使用 Prolog 优化约束逻辑编程中的寻路

Question

我正在开发一个小型 prolog 应用程序来解决 Skyscrapers and Fences 难题。

一个未解之谜：

一个已解决的谜题：

当我通过已解决难题的程序时，它会很快，几乎是瞬间为我验证它。当我通过程序非常小的谜题（例如 2x2，当然要修改规则）时，找到解决方案的速度也相当快。

问题在于计算“原生”大小为 6x6 的谜题。在中止它之前，我已经让它运行了 5 个小时左右。时间太长了。

我发现耗时最长的部分是“栅栏”，而不是“摩天大楼”。单独运行“摩天大楼”会产生快速的解决方案。

这是我的栅栏算法：

• 顶点用数字表示，0 表示路径不经过该特定顶点，> 1 表示该顶点在路径中的顺序。
• 限制每个单元格周围有适当数量的线条。
  • 这意味着如果两个顶点具有连续编号，则它们是连接的，例如，1 -> 2, 2 -> 1, 1 -> Max, Max -> 1（Max 是路径中的最后一个顶点。通过maximum/2 计算）
• 确保每个非零顶点至少有两个具有连续编号的相邻顶点
• 将Max 约束为等于(BoardWidth + 1)^2 - NumberOfZeros（BoardWidth+1 是沿边的顶点数，NumberOfZeros 是通过count/4 计算的）。
• 使用nvalue(Vertices, Max + 1) 确保Vertices 中不同值的数量为Max（即路径中的顶点数量）加上1（零值）
• 查找包含3 的第一个单元格并强制路径从那里开始和结束，以提高效率

我可以做些什么来提高效率？代码包含在下面以供参考。

skyscrapersinfences.pro

:-use_module(library(clpfd)).
:-use_module(library(lists)).

:-ensure_loaded('utils.pro').
:-ensure_loaded('s1.pro').

print_row([]).

print_row([Head|Tail]) :-
    write(Head), write(' '),
    print_row(Tail).

print_board(Board, BoardWidth) :-
    print_board(Board, BoardWidth, 0).

print_board(_, BoardWidth, BoardWidth).

print_board(Board, BoardWidth, Index) :-
    make_segment(Board, BoardWidth, Index, row, Row),
    print_row(Row), nl,
    NewIndex is Index + 1,
    print_board(Board, BoardWidth, NewIndex).

print_boards([], _).
print_boards([Head|Tail], BoardWidth) :-
    print_board(Head, BoardWidth), nl,
    print_boards(Tail, BoardWidth).

get_board_element(Board, BoardWidth, X, Y, Element) :-
    Index is BoardWidth*Y + X,
    get_element_at(Board, Index, Element).

make_column([], _, _, []).

make_column(Board, BoardWidth, Index, Segment) :-
    get_element_at(Board, Index, Element),
    munch(Board, BoardWidth, MunchedBoard),
    make_column(MunchedBoard, BoardWidth, Index, ColumnTail),
    append([Element], ColumnTail, Segment).

make_segment(Board, BoardWidth, Index, row, Segment) :-
    NIrrelevantElements is BoardWidth*Index,
    munch(Board, NIrrelevantElements, MunchedBoard),
    select_n_elements(MunchedBoard, BoardWidth, Segment).

make_segment(Board, BoardWidth, Index, column, Segment) :-
    make_column(Board, BoardWidth, Index, Segment).

verify_segment(_, 0).
verify_segment(Segment, Value) :-
    verify_segment(Segment, Value, 0).

verify_segment([], 0, _).
verify_segment([Head|Tail], Value, Max) :-
    Head #> Max #<=> B, 
    Value #= M+B,
    maximum(NewMax, [Head, Max]),
    verify_segment(Tail, M, NewMax).

exactly(_, [], 0).
exactly(X, [Y|L], N) :-
    X #= Y #<=> B,
    N #= M  +B,
    exactly(X, L, M).

constrain_numbers(Vars) :-
    exactly(3, Vars, 1),
    exactly(2, Vars, 1),
    exactly(1, Vars, 1).

iteration_values(BoardWidth, Index, row, 0, column) :-
    Index is BoardWidth - 1.

iteration_values(BoardWidth, Index, Type, NewIndex, Type) :-
    \+((Type = row, Index is BoardWidth - 1)),
    NewIndex is Index + 1.

solve_skyscrapers(Board, BoardWidth) :-
    solve_skyscrapers(Board, BoardWidth, 0, row).

solve_skyscrapers(_, BoardWidth, BoardWidth, column).

solve_skyscrapers(Board, BoardWidth, Index, Type) :-
    make_segment(Board, BoardWidth, Index, Type, Segment),

    domain(Segment, 0, 3),
    constrain_numbers(Segment),

    observer(Type, Index, forward, ForwardObserver),
    verify_segment(Segment, ForwardObserver),

    observer(Type, Index, reverse, ReverseObserver),
    reverse(Segment, ReversedSegment),
    verify_segment(ReversedSegment, ReverseObserver),

    iteration_values(BoardWidth, Index, Type, NewIndex, NewType),
    solve_skyscrapers(Board, BoardWidth, NewIndex, NewType).

