Reasoning With Qualitative Probabilities Can Be Tractable

TL;DR

This paper improves the computational understanding of reasoning with qualitative probabilities, showing polynomial complexity for certain cases and integrating belief revision within a formalism based on extreme conditional probabilities.

Contribution

It demonstrates that computing the priority function Z+ is polynomial, refuting previous conjectures of exponential complexity, and extends the formalism to handle imprecise observations and belief revision.

Findings

Computing Z+ requires polynomial number of satisfiability tests.

Reasoning with imprecise observations can be incorporated into the formalism.

Belief revision and epistemic entrenchment are naturally embodied and tractable.

Abstract

We recently described a formalism for reasoning with if-then rules that re expressed with different levels of firmness [18]. The formalism interprets these rules as extreme conditional probability statements, specifying orders of magnitude of disbelief, which impose constraints over possible rankings of worlds. It was shown that, once we compute a priority function Z+ on the rules, the degree to which a given query is confirmed or denied can be computed in O(log n`) propositional satisfiability tests, where n is the number of rules in the knowledge base. In this paper, we show that computing Z+ requires O(n2 X log n) satisfiability tests, not an exponential number as was conjectured in [18], which reduces to polynomial complexity in the case of Horn expressions. We also show how reasoning with imprecise observations can be incorporated in our formalism and how the popular notions of belief revision and epistemic entrenchment are embodied naturally and tractably.