Basically I am trying to implement a program in prolog that can solve a game called Yin Yang, independente of the size of the board.
The game is basically to paint every cell of the board of a specific color (either black or white) and all cells of same color are connected to each other, vertically or horizontally. I am representing this with a list of lists. Example initial->solution: https://i.gyazo.com/24a70d868934dfbf1540343e89d14c4b.png
However there is a rule I'm having trouble with: "No 2X2 group of cells can contain circles of a single color."
Any ideas how I can assure this doesn't happen using the clpfd library?
List representing board example:
[[0,0,0,0,0,0],
[0,0,0,1,0,0],
[0,0,1,1,2,0],
[0,2,0,0,2,2],
[1,0,2,0,1,0],
[0,1,0,0,0,1]]
I managed to do a variant of the problem :
get-elem(B, R, C, E) :- get-line(B, R, Row), get-line(Row, C, E).
get-line([R|_], 0, R).
get-line([_|A], N, R) :- get-line(A, M, R), N is M + 1.
twoXTwo(B, R, C) :- \+ (R1 is R+ 1, C1 is C+1,
get-elem(B, R, C, E1),
get-elem(B, R1, C, E2), E1 = E2,
get-elem(B, R, C1, E3), E2 = E3,
get-elem(B, R1, C1, E4), E3 = E4).
where I check the element at positions [(r+1,c), (r,c+1), (r+1,c+1)] are not the same element. I am also not sure what the numbers mean given you have either black (1) or white (0).
I solved the issue: This is the code I used for solving the problem described above.
twoByTwo([_],[_]).
twoByTwo([F1,S1|T1],[F2,S2|T2]):-
F1 + S1 + F2 + S2 #> 4,
F1 + S1 + F2 + S2 #< 8,
twoByTwo([S1|T1],[S2|T2]).
Basically twoByTwo receives a line of the board (in order) on each argument.
So basically this function is called recursively: twoByTwo(Line1,Line2), twoByTwo(Line2,Line3), etc.
Related
I have a problem with coding the program described below
Consider the following game. A board with three black stones, three white stones and an empty space is given. The goal of the game is to swap places of black pawns with white pawns. Moving rules define the following sentences: Move the white and black pieces alternately. Each pawn can move vertically or horizontally taking up an empty space. Each piece can jump vertically or horizontally over another piece (of any color). Write a program in Prolog to find all possible ways to find a winning sequence. For example, if we ask the question:
? - play (w, s (w, w, w, e, b, b, b), s (b, b, b, e, w, w, w), S, R ).
The prologue should answer, for example:
S = [s (w, w, w, e, b, b, b), s (w, e, w, w, b, b, b), ..., s (b, b, b, e, w, w, w)] R = [[w, 2,4], [b, 6,2], [w, 4,6], ..., [w, 4,6]]
Here [ w, 2,4] means moving the white pawn from position 2 to position 4. Of course Prolog should return both letters S and R in full (without "...").
What is the maximum number of different pawn settings possible on the board? Check the query:
? - play (_, s (w, w, w, e, b, b, b), s (b, b, e, w, w, b, w), _, _).
What does Prolog's answer mean? Hint: solve the problem for play/4 without R first
There's also a game board that looks like this:
I have no clue at all even where to start? How can I do that? Could you guys, help me with this one?
This is a standard state space search, a standard paradigm of GOFAI since the mid 50s at least.
The barebones algorithm:
search(State,Path,Path) :- is_final(State),!. % Done, bounce "Path" term
search(State,PathSoFar,PathOut) :-
generate_applicable_operators(State,Operators),
(is_empty(Operators) -> fail ; true),
select_operator(Operators,Op,PathSoFar),
apply_operator(State,Op,NextState), % depth-first / best first
search(NextState,[[NextState,Op]|PathSoFar],PathOut).
% Called like this, where Path will contain the reverse Path through
% State Space by which one may reach a final state:
search(InitialState,[[InitialState,nop]],Path).
First you need to represent a given state in this case the state of the board (at some time t).
We can either list the board positions and their content (w for white, b for black, e for empty token) or list the tokens and their positions. Let's list the board positions.
In Prolog, a term that can be easily pattern-matched is appropriate. The question already provides something: (w, w, w, e, b, b, b). This seems to be inspired by LISP and is not well adapted to Prolog. Let's use a list instead: [w, w, w, e, b, b, b]
The mapping of board positions to list positions shall be:
+---+---+
| 0 | 1 |
+---+---+---+
| 2 | 3 | 4 |
+---+---+---+
| 5 | 6 |
+---+---+
And we are done with setting up a state description!
