We study the problem of counting answers to unions of conjunctive queries (UCQs) under structural restrictions on the input query. Concretely, given a class \(C\) of UCQs, the problem \(\#\text{UCQ}(C)\) provides as input a UCQ \(\Psi\in C\) and a database \(\mathcal{D}\) and the problem is to compute the number of answers of \(\Psi\) in \(\mathcal{D}\) . Chen and Mengel [PODS’16] have shown that for any recursively enumerable class \(C\) , the problem \(\#\text{UCQ}(C)\) is either fixed-parameter tractable or hard for one of the parameterised complexity classes \(\mathrm{W}[1]\) or \(\#\mathrm{W}[1]\) . However, their tractability criterion is unwieldy in the sense that, given any concrete class \(C\) of UCQs, it is not easy to determine how hard it is to count answers to queries in \(C\) . Moreover, given a single specific UCQ \(\Psi\) , it is not easy to determine how hard it is to count answers to \(\Psi\) . In this work, we address the question of finding a natural tractability criterion: The combined conjunctive query of a UCQ \(\Psi=\varphi_{1}\vee\dots\vee\varphi_{\ell}\) is the conjunctive query \(\boldsymbol{\wedge}\left(\Psi\right)=\varphi_{1}\wedge\dots\wedge\varphi_{\ell}\) . We show that under natural closure properties of \(C\) , the problem \(\#\text{UCQ}(C)\) is fixed-parameter tractable if and only if the combined conjunctive queries of UCQs in \(C\) , and their contracts, have bounded treewidth. A contract of a conjunctive query is an augmented structure, taking into account how the quantified variables are connected to the free variables — if all variables are free, then a conjunctive query is equal to its contract; in this special case the criterion for fixed-parameter tractability of \(\#\text{UCQ}(C)\) thus simplifies to the combined queries having bounded treewidth. Finally, we give evidence that a closure property on \(C\) is necessary for obtaining a natural tractability criterion: We show that even for a single UCQ \(\Psi\) , the meta problem of deciding whether \(\#\text{UCQ}(\{\Psi\})\) can be solved in time \(O(|\mathcal{D}|^{d})\) is \(\mathrm{NP}\) -hard for any fixed \(d\geq 1\) . Moreover, we prove that a known exponential-time algorithm for solving the meta problem is optimal under assumptions from fine-grained complexity theory. As a corollary of our reduction, we also establish that approximating the Weisfeiler-Leman-Dimension of a UCQ is \(\mathrm{NP}\) -hard.

Higher-order logic HOL offers a very simple syntax and semantics for knowledge representation and reasoning in various particular domains, including in particular representing and reasoning about typed data structures. But its type system lacks advanced features where types may depend on terms. Dependent type theory offers such a rich type system, but has rather substantial conceptual differences to HOL, as well as comparatively poor proof automation support. We introduce a dependently-typed extension DHOL of HOL that retains the style and conceptual framework of HOL. Moreover, we build a translation from DHOL to HOL and implement it as a preprocessor to HOL theorem provers able to parse TPTP, thereby making all such provers able to run on DHOL problems.

We study descriptive complexity of counting complexity classes in the range from #P to \({\text{#}\!\cdot\!\text{NP}}\) . The proof of Fagin’s characterization of NP by existential second-order logic generalizes to the counting setting in the following sense: The class #P can be logically described as the class of functions counting satisfying assignments to free relation variables in first-order formulae. This was first observed by Saluja et al. (1995). In this paper we extend this study to classes beyond #P and extensions of first-order logic with team semantics. These team-based logics are closely related to existential second-order logic and its fragments, hence our results also shed light on the complexity of counting for extensions of first-order logic in Tarski’s semantics. Our results show that the class \({\text{#}\!\cdot\!\text{NP}}\) can be logically characterized by independence logic and existential second-order logic, whereas dependence logic and inclusion logic give rise to subclasses of \({\text{#}\!\cdot\!\text{NP}}\) and #P , respectively. We further relate the class obtained from inclusion logic to the complexity class \({\text{TotP}} \subseteq{\text{#P}}\) .

