We propose a generic framework for establishing the decidability of a wide range of logical entailment problems (briefly called querying), based on the existence of countermodels that are structurally simple, gauged by certain types of width measures (with treewidth and cliquewidth as popular examples). As an important special case of our framework, we identify logics exhibiting width-finite finitely universal model sets, warranting decidable entailment for a wide range of homomorphism-closed queries, subsuming a diverse set of practically relevant query languages. As a particularly powerful width measure, we propose to employ Blumensath's partitionwidth, which subsumes various other commonly considered width measures and exhibits highly favorable computational and structural properties. Focusing on the formalism of existential rules as a popular showcase, we explain how finite partitionwidth sets of rules subsume other known abstract decidable classes but - leveraging existing notions of stratification - also cover a wide range of new rulesets. We expose natural limitations for fitting the class of finite unification sets into our picture and suggest several options for remedy.

In our pursuit of generic criteria for decidable ontology-based querying, we introduce finite-cliquewidth sets (fcs) of existential rules, a model-theoretically defined class of rule sets, inspired by the cliquewidth measure from graph theory. By a generic argument, we show that fcs ensures decidability of entailment for a sizable class of queries (dubbed DaMSOQs) subsuming conjunctive queries (CQs). The fcs class properly generalizes the class of finite-expansion sets (fes), and for signatures of arity ≤ 2, the class of bounded-treewidth sets (bts). For higher arities, bts is only indirectly subsumed by fcs by means of reification. Despite the generality of fcs, we provide a rule set with decidable CQ entailment (by virtue of first-order-rewritability) that falls outside fcs, thus demonstrating the incomparability of fcs and the class of finite-unification sets (fus). In spite of this, we show that if we restrict ourselves to single-headed rule sets over signatures of arity ≤ 2, then fcs subsumes fus.

Predicate logic is the premier choice for specifying classes of relational structures. Homomorphisms are key to describing correspondences between relational structures. Questions concerning the interdependencies between these two means of characterizing (classes of) structures are of fundamental interest and can be highly non-trivial to answer. We investigate several problems regarding the homomorphism closure (homclosure) of the class of all (finite or arbitrary) models of logical sentences: membership of structures in a sentence's homclosure; sentence homclosedness; homclosure characterizability in a logic; normal forms for homclosed sentences in certain logics. For a wide variety of fragments of first- and second-order predicate logic, we clarify these problems' computational properties.