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Probabilistic logic under coherence: complexity and algorithms

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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.

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Author notes

  1. Thomas Lukasiewicz

    Present address: Institut für Informationssysteme, Technische Universität Wien, Favoritenstraße 9-11, A-1040, Vienna, Austria

Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università degli Studi di Catania, Città Universitaria, Viale A. Doria 6, I-95152, Catania, Italy

    Veronica Biazzo

  2. Dipartimento di Metodi e Modelli Matematici, Università di Roma “La Sapienza”, Via A. Scarpa 16, I-00161, Rome, Italy

    Angelo Gilio

  3. Dipartimento di Informatica e Sistemistica, Università di Roma “La Sapienza”, Via Salaria 113, I-00198, Rome, Italy

    Thomas Lukasiewicz

  4. Dipartimento di Scienze Statistiche e Matematiche, Università degli Studi di Palermo, Viale delle Scienze, I-90128, Palermo, Italy

    Giuseppe Sanfilippo

Authors
  1. Veronica Biazzo
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  2. Angelo Gilio
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  3. Thomas Lukasiewicz
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  4. Giuseppe Sanfilippo
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Corresponding author

Correspondence to Veronica Biazzo.

Additional information

This paper is a substantially extended and revised version of a preliminary paper that appeared in: Proceedings of the Second International Symposium on Imprecise Probabilities and Their Applications (ISIPTA '01), pp. 51–61, 2001.

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Biazzo, V., Gilio, A., Lukasiewicz, T. et al. Probabilistic logic under coherence: complexity and algorithms. Ann Math Artif Intell 45, 35–81 (2005). https://doi.org/10.1007/s10472-005-9005-y

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