Then you need to represent/define the operators (operations?) that can be applied to a state: they transform a valid state into another valid state.
An operator corresponds to "moving a token" and of course not all operators apply to a given state (you cannot move a token from field 1 if there is no token there; you cannot move a token to field 1 if there already is a token there).
So you want to write a predicate that links a board state to the operators applicable to that state: generate_applicable_operators/2
Then you need to select the operator that you want to apply. This can be done randomly, exhaustively, according to some heuristic (for example A*), but definitely needs to examine the path taken through the state space till now to avoid cycles: select_operator/3.
Then you apply the operator to generate the next state: apply_operator/3.
And finally recursively call search/3 to find the next move. This continues until the "final state", in this case [b, b, b, e, w, w, w] has been reached!
You can also use Iterative Deepening if you want to perform "breadth-first search" instead, but for that the algorithm structure must be modified.
And that's it.
I have to determine whether two rectangles overlap or not, I can do this but I am struggling with figuring out how to grab my given data, and compare it to eachother to determine larger values.
%This is :what would be happening :
%separate(rectangle(0,10,10,0), rectangle(4,6,6,4))
separate(R1,R2) :-
%I Have to figure out how to take the values from R1 and R2 and compare
%them to one another.
.
It is called "pattern matching".
separated(R1, R2) :-
R1 = rectangle(A1, B1, C1, D1),
R2 = rectangle(A2, B2, C2, D2),
/* now just use your As and Bs */
and in many cases it is better to write straight away:
separated(rectangle(A1, B1, C1, D1), rectangle(A2, B2, C2, D2)) :-
/* now just use your As and Bs */
So I am trying to solve this Booth arrangement problem given here. It is basically a sliding tile puzzle where one (booth)tile has to reach a target spot and in the end all other (booths)tiles should be in their original location. Each tile/booth has a dimension and following are the input fact and relation descriptions:
One fact of the form room(W,H), which specifies the width W and
height H of the room (3 ≤ W, H ≤ 20).
One fact booths(B), which
specifies the number of booths (1 ≤ B ≤ 20).
A relation that consists
of facts of the form dimension(B, W, H), which specifies the width W
and height H of booth B.
A relation consisting of facts of the form
position(B, W, H), specifying the initial position (W, H) of booth B.
One fact target(B, W, H), specifying the destination (W, H) of the
target booth B.
An additional fact horizon(H) gives an upper bound on
the number of moves to be performed.
The program is supposed to read input facts from a file but I am just trying to do the solving so I have just copy pasted one possible input for now, and I have written some basic clauses:
room(3, 3).
booths(3).
dimension(1, 2, 1).
dimension(2, 2, 1).
dimension(3, 1, 1).
position(1, 0, 1).
position(2, 1, 2).
position(3, 0, 0).
target(3, 0, 2).
horizon(10).
xlim(X) :- room(X,_).
ylim(X) :- room(_,X).
sum(X,Y,Z) :- Z is X+Y .
do(position(B,X,Y),movedown,position(B,X,Z)) :- Y > 0 , sum(Y,-1,Z) .
do(position(B,X,Y),moveup,position(B,X,Z)) :- ylim(L), Y < L , sum(Y,1,Z) .
do(position(B,X,Y),moveleft,position(B,Z,Y)) :- X > 0 , sum(X,-1,Z) .
do(position(B,X,Y),moveright,position(B,Z,Y)) :- xlim(L), X < L, sum(X,1,Z) .
noverlap(B1,B2) :-
position(B1,X1,Y1),
position(B2,X2,Y2),
ends(Xe1,Ye1,B1),
ends(Xe2,Ye2,B2),
( Xe1 < X2 ;
Xe2 < X1 ;
Ye1 < Y2 ;
Ye2 < Y1 ).
ends(Xe,Ye,B) :-
dimension(B,W,H),
position(B,X,Y),
Xe is X+W-1,
Ye is Y+H-1.
between(X,Y,Z) :-
X > Y ,
X < Z .
validMove(M,B) :- do(position(B,X,Y),M,position(B,Xn,Yn)) .
I am new to Prolog and I am stuck on how to go from here, I have the no_overlap rule so I can test if a move is valid or not but I am not sure how with the current clauses that I have. My current clauses for moves do/3 probably needs some modification. Any pointers?.
You need to express the task in terms of relations between states of the puzzle. Your current clauses determine the validity of a single move, and can also generate possible moves.
