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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.
Given an undirected graph G=V,E, 2 vertices: x, y and an edge e,
I would like to check if there is a path from x to y that contains the given edge e.
What I thought: Solve this by defining a network flow where x and y are source and sink and check if the flow in e is more than 0 it means that there is a path.
BUT there are 2 problems:
I don't know how to direct to edges
What would be the capacity of every edge?
So I'm guessing it's not the right approach... If someone can give an idea it'd be great.
One approach could be the following:
Remove the edge e = (u,v) from E(G).
If any x-y path contains e then the path will either be one of the two following forms: (1) x *-> u -> v *-> y or (2) x *-> v -> u *-> y where *-> means reachable. Use this fact to check if either of the following cases holds true
2.1. x is reachable from u and y is reachable from v.
2.2. x is reachable from v and y is reachable from u.
We can use BFS to find reachablility.
I am trying to implement a depth first iterative deepening search of a state space graph.
I have a graph with three vertices and their are two activating edges and two inhibition edges. Each node has a binary value, collectively this is the state of the graph. The graph can transition to a new state by seeing if one of the nodes is above a threshold or below a threshold (calculated from summing all the incoming nodes). At most one node will change at each transition. As their are three nodes, their are three state transition edges leaving each state in the state transition graph.
I think my state_change/3 works correctly, for instance I can query:
?-g_s_s(0,1,1,Begin),node(Arc),state_change(g_s(Begin),Second,Arc).
And it gives me the three correct answers:
Begin = [node(v1, 0), node(v2, 1), node(v3, 1)],
Arc = v1,
Second = g_s([node(v1, 1), node(v2, 1), node(v3, 1)]) ;
Begin = [node(v1, 0), node(v2, 1), node(v3, 1)],
Arc = v2,
Second = g_s([node(v1, 0), node(v2, 0), node(v3, 1)]) ;
Begin = [node(v1, 0), node(v2, 1), node(v3, 1)],
Arc = v3,
Second = g_s([node(v1, 0), node(v2, 1), node(v3, 0)])
I am trying to use the predicate id_path given in Bratkos Prolog for A.I book, the solution to question 11.3 but I am having problems using/adapting it. I want to create a path from a start node to the other nodes, with out getting into loops- I don't want it to have repeat elements or to get stuck when a path does not exist. I want the path to say the starting state, then a succession of states you can visit from the start state. If there is a self loop I want this to be included once for every way of getting there. Ie I want to keep track of the way that I got to the state space and make this unique not just that the state space is unique in the path.
For instance from 011 I want all three paths of length one to be found with the arcs.
?-id_path(g_s([node(v1,0),node(v2,1),node(v3,1)],Last,[Temp],Path).
Path = [[node(v1,0),node(v2,1),node(v3,1)],to([node(v1,1),node(v2,1),node(v3,1)],v1)];
Path =[[node(v1,0),node(v2,1),node(v3,1)], to([node(v1,0),node(v2,0),node(v3,1)],v2)];
Path=[[node(v1,0),node(v2,1),node(v3,1)],to([node(v1,1),node(v2,1),node(v3,0)],v3)];
and then at the next level all the paths with three nodes, showing the two arcs it needs to get to the nodes, then at the next level all the paths with fours nodes showing the three arcs it needs etc
I have also put my code in SWISH if this is helpful? (Trying this for the first time?!)
http://pengines.swi-prolog.org/apps/swish/p/HxBzEwLb.pl#&togetherjs=xydMBkFjQR
a(v1,v3). %a activating edge
a(v3,v1).
i(v1,v2). %a inhibition edge
i(v2,v3).
nodes([v1,v2,v3]).
node(X):- nodes(List),member(X,List). %to retrieve a node in graph a) or an arc in graph b)
g_s_s(X,Y,Z,g_s([node(v1,X),node(v2,Y),node(v3,Z)])). %graph_state_simple - I use this to simply set a starting graph state.
sum_list([], 0).
sum_list([H|T], Sum) :-
sum_list(T, Rest),
Sum is H + Rest.
invert(1,0).
invert(0,1).
state_of_node(Node,g_s(List),State):-
member(node(Node,State),List).
