Find an input to the system to achieve desired output - prolog

Question

I have the following equation:
y(t) = x(t) and not x(t-1)
x, y are boolean variables
t is discrete time
which is an edge detector.
I further know that
y(0) = 0
y(1) = 0
y(2) = 1
y(3) = 0
y(4) = 0
and initial conditions
x(0) = 1
x(4) = 0
I would like to solve the equation so that i find satisfactory values for x(1), x(2), x(3). For this example I expect one of the solutions to be:
x(1) = 0
x(2) = 1
x(3) = 1
I am Prolog beginner, but I think that Prolog is exactly the thing usable for solving tasks like this. I would write the expected output and initial conditions as facts:
y(0,0).
y(1,0).
y(2,1).
y(3,0).
y(4,0).
x(0,1).
x(4,0).
and the equation self as a predicate. How to rewrite the equation as a Prolog predicate and how to query for x ? Is it possible to obtain all soulutions?
Many thanks for every idea!
Best wishes.
After reading the 1st answer
There is a mistake in the constraints (if I use this term well).
The equation (system) as stimulated by x(t) will give particular y(t)
x(t): 1 0 1 0 0
y(t): X 0 1 0 0
t : 0 1 2 3 4
where X means "we do not know" because x(-1) needs to be known for y(0) computation.
The task is to compute x(t) for t in 1..4 when y(t) is known.
x(0) and x(4) can be requested to have particular values or not. This only affects the number of solutions, but it is not crucial for the essence of the task, as I think.
For given
y(t): X 0 1 0 0
t : 0 1 2 3 4
there are imho two solutions:
S1:
x(t): 1 0 1 0 0
y(t): X 0 1 0 0
t : 0 1 2 3 4
S2:
x(t): 1 0 1 1 0
y(t): X 0 1 0 0
t : 0 1 2 3 4
If there are no initial/final constraints for x(t) there are at least these solutions in addition:
S3:
x(t): 1 0 1 1 1
y(t): X 0 1 0 0
t : 0 1 2 3 4
S4:
x(t): 0 0 1 0 0
y(t): X 0 1 0 0
t : 0 1 2 3 4
S5:
x(t): 0 0 1 1 0
y(t): X 0 1 0 0
t : 0 1 2 3 4
S6:
x(t): 0 0 1 1 1
y(t): X 0 1 0 0
t : 0 1 2 3 4
I would like to solve this even for longer series and for more complicated equations, this example is very simplified.

Answer 1

Consider using a constraint solver over Boolean variables or finite domains. For example, with SWI-Prolog:
:- use_module(library(clpfd)).
edge(Xs, Ys) :-
Xs = [X0,X1,X2,X3,X4],
Ys = [_Y0,Y1,Y2,Y3,Y4],
Xs ins 0..1,
Ys ins 0..1,
X0 = 1,
X4 = 0,
Ys = [0,0,1,0,0],
Y1 #= X1 #/\ Y1 #\= X0,
Y2 #= X2 #/\ Y2 #\= X1,
Y3 #= X3 #/\ Y3 #\= X2,
Y4 #= X4 #/\ Y4 #\= X3.
I get:
?- edge(Xs, Ys).
false.
So assuming I understood and modeled this correctly, your constraints do not admit a solution. When I comment out the last constraint, I get a unique solution:
?- edge(Xs, Ys).
Xs = [1, 0, 1, 0, 0],
Ys = [0, 0, 1, 0, 0].
It is clear from this that no Y4 can satisfy the final constraint with these given Xs, since it would have to be both 0 and distinct from 0 (the final two elements in Xs). You can use label/1 to enumerate all concrete solutions when there are still variables in your answers (which is not the case above).