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Marco Wilhelm

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9 papers

2 author rows

KER Journal 2026 Journal Article

c-Core closure and syntax splitting for conditional belief bases

Marco Wilhelm
Gabriele Kern-Isberner
Christoph Beierle

With core c-representations we develop a new class of ranking models for conditional belief bases that combine the advantages of c-representations and System Z. On the one hand, they exhibit high-quality inferential behavior, just like c-representations, and on the other hand, they are stratified like the System Z ranking function, and can thus be constructed layer by layer. This allows for the identification of a unique minimal core c-representation from which we derive a new inductive inference operator, the c-core closure operator. This inference operator features conditional syntax splitting, like skeptical c-inference, and therefore does not suffer from the drowning problem, in contrast to System Z. Additionally, c-core closure satisfies rational monotony and inductive enforcement, and belongs to the class of basic defeasible entailment operators.

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KR Conference 2024 Conference Paper

Conditional Splittings of Belief Bases and Nonmonotonic Inference with c-Representations

Christoph Beierle
Lars-Phillip Spiegel
Jonas Haldimann
Marco Wilhelm
Jesse Heyninck
Gabriele Kern-Isberner

The concept of conditional syntax splitting for inductive inference from conditional belief bases has been proposed as a generalization of syntax splitting which also covers cases where the conditionals in the subbases share some atoms. p-Entailment and system Z fail to satisfy conditional syntax splitting, and up to now, only two inductive inference operators, lexicographic inference and system W, have been shown to satisfy this property. In this paper, we introduce the concept of conditional semantic splitting. We show that c-representations satisfy a core postulate relating conditional splittings on the syntax and the semantic level. Based on these findings, we investigate conditional syntax splitting for nonmonotonic inference with c-representations. Regarding single c-representations, we utilize the concept of selection strategies, and show that a straightforward property of the selection strategy leads to inference operators satisfying conditional syntax splittings. Furthermore, we show that c-inference taking all c-representations of a belief base into account also fully complies with conditional syntax splitting.

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AAAI Conference 2024 Conference Paper

Decomposing Constraint Networks for Calculating c-Representations

Marco Wilhelm
Gabriele Kern-Isberner

It is well-known from probability theory that network-based methods like Bayesian networks constitute remarkable frameworks for efficient probabilistic reasoning. In this paper, we focus on qualitative default reasoning based on Spohn’s ranking functions for which network-based methods have not yet been studied satisfactorily. With constraint networks, we develop a framework for iterative calculations of c-representations, a family of ranking models of conditional belief bases which show outstanding properties from a commonsense and formal point of view, that are characterized by assigning possible worlds a degree of implausibility via penalizing the falsification of conditionals. Constraint networks unveil the dependencies among these penalty points (and hence among the conditionals) and make it possible to compute the penalty points locally on so-called safe sub-bases. As an application of our framework, we show that skeptical c-inferences can be drawn locally from safe sub-bases without losing validity.

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NMR Workshop 2024 Conference Paper

Extraction of Conditional Belief Bases and the System Z Ranking Model From Multilayer Perceptrons for Binary Classification

Marco Wilhelm
Alexander Hahn 0001
Gabriele Kern-Isberner

We extract propositional conditional belief bases from multilayer perceptrons, a basic type of feedforward neural networks, and investigate the relation between these two prevalent formalisms from knowledge representation and reasoning (KRR) and machine learning (ML), respectively. The ultimate goal of our work is to imitate with the extracted belief base the main information flow in the original multilayer perceptron detached from specific input data. For this, we introduce a notion of sufficient (in)activators of neurons which reflect the most relevant connections within the multilayer perceptron that lead to the (in)activation of the subsequent neurons. While focusing on the binary multi-class classification task, we show that our approach produces consistent belief bases from which principled inferences can be drawn, for instance under System Z. In particular, no inferences are invented by the System Z ranking model that are not in accordance with the initial neural network.

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FLAP Journal 2023 Journal Article

Activation-based Conditional Inference.

