Is it possible to declare an ascending list?

Question

I can make lists of ascending integer like so:

?- findall(L,between(1,5,L),List).

I know I can also generate values using:

?- length(_,X).

But I don't think I can use this in a findall, as things like the following loop:

?- findall(X,(length(_,X),X<6),Xs).

I can also generate a list using clpfd.

:- use_module(library(clpfd)).

list_to_n(N,List) :-
   length(List,N),
   List ins 1..N,
   all_different(List),
   once(label(List)).

or

list_to_n2(N,List) :-
   length(List,N),
   List ins 1..N,
   chain(List,#<),
   label(List).

The last method seems best to me as it is the most declarative, and does not use once/1 or between/3 or findall/3 etc.

Are there other ways to do this? Is there a declarative way to do this in 'pure' Prolog? Is there a 'best' way?

Answer 1

The "best" way depends on your concrete use cases! Here's another way to do it using clpfd:

:- use_module(library(clpfd)).

We define predicate equidistant_stride/2 as suggested by @mat in a comment to a previous answer of a related question:

equidistant_stride([],_).
equidistant_stride([Z|Zs],D) :- 
   foldl(equidistant_stride_(D),Zs,Z,_).

equidistant_stride_(D,Z1,Z0,Z1) :-
   Z1 #= Z0+D.

Based on equidistant_stride/2, we define:

consecutive_ascending_integers(Zs) :-
   equidistant_stride(Zs,1).

consecutive_ascending_integers_from(Zs,Z0) :-
   Zs = [Z0|_],
   consecutive_ascending_integers(Zs).

consecutive_ascending_integers_from_1(Zs) :-
   consecutive_ascending_integers_from(Zs,1).

Let's run some queries! First, your original use case:

?- length(Zs,N), consecutive_ascending_integers_from_1(Zs).
  N = 1, Zs = [1]
; N = 2, Zs = [1,2]
; N = 3, Zs = [1,2,3]
; N = 4, Zs = [1,2,3,4]
; N = 5, Zs = [1,2,3,4,5]
...

With clpfd, we can ask quite general queries—and get logically sound answers, too!

?- consecutive_ascending_integers([A,B,0,D,E]).
A = -2, B = -1, D = 1, E = 2.

?- consecutive_ascending_integers([A,B,C,D,E]).
A+1#=B, B+1#=C, C+1#=D, D+1#=E.

---

An alternative implementation of equidistant_stride/2:

I hope the new code makes better use of constraint propagation.

Thanks to @WillNess for suggesting the test-cases that motivated this rewrite!

equidistant_from_nth_stride([],_,_,_).
equidistant_from_nth_stride([Z|Zs],Z0,N,D) :-
   Z  #= Z0 + N*D,
   N1 #= N+1,
   equidistant_from_nth_stride(Zs,Z0,N1,D).

equidistant_stride([],_).
equidistant_stride([Z0|Zs],D) :-
   equidistant_from_nth_stride(Zs,Z0,1,D).

Comparison of old vs new version with @mat's clpfd:

First up, the old version:

?- equidistant_stride([1,_,_,_,14],D).
_G1133+D#=14,
_G1145+D#=_G1133,
_G1157+D#=_G1145,
1+D#=_G1157.                               % succeeds with Scheinlösung

?- equidistant_stride([1,_,_,_,14|_],D).
  _G1136+D#=14, _G1148+D#=_G1136, _G1160+D#=_G1148, 1+D#=_G1160
; 14+D#=_G1340, _G1354+D#=14, _G1366+D#=_G1354, _G1378+D#=_G1366, 1+D#=_G1378
...                                        % does not terminate universally

Now let's switch to the new version and ask the same queries!

?- equidistant_stride([1,_,_,_,14],D).      
false.                                     % fails, as it should

?- equidistant_stride([1,_,_,_,14|_],D).
false.                                     % fails, as it should

---

More, now, again! Can we fail earlier by tentatively employing redundant constraints?