build_vertex_list(_, Vertices, BoardWidth, X, Y, List) :-
    V1X is X, V1Y is Y, V1Index is V1X + V1Y*(BoardWidth+1),
    V2X is X+1, V2Y is Y, V2Index is V2X + V2Y*(BoardWidth+1),
    V3X is X+1, V3Y is Y+1, V3Index is V3X + V3Y*(BoardWidth+1),
    V4X is X, V4Y is Y+1, V4Index is V4X + V4Y*(BoardWidth+1),
    get_element_at(Vertices, V1Index, V1),
    get_element_at(Vertices, V2Index, V2),
    get_element_at(Vertices, V3Index, V3),
    get_element_at(Vertices, V4Index, V4),
    List = [V1, V2, V3, V4].

build_neighbors_list(Vertices, VertexWidth, X, Y, [NorthMask, EastMask, SouthMask, WestMask], [NorthNeighbor, EastNeighbor, SouthNeighbor, WestNeighbor]) :-
    NorthY is Y - 1,
    EastX is X + 1,
    SouthY is Y + 1,
    WestX is X - 1,
    NorthNeighborIndex is (NorthY)*VertexWidth + X,
    EastNeighborIndex is Y*VertexWidth + EastX,
    SouthNeighborIndex is (SouthY)*VertexWidth + X,
    WestNeighborIndex is Y*VertexWidth + WestX,
    (NorthY >= 0, get_element_at(Vertices, NorthNeighborIndex, NorthNeighbor) -> NorthMask = 1 ; NorthMask = 0),
    (EastX < VertexWidth, get_element_at(Vertices, EastNeighborIndex, EastNeighbor) -> EastMask = 1 ; EastMask = 0),
    (SouthY < VertexWidth, get_element_at(Vertices, SouthNeighborIndex, SouthNeighbor) -> SouthMask = 1 ; SouthMask = 0),
    (WestX >= 0, get_element_at(Vertices, WestNeighborIndex, WestNeighbor) -> WestMask = 1 ; WestMask = 0).

solve_path(_, VertexWidth, 0, VertexWidth) :-
    write('end'),nl.

solve_path(Vertices, VertexWidth, VertexWidth, Y) :-
    write('switch row'),nl,
    Y \= VertexWidth,
    NewY is Y + 1,
    solve_path(Vertices, VertexWidth, 0, NewY).

solve_path(Vertices, VertexWidth, X, Y) :-
    X >= 0, X < VertexWidth, Y >= 0, Y < VertexWidth,
    write('Path: '), nl,
    write('Vertex width: '), write(VertexWidth), nl,
    write('X: '), write(X), write(' Y: '), write(Y), nl,
    VertexIndex is X + Y*VertexWidth,
    write('1'),nl,
    get_element_at(Vertices, VertexIndex, Vertex),
    write('2'),nl,
    build_neighbors_list(Vertices, VertexWidth, X, Y, [NorthMask, EastMask, SouthMask, WestMask], [NorthNeighbor, EastNeighbor, SouthNeighbor, WestNeighbor]),
    L1 = [NorthMask, EastMask, SouthMask, WestMask],
    L2 = [NorthNeighbor, EastNeighbor, SouthNeighbor, WestNeighbor],
    write(L1),nl,
    write(L2),nl,
    write('3'),nl,
    maximum(Max, Vertices),
    write('4'),nl,
    write('Max: '), write(Max),nl,
    write('Vertex: '), write(Vertex),nl,
    (Vertex #> 1 #/\ Vertex #\= Max) #=> (
                        ((NorthMask #> 0 #/\ NorthNeighbor #> 0) #/\ (NorthNeighbor #= Vertex - 1)) #\
                        ((EastMask #> 0 #/\ EastNeighbor #> 0) #/\ (EastNeighbor #= Vertex - 1)) #\
                        ((SouthMask #> 0 #/\ SouthNeighbor #> 0) #/\ (SouthNeighbor #= Vertex - 1)) #\
                        ((WestMask #> 0 #/\ WestNeighbor #> 0) #/\ (WestNeighbor #= Vertex - 1))
                    ) #/\ (
                        ((NorthMask #> 0 #/\ NorthNeighbor #> 0) #/\ (NorthNeighbor #= Vertex + 1)) #\
                        ((EastMask #> 0 #/\ EastNeighbor #> 0) #/\ (EastNeighbor #= Vertex + 1)) #\
                        ((SouthMask #> 0 #/\ SouthNeighbor #> 0) #/\ (SouthNeighbor #= Vertex + 1)) #\
                        ((WestMask #> 0 #/\ WestNeighbor #> 0) #/\ (WestNeighbor #= Vertex + 1))
                    ),
    write('5'),nl,
    Vertex #= 1 #=> (
                        ((NorthMask #> 0 #/\ NorthNeighbor #> 0) #/\ (NorthNeighbor #= Max)) #\
                        ((EastMask #> 0 #/\ EastNeighbor #> 0) #/\ (EastNeighbor #= Max)) #\
                        ((SouthMask #> 0 #/\ SouthNeighbor #> 0) #/\ (SouthNeighbor #= Max)) #\
                        ((WestMask #> 0 #/\ WestNeighbor #> 0) #/\ (WestNeighbor #= Max))
                    ) #/\ (
                        ((NorthMask #> 0 #/\ NorthNeighbor #> 0) #/\ (NorthNeighbor #= 2)) #\
                        ((EastMask #> 0 #/\ EastNeighbor #> 0) #/\ (EastNeighbor #= 2)) #\
                        ((SouthMask #> 0 #/\ SouthNeighbor #> 0) #/\ (SouthNeighbor #= 2)) #\
                        ((WestMask #> 0 #/\ WestNeighbor #> 0) #/\ (WestNeighbor #= 2))
                    ),

