Any problem-solving can be modelled by actions or methods by which from resources or data, one agent makes an action to obtain a result or arrive at a task. A network of actions can be used as a model of the behaviour of the agents. Any sink in the network is a final goal or task. The other tasks are only intermediate tasks. Any source in the network is a primitive resource from which we can begin to obtain results or tasks. Cycles in the network are self-generated resources from the tasks. Now we denote “agent” as the first order that any agent can make one action or can run a method. Now we argue that there also exist agents at the second and at the more high order. Agents that copy the agents of the first order are agents of the second order. To copy one agent of the first order means to copy all the properties of one agent or part of the properties. This is similar to the offspring for animals. The network of resources, action, and tasks in the new agent, has the same properties or part of the properties of the original network. In the copy process, it is possible that the new agent has new properties that are not present in the prototype agent. This is similar to the genetic process of the cross over. The agent of the second order uses the prototype agent as a reference to create new a agent in which all the properties or part of the properties of the original agent are present. When all properties of the prototype network of agents are copied in the new network of agents, we have a symmetry between the prototype network of agents in the new network. When agents are permuted in the same network, agents can change their type of activities without losing the global properties of the network. The properties are invariant for the copy operation as permutation. We remember that also, if two networks of agents have the same properties, they are not equal. When in the copy process, only part of the properties do not change, and new properties appear; in this case we say that we have a break of symmetry. For example, in the animals in the clone process, one cellule is generated from another. The new cell has the same properties of the old cell. In this case, we have symmetry among cells. In fact, because any cell is considered as a network of internal agents (enzymes), two cells are in a symmetric position when the internal network of both the cells have the same properties. With the sexual copy process, it is possible that we lose properties or we generate new properties. In this case, the cellular population assume or lose properties. We break the symmetry in the cellular population. The adaptation process can be considered as a copy process triggered by the environment. For example, to play chess is a network of possible actions with resources and tasks. Any player is an agent of the second order that can change the network of the possible actions. The player can copy the schemes or network of actions located in the external environment in his mind. A physician that makes a model of the nature is a second order agent that makes a copy of the agent’s network of actions in a physical nature into the symbolic domain of the mathematical expressions. Agents as ants can share resources from one field generated by other agents or ants. This field is a global memory that is used by the agents. For example, an ant pheromone field, generated by any ant, is used by all ants. In this way, ants are guided to obtain their task (minimum path). In this case, the pheromone field is an example of global memory resource. The network of connection among ants and its field is shown in the article. Agents that take care to copy one network of agents in another network of agents are agents of the second order. Because we can also copy the network of agents of the second order by agents, these agents are at the third order. In this way, agents of any order can control and adapt a network of agents at a lesser order.
THE n-r.e. DEGREES: UNDECIDABILITY AND Σ1 SUBSTRUCTURES