However, that is not sufficient: You need to express more than just a single move and its effect on a single tile. You need to encode, in some way, the state of the whole puzzle, and also encode how a single move changes the state of the whole task.
For a start, I recommend you think about a relation like:
world0_move_world(W0, M, W) :- ...
and express the relation between a given "world" W0, a possible move M, and the resulting world W. This relation should be so general as to generate, on backtracking, each move M that is possible in W0. Ideally, it should even work if W0 is a free variable, and for this you may find clpfd useful: Constraints allow you to express arithmetic relations in a much more general way than you are currently using.
Once you have such a relation, the whole task is to find a sequence Ms of moves such that any initial world W0 is transformed to a desired state W.
Assuming you have implemented world0_move_world/3 as a building block, you can easily lift this to lists of moves as follows (using dcg):
moves(W0) --> { desired_world(W0) }.
moves(W0) --> [M], { world0_move_world(W0, M, W) }, moves(W).
You can then use iterative deepening to find a shortest sequence of moves that solves the puzzle:
?- length(Ms, _), initial_world(W0), phrase(moves(W0), Ms).
I have a function that uses 4 parameters, called tile . It is designed to work the following way :
tile(?E, ?S, ?W, ?N, ?ID)
I would like a getter function that given an ID, it returns the first 4 parameters: E, S, W and N.
I have tried something like:
coordonates(tile(E,S,W,N,L), (E,S,W,N)).
But it does not return the actual values, only true.
If I type tile(E, S, W, N, #1) in the terminal I get the desired result but I do not know what exactly is returned (a list maybe?).
Let's suppose our facts describing tile looks as follows:
tile(p1,p2,p3,p4,id1).
tile(q1,q2,q3,q4,id2).
tile(r1,r2,r3,r4,id3).
In this we have a finite number of facts. That can be checked by the most general query for tile:
?- tile(E,S,W,N,I).
E = p1,
S = p2,
W = p3,
N = p4,
I = id1 ; % <---- user input ; to continue
E = q1,
S = q2,
W = q3,
N = q4,
I = id2 ; % <---- user input ; to continue
E = r1,
S = r2,
W = r3,
N = r4,
I = id3. % <---- toplevel outputs . -- we're done
So in theory, we could define coordonates as follows:
coordonates(id1, t(p1, p2, p3, p4)).
coordonates(id2, t(q1, q2, q3, q4)).
coordonates(id3, t(r1, r2, r3, r4)).
which could be queried for id2 as follows:
?- coordonates(id2,X).
X = t(q1, q2, q3, q4).
I used the functor t to group the solution, to make clear that it is not the predicate tile we defined earlier. There's also a lot of repetition in this definition which is already a hint, that we can do better. What we are looking for is a rule which tells us how, given we have a answer for tile, we can describe coordonates. In logical terms, this is written as an implication of the form: goal1 ∧ ... ∧ goalN → head. which means "Suppose I know that goal1 to goalN is true, then I also know that head is true." In Prolog, this is written backwards:
head :-
goal1,
% ...
goalN.
Let's go back to our task: we know something about a tile and we want to describe how the projection looks like. This means, our code looks as follows:
coordonates( ... ) :-
% ...
tile(E,S,W,N,I).
The body tile(E,S,W,N,I) is the most general form we can write (see our query above) and can be read as "suppose I have any tile at coordinates E S W N with id I". Now we only need to fill in, how coordonates should look like. We know it has two arguments, because it relates the id with the four other elements. Lets give them names, say Id and Coords:
coordonates(Id, Coords) :-
% ...
tile(E,S,W,N,I).
Now we only need to find out how to relate E,S,E,N and I with Id and Coords. One is easy: Id is just I. The other one is also not too hard, we just need to group the coordinates into one term. We can pick an arbitrary one, but already above decided to take t, so we will stick with it:
coordonates(Id, Coords) :-
Id = I,
Coords = t(E,S,W,N),
tile(E,S,W,N,I).
This already works as we expect:
?- coordonates(X,Y).
X = id1,
Y = t(p1, p2, p3, p4) ;
X = id2,
Y = t(q1, q2, q3, q4) ;
X = id3,
Y = t(r1, r2, r3, r4).
Now we can make one observation: if two terms are equal, we can use one instead of the other. So instead of writing Id = I, we can just reuse Id. The same goes for Coords and t(E,S,W,N):
coordonates(I, t(E,S,W,N)) :-
tile(E,S,W,N,I).