    write('6'),nl,
    Vertex #= Max #=> (
                        ((NorthMask #> 0 #/\ NorthNeighbor #> 0) #/\ (NorthNeighbor #= 1)) #\
                        ((EastMask #> 0 #/\ EastNeighbor #> 0) #/\ (EastNeighbor #= 1)) #\
                        ((SouthMask #> 0 #/\ SouthNeighbor #> 0) #/\ (SouthNeighbor #= 1)) #\
                        ((WestMask #> 0 #/\ WestNeighbor #> 0) #/\ (WestNeighbor #= 1))
                    ) #/\ (
                        ((NorthMask #> 0 #/\ NorthNeighbor #> 0) #/\ (NorthNeighbor #= Max - 1)) #\
                        ((EastMask #> 0 #/\ EastNeighbor #> 0) #/\ (EastNeighbor #= Max - 1)) #\
                        ((SouthMask #> 0 #/\ SouthNeighbor   #> 0) #/\ (SouthNeighbor #= Max - 1)) #\
                        ((WestMask #> 0 #/\ WestNeighbor #> 0) #/\ (WestNeighbor #= Max - 1))
                    ),

    write('7'),nl,
    NewX is X + 1,
    solve_path(Vertices, VertexWidth, NewX, Y).

solve_fences(Board, Vertices, BoardWidth) :-
    VertexWidth is BoardWidth + 1,
    write('- Solving vertices'),nl,
    solve_vertices(Board, Vertices, BoardWidth, 0, 0),
    write('- Solving path'),nl,
    solve_path(Vertices, VertexWidth, 0, 0).

solve_vertices(_, _, BoardWidth, 0, BoardWidth).

solve_vertices(Board, Vertices, BoardWidth, BoardWidth, Y) :-
    Y \= BoardWidth,
    NewY is Y + 1,
    solve_vertices(Board, Vertices, BoardWidth, 0, NewY).

solve_vertices(Board, Vertices, BoardWidth, X, Y) :-
    X >= 0, X < BoardWidth, Y >= 0, Y < BoardWidth,
    write('process'),nl,
    write('X: '), write(X), write(' Y: '), write(Y), nl,
    build_vertex_list(Board, Vertices, BoardWidth, X, Y, [V1, V2, V3, V4]),
    write('1'),nl,
    get_board_element(Board, BoardWidth, X, Y, Element),
    write('2'),nl,
    maximum(Max, Vertices),
    (V1 #> 0 #/\ V2 #> 0 #/\ 
        (
            (V1 + 1 #= V2) #\ 
            (V1 - 1 #= V2) #\ 
            (V1 #= Max #/\ V2 #= 1) #\
            (V1 #= 1 #/\ V2 #= Max) 
        ) 
    ) #<=> B1,
    (V2 #> 0 #/\ V3 #> 0 #/\ 
        (
            (V2 + 1 #= V3) #\ 
            (V2 - 1 #= V3) #\ 
            (V2 #= Max #/\ V3 #= 1) #\
            (V2 #= 1 #/\ V3 #= Max) 
        ) 
    ) #<=> B2,
    (V3 #> 0 #/\ V4 #> 0 #/\ 
        (
            (V3 + 1 #= V4) #\ 
            (V3 - 1 #= V4) #\ 
            (V3 #= Max #/\ V4 #= 1) #\
            (V3 #= 1 #/\ V4 #= Max) 
        ) 
    ) #<=> B3,
    (V4 #> 0 #/\ V1 #> 0 #/\ 
        (
            (V4 + 1 #= V1) #\ 
            (V4 - 1 #= V1) #\ 
            (V4 #= Max #/\ V1 #= 1) #\
            (V4 #= 1 #/\ V1 #= Max) 
        ) 
    ) #<=> B4,
    write('3'),nl,
    sum([B1, B2, B3, B4], #= , C),
    write('4'),nl,
    Element #> 0 #=> C #= Element,
    write('5'),nl,
    NewX is X + 1,
    solve_vertices(Board, Vertices, BoardWidth, NewX, Y).

sel_next_variable_for_path(Vars,Sel,Rest) :-
    % write(Vars), nl,
    findall(Idx-Cost, (nth1(Idx, Vars,V), fd_set(V,S), fdset_size(S,Size), fdset_min(S,Min),  var_cost(Min,Size, Cost)), L), 
    min_member(comp, BestIdx-_MinCost, L),
    nth1(BestIdx, Vars, Sel, Rest),!.

var_cost(0, _, 1000000) :- !.
var_cost(_, 1, 1000000) :- !.
var_cost(X, _, X).

%build_vertex_list(_, Vertices, BoardWidth, X, Y, List)

constrain_starting_and_ending_vertices(Vertices, [V1,V2,V3,V4]) :-
    maximum(Max, Vertices),
    (V1 #= 1 #/\        V2 #= Max #/\       V3 #= Max - 1 #/\   V4 #= 2         ) #\
    (V1 #= Max #/\      V2 #= 1 #/\         V3 #= 2 #/\         V4 #= Max - 1   ) #\
    (V1 #= Max - 1 #/\  V2 #= Max #/\       V3 #= 1 #/\         V4 #= 2         ) #\
    (V1 #= 2 #/\        V2 #= 1 #/\         V3 #= Max #/\       V4 #= Max - 1   ) #\
    (V1 #= 1 #/\        V2 #= 2 #/\         V3 #= Max - 1 #/\   V4 #= Max       ) #\
    (V1 #= Max #/\      V2 #= Max - 1 #/\   V3 #= 2 #/\         V4 #= 1         ) #\
    (V1 #= Max - 1 #/\  V2 #= 2 #/\         V3 #= 1 #/\         V4 #= Max       ) #\
    (V1 #= 2 #/\        V2 #= Max - 1 #/\   V3 #= Max #/\       V4 #= 1         ).

