In this example (an earlier level) the goal is to create fish. We can transmute
or exchange (I will only write about exchange from now on) an apple and a pair
of cherries for a fish. Apple and cherry we have, or are created from thin air,
in the context of the game. As you can see in the other example, this can
get very complex.

The question is, given some input we have, in what order do we have to use
the stations for exchange to create some output. How do we best model this
problem?

## Modeling the Problem

From a previous article we already know
that we can encode numbers in Prolog terms using Peano numbers.
Using these we can match terms that represent at least or exactly some number.
Exactly 3 would be `s(s(s(z)))` while at least 3 would be `s(s(s(X)))` with
the variable `X` representing the rest of the term for some \(n\) so that \(3 + x = n\).

Let's start with a small example: we have an aunt that gives us apples. We have
bob who is willing to exchange an apple for two berries. And we have a cousin
who will bake a cake if we give them two apples and five berries.

Let's formalize our rules in some way:

a: / -> 1A
b: A -> 2B
c: 2A, 5B -> 1C

We model this as some kind of `inventory(A, B, C)` with some slots for Peano encoded
numbers. Meaning that `inventory(s(z), z, s(C))` means an inventory of exactly
one apple, zero berries, and at least one cake. Lucky us.

Let's define our possible exchanges as Prolog facts with `exchange(Kind, Pre, Post)`
meaning we use the exchange method `Kind` and we get from `inventory` state
`Pre` to `inventory` state `Post`. If we lose an item, we have it in
`Pre` but don't in `Post`. If we gain one, it's the other way around.
Of course, the rest of our inventory should stay the same.

exchange(a, inventory(A, B, C), inventory(s(A), B, C)).
exchange(b, inventory(s(A),B,C), inventory(A, s(s(B)), C)).
exchange(c, inventory(s(s(A)), s(s(s(s(s(B))))), C), inventory(A, B, s(C))).

As we can see, we can encode basically any numeric condition that uses only
"just exactly" and "at least", as well as addition and subtraction and "setting"
fields to specific values. If we have to give at least three apples, but all of them,
to get a cake, the rule would be `exchange(c, inventory(s(s(s(_))), B, C), inventory(z, B, s(C))).`.

## Programming the Solver

We want to create a predicate `trades(Start, Goal, Path)` where we give a
`Start` inventory and a `Goal` inventory and want to get a `Path` of
exchanges. The most obvious approach is the following:

trades(Goal, Goal, []).
trades(Start, Goal, [H|T]) :- exchange(H, Start, Temp), trades(Temp, Goal, T).

So we're done when our `Goal` is reached. And if not, we have to try some
`exchange`. We try it and we quickly learn that...

?- trades(inventory(z, z, z), inventory(_, _, s(z)), Path).
ERROR: Out of local stack

... it doesn't work.

The problem is, that calling the `exchange` predicate in the recursive call
always tries the first rule first and since that basically always works, we get
into an infinite loop. Let's reorder the rules and put the generating rule first.

?- trades(inventory(z, z, z), inventory(_, _, s(z)), Path).
ERROR: Out of local stack

It still doesn't work. Let's look at the `trace`.

[trace] ?- trades(inventory(z, z, z), inventory(_, _, s(z)), Path).
Call: (8) trades(inventory(z, z, z), inventory(_700, _702, s(z)), _712) ? creep
Call: (9) exchange(_960, inventory(z, z, z), _984) ? creep
Exit: (9) exchange(a, inventory(z, z, z), inventory(s(z), z, z)) ? creep
Call: (9) trades(inventory(s(z), z, z), inventory(_700, _702, s(z)), _962) ? creep
Call: (10) exchange(_978, inventory(s(z), z, z), _1002) ? creep
Exit: (10) exchange(b, inventory(s(z), z, z), inventory(z, s(s(z)), z)) ? creep
Call: (10) trades(inventory(z, s(s(z)), z), inventory(_700, _702, s(z)), _980) ? creep
Call: (11) exchange(_1000, inventory(z, s(s(z)), z), _1024) ? creep
Exit: (11) exchange(a, inventory(z, s(s(z)), z), inventory(s(z), s(s(z)), z)) ? creep
Call: (11) trades(inventory(s(z), s(s(z)), z), inventory(_700, _702, s(z)), _1002) ? creep
Call: (12) exchange(_1018, inventory(s(z), s(s(z)), z), _1042) ? creep
Exit: (12) exchange(b, inventory(s(z), s(s(z)), z), inventory(z, s(s(s(s(z)))), z)) ? creep
Call: (12) trades(inventory(z, s(s(s(s(z)))), z), inventory(_700, _702, s(z)), _1020) ? creep
Call: (13) exchange(_1040, inventory(z, s(s(s(s(z)))), z), _1064) ? creep
Exit: (13) exchange(a, inventory(z, s(s(s(s(z)))), z), inventory(s(z), s(s(s(s(z)))), z)) ? creep
Call: (13) trades(inventory(s(z), s(s(s(s(z)))), z), inventory(_700, _702, s(z)), _1042) ? creep
Call: (14) exchange(_1058, inventory(s(z), s(s(s(s(z)))), z), _1082) ? creep
Exit: (14) exchange(b, inventory(s(z), s(s(s(s(z)))), z), inventory(z, s(s(s(s(s(s(z)))))), z)) ? creep
Call: (14) trades(inventory(z, s(s(s(s(s(s(z)))))), z), inventory(_700, _702, s(z)), _1060) ? creep
Call: (15) exchange(_1080, inventory(z, s(s(s(s(s(s(z)))))), z), _1104) ? creep
Exit: (15) exchange(a, inventory(z, s(s(s(s(s(s(z)))))), z), inventory(s(z), s(s(s(s(s(s(z)))))), z)) ? creep
Call: (15) trades(inventory(s(z), s(s(s(s(s(s(z)))))), z), inventory(_700, _702, s(z)), _1082) ? creep
Call: (16) exchange(_1098, inventory(s(z), s(s(s(s(s(s(z)))))), z), _1122) ? creep

