Browsing through the awesome On-Line Encyclopedia of Integer Sequences (cf. en.wikipedia.org), I stumbled upon the following integer sequence:
A031877: Nontrivial reversal numbers (numbers which are integer multiples of their reversals), excluding palindromic numbers and multiples of 10.
By re-using some code I wrote for my answer to the related question "Faster implementation of verbal arithmetic in Prolog" I could write down a solution quite effortlessly—thanks to clpfd!
:- use_module(library(clpfd)).
We define the core relation a031877_ndigits_/3 based on
digits_number/2 (defined earlier):
a031877_ndigits_(Z_big,N_digits,[K,Z_small,Z_big]) :-
K #> 1,
length(D_big,N_digits),
reverse(D_small,D_big),
digits_number(D_big,Z_big),
digits_number(D_small,Z_small),
Z_big #= Z_small * K.
The core relation is deterministic and terminates universally whenever
N_digit is a concrete integer. See for yourself for the first 100 values of N_digit!
?- time((N in 0..99,indomain(N),a031877_ndigits_(Z,N,Zs),false)).
% 3,888,222 inferences, 0.563 CPU in 0.563 seconds (100% CPU, 6903708 Lips)
false
Let's run some queries!
?- a031877_ndigits_(87912000000087912,17,_). true % succeeds, as expected ; false. ?- a031877_ndigits_(87912000000987912,17,_). false. % fails, as expected
Next, let's find some non-trivial reversal numbers comprising exactly four decimal-digits:
?- a031877_ndigits_(Z,4,Zs), labeling([],Zs).
Z = 8712, Zs = [4,2178,8712]
; Z = 9801, Zs = [9,1089,9801]
; false.
OK! Let's measure the runtime needed to prove universal termination of above query!
?- time((a031877_ndigits_(Z,4,Zs),labeling([],Zs),false)).
% 11,611,502 inferences, 3.642 CPU in 3.641 seconds (100% CPU, 3188193 Lips)
false. % terminates universally
Now, that's way too long!
What can I do to speed things up? Use different and/or other constraints? Maybe even redundant ones? Or maybe identify and eliminate symmetries which slash the search space size? What about different clp(*) domains (b,q,r,set)? Or different consistency/propagation techniques? Or rather Prolog style coroutining?
Got ideas? I want them all! Thanks in advance.
So far ... no answers:(
I came up with the following...
How about using different variables for labeling/2?
a031877_ndigitsNEW_(Z_big,N_digits,/*[K,Z_small,Z_big]*/ [K|D_big]) :- K #> 1, length(D_big,N_digits), reverse(D_small,D_big), digits_number(D_big,Z_big), digits_number(D_small,Z_small), Z_big #= Z_small * K.
Let's measure some runtimes!
?- time((a031877_ndigits_(Z,4,Zs),labeling([ff],Zs),false)).
% 14,849,250 inferences, 4.545 CPU in 4.543 seconds (100% CPU, 3267070 Lips)
false.
?- time((a031877_ndigitsNEW_(Z,4,Zs),labeling([ff],Zs),false)).
% 464,917 inferences, 0.052 CPU in 0.052 seconds (100% CPU, 8962485 Lips)
false.
Better! But can we go further?
?- time((a031877_ndigitsNEW_(Z,5,Zs),labeling([ff],Zs),false)).
% 1,455,670 inferences, 0.174 CPU in 0.174 seconds (100% CPU, 8347374 Lips)
false.
?- time((a031877_ndigitsNEW_(Z,6,Zs),labeling([ff],Zs),false)).
% 5,020,125 inferences, 0.614 CPU in 0.613 seconds (100% CPU, 8181572 Lips)
false.
?- time((a031877_ndigitsNEW_(Z,7,Zs),labeling([ff],Zs),false)).
% 15,169,630 inferences, 1.752 CPU in 1.751 seconds (100% CPU, 8657015 Lips)
false.
There is still lots of room for improvement, for sure! There must be...