Exhaustive search is a method in AI planning to find a solution for so-called Constraint Satisfaction Problems. (CSP). Those are problems that have some conditions to fulfill and the solver is trying out all the alternatives. An example CSP problem is the 8-queens problem which has geometrical constraints. The standard method in finding a solution for the 8-queens problem is a backtracking solver. That is an algorithm that generates a tree for the state space to search inside the graph.
Apart from practical applications of backtracking search, there are some logic-oriented discussions available which are asking on a formal level which kind of problems have a solution and which not. For example to find a solution for the 8-queen problem many millions of iterations of the algorithm are needed. The question is now: which problems are too complex to find a solution. The second problem is, that sometimes the problem itself has no solution, even the complete state space was searched fully.
Let us take an example. At first, we construct a problem in which the constraints are so strict that even a backtracking search won't find a solution. One example would be to prove that “1+1=3” another example would be to find a chess sequence if the game is lost or it is also funny to think about how to arrange nine! queen on a chess table so that they don't hurt.
Is there any literature available which is describing Constraint Satisfaction Problems on a theoretical basis in which the constraints of the problem are too strict?
Just wondering - like with an 8-queens problem. If we change it to a 9-queens problem and do an exhaustive search, we will see that there is no solution. Is there a problem in which the search fails to show that a solution does not exist?