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Researcher profile

Anselm Haak

Anselm Haak contributes to research discovery and scholarly infrastructure.

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Logic in Computer ScienceComputational ComplexityArtificial Intelligence

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Published work

6 published item(s)

preprint2026arXiv

ABox Abduction for Inconsistent Knowledge Bases under Repair Semantics

Given a knowledge base (KB) with a non-entailed fact, the ABox abduction problem asks for possible extensions of the KB that would entail this fact. This problem has many applications, ranging from diagnosis to explainability and repair. ABox abduction has been well-investigated for consistent KBs and classical semantics, but little is known for the case of inconsistent KBs, which can be caused by erroneous data. In this paper we define suitable notions of abduction in this setting and propose criteria that guide abduction towards "useful" hypotheses. To regain meaningful reasoning in the presence of inconsistencies, we use well-established repair semantics. We provide a comprehensive landscape of the complexity of ABox abduction under repair semantics, treating different variants of the abduction problem for the light-weight description logics DL-Lite and EL_bot.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

Parameterised Counting in Logspace

In this paper, we introduce a new framework for parameterised counting in logspace, inspired by the parameterised space bounded models developed by Elberfeld, Stockhusen and Tantau (IPEC 2013, Algorithmica 2015). They defined the operators paraW and paraBeta for parameterised space complexity classes by allowing bounded nondeterminism with multiple-read and read-once access, respectively. Using these operators, they characterised the parameterised complexity of natural problems on graphs. In the spirit of the operators paraW and paraBeta by Stockhusen and Tantau, we introduce variants based on tail-nondeterminism, paraW[1] and paraBeta-Tail. Then, we consider counting versions of all four operators applied to logspace and obtain several natural complete problems for the resulting classes: counting of paths in digraphs, counting first-order models for formulas, and counting graph homomorphisms. Furthermore, we show that the complexity of a parameterised variant of the determinant function for (0,1)-matrices is #paraBeta-Tail-L-hard and can be written as the difference of two functions in #paraBetaTail-L. For example, we show that the closure of #paraBetaTail-L under parameterised logspace parsimonious reductions coincides with #paraBeta-L, that is, modulo parameterised reductions, tail-nondeterminism with read-once access is the same as read-once nondeterminism. We show that all introduced classes are closed under addition and multiplication, and those without tail-nondeterminism are closed under parameterised logspace parsimonious reductions. Finally, we underline the significance of this topic by providing a promising outlook showing several open problems and options for further directions of research.

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preprint2020arXiv

Counting of Teams in First-Order Team Logics

We study descriptive complexity of counting complexity classes in the range from #P to #$\cdot$NP. A corollary of Fagin's characterization of NP by existential second-order logic is that #P can be logically described as the class of functions counting satisfying assignments to free relation variables in first-order formulae. 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 FO in Tarski's semantics. Our results show that the class #$\cdot$NP can be logically characterized by independence logic and existential second-order logic, whereas dependence logic and inclusion logic give rise to subclasses of #$\cdot$NP and #P , respectively. Our main technical result shows that the problem of counting satisfying assignments for monotone Boolean $Σ_1$-formulae is #$\cdot$NP-complete as well as complete for the function class generated by dependence logic.

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preprint2020arXiv

Enumerating Teams in First-Order Team Logics

We start the study of the enumeration complexity of different satisfiability problems in first-order team logics. Since many of our problems go beyond DelP, we use a framework for hard enumeration analogous to the polynomial hierarchy, which was recently introduced by Creignou et al. (Discret. Appl. Math. 2019). We show that the problem to enumerate all satisfying teams of a fixed formula in a given first-order structure is DelNP-complete for certain formulas of dependence logic and independence logic. For inclusion logic formulas, this problem is even in DelP. Furthermore, we study the variants of this problems where only maximal, minimal, maximum and minimum solutions, respectively, are considered. For the most part these share the same complexity as the original problem. An exception is the minimum-variant for inclusion logic, which is DelNP-complete.

0 citations0 reviewsNo code linkedOpen access
preprint2017arXiv

A Model-Theoretic Characterization of Constant-Depth Arithmetic Circuits

We study the class $\textrm{AC}^0$ of functions computed by constant-depth polynomial-size arithmetic circuits of unbounded fan-in addition and multiplication gates. No model-theoretic characterization for arithmetic circuit classes is known so far. Inspired by Immerman's characterization of the Boolean class $\textrm{AC}^0$, we remedy this situation and develop such a characterization of $\textrm{AC}^0$. Our characterization can be interpreted as follows: Functions in $\textrm{AC}^0$ are exactly those functions counting winning strategies in first-order model checking games. A consequence of our results is a new model-theoretic characterization of $\textrm{TC}^0$, the class of languages accepted by constant-depth polynomial-size majority circuits.

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preprint2016arXiv

Descriptive Complexity of $\#\textrm{AC}^0$ Functions

We introduce a new framework for a descriptive complexity approach to arithmetic computations. We define a hierarchy of classes based on the idea of counting assignments to free function variables in first-order formulae. We completely determine the inclusion structure and show that #P and #AC^0 appear as classes of this hierarchy. In this way, we unconditionally place #AC^0 properly in a strict hierarchy of arithmetic classes within #P. We compare our classes with a hierarchy within #P defined in a model-theoretic way by Saluja et al. We argue that our approach is better suited to study arithmetic circuit classes such as #AC^0 which can be descriptively characterized as a class in our framework.

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