Christoph Beierle

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.

IJCAI Conference 2025 Conference Paper

Generalized Safe Conditional Syntax Splitting of Belief Bases

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

Splitting techniques in knowledge representation help focus on relevant parts of a belief base and reduce the complexity of reasoning generally. In this paper, we propose a generalization of safe conditional syntax splittings that broadens the applicability of splitting postulates for inductive inference from belief bases. In contrast to safe conditional syntax splitting, our generalized notion supports syntax splittings of a belief base ∆ where the subbases of ∆ may share atoms and nontrivial conditionals. We illustrate how this new notion overcomes limitations of previous splitting concepts, and we identify genuine splittings, separating them from simple splittings that do not provide benefits for inductive inference from ∆. We introduce adjusted inference postulates based on our generalization of conditional syntax splitting. We evaluate several inductive inference operators with respect to these postulates, and show that generalized safe conditional syntax splitting is a strictly stronger requirement for inductive inference operators, covering more syntax splitting applications.

KER Journal 2025 Journal Article

Inductive inference from weakly consistent belief bases

• Jonas Philipp Haldimann
• Christoph Beierle
• Gabriele Kern-Isberner

Abstract We consider nonmonotonic inferences from belief bases that contain conditionals enforcing some of the possible worlds to be infeasible and thus completely implausible. In contrast to belief bases satisfying the strong notion of consistency requiring every world to be at least somewhat plausible, we call such belief bases weakly consistent. First, we review the treatment of weakly consistent belief bases by the seminal approaches of p-entailment, which coincides with system P, and of system Z, which coincides with rational closure. Then we focus on c-inference, an inductive inference operator that has been shown to exhibit many desirable properties put forward for nonmonotonic reasoning. It is based on c-representations, which are a special kind of ranking model ordering worlds according to their plausibility. While c-representation is defined for strongly consistent belief bases only, in this article, we extend the notions of c-representation and of c-inference to cover also weakly consistent belief bases. We adapt a constraint satisfaction problem (CSP) characterizing c-representations to capture extended c-representations, and we show how this extended CSP can be used to characterize extended c-inference, providing a basis for its implementation. We show various properties of extended c-inference and in particular, we prove that also the extended notion of c-inference fully satisfies syntax splitting. Furthermore, we extend and evaluate credulous and weakly skeptical c-inference to weakly consistent belief bases and provide characterizations for them as CSPs.

NMR Workshop 2025 Conference Paper

Kinematics Principles for Inductive Reasoning from Conditional Belief Bases

• Alexander Hahn 0001
• Gabriele Kern-Isberner
• Lars-Phillip Spiegel
• Christoph Beierle

The kinematics principle, originating from probability theory, captures the idea that conditional beliefs should be independent from changes in the plausibility of facts. Furthermore, conditional information with respect to exclusive cases should be relevant only to the respective case, but not influence others. This principle was recently adapted to belief revision of ranking functions and total preorders. In this paper, we propose a kinematics principle for non-monotonic inference relations induced from conditional belief bases. We derive this principle from the connection between inductive reasoning and belief revision of both total preorders and ranking functions. Moreover, we evaluate several inference operators from the literature with respect to this new kinematics principle for inductive reasoning.

NMR Workshop 2025 Conference Paper

Merging Marginalized Total Preorders

• Alexander Hahn 0001
• Gabriele Kern-Isberner
• Lars-Phillip Spiegel
• Christoph Beierle

Total preorders over possible worlds are often used to represent epistemic states, with the minimal worlds representing an agent’s beliefs. Marginalization refers to reducing the signature of an epistemic state, resulting in a smaller total preorder over possible worlds over the chosen subset of the original signature. Together with the syntax splitting principle, which states that only syntactically relevant parts of the epistemic state should be modified during revision, marginalization can be used to make belief revision more efficient: The original preorder is marginalized into smaller preorders, which are revised independently, and then merged back together. This last merging step offers a challenge, however, since some information about the original ordering may be lost during marginalization. In this paper, we explore how this merging step may be performed while preserving as much information as possible, in particular the original syntax splitting.

NMR Workshop 2025 Conference Paper

Strategic Principles for Revising Ranking Functions

• Gabriele Kern-Isberner
• Alexander Hahn 0001
• Lars-Phillip Spiegel
• Christoph Beierle

Rational belief revision has been characterized by numerous postulates, starting with the AGM axioms for propositional revision. The perspective on belief revision has been broadened significantly by Darwiche and Pearl who proposed a framework for iterated revision of epistemic states, extending the AGM framework. In view of representation results for AGM, most postulates for iterated AGM revision naturally correpond to conditions on how specific total preorders are modified by specific propositions. In this paper, we propose strategic principles for iterated revision that establish links among revisions from different priors by different (propositional or conditional) inputs. We start with reinterpreting a strategic principle of Chandler and Booth that expresses an independence of irrelevant alternatives (IIA principle) in the framework of Spohn’s ranking functions. We combine this ranking-based principle with Kern-Isberner’s principle of conditional preservation (PCP principle), yielding strategic principles significantly extending both IIA and PCP principles. Moreover, we elaborate the consequences of these postulates for strategic c-revisions, presenting classes of revision operators for ranking functions that comply with these postulates.

