The complexity of agent design problems: determinism and history dependence

The agent design problem is as follows: given a specification of an environment, together with a specification of a task, is it possible to construct an agent that can be guaranteed to successfully accomplish the task in the environment? In this article, we study the computational complexity of the agent design problem for tasks that are of the form Bachieve this state of affairs ^ or Bmaintain this state of affairs. ^ We consider three general formulations of these problems (in both non-deterministic and deterministic environments) that differ in the nature of what is viewed as an Bacceptable ^ solution: in the least restrictive formulation, no limit is placed on the number of actions an agent is allowed to perform in attempting to meet the requirements of its specified task. We show that the resulting decision problems are intractable, in the sense that these are non-recursive (but recursively enumerable) for achievement tasks, and non-recursively enumerable for maintenance tasks. In the second formulation, the decision problem addresses the existence of agents that have satisfied their specified task within some given number of actions. Even in this more restrictive setting the resulting decision problems are either PSPACE-complete or NPcomplete. Our final formulation requires the environment to be history independent and bounded. In these cases polynomial time algorithms exist: for deterministic environments the decision problems are NL-complete; in non-deterministic environments, P-complete