The Amb library is part of the continuations talk that Chad and I gave

at last years Ruby conference. Amb is a library that allows you to

choose from a number of possibilities, e.g.:

```
A = Amb.new
x = A.choose(1,2,3,4)
# at this point, x may be 1, 2, 3, or possibly 4
```

And then assert things that must be true about the choice:

```
A.assert (x % 2)
# at this point, x can only be 2 or 4.
```

If the assertion is not true, Amb will rewind the program back to the

point of choosing, and make a different choice.

Here is the solution to part one of the quiz using the Amb library:

–BEGIN-----------------------------------------------------------

require ‘amb’

A = Amb.new

begin

a = A.choose(*(0…4))
b = A.choose(*(0…4))

c = A.choose(*(0…4))

A.assert a < b

A.assert a + b == c

puts “a=#{a}, b=#{b}, c=#{c}”

A.failure

rescue Amb::ExhaustedError

puts “No More Solutions”

end

–END-------------------------------------------------------------

Here’s the output of the solution:

$ ruby quiz70.rb

a=0, b=1, c=1

a=0, b=2, c=2

a=0, b=3, c=3

a=0, b=4, c=4

a=1, b=2, c=3

a=1, b=3, c=4

No More Solutions

The magic square solution is a bit more involved for two reasons. (1) I

created a magic square object that allowed me to express the constraints

easily, and the class turned out to be a bit wordy. (2) In order to

reduced the number of states searched for a solution, it is necessary to

assert constraints about the magic square as early as possible. For

example, we assert that a new number is unique as soon as we add it to

the square, rather than building the entire square and asserted that all

members are unique. This greatly reduced the number of possibilities

that needed to be tested and allowed the program to finish quite

quickly.

Here is the magic square solution:

–BEGIN------------------------------------------------------

#!/usr/bin/env ruby

# Ruby Q. # 70 – Constraint Solving

# Copyright 2006 by Jim W.

# Again, using Amb to solve the magic square constraint problem. The

# key to getting this to run half-way efficiently is to do the

# assertions as soon as possible, before additional choices have been

# made. This minimizes the amount of backtracking needed.

# See the following for details about Amb:

http://www.ccs.neu.edu/home/dorai/t-y-scheme/t-y-scheme-Z-H-16.html#node_chap_14

require ‘amb’

# A Magic Square class, mostly to get convenient column/row/diagonal

# summations and detecting previously used values.

class MagicSquare

def initialize(side)

@side = side

@values = Array.new(side**2) { 0 }

end

# Access square value

def [](col, row)

@values[col + @side*row]

end

# Set square value

def []=(col, row, value)

@values[index(col, row)] = value

end

# Calculate the sum of value along row +row+.

def row_sum(row)

(0…SIDE).inject(0) { |sum, col| sum + self[col,row] }

end

# Calculate the sum of values in column +col+.

def col_sum(col)

(0…SIDE).inject(0) { |sum, row| sum + self[col,row] }

end

# Calculate the sum of values along the major diagonal (i.e. r ==

# c).

def diagonal_sum

(0…SIDE).inject(0) { |sum, i| sum + self[i,i] }

end

# Calculate the sum of values along the other diagnol (i.e. r ==

# SIDE-c-1).

def other_diagonal_sum

(0…SIDE).inject(0) { |sum, i| sum + self[i,SIDE-i-1] }

end

# Has this value been previously used in the square in any previous

# row, or any previous column of the current row?

def previously_used?(col, row, value)

(0…index(col, row)).any? { |i| @values[i] == value }

end

# Convert to string for display.

def to_s

(0…SIDE).collect { |r|

(0…SIDE).collect { |c| self[c,r] }.join(’ ')

}.join("\n")

end

private

def index(col, row)

col + @side*row

end

end

SIDE = 3

MAX = SIDE**2

SUM = (MAX*(MAX+1))/(2*SIDE)

A = Amb.new

square = MagicSquare.new(SIDE)

# Build up the square position by position. To minimize backtracking,

# assert everything you know about a position as early as possible.

count = 0

begin

(0…SIDE).each do |r|

(0…SIDE).each do |c|

value = A.choose(*(1…MAX))

A.assert ! square.previously_used?(c, r, value)

square[c,r] = value

A.assert( (r < SIDE-1) || square.col_sum© == SUM)

end

A.assert square.row_sum® == SUM

end

A.assert square.diagonal_sum == SUM

A.assert square.other_diagonal_sum == SUM

count += 1

puts “SOLUTION #{count}:”

puts square

# Explicitly force a failure and make the program search for another

# solution.

A.failure

rescue Amb::ExhaustedError

puts “No More Solutions”

end

–END--------------------------------------------------------

And the output of the magic squares program:

$ ruby magic_square.rb

SOLUTION 1:

2 7 6

9 5 1

4 3 8

SOLUTION 2:

2 9 4

7 5 3

6 1 8

SOLUTION 3:

4 3 8

9 5 1

2 7 6

SOLUTION 4:

4 9 2

3 5 7

8 1 6

SOLUTION 5:

6 1 8

7 5 3

2 9 4

SOLUTION 6:

6 7 2

1 5 9

8 3 4

SOLUTION 7:

8 1 6

3 5 7

4 9 2

SOLUTION 8:

8 3 4

1 5 9

6 7 2

No More Solutions

Finally, we get to the Amb library. It is a direct Ruby port of the

Scheme amb function described in Teach yourself Scheme in fixnum days.

In brief, Amb saves a continuation whenever a choice has to be made.

Whenever an Amb assertion fails, the last choice is redone with a

different result by invoking the latest continuation. If all choices

are used up, then we go to the previous choice and undo retry it.

If you want a more indepth description of Amb, see

http://www.ccs.neu.edu/home/dorai/t-y-scheme/t-y-scheme-Z-H-16.html#node_chap_14

Here is the amb.rb library:

–BEGIN (amb.rb) ------------------------------------------------

class Amb

class ExhaustedError < RuntimeError; end

def initialize

@fail = proc { fail ExhaustedError, “amb tree exhausted” }

end

def choose(*choices)

prev_fail = @fail

callcc { |sk|

choices.each { |choice|

callcc { |fk|

@fail = proc {

@fail = prev_fail

fk.call(:fail)

}

if choice.respond_to? :call

sk.call(choice.call)

else

sk.call(choice)

end

}

}

@fail.call

}

end

def failure

choose

end

def assert(cond)

failure unless cond

end

end

–END------------------------------------------------------------

–

– Jim W.