If we want to achieve a goal such as becoming rich or getting from here to Skørping, we use the knowledge and beliefs we have about the world to lay down a strategy for attaining this goal. Our knowledge might include facts about how to obtain information on public transportation, how to get to the nearest train station, etc. Knowledge based AI (artificial intelligence) is based on the same idea: our AI systems should have knowledge and beliefs about the environment they inhabit, which they can use for planning and reasoning.

The knowledge and beliefs that a human being or an AI system has of its world forms a model of this world. We can think of our brains as containing a "map" of the world -- the world as we see it. We use our model of the world to predict the consequences of performing various possible actions, and thus in planning which actions to take in a given situation (for instance in planning the next move in a game of chess).

When we want to equip AI systems with models of the world that surrounds them, we encounter several theoretical challenges concerning the relation between model and reality (between that which models and that which is being modelled). The picture above gives a very simple example of a model (the Lego model of Big Ben in Legoland, Billund, Denmark) and the corresponding reality (Big Ben in London).

One of the major challenges in constructing AI systems is to give these self-awareness and the ability to reflect on themselves and their own models of the world. Having such abilities seems to be of cruical importance in obtaining powerful AI systems, since much intelligent behavior is known to involve an ability to model not only one's external environment but also oneself and one's own reasoning. For instance, if we want to get better at playing chess we would reflect on ourselves and the way we play at present to see where and how to improve our skills. An AI system that can reflect on itself must have a model which is modelling a part of reality containing the model itself. This presents us with a fundamentally different situation from the simple one in which the model and the thing being modelled are completely separate (as in the Big Ben case above). To get an idea about the kind of problems that can occur if we want to construct a model modelling a reality containing the model itself, consider the following situation.

In Legoland there is a Lego model of different cities around the world. Assume it is decided to extend Legoland in Billund to contain a Lego model of the entire globe. Then this Lego model will contain a model of Legoland itself, since Legoland is on the globe. Assume I one day decide to go from Copenhagen to Billund to visit this new Legoland. The first thing I do in Legoland is to go to Lego-Copenhagen to try to find the miniature model of my own house. After seeing the house, I come up with the idea of trying to take the same trip in Legoland as I did in the real world the same morning to get from home to Billund. So I walk from Lego-Copenhagen to Lego-Billund and further to the entrance of Lego-Legoland and walk in. But ... where am I now? I am in Legoland's model of Legoland itself! I get the following thought: let me try to take the same trip in Lego-Legoland as I just did in Legoland. So I walk up to my house. But now this house is no longer a (direct) model of my house in Copenhagen but rather a model of the Lego model of my house. Thus it must be much smaller than the Lego model of my house I saw previously. The point is that I could now walk from this house to Lego-Lego-Billund and further into Lego-Lego-Legoland, and this process never stops.

If Legoland contains a model of Legoland itself, then it will contain infinitely many such models of itself nested within each other. Thus we would need to use infinitely many Lego bricks having no lower bound on their sizes to build this Legoland.

When we want to construct AI systems with the ability to reflect on themselves and their own models of the world we are faced with a similar situation of having infinitely many models nested within each other. To actually represent this situation in an appropriate manner in an AI system (computer, robot) is an interesting and important challenge in AI.

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