%all activating nodes in a graph state for a node
all_a(Node,As,Ss,g_s(NodeList)):-
findall(A, a(A,Node),As),
findall(S,(member(M,As),member(node(M,S),NodeList)),Ss).
%all inhibiting nodes in a graph state for a node
all_i(Node,Is,Ss,g_s(NodeList)):-
findall(I, i(I,Node),Is),
findall(S,(member(M,Is),member(node(M,S),NodeList)),Ss).
%sum of activating nodes of a node in a state
sum_a(Node,g_s(NodeList),Sum):-
all_a(Node,_As,Ss,g_s(NodeList)),
sum_list(Ss,Sum).
%sum of inhibiting nodes of a node in a state
sum_i(Node,g_s(NodeList),Sum):-
all_i(Node,_Is,Ss,g_s(NodeList)),
sum_list(Ss,Sum).
above_threshold(Threshold,Node,g_s(NodeList),TrueFalse):-
sum_a(Node,g_s(NodeList),Sum_A),
sum_i(Node,g_s(NodeList),Sum_I),
TrueFalse = true,
Threshold < (Sum_A-Sum_I),
!.
above_threshold(Threshold,Node,g_s(NodeList),TrueFalse):-
sum_a(Node,g_s(NodeList),Sum_A),
sum_i(Node,g_s(NodeList),Sum_I),
TrueFalse = false,
Threshold >= (Sum_A-Sum_I).
%arc needs to be instantiated
state_change(g_s(State1),g_s(State1),Arc):-
above_threshold(0,Arc,g_s(State1),true),
state_of_node(Arc,g_s(State1),1).
state_change(g_s(State1),g_s(State2),Arc):-
above_threshold(0,Arc,g_s(State1),false),
state_of_node(Arc,g_s(State1),1),
my_map(State1,State2,Arc).
state_change(g_s(State1),g_s(State2),Arc):-
above_threshold(0,Arc,g_s(State1),true),
state_of_node(Arc,g_s(State1),0),
my_map(State1,State2,Arc).
state_change(g_s(State1),g_s(State1),Arc):-
above_threshold(0,Arc,g_s(State1),false),
state_of_node(Arc,g_s(State1),0).
%
my_map([],[],_).
my_map([X|T],[Y|L],Arc):-
X= node(Node,Value1),
Node =Arc,
invert(Value1,Value2),
Y = node(Node,Value2),
my_map(T,L,Arc).
my_map([X|T],[Y|L],Arc):-
X= node(Node,Value1),
Node \= Arc,
Y = node(Node,Value1),
my_map(T,L,Arc).
%this is the def in the book which I can not adapt.
path(Begin,Begin,[start(Begin)]).
path(First, Last,[First,Second|Rest]):-
state_change(First,Second,Arc),
path(Second,Last,[Second|Rest]).
%this is the def in the book which I can not adapt.
id_path(First,Last,Template,Path):-
Path = Template,
path(First,Last,Path)
; copy_term(Template,P),
path(First,_,P),
!,
id_path(First,Last,[_|Template],Path).
Since the state space is finite, there will be only finitely many minimal loops or terminal paths. The following options come to mind to represent minimal loops in a graph.
- Rational Terms: Some Prolog systems support rational terms, so a repeating path [0,1,2,2,2,...] can be represented as X = [0,1|Y], Y=[2|Y].
- Non-Rational Terms: You could of course represent a repeating path also as a pair. The previous example would then be ([0,1], [2]).
Detecting a loop pattern and find not only whether something is loop, but also the part that is looping, can be archived by the following code. The append predicate will do the search:
?- append(X, [2|Y], [0,1,2,3]).
X = [0, 1],
Y = [3]
So we know that when we have already found a path [0,1,2,3], and when we see a node 2, that we have found a loop, and we can expressed the found loop with an Omega word as follows [0,1] [2,3]ω. Here is a simple backtracking code:
path(P, P).
path((C,[]), P) :-
last(C, X), edge(X, Y),
extend(C, Y, A, B), path((A,B), P).
extend(C, Y, A, [Y|B]) :-
append(A, [Y|B], C), !.
extend(C, Y, A, []) :-
append(C, [Y], A).
Here is an example run:
?- path(([s(0,1,1)],[]), X).
