#+TITLE: 4.3 - Variations on a Scheme — Nondeterministic Computing
#+STARTUP: indent
#+OPTIONS: num:nil

* COMMENT Set up source file
#+BEGIN_SRC scheme :tangle yes
  ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
  ;; 4.3 - Variations on a Scheme — Nondeterministic Computing
  ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

  (load "4-2.scheme")

#+END_SRC

#+BEGIN_QUOTE
Just as the lazy evaluator freed the programmer from the
details of how values are delayed and forced, the nondeterministic
program evaluator will free the programmer from the details of how
choices are made.
#+END_QUOTE
* <<4.3.1>> Amb and Search

The new special form ~amb~ returns the value of one of the expressions
passed to it "ambiguously".

#+BEGIN_SRC scheme :tangle yes
  (define (require p)
    (if (not p) (amb)))

  (define (an-element-of items)
    (require (not (null? items)))
    (amb (car items) (an-element-of (cdr items))))

  (define (an-integer-starting-from n)
    (amb n (an-integer-starting-from (+ n 1))))
#+END_SRC

** Exercise 4.35
Write a procedure `an-integer-between' that returns an integer between
two given bounds.  This can be used to implement a procedure that
finds Pythagorean triples, i.e., triples of integers (i,j,k) between
the given bounds such that i <= j and i^2 + j^2 = k^2, as follows:

#+BEGIN_SRC scheme
  (define (a-pythagorean-triple-between low high)
    (let ((i (an-integer-between low high)))
      (let ((j (an-integer-between i high)))
        (let ((k (an-integer-between j high)))
          (require (= (+ (* i i) (* j j)) (* k k)))
          (list i j k)))))
#+END_SRC

----------------------------------------------------------------------

#+BEGIN_SRC scheme
  (define (an-integer-between low high)
    (let ((i (an-integer-starting-from (+ low 1))))
      (require (< i high))
      i))
#+END_SRC
** Exercise 4.36
*Note Exercise 3-69:: discussed how to generate the stream of _all_
Pythagorean triples, with no upper bound on the size of the integers
to be searched.  Explain why simply replacing `an-integer-between' by
`an-integer-starting-from' in the procedure in *Note Exercise 4-35::
is not an adequate way to generate arbitrary Pythagorean triples.
Write a procedure that actually will accomplish this.  (That is, write
a procedure for which repeatedly typing `try-again' would in principle
eventually generate all Pythagorean triples.)

** Exercise 4.37
Ben Bitdiddle claims that the following method for generating
Pythagorean triples is more efficient than the one in *Note Exercise
4-35::.  Is he correct?  (Hint: Consider the number of possibilities
that must be explored.)

#+BEGIN_SRC scheme
  (define (a-pythagorean-triple-between low high)
    (let ((i (an-integer-between low high))
          (hsq (* high high)))
      (let ((j (an-integer-between i high)))
        (let ((ksq (+ (* i i) (* j j))))
          (require (>= hsq ksq))
          (let ((k (sqrt ksq)))
            (require (integer? k))
            (list i j k))))))
#+END_SRC

* <<4.3.2>> Examples of Nondeterministic Programs
** Exercise 4.41
Write an ordinary Scheme program to solve the
multiple dwelling puzzle.

----------------------------------------------------------------------

*/Nope./*
* Links

The following is a set of semi-relevant links that came up during the discussion:

- https://aphyr.com/posts/314-computational-techniques-in-knossos
- https://bernardopires.com/2013/10/try-logic-programming-a-gentle-introduction-to-prolog/
- http://www.cs.bham.ac.uk/~hxt/research/Logiccolumn8.pdf
  Continuation, functions and jumps
- http://double.co.nz/pdf/continuations.pdf
  Web programming with continuations
- http://www.codersatwork.com/