Approximate reasoning in the algebra of bounded rational agents

In this paper, we generalize the utility theory to allow to use various performance measures, including utilities, costs and fitness, and probability theory we extend to uncertainty theory, including probabilities, fuzzy sets and rough sets. The decision theory is defined typically as the combination of utility theory and probability theory. We generalize the decision theory as the performance measure theory and uncertainty theory. Bounded rational agents look for approximate optimal decisions under bounded resources and uncertainty. The $-calculus process algebra for problem solving applies the cost performance measures to converge to optimal solutions with minimal problem solving costs, and allows to incorporate probabilities, fuzzy sets and rough sets to deal with uncertainty and incompleteness. The approach is illustrated to find the optimal solutions with or without uncertainty. The same approach can be used to find solutions of the totally optimization problem, representing the tradeoff between the best quality and least costly solutions.