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.

KR Conference 2024 Conference Paper

Total Preorders vs Ranking Functions under Belief Revision – the Dynamics of Empty Layers

• Gabriele Kern-Isberner
• Alexander Hahn
• Jonas Haldimann
• Christoph Beierle

Total preorders and Spohn’s ranking functions are most popular semantic structures in nonmonotonic reasoning and belief revision. Each ranking function uniquely induces a total preorder, while each total preorder corresponds to infinitely many ranking functions because of the empty layers that ranking functions may have. In this paper, we adopt a dynamic perspective and investigate the role of empty layers in belief revision scenarios. We strengthen the notion of (inferential) equivalence of ranking functions by introducing revision equivalence which postulates the equivalence of ranking functions after (most general) revision operations. Moreover, we single out so-called linearly equivalent ranking functions as prototypes of ranking functions with regularly inserted empty layers. Such ranking functions are most suitable to provide an invariance property for revision equivalence which claims that linear equivalence should be preserved. We show that strategic c-revisions ensure (conditional) revision equivalence of linearly equivalent ranking functions if the strategies are adequately chosen, whereas the Darwiche-Pearl postulates for iterated revision alone are not enough to guarantee revision equivalence of ranking functions. We evaluate various other iterated revision approaches from the literature with respect to revision equivalence and preserving linear equivalence under revision. Furthermore, we present an approach to defining equivalence preserving revision operators for ranking functions from revision operators for total preorders.

NMR Workshop 2024 Conference Paper

Using SAT and Partial MaxSAT for Reasoning with System Z and System W

• Christoph Beierle
• Aron Spang
• Jonas Haldimann

Nonmonotonic reasoning from conditional belief bases typically depends on a structure over possible worlds that relies on the verification and falsification of conditionals. A major challenge in implementing such reasoning approaches is that the number of worlds in these structures grows exponentially with the number of propositional variables occurring in the belief base. For addressing this problem by using the power of current solvers, recently an implementation of reasoning with system W using Partial MaxSAT problems has been proposed. In this paper, we investigate this approach in more detail, present a formal correctness proof of the system W inference algorithm SWinf, and extend its empirical evaluation. Furthermore, we show that the approach can be transferred to implementing Pearl’s system Z by using SAT problems, and prove the correctness of the resulting system Z inference algorithm SZinf. Our implementations of system Z and system W demonstrate that they outperform previous implementations and allow for signature and knowledge base sizes that have been infeasible before.

AIJ Journal 2023 Journal Article

A kinematics principle for iterated revision

• Gabriele Kern-Isberner
• Meliha Sezgin
• Christoph Beierle

In probabilistic belief revision, the kinematics principle is a well-known and powerful principle which ensures that changing the probabilities of facts does not change unnecessarily conditional probabilities. A related principle, the principle of conditional preservation, has also been one of the main guidelines for the axioms of iterated belief revision in the seminal paper by Darwiche and Pearl. However, to date, a fully elaborated kinematics principle for iterated revision has not been presented. We aim to fill this gap in this paper by proposing a qualitative kinematics principle for iterated revision of epistemic states represented by total preorders. As new information, we allow sets of conditional beliefs, going far beyond the current state of the art of belief revision. We introduce a qualitative conditioning operator for total preorders which is compatible with conditioning for Spohn's ranking functions as far as possible, and transfer the technique of c-revisions to total preorders to provide a proof of concept for our kinematics principle at least for special revision scenarios.

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.

AAAI Conference 2023 Conference Paper

Conditional Syntax Splitting for Non-monotonic Inference Operators

• Jesse Heyninck
• Gabriele Kern-Isberner
• Thomas Meyer
• Jonas Philipp Haldimann
• Christoph Beierle

Syntax splitting is a property of inductive inference operators that ensures we can restrict our attention to parts of the conditional belief base that share atoms with a given query. To apply syntax splitting, a conditional belief base needs to consist of syntactically disjoint conditionals. This requirement is often too strong in practice, as conditionals might share atoms. In this paper we introduce the concept of conditional syntax splitting, inspired by the notion of conditional independence as known from probability theory. We show that lexicographic inference and system W satisfy conditional syntax splitting, and connect conditional syntax splitting to several known properties from the literature on non-monotonic reasoning, including the drowning effect.

