Annals of Mathematics and Artiﬁcial Intelligence 35: 125–149, 2002.  2002 Kluwer Academic Publishers. Printed in the Netherlands. Locally strong coherence in inference processes Andrea Capotorti a and Barbara Vantaggi b,∗ a Dipartimento di Matematica e Informatica, Università di Perugia, via Vanvitelli 1, 06123 Perugia, Italy E-mail: capot@dipmat.unipg.it b Dipartimento di Metodi e Modelli Matematici, Università “La Sapienza”, via Scarpa 16, 00161 Roma, Italy E-mail: vantaggi@dmmm.uniroma1.it In this paper we deal with probabilistic inference in the most general form of coherent conditional probability assessments. In particular, our aim is to reduce computational difﬁcul- ties that could arise with a direct application of the main characterization results. We reach our goal by introducing the notion of locally strong coherence and characterizing it by logical conditions. Hence, some of the numerical constraints are replaced by Boolean satisﬁability conditions. An automatic procedure is proposed and its efﬁciency is proved. Some examples are reported to make easier the understanding of the machinery and to show its effectiveness. Keywords: inference, conditional probability assessment, coherence, locally strong coherence AMS subject classiﬁcation: 60Axx, 62Cxx, 91-08 1. Introduction Probability is usually based on a measure-theoretic framework, so that the relevant models refer to a joint probability completely speciﬁed on a well-structured (algebra) set of events. Such joint probability is given either explicitly or (e.g., in graphical models) it comes from conditional independence statements and from the relevant conditional probabilities. Inside this framework, conditional probability P(E|H) requires that the conditioning event H has positive probability, so that it is computed as a simple ratio between the joint P(E ∧ H) and the marginal P(H). However, such models are unsuitable and not ﬂexible to manage partial information on arbitrary domains. Clearly, especially for real applications (e.g., decision process, medical diagnosis), it is signiﬁcant not to assume that the probability be completely spec- iﬁed on a structured set. In fact, a ﬁeld expert or an agent is not able to give categorical answers about all the events (i.e., sentences or statements) constituting his environment, and he must act under partial knowledge. “Ad hoc” methods for each speciﬁc situation would be needed to manage uncertainty on arbitrary domains within usual probabilistic theory. On the other hand, “alternative” approaches have been introduced in literature to model incomplete probabilistic evaluations, giving up any “ad hoc” assumption, and ∗ Corresponding author.