set_starting_and_ending_vertices(Board, Vertices, BoardWidth) :-
    set_starting_and_ending_vertices(Board, Vertices, BoardWidth, 0, 0).

set_starting_and_ending_vertices(Board, Vertices, BoardWidth, BoardWidth, Y) :-
    Y \= BoardWidth,
    NewY is Y + 1,
    solve_path(Board, Vertices, BoardWidth, 0, NewY).

set_starting_and_ending_vertices(Board, Vertices, BoardWidth, X, Y) :-
    X >= 0, X < BoardWidth, Y >= 0, Y < BoardWidth,
    build_vertex_list(_, Vertices, BoardWidth, X, Y, List),
    get_board_element(Board, BoardWidth, X, Y, Element),
    (Element = 3 -> 
        constrain_starting_and_ending_vertices(Vertices, List) 
        ; 
            NewX is X + 1,
        set_starting_and_ending_vertices(Board, Vertices, BoardWidth, NewX, Y)).

solve(Board, Vertices, BoardWidth) :-
    write('Skyscrapers'), nl,
    solve_skyscrapers(Board, BoardWidth),
    write('Labeling'), nl,
    labeling([ff], Board), !, 
    write('Setting domain'), nl,
    NVertices is (BoardWidth+1)*(BoardWidth+1),
    domain(Vertices, 0, NVertices),
    write('Starting and ending vertices'), nl,
    set_starting_and_ending_vertices(Board, Vertices, BoardWidth),
    write('Setting maximum'), nl,
    maximum(Max, Vertices),
    write('1'),nl,
    Max #> BoardWidth + 1,
    write('2'),nl,
    Max #< NVertices,
    count(0, Vertices, #=, NZeros),
    Max #= NVertices - NZeros,
    write('3'),nl,
    write('Calling nvalue'), nl,
    ValueCount #= Max + 1,
    nvalue(ValueCount, Vertices),
    write('Solving fences'), nl,
    solve_fences(Board, Vertices, BoardWidth),
    write('Labeling'), nl,
    labeling([ff], Vertices).

main :-
    board(Board),
    board_width(BoardWidth),
    vertices(Vertices),

    solve(Board, Vertices, BoardWidth),

    %findall(Board,
    %   labeling([ff], Board),
    %   Boards
    %),

    %append(Board, Vertices, Final),

    write('done.'),nl,
    print_board(Board, 6), nl,
    print_board(Vertices, 7).

utils.pro

get_element_at([Head|_], 0, Head).

get_element_at([_|Tail], Index, Element) :-
  Index \= 0,
  NewIndex is Index - 1,
  get_element_at(Tail, NewIndex, Element).

reverse([], []).

reverse([Head|Tail], Inv) :-
  reverse(Tail, Aux),
  append(Aux, [Head], Inv).

munch(List, 0, List).

munch([_|Tail], Count, FinalList) :-
    Count > 0,
    NewCount is Count - 1,
    munch(Tail, NewCount, FinalList).

select_n_elements(_, 0, []).

select_n_elements([Head|Tail], Count, FinalList) :-
    Count > 0,
    NewCount is Count - 1,
    select_n_elements(Tail, NewCount, Result),
    append([Head], Result, FinalList).

generate_list(Element, NElements, [Element|Result]) :-
  NElements > 0,
  NewNElements is NElements - 1,
  generate_list(Element, NewNElements, Result).

generate_list(_, 0, []).

s1.pro

% Skyscrapers and Fences puzzle S1

board_width(6).

%observer(Type, Index, Orientation, Observer),
observer(row, 0, forward, 2).
observer(row, 1, forward, 2).
observer(row, 2, forward, 2).
observer(row, 3, forward, 1).
observer(row, 4, forward, 2).
observer(row, 5, forward, 1).

observer(row, 0, reverse, 1).
observer(row, 1, reverse, 1).
observer(row, 2, reverse, 2).
observer(row, 3, reverse, 3).
observer(row, 4, reverse, 2).
observer(row, 5, reverse, 2).

observer(column, 0, forward, 2).
observer(column, 1, forward, 3).
observer(column, 2, forward, 0).
observer(column, 3, forward, 2).
observer(column, 4, forward, 2).
observer(column, 5, forward, 1).

observer(column, 0, reverse, 1).
observer(column, 1, reverse, 1).
observer(column, 2, reverse, 2).
observer(column, 3, reverse, 2).
observer(column, 4, reverse, 2).
observer(column, 5, reverse, 2).

board(
    [
        _, _, 2, _, _, _,
        _, _, _, _, _, _,
        _, 2, _, _, _, _,
        _, _, _, 2, _, _,
        _, _, _, _, _, _,
        _, _, _, _, _, _
    ]
).

vertices(
    [
        _, _, _, _, _, _, _,
        _, _, _, _, _, _, _,
        _, _, _, _, _, _, _,
        _, _, _, _, _, _, _,
        _, _, _, _, _, _, _,
        _, _, _, _, _, _, _,
        _, _, _, _, _, _, _
    ]
).

