The present paper establishes systematic connections among the first-order correspondents of Sahlqvist modal reduction principles in various relational semantic settings, including crisp and many-valued Kripke frames, and crisp and many-valued polarity-based frames (aka enriched formal contexts). Building on unified correspondence theory, we aim at introducing a theoretical environment which makes it possible to: (a) compare and inter-relate the various frame correspondents (in different relational settings) of any given Sahlqvist modal reduction principle; (b) recognize when first-order sentences in the frame-correspondence languages of different types of relational structures encode the same “modal content”; (c) meaningfully transfer and represent well known relational properties such as reflexivity, transitivity, symmetry, seriality, confluence, density, across different semantic contexts. These results can be understood as a first step in a research program aimed at making correspondence theory not just (methodologically) unified, but also (effectively) parametric.

One problem to solve in the context of information fusion, decision-making, and other artificial intelligence challenges is to compute justified beliefs based on evidence. In real-life examples, this evidence may be inconsistent, incomplete, or uncertain, making the problem of evidence fusion highly non-trivial. In this paper, we propose a new model for measuring degrees of beliefs based on possibly inconsistent, incomplete, and uncertain evidence, by combining tools from Dempster-Shafer Theory and Topological Models of Evidence. Our belief model is more general than the aforementioned approaches in two important ways: (1) it can reproduce them when appropriate constraints are imposed, and, more notably, (2) it is flexible enough to compute beliefs according to various standards that represent agents' evidential demands. The latter novelty allows the users of our model to employ it to compute an agent's (possibly) distinct degrees of belief, based on the same evidence, in situations when, e.g, the agent prioritizes avoiding false negatives and when it prioritizes avoiding false positives. Finally, we show that computing degree of belief with this model is #P-complete in general.

Dempster’s rule of combination allows us to combine various independent pieces of evidence that each have a certain degree of uncertainty. This provides a useful way for dealing with uncertain evidence, but the rule is computationally intractable. In this paper, we analyze the complexity of this rule for differently structured bodies of evidence and we consider a known algorithm by Shafer and Logan to compute this rule efficiently over a hierarchical set of evidence. We show that one can check in polynomial time whether an arbitrary set of evidence has a hierarchical shape, enabling the use of Shafer and Logan’s algorithm. Moreover, we consider two different approaches to deal with non-hierarchical sets of evidence: (i) considering hierarchical subsets and (ii) taking advantage of internal hierarchical structures in the overall set. For the former case, we conclude that getting different hierarchies from an arbitrary set of pieces of evidence corresponds to the VERTEX COVER problem and we present algorithms for obtaining these hierarchies based on this correspondence. For the latter case, we present a fixed-parameter tractable algorithm which computes the belief function of any piece of evidence included in the set.