Twin-width is a powerful graph invariant that supports the efficient solution of various NP-hard problems when the input graph has a bounded twin-width. First-order model checking is fixed-parameter tractable on graph classes of bounded twin-width [11]. This work introduces two algorithmic strategies for exact twin-width computation: SAT encodings and a Branch & Bound approach. The SAT encodings explore distinct formulations of twin-width, enhancing performance across different instance types; the Branch & Bound algorithm leverages cached partial solutions for improved efficiency on larger graphs. We propose a verification framework combining these methods and yield verifiable proofs for computed twin-width. Our research contributes conceptual insights into twin-width computation, including new contraction orderings and lower and upper bound techniques that can be of independent interest. We accompany our theoretical developments with a rigorous experimental evaluation.

Das, van der Giessen, and Marin recently introduced \(\mathsf{IGL}\) , an intuitionistic version of Gödel-Löb logic. Their proof systems involves ill-founded proofs with a progressiveness condition. Their completeness proof uses the principle of \(\Sigma^{1}_{1}\) -determinacy; which is not provable in \(\mathsf{ZFC}\) . We define a cyclic proof system for \(\mathsf{IGL}\) and give a proof of its completeness theorem avoiding \(\Sigma^{1}_{1}\) -determinacy.

In this article, we consider the interplay of generalized quantifiers and built-in relations over finite structures, in particular, in the range of logics capturing the circuit complexity classes \(\mathrm{AC^{0}}\) and \(\mathrm{TC^{0}}\) . It is well known that for capturing \(\mathrm{AC^{0}}\) first-order logic has to be equipped with order and, e.g., predicates for addition and multiplication, whereas for \(\mathrm{TC^{0}}\) generalized quantifiers such as majority quantifiers are necessary. The sharp division between the classes \(\mathrm{AC^{0}}\) and \(\mathrm{TC^{0}}\) can be explained by the fact that \(\mathrm{AC^{0}}\) is not closed under restricting \(\mathrm{AC^{0}}\) -computable queries into simple subsequences of the input, whereas \(\mathrm{TC^{0}}\) is closed under such relativization as its queries can be expressed in terms of first-order formulas using universe-independent generalized quantifiers and order as the only built-in relation. In the terminology of abstract logics, the above means that logics capturing \(\mathrm{AC^{0}}\) do not have the relativization property, and hence, they are not regular logics unlike the logics capturing \(\mathrm{TC^{0}}\) . This weakness of \(\mathrm{AC^{0}}\) has been also elaborated in the line of research on the Crane Beach Conjecture. The conjecture (which was refuted by Barrington et al.) was that if a language \( L \) has a neutral letter, then \( L \) can be defined in \(\operatorname{FO}_{\mathcal{A}}\) , first-order logic with the collection of all numerical built-in relations \(\mathcal{A}\) , if and only if \( L \) can be already defined in \(\operatorname{FO}_{\leq}\) . Our approach is two-fold. First, we study universe-independent cardinality quantifiers \(\operatorname{\mathsf{Q}}\) defined by a parameter set \(S\subseteq\mathbb{N}\) and formulate a combinatorial criterion for \( S \) implying that all languages in \(\mathrm{DLOGTIME}\) -uniform \(\mathrm{TC^{0}}\) can be defined in \(\operatorname{FO}_{\leq}(\operatorname{\mathsf{Q}})\) . For instance, this criterion is satisfied if \( S \) is the range of some polynomial with positive integer coefficients of degree at least two. Second, by adapting the key properties of abstract logics to accommodate built-in relations, we define the regular interior \(\operatorname{\mathcal{R}-int}(\mathcal{L})\) (the largest regular \(\mathcal{L}^{*}\) such that \(\mathcal{L}^{*}\subseteq\mathcal{L}\) ) and regular closure \(\operatorname{\mathcal{R}-cl}(\mathcal{L})\) (the least regular \(\mathcal{L}^{*}\) such that \(\mathcal{L}\subseteq\mathcal{L}^{*}\) ), of a logic \(\mathcal{L}\) with built-in relations, and show that the Crane Beach Conjecture can be interpreted as a statement concerning the regular interior of \(\mathcal{L}\) . By extending the results of Barrington et al., we further show that if \(\mathcal{B}=\{+\}\) , or \(\mathcal{B}\) contains only unary relations besides \(\leq\) , then \(\operatorname{\mathcal{R}-int}(\operatorname{FO}_{\mathcal{B}})\equiv \operatorname{FO}_{\leq}\) . In contrast, our results from the first part of the article imply that if \(\mathcal{B}\) contains \(\leq\) and the range of a polynomial of degree at least two, then \(\operatorname{\mathcal{R}-cl}(\operatorname{FO}_{\mathcal{B}})\) includes all languages in \(\mathrm{DLOGTIME}\) -uniform \(\mathrm{TC^{0}}\) .