X = ([s(0,1,1)],[]) ;
X = ([s(0,1,1),s(0,1,0)],[]) ;
X = ([s(0,1,1),s(0,1,0),s(0,0,0)],[]) ;
X = ([s(0,1,1),s(0,1,0)],[s(0,0,0)]) ;
X = ([s(0,1,1)],[s(0,1,0)]) ;
X = ([s(0,1,1),s(1,1,1)],[]) ;
...
Okay, so I've been trying to teach myself Prolog recently, and am having a hard time wrapping my head around finding a "Shortest Path" between two (defined) elements in a list of lists. It may not be the most effective way of representing a Grid or finding a Shortest Path, but I'd like to try it this way.
For example:
[[x,x,x,x,x,x,x],
[x,1,o,o,o,o,x],
[x,-,-,-,o,-,x],
[x,-,-,o,o,-,x],
[x,o,o,o,o,2,x],
[x,o,-,-,o,o,x],
[x,x,x,x,x,x,x]]
A few assumptions I can make (either given or based on checking before path-finding):
The grid is square
Their will always exist a path from 1 to 2
'1' can pass through anything except '-' (walls) or 'x' (borders)
The goal is for '1' to find a shortest path to '2'.
In the instance of:
[[x,x,x,x,x,x,x],
[x,o,o,1,o,o,x],
[x,-,o,o,o,-,x],
[x,-,o,-,o,-,x],
[x,o,o,2,o,o,x],
[x,o,-,-,-,o,x],
[x,x,x,x,x,x,x]]
Notice, there are two "Shortest paths":
[d,l,d,d,r]
and
[d,r,d,d,l]
In Prolog, I'm trying to make the function (if that's the proper name):
shortestPath(Grid,Path)
I've made a function to find elements '1' and '2', and a function that verifies that the grid is valid, but I can't even begin how to start constructing a function to find a shortest path from '1' to '2'.
Given a defined Grid, I'd like the output of Path to be the shortest path. Or, given a defined Grid AND a defined Path, I'd like to check if it's indeed a shortest path.
Help would be much appreciated! If I missed anything, or was unclear, let me know!
not optimized solution
shortestPath(G, S) :-
findall(L-P, (findPath(G,P), length(P,L)), All),
keysort(All, [_-S|_]).
findPath(G, Path) :-
pos(G, (Rs,Cs), 1),
findPath(G, [(Rs,Cs)], [], Path).
findPath(G, [Act|Rest], Trail, Path) :-
move(Act,Next,Move),
pos(G, Next, Elem),
( Elem == 2
-> reverse([Move|Trail], Path)
; Elem == o
-> \+ memberchk(Next, Rest),
findPath(G, [Next,Act|Rest], [Move|Trail], Path)
).
move((R,C), (R1,C1), M) :-
R1 is R-1, C1 is C , M = u;
R1 is R , C1 is C-1, M = l;
R1 is R+1, C1 is C , M = d;
R1 is R , C1 is C+1, M = r.
pos(G, (R,C), E) :- nth1(R, G, Row), nth1(C, Row, E).
grid(1,
[[x,x,x,x,x,x,x],
[x,1,o,o,o,o,x],
[x,-,-,-,o,-,x],
[x,-,-,o,o,-,x],
[x,o,o,o,o,2,x],
[x,o,-,-,o,o,x],
[x,x,x,x,x,x,x]]).
grid(2,
[[x,x,x,x,x,x,x],
[x,o,o,1,o,o,x],
[x,-,o,o,o,-,x],
[x,-,o,-,o,-,x],
[x,o,o,2,o,o,x],
[x,o,-,-,-,o,x],
[x,x,x,x,x,x,x]]).
I would like to know if there is a way to extract the X Value when I'm doing this :
aggregate_all(min(V), simulate(P, Color, V, X), Value)
The simulate predicate is used with P and Color as inputs and V and X as outputs.
For now, this works well to get the min value of V, but what I actually want is to get the value of X when V is at its minimum.
Is there a way to do that ? Any idea about how should I proceed ?
aggregate library supports a 'Witness' on min/max scalar aggregates: then this should work
aggregate_all(min(V,X), simulate(P, Color, V, X), min(Value,X))