Marco Wilhelm
Diana Howey
Gabriele Kern-Isberner
Kai Sauerwald
Christoph Beierle

Activation-based conditional inference (ActInf) combines conditional rea- soning and ACT-R, a cognitive architecture developed to formalize human rea- soning, and therewith provides a powerful inference formalism which makes it possible to integrate several aspects of human reasoning, such as focusing, forgetting, and remembering, into formal uncertain reasoning. The basic idea of activation-based conditional inference is to determine a reasonable, cogni- tively adequate subset of a conditional belief base before drawing inductive inferences. Central to activation-based conditional inference is the activation function which assigns to the conditionals in the belief base a degree of acti- vation mainly based on the conditional’s relevance for the current query and its usage history. Here, we develop a blueprint for activation-based conditional inference and illustrate how focusing, forgetting, and remembering are included within our framework.

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KR Conference 2023 Conference Paper

Integrating Linear Arithmetic Constraints Into Conditional Maximum Entropy Reasoning

Marco Wilhelm

The principle of maximum entropy (MaxEnt principle) constitutes a valuable methodology for probabilistic commonsense reasoning by adding missing information to probabilistic conditional belief bases in an information theoretically optimal way. In this paper, we integrate linear arithmetic constraints over the integers and reals into propositional probabilistic conditionals in order to be able to formalize uncertain beliefs about arithmetic expressions. The satisfiability of (sets of) constraints is decided modulo theory such that probabilistic reasoning stays finite although the constraints range over infinite domains. Therewith, we provide a novel extension of the MaxEnt principle to beliefs about infinite domains.

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JELIA Conference 2023 Conference Paper

Splitting Techniques for Conditional Belief Bases in the Context of c-Representations

Marco Wilhelm
Meliha Sezgin
Gabriele Kern-Isberner
Jonas Haldimann
Christoph Beierle
Jesse Heyninck

Abstract Splitting belief bases is fundamental for efficient reasoning and for better understanding interrelationships among the knowledge entities. In this paper, we survey the most important splitting techniques for conditional belief bases in the context of c-representations which constitute a specific class of ranking models with outstanding behavior not only with respect to belief base splitting, as shown in recent papers. We provide a splitting hierarchy, in particular by proving that safe conditional syntax splittings and case splittings are so-called CSP-constraint splittings. We advance the level of knowledge about CSP-constraint splittings and present an algorithm for computing CSP-constraint splittings.

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AAAI Conference 2021 Conference Paper

Focused Inference and System P

Marco Wilhelm
Gabriele Kern-Isberner

We bring in the concept of focused inference into the field of qualitative nonmonotonic reasoning by applying focused inference to System P. The idea behind drawing focused inferences is to concentrate on knowledge which seems to be relevant for answering a query while completely disregarding the remaining knowledge even at the risk of missing some meaningful information. Focused inference is motivated by mimicking snap decisions of human reasoners and aims on rapidly drawing still reasonable inferences from large sets of knowledge. In this paper, we define a series of query-dependent, syntactically-driven focused inference relations, elaborate on their formal properties, and show that the series converges against System P. We take advantage of this result in form of an anytime algorithm for drawing inferences which is accompanied by a thorough complexity analysis.

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JELIA Conference 2019 Conference Paper

Counting Strategies for the Probabilistic Description Logic 𝓐ℒ𝒞 ME Under the Principle of Maximum Entropy

Marco Wilhelm
Gabriele Kern-Isberner
Andreas Ecke
Franz Baader

Abstract We present \(\mathcal {ALC}^\mathsf {ME}\), a probabilistic variant of the Description Logic \(\mathcal {ALC}\) that allows for representing and processing conditional statements of the form “if E holds, then F follows with probability p ” under the principle of maximum entropy. Probabilities are understood as degrees of belief and formally interpreted by the aggregating semantics. We prove that both checking consistency and drawing inferences based on approximations of the maximum entropy distribution is possible in \(\mathcal {ALC}^\mathsf {ME}\) in time polynomial in the domain size. A major problem for probabilistic reasoning from such conditional knowledge bases is to count models and individuals. To achieve our complexity results, we develop sophisticated counting strategies on interpretations aggregated with respect to the so-called conditional impacts of types, which refine their conditional structure.

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