It couldn't be much shorter :-)
You have to declare 'E, S, W and N' so that prolog can unify those parameters with the input when you make the query. Something like (In the most basic case):
tile(['cordE1','cordS1','cordW1','cordN1'],1).
tile(['cordE2','cordS2','cordW2','cordN2'],2).
tile(['cordE3','cordS3','cordW3','cordN3'],3).
Query:
?- tile(C,2).
C = [cordE2, cordS2, cordW2, cordN2].
?- tile(C,1).
C = [cordE1, cordS1, cordW1, cordN1].
?- tile(C,3).
C = [cordE3, cordS3, cordW3, cordN3].
I have the following task:
Write a method that will add two polynoms. I.e 0+2*x^3 and 0+1*x^3+2*x^4 will give 0+3*x^3+2*x^4.
I also wrote the following code:
add_poly(+A1*x^B1+P1,+A2*x^B2+P2,+A3*x^B3+P3):-
(
B1=B2,
B3 = B2,
A3 is A1+A2,
add_poly(P1,P2,P3)
;
B1<B2,
B3=B1,
A3=A1,
add_poly(P1,+A2*x^B2+P2,P3)
;
B1>B2,
B3=B2,
A3=A2,
add_poly(+A1*x^B1+P1,P2,P3)
).
add_poly(X+P1,Y+P2,Z+P3):-
Z is X+Y,
add_poly(P1,P2,P3).
My problem is that I don't know how to stop. I would like to stop when one the arguments is null and than to append the second argument to the third one. But how can I check that they are null?
Thanks.
Several remarks:
Try to avoid disjunctions (;)/2 in the beginning. They need special indentation to be readable. And they make reading a single rule more complex — think of all the extra (=)/2 goals you have to write and keep track of.
Then, I am not sure what you can assume about your polynomials. Can you assume they are written in canonical form?
And for your program: Consider the head of your first rule:
add_poly(+A1*x^B1+P1,+A2*x^B2+P2,+A3*x^B3+P3):-
I will generalize away some of the arguments:
add_poly(+A1*x^B1+P1,_,_):-
and some of the subterms:
add_poly(+_+_,_,_):-
This corresponds to:
add_poly(+(+(_),_),_,_) :-
Not sure you like this.
So this rule applies only to terms starting with a prefix + followed by an infix +. At least your sample data did not contain a prefix +.
Also, please remark that the +-operator is left associative. That means that 1+2+3+4 associates to the left:
?- write_canonical(1+2+3+4).
+(+(+(1,2),3),4)
So if you have a term 0+3*x^3+2*x^4 the first thing you "see" is _+2*x^4. The terms on the left are nested deeper.
For your actual question (how to stop) - you will have to test explicitly that the leftmost subterm is an integer, use integer/1 - or maybe a term (*)/2 (that depends on your assumptions).
I assume that polynomials you are speaking of are in 1 variable and with integer exponents.
Here a procedure working on normal polynomial form: a polynomial can be represented as a list (a sum) of factors, where the (integer) exponent is implicitly represented by the position.
:- [library(clpfd)].
add_poly(P1, P2, Sum) :-
normalize(P1, N1),
normalize(P2, N2),
append(N1, N2, Nt),
aggregate_all(max(L), (member(M, Nt), length(M, L)), LMax),
maplist(rpad(LMax), Nt, Nn),
clpfd:transpose(Nn, Tn),
maplist(sumlist, Tn, NSum),
denormalize(NSum, Sum).
rpad(LMax, List, ListN) :-
length(List, L),
D is LMax - L,
zeros(D, Z),
append(List, Z, ListN).
% the hardest part is of course normalization: here a draft
normalize(Ts + T, [N|Ns]) :-
normalize_fact(T, N),
normalize(Ts, Ns).
normalize(T, [N]) :-
normalize_fact(T, N).
% build a list with 0s left before position E
normalize_fact(T, Normal) :-
fact_exp(T, F, E),
zeros(E, Zeros),
nth0(E, Normal, F, Zeros).
zeros(E, Zeros) :-
length(Zeros, E),
maplist(copy_term(0), Zeros).
fact_exp(F * x ^ E, F, E).
fact_exp(x ^ E, 1, E).
fact_exp(F * x, F, 1).
fact_exp(F, F, 0).
% TBD...
denormalize(NSum, NSum).
test:
?- add_poly(0+2*x^3, 0+1*x^3+2*x^4, P).
P = [0, 0, 0, 3, 2]
the answer is still in normal form, denormalize/2 should be written...