Probabilistic logic under coherence: complexity and algorithms

Abstract

In previous work [V. Biazzo, A. Gilio, T. Lukasiewicz and G. Sanfilippo, Probabilistic logic under coherence, model-theoretic probabilistic logic, and default reasoning in System P, Journal of Applied Non-Classical Logics 12(2) (2002) 189-213.], we have explored the relationship between probabilistic reasoning under coherence and model-theoretic probabilistic reasoning. In particular, we have shown that the notions of g-coherence and of g-coherent entailment in probabilistic reasoning under coherence can be expressed by combining notions in model-theoretic probabilistic reasoning with concepts from default reasoning. In this paper, we continue this line of research. Based on the above semantic results, we draw a precise picture of the computational complexity of probabilistic reasoning under coherence. Moreover, we introduce transformations for probabilistic reasoning under coherence, which reduce an instance of deciding g-coherence or of computing tight intervals under g-coherent entailment to a smaller problem instance, and which can be done very efficiently. Furthermore, we present new algorithms for deciding g-coherence and for computing tight intervals under g-coherent entailment, which reformulate previous algorithms using terminology from default reasoning. They are based on reductions to standard problems in model-theoretic probabilistic reasoning, which in turn can be reduced to linear optimization problems. Hence, efficient techniques for model-theoretic probabilistic reasoning can immediately be applied for probabilistic reasoning under coherence (for example, column generation techniques). We describe several such techniques, which transform problem instances in model-theoretic probabilistic reasoning into smaller problem instances. We also describe a technique for obtaining a reduced set of variables for the associated linear optimization problems in the conjunctive case, and give new characterizations of this reduced set as a set of non-decomposable variables, and using the concept of random gain

A Propositional CONEstrip Algorithm

“…This could provide efficiency gains similar to those obtained by algorithms for probabilistic reasoning from the Italian school [1,3]. …”

Coherence checking and propagation of lower probability bounds

“…Further work is needed to determine some algorithms which compute T in a direct way. A relevant example in this sense is the procedure proposed in [30] for the case of conjunctive conditional events, which has been characterized from the point of view of coherence in [5]. As a positive note, we observe that the ''small'' systems resulting from the process of elimination of unknowns are easily checked for their solvability.…”

“…Notice that our method generalizes the procedure proposed in [30] and provides it with a semantics based on the betting criterion (cf. [33,5]). We recall that the basic idea of exploiting zero probabilities has been suggested in [17] and has been studied in [9][10][11], where some concepts of local and strong local coherence have been developed, with that aim of splitting the problem into subproblems.…”

Conjunction of Conditional Events and t-Norms

“…In this paper we use the coherence-based approach to probability of de Finetti ( [1,2,7,9,10,13,14,16,17,18,22,25]). We use a notion of conjunction which, differently from other authors, is defined as a suitable conditional random quantity with values in the unit interval (see, e.g.…”