How to find the number of inversions in a list in Prolog

Question

As someone who\'s new to Prolog, I\'m looking to find out what a good way to count the number of inversions in a list.

I know how to flatten a matrix using flatten

Answer 1

Here's an alternative to my previous answer. It is based on clpfd and meta-predicate mapadj/3:

:- use_module(library(clpfd)).

Using meta-predicate tfilter/3, bool01_t/2, and clpfd sum/3 we define:

z_z_momsign(Z0,Z1,X) :-
   X #= max(-1,min(1,Z1-Z0)).

z_z_absmomsign(Z0,Z1,X) :-
   X #= min(1,abs(Z1-Z0)).

#\=(X,Y,Truth) :-
   X #\= Y #<==> B,
   bool01_t(B,Truth).

Finally, we define zs_invcount/2 like so:

zs_invcount(Zs,N) :-
   mapadj(z_z_momsign,Zs,Ms0),
   tfilter(#\=(0),Ms0,Ms),
   mapadj(z_z_absmomsign,Ms,Ds),
   sum(Ds,#=,N).

Sample use:

?- zs_invcount([1,2,3],0),    
   zs_invcount([1,2,3,2],1),    
   zs_invcount([1,2,3,3,2],1),               % works with duplicate items, too
   zs_invcount([1,2,3,3,2,1,1,1],1),
   zs_invcount([1,2,3,3,2,1,1,1,4,6],2),
   zs_invcount([1,2,3,3,2,1,1,1,4,6,9,1],3),
   zs_invcount([1,2,3,3,2,1,1,1,4,6,9,1,1],3).
true.

---

Edit

Consider the execution of following sample query in more detail:

?- zs_invcount([1,2,4,3,2,3,3,4,5,6,7,6,6,6,5,8],N).

Let's proceed step-by-step!

1. For all adjacent list items, calculate the sign of their "momentum":

   ?- Zs = [1,2,4,3,2,3,3,4,5,6,7,6,6,6,5,8], mapadj(z_z_momsign,Zs,Ms0).
   Zs  = [1,2, 4,3, 2,3,3,4,5,6,7, 6,6,6, 5,8],
   Ms0 = [ 1,1,-1,-1,1,0,1,1,1,1,-1,0,0,-1,1 ].
   

2. Eliminate all sign values of 0:

   ?- Ms0 = [1,1,-1,-1,1,0,1,1,1,1,-1,0,0,-1,1], tfilter(#\=(0),Ms0,Ms).
   Ms0 = [1,1,-1,-1,1,0,1,1,1,1,-1,0,0,-1,1],
   Ms  = [1,1,-1,-1,1,  1,1,1,1,-1,    -1,1].
   

3. Get the "momentum inversions", i.e., absolute signs of the momentum of momentums.

   ?- Ms = [1,1,-1,-1,1,1,1,1,1,-1,-1,1], mapadj(z_z_absmomsign,Ms,Ds).
   Ms = [1,1,-1,-1,1,1,1,1,1,-1,-1,1],
   Ds = [ 0,1, 0, 1,0,0,0,0,1, 0, 1 ].
   

4. Finally, sum up the number of "momentum inversions" using sum/3:

   ?- Ds = [0,1,0,1,0,0,0,0,1,0,1], sum(Ds,#=,N).
   N = 4, Ds = [0,1,0,1,0,0,0,0,1,0,1].
   

Or, alternatively, all steps at once:

:- Zs  = [1,2,4, 3, 2,3,3,4,5,6,7, 6,6,6, 5,8], mapadj(z_z_momsign,Zs,Ms0),
   Ms0 = [ 1,1,-1,-1,1,0,1,1,1,1,-1,0,0,-1,1 ], tfilter(#\=(0),Ms0,Ms),
   Ms  = [ 1,1,-1,-1,1,  1,1,1,1,-1,    -1,1 ], mapadj(z_z_absmomsign,Ms,Ds),
   Ds  = [  0,1, 0, 1, 0, 0,0,0,1,   0,   1  ], sum(Ds,#=,N),
   N   = 4.