The set of binary decision diagrams , an efficient data structure representing Boolean functions, is extensively used in many distinct contexts like model verification, machine learning, cryptography or also resolution of combinatorial problems. The most famous variant, called reduced ordered binary decision diagram ( robdd for short), can be viewed as the result of a specific compaction of a complete decision tree. A great property is that, once an order over the Boolean variables is fixed, each Boolean function is represented by exactly one robdd . In this paper we aim at computing the exact distribution of the Boolean functions in \(k\) variables according to the robdd size . Recall the number of Boolean functions with \(k\) variables is equal to \(2^{2^{k}}\) , which is of double exponential growth with respect to the number of variables. The maximal size of an robdd with \(k\) variables is \(M_{k}\approx 2^{k}/k\) . In this paper, we develop the first polynomial algorithm to derive the distribution of Boolean functions over \(k\) variables with respect to robdd size denoted by \(n\) . It performs \(O(k\;n^{3}\log n)\) arithmetic operations on integers and necessitates to store \(O(n^{2})\) integers in memory storage; note that the maximal size of integers involved in the computations is \(O(k\;2^{k})\) bits. Our new approach relies on a decomposition of robdd s layer by layer and on an enumerative inclusion-exclusion argument.

We characterise the sentences in Monadic Second-Order Logic (MSO) that are over finite structures equivalent to a Datalog program, in terms of an existential pebble game. We also show that for every class \({\mathcal{C}}\) of finite structures that can be expressed in MSO and is closed under homomorphisms, and for all \(\ell,k\in{\mathbb{N}}\) , there exists a canonical Datalog program \(\Pi\) of width \((\ell,k)\) in the sense of Feder and Vardi. The same characterisations also hold for Guarded Second-Order Logic (GSO), which properly extends MSO. To prove our results, we show that every class \({\mathcal{C}}\) in GSO whose complement is closed under homomorphisms is a finite union of Constraint Satisfaction Problems (CSPs) of \(\omega\) -categorical structures. The intersection of MSO and Datalog is known to contain the class of nested monadically defined queries (Nemodeq) ; likewise, we show that the intersection of GSO and Datalog contains all problems that can be expressed by the more expressive language of nested guarded queries (GQ \({}^{+}\) ) . Yet, by exploiting our results, we can show that neither of the two query languages can serve as a characterisation, as we exhibit a CSP whose complement corresponds to a query in the intersection of MSO and Datalog that is not expressible in GQ \({}^{+}\) .

Circular (or cyclic ) proofs have received increasing attention in recent years, and have been proposed as an alternative setting for studying (co)inductive reasoning. In particular, now several type systems based on circular reasoning have been proposed. However, little is known about the complexity theoretic aspects of circular proofs, which exhibit sophisticated loop structures atypical of more common ‘recursion schemes’. This paper attempts to bridge the gap between circular proofs and implicit computational complexity (ICC). Namely we introduce a circular proof system based on Bellantoni and Cook’s famous safe-normal function algebra, and we identify proof theoretical constraints, inspired by ICC, to characterise the polynomial-time and elementary computable functions. Along the way we introduce new recursion theoretic implicit characterisations of these classes that may be of interest in their own right.

A canonical result in model theory is the homomorphism preservation theorem (h.p.t.) which states that a first-order formula is preserved under homomorphisms on all structures if and only if it is equivalent to an existential-positive formula, standardly proved via a compactness argument. Rossman (2008) established that the h.p.t. remains valid when restricted to finite structures. This is a significant result in the field of finite model theory. It stands in contrast to the other preservation theorems proved via compactness where the failure of the latter also results in the failure of the former [ 2 ], [ 27 ]. Moreover, almost all results from traditional model theory that do survive to the finite are those whose proofs work just as well when considering finite structures. Rossman’s result is interesting as an example of a result which remains true in the finite but whose proof uses entirely different methods. It is also of importance to the field of constraint satisfaction due to the equivalence of existential-positive formulas and unions of conjunctive queries [ 7 ]. Adjacently, Dellunde and Vidal (2019) established a version of the h.p.t. holds for a collection of first-order many-valued logics, namely those whose structures (finite and infinite) are defined over a fixed finite MTL-chain. In this paper we unite these two strands. We show how one can extend Rossman’s proof of a finite h.p.t. to a very wide collection of many-valued predicate logics. In doing so, we establish a finite variant to Dellunde and Vidal’s result, one which not only applies to structures defined over algebras more general than MTL-chains but also where we allow for those algebra to vary between models. We identify the fairly minimal critical features of classical logic that enable Rossman’s proof from a model-theoretic point of view, and demonstrate how any non-classical logic satisfying them will inherit an appropriate finite h.p.t. This investigation provides a starting point in a wider development of finite model theory for many-valued logics and, just as the classical finite h.p.t. has implications for constraint satisfaction, the many-valued finite h.p.t. has implications for valued constraint satisfaction problems.

We consider a class of formula equations in first-order logic, Horn formula equations, which are defined by a syntactic restriction on the occurrences of predicate variables. Horn formula equations play an important role in many applications in computer science. We state and prove a fixed-point theorem for Horn formula equations in first-order logic with a least fixed-point operator. Our fixed-point theorem is abstract in the sense that it applies to an abstract semantics which generalises standard semantics. We describe several corollaries of this fixed-point theorem in various areas of computational logic, ranging from the logical foundations of program verification to inductive theorem proving.

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.