Like Pearl’s System Z, c-representations provide a constructive approach to compute a ranking function from a conditional belief base from which further (conditional) beliefs can be derived, meeting major quality standards of nonmonotonic reasoning. This paper proposes a network-based structure for c-representations that allows for cutting down the complexity of reasoning significantly by decomposing the conditional belief base over a hypertree. We introduce c-LEG networks capturing the interactions among conditionals on a syntactical basis in full compatibility with the semantics of c-representations. This allows for reasoning in much smaller local contexts while still complying with the global information provided by the full conditional belief base. Moreover, we generalize the so-called safety property, which was recently presented in the context of conditional syntax splitting, to ensure that local c-representations of subbases over the hyperedges can be merged to yield global c-representations of the full conditional belief base. This allows for computing global c-representations step by step in local contexts, following the structure of the hypertree.

It is well-known from probability theory that network-based methods like Bayesian networks constitute remarkable frameworks for efficient probabilistic reasoning. In this paper, we focus on qualitative default reasoning based on Spohn’s ranking functions for which network-based methods have not yet been studied satisfactorily. With constraint networks, we develop a framework for iterative calculations of c-representations, a family of ranking models of conditional belief bases which show outstanding properties from a commonsense and formal point of view, that are characterized by assigning possible worlds a degree of implausibility via penalizing the falsification of conditionals. Constraint networks unveil the dependencies among these penalty points (and hence among the conditionals) and make it possible to compute the penalty points locally on so-called safe sub-bases. As an application of our framework, we show that skeptical c-inferences can be drawn locally from safe sub-bases without losing validity.

The concept of conditional syntax splitting for inductive inference from conditional belief bases has been proposed as a generalization of syntax splitting which also covers cases where the conditionals in the subbases share some atoms. p-Entailment and system Z fail to satisfy conditional syntax splitting, and up to now, only two inductive inference operators, lexicographic inference and system W, have been shown to satisfy this property. In this paper, we introduce the concept of conditional semantic splitting. We show that c-representations satisfy a core postulate relating conditional splittings on the syntax and the semantic level. Based on these findings, we investigate conditional syntax splitting for nonmonotonic inference with c-representations. Regarding single c-representations, we utilize the concept of selection strategies, and show that a straightforward property of the selection strategy leads to inference operators satisfying conditional syntax splittings. Furthermore, we show that c-inference taking all c-representations of a belief base into account also fully complies with conditional syntax splitting.

The principle of maximum entropy (MaxEnt principle) constitutes a valuable methodology for probabilistic commonsense reasoning by adding missing information to probabilistic conditional belief bases in an information theoretically optimal way. In this paper, we integrate linear arithmetic constraints over the integers and reals into propositional probabilistic conditionals in order to be able to formalize uncertain beliefs about arithmetic expressions. The satisfiability of (sets of) constraints is decided modulo theory such that probabilistic reasoning stays finite although the constraints range over infinite domains. Therewith, we provide a novel extension of the MaxEnt principle to beliefs about infinite domains.