Answer 1

我也和 twinterer 一样，喜欢这个谜题。但作为一个负责人，我必须首先为摩天大楼和栅栏部分找到合适的策略，然后对后者进行深入调试，导致复制变量问题使我锁定了好几个小时。

一旦解决了这个错误，我就面临着第一次尝试的低效率。我在普通的 Prolog 中重新设计了一个类似的模式，只是为了验证它的效率有多低。

至少，我了解如何更有效地使用 CLP(FD) 来模拟问题（在 twinterer 的回答的帮助下），现在程序很快（0.2 秒）。所以现在我可以提示您有关您的代码：所需的约束远比您编写的那些更简单：对于栅栏部分，即固定建筑物的位置，我们有 2 个约束：高度的边数> 0，并将边连接在一起：当使用边时，相邻的总和必须为 1（两边）。

这是我使用 SWI-Prolog 开发的代码的最后一个版本。

/*  File:    skys.pl
    Author:  Carlo,,,
    Created: Dec 11 2011
    Purpose: questions/8458945 on http://stackoverflow.com
        http://stackoverflow.com/questions/8458945/optimizing-pathfinding-in-constraint-logic-programming-with-prolog
*/

:- module(skys, [skys/0, fences/2, draw_path/2]).
:- [index_square,
    lambda,
    library(clpfd),
    library(aggregate)].

puzzle(1,
  [[-,2,3,-,2,2,1,-],
   [2,-,-,2,-,-,-,1],
   [2,-,-,-,-,-,-,1],
   [2,-,2,-,-,-,-,2],
   [1,-,-,-,2,-,-,3],
   [2,-,-,-,-,-,-,2],
   [1,-,-,-,-,-,-,2],
   [-,1,1,2,2,2,2,-]]).

skys :-
    puzzle(1, P),
    skyscrapes(P, Rows),

    flatten(Rows, Flat),
    label(Flat),

    maplist(writeln, Rows),

    fences(Rows, Loop),

    writeln(Loop),
    draw_path(7, Loop).

%%  %%%%%%%%%%
%   skyscrapes part
%   %%%%%%%%%%

skyscrapes(Puzzle, Rows) :-

    % massaging definition: separe external 'visibility' counters
    first_and_last(Puzzle, Fpt, Lpt, Wpt),
    first_and_last(Fpt, -, -, Fp),
    first_and_last(Lpt, -, -, Lp),
    maplist(first_and_last, Wpt, Lc, Rc, InnerData),

    % InnerData it's the actual 'playground', Fp, Lp, Lc, Rc are list of counters
    maplist(make_vars, InnerData, Rows),

    % exploit symmetry wrt rows/cols
    transpose(Rows, Cols),

    % each row or col contains once 1,2,3
    Occurs = [0-_, 1-1, 2-1, 3-1],  % allows any grid size leaving unspecified 0s
    maplist(\Vs^global_cardinality(Vs, Occurs), Rows),
    maplist(\Vs^global_cardinality(Vs, Occurs), Cols),

    % apply 'external visibility' constraint
    constraint_views(Lc, Rows),
    constraint_views(Fp, Cols),

    maplist(reverse, Rows, RRows),
    constraint_views(Rc, RRows),

    maplist(reverse, Cols, RCols),
    constraint_views(Lp, RCols).

first_and_last(List, First, Last, Without) :-
    append([[First], Without, [Last]], List).

make_vars(Data, Vars) :-
    maplist(\C^V^(C \= (-) -> V #= C ; V in 0..3), Data, Vars).

constraint_views(Ns, Ls) :-
    maplist(\N^L^
    (   N \= (-)
    ->  constraint_view(0, L, Rs),
        sum(Rs, #=, N)
    ;   true
    ), Ns, Ls).

constraint_view(_, [], []).
constraint_view(Top, [V|Vs], [R|Rs]) :-
    R #<==> V #> 0 #/\ V #> Top,
    Max #= max(Top, V),
    constraint_view(Max, Vs, Rs).

%%  %%%%%%%%%%%%%%%
%   fences part
%   %%%%%%%%%%%%%%%

fences(SkyS, Ps) :-

    length(SkyS, D),

    % allocate edges
    max_dimensions(D, _,_,_,_, N),
    N1 is N + 1,
    length(Edges, N1),
    Edges ins 0..1,

    findall((R, C, V),
        (nth0(R, SkyS, Row), nth0(C, Row, V), V > 0),
        Buildings),
    maplist(count_edges(D, Edges), Buildings),

    findall((I, Adj1, Adj2),
        (between(0, N, I), edge_adjacents(D, I, Adj1, Adj2)),
        Path),
    maplist(make_path(Edges), Path, Vs),

    flatten([Edges, Vs], Gs),
    label(Gs),

    used_edges_to_path_coords(D, Edges, Ps).

count_edges(D, Edges, (R, C, V)) :-
    cell_edges(D, (R, C), Is),
    idxs0_to_elems(Is, Edges, Es),
    sum(Es, #=, V).

make_path(Edges, (Index, G1, G2), [S1, S2]) :-

    idxs0_to_elems(G1, Edges, Adj1),
    idxs0_to_elems(G2, Edges, Adj2),
    nth0(Index, Edges, Edge),

    [S1, S2] ins 0..3,
    sum(Adj1, #=, S1),
    sum(Adj2, #=, S2),
    Edge #= 1 #<==> S1 #= 1 #/\ S2 #= 1.

