var exp) (cadr exp))

(define (for-data exp) (caddr exp))

(define (for-body exp) (cdddr exp))

(define (for->let exp)
  (define body
	(list 'define 'body (make-lambda
						 (list (for-var exp))
						 (for-body exp))))
  (define cond-exp '(cond ((null? l) '())
		((not (pair? l)) (error "invalid data- for loop expects list"))
		((null? (cdr l)) (body (car l)))
		(else (body (car l)) (iter (cdr l)))))
  (make-named-let 'iter
				  (list (list 'l (for-data exp)))
				  (list body cond-exp)))

;;;SECTION 4.1.3

(define (true? x)
  (not (eq? x false)))

(define (false? x)
  (eq? x false))

(define (make-procedure parameters body env)
  (list 'procedure parameters body env))

(define (compound-procedure? p)
  (tagged-list? p 'procedure))


(define (procedure-parameters p) (cadr p))
(define (procedure-body p) (caddr p))
(define (procedure-environment p) (cadddr p))


(define (enclosing-environment env) (cdr env))

(define (first-frame env) (car env))

(define the-empty-environment '())

;; we need cons to contain list of pairs.
;; the reason is same as when we build tables in ch-3:
;; if we add at beginning of list or pairs
;; it will change the initial point of the pairs.
;; thus we need a place to point to the head of this list. 
(define (make-frame variables values)
  (cons 'table (map cons variables values)))

;;(define (frame-variables frame) (car frame))
;;(define (frame-values frame) (cdr frame))

(define (frame-pairs frame) (cdr frame))

(define (find contains? next data)
  (if (null? data)
	  #f
	  (let ((found (contains? data)))
		(if found
			found
			(find contains? next (next data))))))

(define (assoc var table)
  (find (lambda(tbl)
		  (let ((pair (car tbl)))
			(if (eq? (car pair) var)
				pair
				#f)))
		 cdr
		 table))

(define (find-pair var env)
  (find (lambda(e)
		  (assoc var (frame-pairs (first-frame e))))
		enclosing-environment
		env))

(define (add-binding-to-frame! var val frame)
  (set-cdr! frame (cons (cons var val) (cdr frame))))

(define (extend-environment vars vals base-env)
  (if (= (length vars) (length vals))
      (cons (make-frame vars vals) base-env)
      (if (< (length vars) (length vals))
          (error "Too many arguments supplied" vars vals)
          (error "Too few arguments supplied" vars vals))))

(define (lookup-variable-value var env)
  (let ((pair (find-pair var env)))
	(if pair
		(cdr pair)
		(error "Unbound variable" var))))

(define (set-variable-value! var val env)
  (let ((pair (find-pair var env)))
	(if pair
		(set-cdr! pair val)
		(error "Unbound variable" var))))

(define (define-variable! var val env)
  (let ((pair (assoc var (frame-pairs (first-frame env)))))
	(if pair
		(set-cdr! pair val)
		(add-binding-to-frame! var val (first-frame env))
		)))

(define (unbound-variable! var env)
  (let ((pair (find-pair var env)))
	(if pair
		(begin (set-car! pair '()) (set-cdr! pair '()))
		(error "Unbound variable" var))))

;;;SECTION 4.1.4

(define (setup-environment)
  (let ((initial-env
         (extend-environment (primitive-procedure-names)
                             (primitive-procedure-objects)
                             the-empty-environment)))
    (define-variable! 'true true initial-env)
    (define-variable! 'false false initial-env)
    initial-env))

;[do later] (define the-global-environment (setup-environment))

(define (primitive-procedure? proc)
  (tagged-list? proc 'primitive))

(define (primitive-implementation proc) (cadr proc))

(define primitive-procedures
  (list (list 'car car)
        (list 'cdr cdr)
        (list 'cons cons)
        (list 'null? null?)
		(list '+ +)
		(list 'assoc assoc)
		(list 'cadr cadr)
		(list 'cddr cddr)
		(list '= =)
		(list '- -)
        (list 'pair? pair?)
        (list 'not not)
        (list 'newline newline)
        (list 'display display)
        (list 'error error)
;;      more primitives
        ))

(define (primitive-procedure-names)
  (map car
       primitive-procedures))

(define (primitive-procedure-objects)
  (map (lambda (proc) (list 'primitive (cadr proc)))
       primitive-procedures))

(define (user-print object)
  (if (compound-procedure? object)
      (display (list 'compound-procedure
                     (procedure-parameters object)
                     (procedure-body object)
                     '<procedure-env>))
      (display object)))

(define (apply-primitive-procedure proc args)
  (apply-in-underlying-scheme
   (primitive-implementation proc) args))

(define input-prompt ";;; M-Eval input:")
(define output-prompt ";;; M-Eval value:")
(define exit-symbol 'exit)

(define (driver-loop)
  (prompt-for-input input-prompt)
  (let ((input (read)))
	(if (eq? input exit-symbol)
		'exiting..
		(let ((output (eval input the-global-environment)))
		  (announce-output output-prompt)
		  (user-print output)
		  (driver-loop)))))

;;[user-print moved before primitive procedure as it is used there]

(define (prompt-for-input string)
  (newline) (newline) (display string) (newline))

(define (announce-output string)
  (newline) (display string) (newline))

;;;Following are commented out so as not to be evaluated when
;;; the file is loaded.

(define the-global-environment (setup-environment))
(driver-loop)

'METACIRCULAR-EVALUATOR-LOADED