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.

Abstract We define LE- $$\mathcal {ALC}$$ , a generalization of the description logic $$\mathcal {ALC}$$ based on the propositional logic of general (i.e. not necessarily distributive) lattices, and semantically interpreted on relational structures based on formal contexts from Formal Concept Analysis (FCA). The description logic LE- $$\mathcal {ALC}$$ allows us to formally describe databases with objects, features, and formal concepts, represented according to FCA as Galois-stable sets of objects and features. We describe ABoxes and TBoxes in LE- $$\mathcal {ALC}$$ , provide a tableaux algorithm for checking the consistency of LE- $$\mathcal {ALC}$$ knowledge bases with acyclic TBoxes, and show its termination, soundness and completeness. Interestingly, consistency checking for LE- $$\mathcal {ALC}$$ with acyclic TBoxes is in PTIME, while the complexity of the consistency checking of classical $$\mathcal {ALC}$$ with acyclic TBoxes is PSPACE-complete.

We report on an investigation aimed at identifying small fragments of set theory (typically, sublanguages of Multi-Level Syllogistic) endowed with polynomial-time satisfiability decision tests, potentially useful for automated proof verification. Leaving out of consideration the membership relator ∈ for the time being, in this paper we provide a complete taxonomy of the polynomial and the NP-complete fragments involving, besides variables intended to range over the von Neumann set-universe, the Boolean operators ∪ ∩ \, the Boolean relators ⊆, ⊈,=, ≠, and the predicates ‘• = Ø’ and ‘Disj(•, •)’, meaning ‘the argument set is empty’ and ‘the arguments are disjoint sets’, along with their opposites ‘• ≠ Ø and ‘¬Disj(•, •)’. We also examine in detail how to test for satisfiability the formulae of six sample fragments: three sample problems are shown to be NP-complete, two to admit quadratic-time decision algorithms, and one to be solvable in linear time.