Previously, we proposed using constraints Z1 #= Z0+D*1, Z2 #= Z0+D*2, Z3 #= Z0+D*3 instead of Z1 #= Z0+D, Z2 #= Z1+D, Z3 #= Z2+D (which the 1st version of code in this answer did).

Again, thanks to @WillNess for motivating this little experiment by noting that the goal equidistant_stride([_,4,_,_,14],D) does not fail but instead succeeds with pending goals:

?- Zs = [_,4,_,_,14], equidistant_stride(Zs,D).
Zs = [_G2650, 4, _G2656, _G2659, 14],
14#=_G2650+4*D,
_G2659#=_G2650+3*D,
_G2656#=_G2650+2*D,
_G2650+D#=4.

Let's add some redundant constraints with equidistantRED_stride/2:

equidistantRED_stride([],_).
equidistantRED_stride([Z|Zs],D) :-
   equidistant_from_nth_stride(Zs,Z,1,D),
   equidistantRED_stride(Zs,D).

Sample query:

?- Zs = [_,4,_,_,14], equidistant_stride(Zs,D), equidistantRED_stride(Zs,D).
false.

Done? Not yet! In general we don't want a quadratic number of redundant constraints. Here's why:

?- Zs = [_,_,_,_,14], equidistant_stride(Zs,D).
Zs = [_G2683, _G2686, _G2689, _G2692, 14],
14#=_G2683+4*D,
_G2692#=_G2683+3*D,
_G2689#=_G2683+2*D,
_G2686#=_G2683+D.

?- Zs = [_,_,_,_,14], equidistant_stride(Zs,D), equidistantRED_stride(Zs,D).
Zs = [_G831, _G834, _G837, _G840, 14],
14#=_G831+4*D,
_G840#=_G831+3*D,
_G837#=_G831+2*D,
_G834#=_G831+D,
14#=_G831+4*D,
_G840#=_G831+3*D,
_G837#=_G831+2*D,
_G834#=_G831+D,
D+_G840#=14,
14#=2*D+_G837,
_G840#=D+_G837,
14#=_G834+3*D,
_G840#=_G834+2*D,
_G837#=_G834+D.

But if we use the double-negation trick, the residuum remains in cases that succeed ...

?- Zs = [_,_,_,_,14], equidistant_stride(Zs,D), \+ \+ equidistantRED_stride(Zs,D).
Zs = [_G454, _G457, _G460, _G463, 14],
14#=_G454+4*D,
_G463#=_G454+3*D,
_G460#=_G454+2*D,
_G457#=_G454+D.

... and ...

?- Zs = [_,4,_,_,14], equidistant_stride(Zs,D), \+ \+ equidistantRED_stride(Zs,D).
false.

... we detect failure in more cases than we did before!

---

Let's dig a little deeper! Can we detect failure early in even more generalized uses?

With code presented so far, these two logically false queries do not terminate:

?- Zs = [_,4,_,_,14|_], \+ \+ equidistantRED_stride(Zs,D), equidistant_stride(Zs,D).
...                                        % Execution Aborted

?- Zs = [_,4,_,_,14|_], equidistant_stride(Zs,D), \+ \+ equidistantRED_stride(Zs,D).
...                                        % Execution Aborted

Got fix? Got hack!

?- use_module(library(lambda)).
true.

?- Zs = [_,4,_,_,14|_], 
   \+ ( term_variables(Zs,Vs), 
        maplist(\X^when(nonvar(X),integer(X)),Vs),
        \+ equidistantRED_stride(Zs,D)),
   equidistant_stride(Zs,D).
false.

The hack doesn't guarantee termination of the redundant constraint "part", but IMO it's not too bad for a quick first shot. The test integer/1 upon instantiation of any variable in Zs is meant to allow the clpfd solver to constrain variable domains to singletons, while the instantiation with cons-pairs (which directly leads to non-termination of list-based predicates) is suppressed.

I do realize that the hack can be broken quite easily in more than one way (e.g., using cyclic terms). Any suggestions and comments are welcome!