Axiomatizing causal reasoning

Causal models defined in terms of a collection of equations, as defined by Pearl, are axiomatized here. Axiomatizations are provided for three success ively more general classes of causal models: (1) the class of recursive the ories (those without feedback), (2) the class of theories where the solutio ns to the equations are unique, (3) arbitrary theories (where the equations may not have solutions and, if they do, they are not necessarily unique). It is shown that to reason about causality in the most general third class, we must extend the language used by Galles and Pearl (1997, 1998). In addi tion, the complexity of the decision procedures is characterized for all th e languages and classes of models considered.