The Descriptive Complexity of Relation Modification Problems

Florian Chudigiewitsch, Marlene Gr\"undel, Christian Komusiewicz, Nils Morawietz, Till Tantau
https://arxiv.org/abs/2603.22043 https://arxiv.org/pdf/2603.22043 https://arxiv.org/html/2603.22043

arXiv:2603.22043v1 Announce Type: new
Abstract: A relation modification problem gets a logical structure and a natural number k as input and asks whether k modifications of the structure suffice to make it satisfy a predefined property. We provide a complete classification of the classical and parameterized complexity of relation modification problems - the latter w. r. t. the modification budget k - based on the descriptive complexity of the respective target property. We consider different types of logical structures on which modifications are performed: Whereas monadic structures and undirected graphs without self-loops each yield their own complexity landscapes, we find that modifying undirected graphs with self-loops, directed graphs, or arbitrary logical structures is equally hard w. r. t. quantifier patterns.
Moreover, we observe that all classes of problems considered in this paper are subject to a strong dichotomy in the sense that they are either very easy to solve (that is, they lie in paraAC^{0\uparrow} or TC^0) or intractable (that is, they contain W[2]-hard or NP-hard problems).

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Modal Fragments

Nick Bezhanishvili, Balder ten Cate, Arunavo Ganguly, Arne Meier
https://arxiv.org/abs/2603.05055 https://arxiv.org/pdf/2603.05055 https://arxiv.org/html/2603.05055

arXiv:2603.05055v1 Announce Type: new
Abstract: We survey systematic approaches to basis-restricted fragments of propositional logic and modal logics, with an emphasis on how expressive power and computational complexity depend on the allowed operators. The propositional case is well-established and serves as a conceptual template: Post's lattice organizes fragments via Boolean clones and supports complexity classifications for standard reasoning tasks. For modal fragments, we then bring together two historically independent lines of investigation: a general framework where modal fragments are parameterized by a basis of "connectives" defined by arbitrary modal formulas (initially proposed and studied by logicians such as Kuznetsov and Ratsa in the 1970s), and the more tractable class of what we call simple modal fragments parameterized by Boolean functions plus selected modal operators, where Post-lattice methods enable systematic decidability and dichotomy results. Along the way, we collect and extend results on teachability and exact learnability from examples for both propositional fragments and simple modal fragments, and we conclude by identifying several open problems.

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