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loading... # Prolog Natural Numbers Arĥimedeς ℳontegasppα ℭacilhας Updated on ・3 min read

λ-calculus is the calculus deals with only one data type: function. The simpliest (thus main) numeric type in it is the natural number.

Its sign might look strange at first sight, but it’s actually easy:

``````0 ≡ λsz.z
1 ≡ λsz.sz
2 ≡ λsz.s(sz)
3 ≡ λsz.s(s(sz))
4 ≡ λsz.s(s(s(sz)))
``````

It goes like this: each function has two arguments, the first (`s`) is the increment (or successor) function, and the second is the zero (`z`).

• The zero function returns zero (`z`);
• The one function return the zero’s successor (`sz`, 1);
• The two function return the one’s successor (`s(sz)`, 2);
• And so on.

Some programming languages brings natural numbers built-in. For example, Idris defines natural numbers like this:

``````data Nat : Type where
Z : Nat
S : Nat -> Nat
``````

`Z` is zero, `S Z` is one, `S \$ S Z` is two, `S \$ S \$ S Z` is three, and so on.

### Prolog

It’s possible to emulate this behaviour using Prolog.

First we need a predicate to define natural numbers according to the λ-calculus. It could be `nat/1`, like in Idris:

``````nat(z).
nat(s(N)) :- nat(N).
``````

It’s solved!

### Casting to integer

Hereat, we’ll convert into and from integer type.

An easy way of casting into integer is:

``````to_int(z, 0).
to_int(s(N), R) :- to_int(N, R1), R is R1 + 1.
``````

It could be a solution, but there’s an issue: `to_int/2` accumulates function stacks, perchange easily leading to a stack overflow.

We can solve it by tail-call optimisation: the `to_int/2` must delegate the procedure to its accumulating version `to_int/3`.

So `to_int/2` becames:

``````to_int(N, R) :- nat(N), to_int(N, 0, R).
``````

And the accumulating version `to_int/3` should be:

``````to_int(z, A, A).
to_int(s(N), A, R) :- succ(A, A1), to_int(N, A1, R).
``````

Or, using DCG:

``````to_int(z) --> '='.
to_int(s(N)) --> succ, to_int(N).
``````

### Casting from integer

For the backward casting, we’ll need a natural successor predicate for `s(N)`:

``````nat_succ(N, s(N)) :- nat(N).
``````

Here’s the `from_int/2` (and its pair `from_int/3`):

``````from_int(I, R) :- integer(I), I >= 0, from_int(I, z, R).

from_int(0) --> '='.
from_int(I) --> { I1 is I - 1 }, nat_succ, from_int(I1).
``````

### Let’s see it working

Save it all into `natural.pl`, and then:

``````sh\$ swipl -q

:- [natural].
true.

:- natural:to_int(z, X).
X = 0.

:- natural:to_int(s(s(s(z))), X).
X = 3.

:- natural:from_int(8, X).
X = s(s(s(s(s(s(s(s(z)))))))).
``````

### Making it executable

Append to the `natural.pl`’s end:

``````go :- current_prolog_flag(argv, [Argv]),
atom_to_term(Argv, I, []),
from_int(I, N),
writeln(N), !.
go :- current_prolog_flag(os_argv, [_, '-x', Path | _]),
file_base_name(Path, Script),
format('use: ~w <integer>~n', [Script]).
``````

Then, compile it:

``````sh\$ swipl -q

:- [library(qsave), natural].
true.

:- qsave_program(natural, [init_file('natural.pl'), goal(natural:go), toplevel(halt)]).
true.

:-
``````

Finally you can run:

``````sh\$ ./natural 12
s(s(s(s(s(s(s(s(s(s(s(s(z))))))))))))
sh\$
``````

### Bonus: even and odd

We can determinate if a natural number is even or odd by:

``````even(z).
even(s(N)) :- odd(N).
odd(N) :- \+ even(N).
``````

Note: `\+` means logic negation.