%%  %%%%%%%%%%%%%%
%   utility: draw a path with arrows
%   %%%%%%%%%%%%%%

draw_path(D, P) :-
    forall(between(1, D, R),
           (   forall(between(1, D, C),
              (   V is (R - 1) * D + C - 1,
                  U is (R - 2) * D + C - 1,
                  (   append(_, [V, U|_], P)
                  ->  write(' ^   ')
                  ;   append(_, [U, V|_], P)
                  ->  write(' v   ')
                  ;   write('     ')
                  )
              )),
           nl,
           forall(between(1, D, C),
              (   V is (R - 1) * D + C - 1,
                  (   V < 10
                  ->  write(' ') ; true
                  ),
                  write(V),
                  U is V + 1,
                  (   append(_, [V, U|_], P)
                  ->  write(' > ')
                  ;   append(_, [U, V|_], P)
                  ->  write(' < ')
                  ;   write('   ')
                  )
              )),
             nl
        )
           ).

% convert from 'edge used flags' to vertex indexes
%
used_edges_to_path_coords(D, EdgeUsedFlags, PathCoords) :-
    findall((X, Y),
        (nth0(Used, EdgeUsedFlags, 1), edge_verts(D, Used, X, Y)),
        Path),
    Path = [(First, _)|_],
    edge_follower(First, Path, PathCoords).

edge_follower(C, Path, [C|Rest]) :-
    (   select(E, Path, Path1),
        ( E = (C, D) ; E = (D, C) )
    ->  edge_follower(D, Path1, Rest)
    ;   Rest = []
    ).

输出：

[0,0,2,1,0,3]
[2,1,3,0,0,0]
[0,2,0,3,1,0]
[0,3,0,2,0,1]
[1,0,0,0,3,2]
[3,0,1,0,2,0]

[1,2,3,4,5,6,13,12,19,20,27,34,41,48,47,40,33,32,39,46,45,38,31,24,25,18,17,10,9,16,23,
22,29,30,37,36,43,42,35,28,21,14,7,8,1]

 0    1 >  2 >  3 >  4 >  5 >  6   
      ^                        v   
 7 >  8    9 < 10   11   12 < 13   
 ^         v    ^         v        
14   15   16   17 < 18   19 > 20   
 ^         v         ^         v   
21   22 < 23   24 > 25   26   27   
 ^    v         ^              v   
28   29 > 30   31   32 < 33   34   
 ^         v    ^    v    ^    v   
35   36 < 37   38   39   40   41   
 ^    v         ^    v    ^    v   
42 < 43   44   45 < 46   47 < 48   

正如我所提到的，我的第一次尝试更“程序化”：它绘制了一个循环，但我无法解决的问题基本上是必须事先知道顶点子集的基数，基于全局约束 all_different。它在缩小的 4*4 拼图上工作起来很痛苦，但我在 6*6 原版上几个小时后停止了它。无论如何，从头开始学习如何使用 CLP(FD) 绘制路径是很有收获的。

t :-
    time(fences([[0,0,2,1,0,3],
             [2,1,3,0,0,0],
             [0,2,0,3,1,0],
             [0,3,0,2,0,1],
             [1,0,0,0,3,2],
             [3,0,1,0,2,0]
            ],L)),
    writeln(L).

fences(SkyS, Ps) :-

    length(SkyS, Dt),
        D is Dt + 1,
    Sq is D * D - 1,

    % min/max num. of vertices
    aggregate_all(sum(V), (member(R, SkyS), member(V, R)), MinVertsT),
    MinVerts is max(4, MinVertsT),
    MaxVerts is D * D,

    % find first cell with heigth 3, for sure start vertex
    nth0(R, SkyS, Row), nth0(C, Row, 3),

    % search a path with at least MinVerts
    between(MinVerts, MaxVerts, NVerts),
    length(Vs, NVerts),

    Vs ins 0 .. Sq,
    all_distinct(Vs),

    % make a loop
    Vs = [O|_],
    O is R * D + C,
    append(Vs, [O], Ps),

    % apply #edges check
    findall(rc(Ri, Ci, V),
        (nth0(Ri, SkyS, Rowi),
         nth0(Ci, Rowi, V),
         V > 0), VRCs),
    maplist(count_edges(Ps, D), VRCs),

    connect_path(D, Ps),
    label(Vs).

count_edges(Ps, D, rc(R, C, V)) :-
    V0 is R * D + C,
    V1 is R * D + C + 1,
    V2 is (R + 1) * D + C,
    V3 is (R + 1) * D + C + 1,
    place_edges(Ps, [V0-V1, V0-V2, V1-V3, V2-V3], Ts),
    flatten(Ts, Tsf),
    sum(Tsf, #=, V).

place_edges([A,B|Ps], L, [R|Rs]) :-
    place_edge(L, A-B, R),
    place_edges([B|Ps], L, Rs).
place_edges([_], _L, []).

place_edge([M-N | L], A-B, [Y|R]) :-
    Y #<==> (A #= M #/\ B #= N) #\/ (A #= N #/\ B #= M),
    place_edge(L, A-B, R).
place_edge([], _, []).

connect(X, D, Y) :-
    D1 is D - 1,
    [R, C] ins 0 .. D1,

    X #= R * D + C,
    ( C #< D - 1, Y #= R * D + C + 1
    ; R #< D - 1, Y #= (R + 1) * D + C
    ; C #> 0, Y #= R * D + C - 1
    ; R #> 0, Y #= (R - 1) * D + C
    ).

connect_path(D, [X, Y | R]) :-
    connect(X, D, Y),
    connect_path(D, [Y | R]).
connect_path(_, [_]).