FLAP Journal 2023 Journal Article

Epistemic State Mappings among Ranking Functions and Total Preorders.

• Jonas Philipp Haldimann
• Christoph Beierle
• Gabriele Kern-Isberner

Ranking functions, also called ordinal conditional functions (OCFs), and to- tal preorders on worlds (TPOs) are two common models for epistemic states that can represent conditional beliefs. To explore the connection between these frameworks, we consider mappings among TPOs and OCFs, i. e. , the models of both frameworks. We formalize this kind of mappings as epistemic state mappings. Furthermore, we introduce postulates concerning the preservation of notable properties under the application of these mappings; a prominent example of such a property is syntax splitting. Other postulates regard the compatibility with operations like marginalization and conditionalization. We evaluate the interrelationships among the postulates for epistemic state map- pings within and across the two frameworks, establishing dependencies as well as incompatibilities among postulates. Our results will be useful in particular for transferring methods and tools developed for OCF-based semantics to the TPO framework and the other way around.

NMR Workshop 2023 Conference Paper

Extending c-Representations and c-Inference for Reasoning with Infeasible Worlds

• Jonas Haldimann
• Christoph Beierle
• Gabriele Kern-Isberner

Inductive inference operators capture the process of completing a conditional belief base to an inference relation. One such operator is c-inference which is based on the c-representations of a belief base, c-representations being a special kind of ranking functions. c-Inference exhibits many desirable properties put forward for nonmonotonic reasoning; for instance, it fully complies with syntax splitting. A characterization of c-inference as a constraint satisfaction problem (CSP) yields a basis for implementing c-inference. However, the definitions of c-representations and of c-inference only take belief bases into account that satisfy a rather strong notion of consistency requiring every possible world to be at least somewhat plausible. In this paper, we extend the definition of c-representations to belief bases that need to satisfy only a weaker notion of consistency where some worlds may be completely infeasible. Based on these extended c-representations, we also extend the definition of c-inference correspondingly, thus covering all weakly consistent belief bases. Furthermore, we develop an adapted CSP characterizing the such extended c-inference that can be used as a basis for an implementation.

KR Conference 2023 Conference Paper

Finest Syntax Splittings of Ranking Functions and Total Preorders on Worlds

• Jonas Philipp Haldimann
• Christoph Beierle

The notion of syntax splitting was initially introduced by Parikh for belief sets, and one key observation is that every belief set has a unique finest syntax splitting, i. e. , a syntax splitting that refines every other syntax splitting of that belief set. Later, the notion of syntax splitting was extended to ranking functions and total preorders on worlds (TPOs), which are two common models for belief states in the context of iterated belief revision. In this paper, we prove that ranking functions also have unique finest syntax splittings, i. e. , every ranking function has a syntax splitting that refines all other syntax splittings of that ranking function. Using this, we can show that the syntax splittings of a ranking function κ are exactly the coarsenings of the finest splitting of κ. For TPOs we show that, in contrast to ranking functions, the coarsening of a syntax splitting of a TPO ⪯ is not necessarily a syntax splitting of ⪯. Despite that we can prove that every TPO has a unique finest syntax splitting that refines all other syntax splittings of that TPO.

FLAP Journal 2023 Journal Article

Joining Formal and Cognitive Reasoning.

• Christoph Beierle
• Marco Ragni
• Frieder Stolzenburg
• Matthias Thimm

JELIA Conference 2023 Conference Paper

Rational Closure Extension in SPO-Representable Inductive Inference Operators

• Jonas Haldimann
• Thomas Andreas Meyer
• Gabriele Kern-Isberner
• Christoph Beierle

Abstract The class of inductive inference operators that extend rational closure, as introduced by Lehmann or via Pearl’s system Z, exhibits desirable inference characteristics. The property that formalizes this, known as (RC Extension), has recently been investigated for basic defeasible entailment relations. In this paper, we explore (RC Extension) for more general classes of inference relations. First, we semantically characterize (RC Extension) for preferential inference relations in general. Then we focus on operators that can be represented with strict partial orders (SPOs) on possible worlds and characterize SPO-representable inductive inference operators. Furthermore, we show that for SPO-representable inference operators, (RC Extension) is semantically characterized as a refinement of the Z-rank relation on possible worlds.

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.

