# Probabilistic Logic under Coherence: Complexity and Algorithms

*Veronica Biazzo‚ Angelo Gilio‚ Thomas Lukasiewicz and Giuseppe Sanfilippo*

### Abstract

We study probabilistic logic under the viewpoint of the coherence principle of de Finetti. In detail, we explore the relationship between coherence-based and classical model-theoretic probabilistic logic. Interestingly, we show that the notions of g-coherence and of g-coherent entailment can be expressed by combining notions in model-theoretic probabilistic logic with concepts from default reasoning. Using these results, we analyze the computational complexity of probabilistic reasoning under coherence. Moreover, we present new algorithms for deciding g-coherence and for computing tight g-coherent intervals, which reduce these tasks to standard reasoning tasks in model-theoretic probabilistic logic. Thus, efficient techniques for model-theoretic probabilistic reasoning can immediately be applied for probabilistic reasoning under coherence, for example, column generation techniques. We then describe two other interesting techniques for efficient model-theoretic probabilistic reasoning in the conjunctive case.