感谢您提出这么有趣的问题。

更多编辑：这里是完整解决方案的主要缺失代码（index_square.pl）

/*  File:    index_square.pl
    Author:  Carlo,,,
    Created: Dec 15 2011
    Purpose: indexing square grid for FD mapping
*/

:- module(index_square,
      [max_dimensions/6,
       idxs0_to_elems/3,
       edge_verts/4,
       edge_is_horiz/3,
       cell_verts/3,
       cell_edges/3,
       edge_adjacents/4,
       edge_verts_all/2
      ]).

%
% index row  : {D}, left to right
% index col  : {D}, top to bottom
% index cell : same as top edge or row,col
% index vert : {(D + 1) * 2}
% index edge : {(D * (D + 1)) * 2}, first all horiz, then vert
%
% {N} denote range 0 .. N-1
%
%  on a 2*2 grid, the numbering schema is
%
%       0   1
%   0-- 0 --1-- 1 --2
%   |       |       |
% 0 6  0,0  7  0,1  8
%   |       |       |
%   3-- 2 --4-- 3 --5
%   |       |       |
% 1 9  1,0  10 1,1  11
%   |       |       |
%   6-- 4 --7-- 5 --8
%
%  while on a 4*4 grid:
%
%       0   1       2       3
%   0-- 0 --1-- 1 --2-- 2 --3-- 3 --4
%   |       |       |       |       |
% 0 20      21      22      23      24
%   |       |       |       |       |
%   5-- 4 --6-- 5 --7-- 6 --8-- 7 --9
%   |       |       |       |       |
% 1 25      26      27      28      29
%   |       |       |       |       |
%   10--8 --11- 9 --12--10--13--11--14
%   |       |       |       |       |
% 2 30      31      32      33      34
%   |       |       |       |       |
%   15--12--16--13--17--14--18--15--19
%   |       |       |       |       |
% 3 35      36      37      38      39
%   |       |       |       |       |
%   20--16--21--17--22--18--23--19--24
%
%   |       |
% --+-- N --+--
%   |       |
%   W  R,C  E
%   |       |
% --+-- S --+--
%   |       |
%

% get range upper value for interesting quantities
%
max_dimensions(D, MaxRow, MaxCol, MaxCell, MaxVert, MaxEdge) :-
    MaxRow is D - 1,
    MaxCol is D - 1,
    MaxCell is D * D - 1,
    MaxVert is ((D + 1) * 2) - 1,
    MaxEdge is (D * (D + 1) * 2) - 1.

% map indexes to elements
%
idxs0_to_elems(Is, Edges, Es) :-
    maplist(nth0_(Edges), Is, Es).
nth0_(Edges, I, E) :-
    nth0(I, Edges, E).

% get vertices of edge
%
edge_verts(D, E, X, Y) :-
    S is D + 1,
    edge_is_horiz(D, E, H),
    (   H
    ->  X is (E // D) * S + E mod D,
        Y is X + 1
    ;   X is E - (D * S),
        Y is X + S
    ).

% qualify edge as horizontal (never fail!)
%
edge_is_horiz(D, E, H) :-
    E >= (D * (D + 1)) -> H = false ; H = true.

% get 4 vertices of cell
%
cell_verts(D, (R, C), [TL, TR, BL, BR]) :-
    TL is R * (D + 1) + C,
    TR is TL + 1,
    BL is TR + D,
    BR is BL + 1.

% get 4 edges of cell
%
cell_edges(D, (R, C), [N, S, W, E]) :-
    N is R * D + C,
    S is N + D,
    W is (D * (D + 1)) + R * (D + 1) + C,
    E is W + 1.

% get adjacents at two extremities of edge I
%
edge_adjacents(D, I, G1, G2) :-
    edge_verts(D, I, X, Y),
    edge_verts_all(D, EVs),
    setof(E, U^V^(member(E - (U, V), EVs), E \= I, (U == X ; V == X)), G1),
    setof(E, U^V^(member(E - (U, V), EVs), E \= I, (U == Y ; V == Y)), G2).

% get all edge_verts/4 for grid D
%
edge_verts_all(D, L) :-
    (   edge_verts_all_(D, L)
    ->  true
    ;   max_dimensions(D, _,_,_,_, S), %S is (D + 1) * (D + 2) - 1,
        findall(E - (X, Y),
            (   between(0, S, E),
            edge_verts(D, E, X, Y)
            ), L),
        assert(edge_verts_all_(D, L))
    ).

:- dynamic edge_verts_all_/2.

%%  %%%%%%%%%%%%%%%%%%%%

:- begin_tests(index_square).

test(1) :-
    cell_edges(2, (0,1), [1, 3, 7, 8]),
    cell_edges(2, (1,1), [3, 5, 10, 11]).

test(2) :-
    cell_verts(2, (0,1), [1, 2, 4, 5]),
    cell_verts(2, (1,1), [4, 5, 7, 8]).

test(3) :-
    edge_is_horiz(2, 0, true),
    edge_is_horiz(2, 5, true),
    edge_is_horiz(2, 6, false),
    edge_is_horiz(2, 9, false),
    edge_is_horiz(2, 11, false).

test(4) :-
    edge_verts(2, 0, 0, 1),
    edge_verts(2, 3, 4, 5),
    edge_verts(2, 5, 7, 8),
    edge_verts(2, 6, 0, 3),
    edge_verts(2, 11, 5, 8).

test(5) :-
    edge_adjacents(2, 0, A, B), A = [6], B = [1, 7],
    edge_adjacents(2, 9, [2, 6], [4]),
    edge_adjacents(2, 10, [2, 3, 7], [4, 5]).

test(6) :-
    cell_edges(4, (2,1), [9, 13, 31, 32]).