IJCAI Conference 2022 Conference Paper

Conditional Independence for Iterated Belief Revision

• Gabriele Kern-Isberner
• Jesse Heyninck
• Christoph Beierle

Conditional independence is a crucial concept for efficient probabilistic reasoning. For symbolic and qualitative reasoning, however, it has played only a minor role. Recently, Lynn, Delgrande, and Peppas have considered conditional independence in terms of syntactic multivalued dependencies. In this paper, we define conditional independence as a semantic property of epistemic states and present axioms for iterated belief revision operators to obey conditional independence in general. We show that c-revisions for ranking functions satisfy these axioms, and exploit the relevance of these results for iterated belief revision in general.

KR Conference 2022 Conference Paper

Inference with System W Satisfies Syntax Splitting

• Jonas Haldimann
• Christoph Beierle

In this paper, we investigate inductive inference with system W from conditional belief bases with respect to syntax splitting. The concept of syntax splitting for inductive inference states that inferences about independent parts of the signature should not affect each other. This was captured in work by Kern-Isberner, Beierle, and Brewka in the form of postulates for inductive inference operators expressing syntax splitting as a combination of relevance and independence; it was also shown that c-inference fulfils syntax splitting, while system P inference and system Z both fail to satisfy it. System W is a recently introduced inference system for nonmonotonic reasoning that captures and properly extends system Z as well as c-inference. We show that system W fulfils the syntax splitting postulates for inductive inference operators by showing that it satisfies the required properties of relevance and independence. This makes system W another inference operator besides c-inference that fully complies with syntax splitting, while in contrast to c-inference, also extending rational closure.

KR Conference 2022 Conference Paper

Iterated Belief Change, Computationally

• Kai Sauerwald
• Christoph Beierle

This paper considers belief change in the Darwiche-Pearl framework. We demonstrate that iterative belief revision is Turing complete by showing how revision operators over ranking functions can simulate every Turing machine. Our result holds even under the condition that the broadly accepted Darwiche-Pearl postulates for iterated revision hold.

JELIA Conference 2021 Conference Paper

Conditional Descriptor Revision and Its Modelling by a CSP

• Jonas Haldimann
• Kai Sauerwald
• Martin von Berg
• Gabriele Kern-Isberner
• Christoph Beierle

Abstract Descriptor revision is a belief change framework that was introduced by Hansson as an alternative to the currently prevailing AGM paradigm. One central idea of descriptor revision is to describe the desired outcome of a belief change. Thus, descriptor revision allows expressing different kinds of belief change operations like revision or contraction in a structured and combined way. In this paper, we investigate the framework of conditional descriptor revision. Conditional descriptor revision is a variation of descriptor revision aimed at the revision of ranking functions in the context of conditional logic. It is obtained by applying descriptor revision to conditional logic and additionally requiring the belief changes to fulfil the principle of conditional preservation. We show how conditional descriptor revision can be characterized by a constraint satisfaction problem (CSP). In contrast to previous work, we cover the full descriptor language over conditionals closed under conjunction, disjunction, and negation. We also line out an implementation of conditional descriptor revision based on its CSP representation. Since propositional logic can be embedded into conditional logic, our approach also provides descriptor revision for propositional logic.

IJCAI Conference 2021 Conference Paper

InfOCF-Web: An Online Tool for Nonmonotonic Reasoning with Conditionals and Ranking Functions

• Steven Kutsch
• Christoph Beierle

InfOCF-Web provides implementations of system P and system Z inference, and of inference relations based on c-representation with respect to various inference modes and different classes of minimal models. It has an easy-to-use online interface for computing ranking models of a conditional knowledge R, and for answering queries and comparing inference results of nonmonotonic inference relations induced by R.

AIJ Journal 2021 Journal Article

Properties and interrelationships of skeptical, weakly skeptical, and credulous inference induced by classes of minimal models

• Christoph Beierle
• Christian Eichhorn
• Gabriele Kern-Isberner
• Steven Kutsch

There are multiple ways of defining nonmonotonic inference relations based on a conditional knowledge base. While the axiomatic system P is an important standard for such plausible nonmonotonic reasoning, inference relations obtained from system Z or from c-representations have been designed which go beyond system P by selecting preferred models for inference. For any class of models M, we propose the notion of weakly skeptical inference, first introduced in an ECAI conference paper this article revises and extends, that lies between skeptical and credulous inference with respect to M. Weakly skeptical c-inference properly extends skeptical c-inference, but avoids disadvantages of a too liberal credulous c-inference. We extend the concepts of skeptical, weakly skeptical, and credulous c-inference modes by taking models obtained from different minimality criteria into account. We illustrate the usefulness of the obtained inference relations and show that they fulfill various desirable properties put forward for nonmonotonic reasoning. Furthermore, we elaborate in detail the interrelationships among the inference relations when taking the different inference modes and various classes of minimal models into account.

JELIA Conference 2021 Conference Paper

Syntax Splitting for Iterated Contractions, