ABSTRACT. In this article, we take the operators that make up variable elimination in probabilistic graphical models and reinterpret them as forgetting operators in the vein of marginalisation for forgetting in logic. As such, we define the inference task of forgetting in probabilistic graphical models and provide operators to solve the task, discussing effects on complexity versus variable elimination for probabilistic inference. In addition, we consider different properties for forgetting in probabilistic graphical models, arguing whether variable elimination as forgetting fulfils them.

ABSTRACT. The conditions of trivial independence, symmetry, decomposition, and mixing are a well-established axiomatisation of independence. These are known to be sound and complete for the usual independence relation among random variables on probability distributions. We lift the semantic definitions needed to establish this result to the level of arbitrary structures of preference. As an example of this, we confirm that the same holds for ordinal conditional functions (OCFs): not only are independence statements over OCFs sound with respect to these axioms, but they are also complete. We also show that a faithful operator exists for OCFs, allowing us to deduce that, as for probability spaces, OCFs characterise these axioms.

ABSTRACT. Ranking functions, also known as ordinal conditional functions (OCFs), and total preorders on worlds (TPOs) are standard representations in belief change and nonmonotonic reasoning. In bounded unstructured spaces, we study how often these representations admit non-trivial syntax splittings, i.e., decompositions induced by partitions of the underlying propositional signature, and how the resulting splitting families are structured. For rank-bounded OCFs, coarsening closure makes the unique finest syntax splitting an exact counting handle. We derive exact counts, prevalence ratios, and finest-shape distributions. The resulting class is small relative to the ambient space and is strongly concentrated on coarse finest shapes. For depth-bounded TPOs, we report exact prevalence counts and analyze the subcorpus admitting non-trivial syntax splittings using Bell baselines, closure deficits, and relative splitting profiles. In the exhaustive |Σ| = 3 series, all such TPOs are Bell-complete up to depth 3, while closure-irregular families first occur at depth 4. A supplementary inspection interface connects these aggregate findings to concrete TPO instances by displaying splitting families, missing coarsenings, and bridge OCFs obtained from absolute marginal sums.

ABSTRACT. Nonmonotonic reasoning with conditional belief bases aims at answering the question “Given a belief base Δ and two formulas 𝐴 and 𝐵, does formula 𝐴 entail formula 𝐵 in the context of Δ?”. Answering these inference tasks is usually quite complex as their semantics involve structures that often grow exponentially in signature size. As a means of localizing such inferences tasks, safe conditional syntax splitting has been introduced in order to split conditional belief bases into smaller subbases, allowing to take only the relevant parts of a given belief base into account. Recently, the kinds of splittings that can be exploited for inductive inference have been refined in two ways: Generalized safe splittings broaden the number of splittings exploitable while genuine splittings identify those splittings that can be meaningfully used for inductive inference. While the theoretical aspect of splittings has been investigated in some detail, their implementation has received far less attention. We present algorithms for computing the different kinds of conditional syntax splittings of a given belief base. These algorithms are capable of computing all conditional syntax splittings and finding all safe, generalized safe, and genuine splittings of a belief base. A modular approach ensures that any combination of safe, generalized safe, and genuine, e.g., the genuine and generalized safe splittings, can be easily computed as well. We evaluate an implementation of these algorithms with respect to the number of splittings computed for each kind and the runtime of each algorithm, analyzing in particular the differences between safe and generalized safe splittings, as well as the benefits of genuine splittings.

ABSTRACT. (We submit a (dense) Extended Abstract) We present a general framework and postulates for defaults and default inference. One issue here is the Exceptional Inheritance Paradox, which reveals a serious conflict between well-recognized axioms for default conditionals and natural desiderata for default inference. Reacting to the trichotomy induced by an impossibility result centered on Splitting, we will identify default formalisms exemplifying each branch, including two new ones.

ABSTRACT. The problem of conditional inference is the problem of establishing which conditional statements ("if A, then normally B") should be said to follow from a finite knowledge base of such statements. A recent solution proposed by Booth and Varzinczak is that we should take the disjunctive rational closure (DRC), which is based on the well-known rational closure construction of Lehmann and Magidor, but which returns, in general, a consequence relation which is representable by an interval order over the set of valuations rather than a total preorder, as usually assumed in the literature. Although the DRC has been shown to satisfy several reasonable properties, there are some properties that fail. In particular, it does not satisfy Cut at the level of conditional inference, i.e., adding to the knowledge base a conditional which is already inferred can lead to additional inferences. It also fails the Relevance property, according to which reasoning under syntactically disjoint knowledge bases should be possible to do in a modular way. In this paper, we propose an alternative interval-order based construction, called the trans-rational closure, which addresses, albeit only partially, these deficiencies.