핵심 개념
The core message of this paper is that the complexity of probabilistic and causal reasoning with summation operators remains equally difficult, but allowing free variables for random variable values results in an undecidable system.
초록
The paper analyzes the complexity of probabilistic and causal reasoning with summation operators. It builds on previous work that axiomatized increasingly expressive languages of causation and probability, and showed that reasoning in each causal language is as difficult as reasoning in its merely probabilistic or "correlational" counterpart.
The key insights are:
Introducing a summation operator to capture common devices like the do-calculus partially extends the earlier complexity results to causal and probabilistic languages with marginalization. The paper completes this extension, fully characterizing the complexity of probabilistic and causal reasoning with summation.
Surprisingly, allowing free variables for random variable values results in a system that is undecidable, so long as the ranges of these random variables are unrestricted. This is due to the fact that the problem of deciding whether a set of conditional independence statements implies another is undecidable in this setting.
The paper axiomatizes these languages featuring marginalization (or more generally summation), resolving open questions posed by previous work.
For the bounded case where variables have finite ranges, the paper shows that the satisfiability problem for the causal language with summation is complete for the complexity class succDR. This extends the previous results on the complexity of causal reasoning.
For the unbounded case, the paper shows that satisfiability for the probabilistic and causal languages with summation is recursively enumerable.