Motivated by rough set theory, we introduce a novel semantics for the basic modal language based on the possibility lower approximation operator of subset approximation structures. The study investigates axiomatization, expressiveness, and invariance results related to this semantics. From a rough set perspective, it provides a formal language for reasoning about the possibility lower approximation operator. Additionally, the axiomatization results obtained here provide characterizing properties of the operator.

Ehrenfeucht-Fraïssé games provide means to characterize elementary equivalence for first-order logic, and by standard translation also for modal logics. We propose a novel generalization of Ehrenfeucht-Fraïssé games to hybrid-dynamic logics which is direct and fully modular: parameterized by the features of the hybrid language we wish to include, for instance, the modal and hybrid language operators as well as first-order existential quantification. We use these games to establish a new modular Fraïssé-Hintikka theorem for hybrid-dynamic propositional logic and its various fragments. We study the relationship between countable game equivalence (determined by countable Ehrenfeucht-Fraïssé games) and bisimulation (determined by countable back-and-forth systems). In general, the former turns out to be weaker than the latter, but under certain conditions on the language, the two coincide. As a corollary we obtain an analogue of the Hennessy-Milner theorem. We also prove that for reachable image-finite Kripke structures elementary equivalence implies isomorphism.

A formal context consists of objects, properties, and the incidence relation between them. Various notions of concepts defined with respect to formal contexts and their associated algebraic structures have been studied extensively, including formal concepts in formal concept analysis (FCA), rough concepts arising from rough set theory (RST), and semiconcepts and protoconcepts for dealing with negation. While all these kinds of concepts are associated with lattices, semiconcepts and protoconcepts additionally yield an ordered algebraic structure, called double Boolean algebras. As the name suggests, a double Boolean algebra contains two underlying Boolean algebras. In this article, we investigate logical and algebraic aspects of the representation and reasoning about different concepts with respect to formal contexts. We first review our previous work on two-sorted modal logic systems KB and KF for the representation and reasoning of rough concepts and formal concepts, respectively. Then, in order to represent and reason about both formal and rough concepts in a single framework, these two logics are unified into a two-sorted Boolean modal logic BM , in which semiconcepts and protoconcepts are also expressible. Based on the logical representation of semiconcepts and protoconcepts, we prove the characterization of double Boolean algebras in terms of their underlying Boolean algebras. Finally, we also discuss the possibilities of extending our logical systems for the representation and reasoning of more fine-grained quantitative information in formal contexts.

We introduce an operator on classes of regular languages, the star-free closure. Our motivation is to generalize standard results of automata theory within a unified framework. Given an arbitrary input class \(\mathscr{C}\) , the star-free closure operator outputs the least class closed under Boolean operations and language concatenation, and containing all languages of \(\mathscr{C}\) as well as all finite languages. We establish several equivalent characterizations of star-free closure: in terms of regular expressions, first-order logic, pure future and future-past temporal logic, and recognition by finite monoids. A key ingredient is that star-free closure coincides with another closure operator, defined in terms of regular operations where Kleene stars are allowed in restricted contexts. A consequence of this first result is that we can decide membership of a regular language in the star-free closure of a class whose separation problem is decidable. Moreover, we prove that separation itself is decidable for the star-free closure of any finite class, and of any class of group languages having itself decidable separation (plus mild additional properties). We actually show decidability of a stronger property, called covering.