We're still running into an infinite loop since we're just getting apples
and trading them for berries.

There's a trick if our solution is a list. If the `length(List, Length)` predicate
is called with both predicates free (`length(L, _)` for example), we get
lists of increasing length as solutions. So if we can bind our solution to
lists of increasing length, we won't get infinite loops of apple-berry-exchanges
without ever trying short solutions.

So our query now looks like this:

?- length(Path,_), trades(inventory(z, z, z), inventory(_, _, s(z)), Path).
Path = [a, b, a, b, a, b, a, a, c] ;
Path = [a, b, a, b, a, a, b, a, c] ;
Path = [a, b, a, b, a, a, a, b, c] ;
Path = [a, b, a, a, b, b, a, a, c] ;
Path = [a, b, a, a, b, a, b, a, c] ;
Path = [a, b, a, a, b, a, a, b, c] ;
Path = [a, b, a, a, a, b, b, a, c] ;
Path = [a, b, a, a, a, b, a, b, c] ;
Path = [a, b, a, a, a, a, b, b, c] ...

As it turns out, there are a few ways to do it, but nothing shorter than 9
trades. We can even look, whether we have anything left over after we got
our cake:

?- length(Path,_), trades(inventory(z, z, z), inventory(A, B, s(z)), Path).
Path = [a, b, a, b, a, b, a, a, c],
A = z,
B = s(z) .

Yeah. As it turns out, we have a berry to spare.

## Conclusion

This is how you do it. And of course, you could simulate a shop with that,
as one inventory slot could be the number of coins. If the inventory gets
too large, you can split it into item types so you don't have to copy every
unchanged item from the pre-condition to the post-condition. Then you can copy
a term that might represent any number of items.

But beware: this algorithm (with Prolog's evaluation scheme) has not so good
complexity. For a solution of length \(n\) and \(m\) rules it
has a run time of \(O(m^n)\). We create every possible solution list permutation
but break early, if no rules are applicable.

Depending on our instantiation of `Start` and `Goal`, this algorithm doesn't terminate (if the system isn't terminating, the algorithm isn't either).
Our system
doesn't terminate as we can always apply `a` again and again.
We're creating increasingly longer partial solutions which might turn out to
be no solution at all. If there is no solution, you won't know. The algorithm
just tries longer and longer lists trying to find a solution.

If you have any other conditions on your solution (some exchanges might be limited
or necessary) you can put most of them after the `trades` call. Some conditions,
like membership, can be put before.

Let's say we have a terminating system by taking away the `a` rule.
If our `Start` is fully instantiated, then our
algorithm terminates. Let's try another query and find out, with how many apples
we need to start to get a cake:

?- trades(inventory(A, z, z), inventory(_, _, s(z)), Path).
ERROR: Out of local stack

Through the unification rules, it is possible that a rule creates terms
on the left side of the rule. In this case, the algorithm tries to give
us increasing amounts of berries to match that with the starting condition
(only rule `b` is applied in reverse, always leaving us with one cake,
which doesn't match the starting condition).
There we need the `length` predicate.

?- length(Path,_), trades(inventory(A, z, z), inventory(_, _, s(z)), Path).
Path = [b, b, b, c],
A = s(s(s(s(s(_2334))))) .

We need at least five apples for a cake. But we knew that.

To recap:

- We saw again how useful it is, to encode the natural numbers in a term structure.
- A solver is easily written but the algorithm isn't fast.
- The rules can be applied in both directions because of Prolog's evaluation scheme. This leaves queries that would intuitively work divergent.
- By using the
`length` predicate we can simulate breadth-first evaluation, while the default evaluation scheme is depth-first.

I think, there's a different approach using linear programming. I might try
that in the future.