# On modelling problem situations with tuples

### Y. B. Karasik, Thoughts Guiding Systems Corp., Ottawa, Canada. e-mail:karasik@sympatico.ca

Many problems can be characterized by a set of tuples (action, the possibility of the action, property of object undergoing the action, the possibility of the property, result, its wantedness). For example, consider the following problem situation. To obtain a wanted result R when action A1 is excerted on an object it has to have property P1, but to obtain a wanted result when action A2 is excerted the object has to have property P2. This situation can be characterized by 2 tuples: (A1, possible, P1, possible, R, wanted) and (A2, possible, P2, possible, R, wanted). The problem is due to object cannot have two properties P1 and P2 at the same time.

Consider another situation: to convert an object with a property P1 into an object with property P it has to be underwent action A1, which is impossible. But in order for a possible action A2 converted an object into object with property P it has to have initial property P2. Here we have 2 tuples: (A1, impossible, P1, possible, R, wanted) and (A2, possible, P2, possible, R, wanted).

Generally, the number of actions involved in a situation could be not 2 but greater. The same goes for the number of object properties and the number of results. Thus, one additional tuple charaterizes a situation. This tuple is (I, J, K, L, M, N), where I is the number of actions, J is the number of impossible actions, K is the number of object properties, L is the number of impossible properties, M is the total number of results, N is the number of unwanted results. In what follows I will call such an additional tuple the characteristic tuple of a problem situation.

The first situation above is characterized by the following additonal tuple (2, 0, 2, 0, 2, 0), whereas the second situation is characterized by an additional tuple (2, 1, 2, 0, 2, 0).

The situation contains a contradicrtion when either J>0, or K>1, or N>0. The both aforementioned situations contain a contradiction.

Here are other possible problem situations containing a contradiction.

1. To obtain a wanted result R1 when an object is undergone action A1 it has to have property P1. But in order for not obtaining a parallel unwanted result R2, it has to have property P2. Thus, the characteristic tuple of such problem situation is (1, 0, 2, 0, 2, 1).
2. Action A1 on an object brings about the wanted result R1. To enhance this result or speed it up action A1 has to be intensified to the level A2. But this also results in an unwanted effect on the object. Thus, we have 3 tuples (A1, possible, P1, possible, R1, wanted), (A2, possible, P1, possible, R2, wanted), and (A1, possible, P1, possible, R3, unwanted). The characteristic tuple is (2, 0, 1, 0, 3, 1).
3. To convert oan object into another object it has to be undergone action A, which is hardly possible. This situation can be modelled by a tuple (A, impossible, P, possible, R, wanted) and a characteristic tuple (1, 1, 1, 0, 1, 0).
4. In order for action A1 of one object on another one there would be no unwanted effect R1, the second object has to have property P1, which is hard to achieve. If the object does not have propery P1, then action A2 does not cause unwanted effect R1 (meaning it causes a wanted effect R2). Thus, we have (A1, possible, P1, impossible, R1, unwanted), (A2, possible, P2, possible, R2, wanted), and (2, 0, 2, 1, 2, 1).

We considered problem situations characterized by tuples

(2, 0, 2, 0, 2, 0)
(2, 1, 2, 0, 2, 0)
(1, 0, 2, 0, 2, 1)
(2, 0, 1, 0, 3, 1)
(1, 1, 1, 0, 1, 0)
(2, 0, 2, 1, 2, 1)

Generally, the number of different actions and object properties can hardly be greater than 2, whereas the number of results could be as high as 4. Thus all problem situations can be characterized by 2 x 3 x 2 x 3 x 4 x 5 = 720 different characteristic tuples. Hence, there are 720 types of problem situations and corresponding physical/logical contradictions.