:- end_tests(index_square).

Answer 2

快速浏览一下您的程序表明您大量使用了具体化。不幸的是，这样的公式意味着 SICStus 等当前系统的一致性很弱。

然而，通常情况下，事物可以更紧凑地表述，从而获得更好的一致性。下面是一个示例，您可以根据自己的需要进行调整。

比如说，你想表达 (X1,Y1) 和 (X2,Y2) 是水平或垂直的邻居。您可以针对每种可能性说 ( X1+1 #= X2 #/\ Y1 #= Y2 ) #\ ...（并检查您的健康保险是否涵盖 RSI）。

或者你可以说abs(X1-X2)+abs(Y1-Y2) #= 1。在过去，SICStus Prolog 曾经有一个对称的差异(--)/2，但我假设您使用的是版本 4。

以上公式保持区间一致性（至少我从我尝试过的示例中得出了这一结论）：

| ?- X1 in 1..9, abs(X1-X2)+abs(Y1-Y2) #= 1.
X1 in 1..9,
X2 in 0..10,...

所以X2 很容易受到约束！

在某些情况下（正如您在回复中指出的那样），您需要具体化的表单来维护其他约束。在这种情况下，您可以考虑同时发布两者。

翻阅手册，有几个组合约束也可能很有趣。作为快速修复，smt/1 可能会有所帮助（4.2.0 中的新功能）。有兴趣听听这个...

另一种可能是使用其他实现：例如 YAP 的 library(clpfd) 或 SWI。

Answer 3

多么漂亮的小拼图啊！为了理解这些属性，我在 ECLiPSe 中实现了一个解决方案。它可以在这里找到：http://pastebin.com/eZbgjgFA（如果您在代码中看到循环，请不要担心：这些可以很容易地转换为标准 Prolog 谓词。不过，还有其他东西，从 ECLiPSe 转换为 Sicstus 并不那么容易）

执行时间比您报告的要快，但可能会更好：

?- snf(L).
L = [[]([]([](0,0,1,1),[](1,1,0,0),[](0,1,0,1),[](0,1,0,0),[](0,1,0,0),[](0,1,1,1)),
        []([](1,1,0,0),[](0,0,1,0),[](1,1,1,0),[](1,0,0,1),[](0,0,1,0),[](1,1,0,1)),
        []([](1,0,0,0),[](0,0,1,1),[](1,0,0,0),[](0,1,1,1),[](1,0,0,0),[](0,1,1,0)),
        []([](1,0,1,0),[](1,1,0,1),[](0,0,1,0),[](1,1,0,0),[](0,0,0,1),[](0,0,1,0)),
        []([](1,0,0,0),[](0,1,1,1),[](1,0,1,0),[](1,0,1,0),[](1,1,1,0),[](1,0,1,0)),
        []([](1,0,1,1),[](1,1,0,0),[](0,0,1,0),[](1,0,1,1),[](1,0,1,0),[](1,0,1,1))),
     ...]
Yes (40.42s cpu, solution 1, maybe more)
No (52.88s cpu)

您在答案中看到的是边矩阵。每个内部项表示拼图中的一个字段，哪个边缘是活动的（左、上、右、下）。我编辑了其余的。

我总共使用了八个数组：HxWx4 边数组 (0/1)、一个 (H+1)x(W+1) 每个字段顶点的活动边数组 (0/2)、一个 HxW 数组活动边的总和 (0..3)、建筑物的 HxW 数组 (0/1)、建筑物高度的两个 [H,W]x3 数组和建筑物位置的两个 [H,W]x3 数组。

必须只有一个路径的要求没有作为约束，而只是在标记过程中找到潜在解决方案后作为检查执行。

约束是：

• sum 数组必须包含每个字段的有效边的总和

• 相邻字段的接触边缘必须包含相同的值

• 顶点必须有两条连接到它们的活动边，或者没有

• 在每一列/行中，必须放置三个建筑物。一些建筑物是按照拼图的定义放置的

• 行/列中的每个建筑物高度必须不同

• 建筑高度对应于该位置的活动边的总和

• 可见建筑物的数量由拼图的定义指定。这限制了建筑物在行/列中出现的顺序。

• 建筑物在行/列中的位置必须按升序给出

• 一旦知道了第一/第二/第三栋建筑的位置，我们就可以推断出一些不能放置建筑的位置。

有了这组约束，我们现在可以进行标注了。标记分两步完成，加快了求解过程。

在第一步中，仅标记建筑物位置。这是最受限制的部分，如果我们在这里找到解决方案，剩下的就容易多了。

在第二步中，所有其他变量都被标记。对于这两个步骤，我都选择了“先失败”作为标记策略，即首先标记具有最小域的变量。

如果不先解决建筑物的位置，程序会花费更长的时间（我总是在几分钟后停止它）。由于我没有第二个拼图实例可用，但我不确定搜索策略是否适用于所有实例

再次查看您的程序，您似乎遵循了类似的策略，即首先放置建筑物。但是，您在设置约束和标签之间进行迭代。这效率不高。在 CLP 中，您应该始终将约束放在前面（除非约束确实取决于部分解决方案的当前状态），并且只有在发布约束时您才搜索解决方案。这样，您可以在搜索过程中检测到所有约束的失败。否则，您可能会找到满足您迄今为止发布的一组约束的部分解决方案，只是发现一旦添加了其他约束，您就无法完成该解决方案。

此外，如果您有不同的变量集，请尝试变量的标记顺序。不过，没有通用的方法。

希望这会有所帮助！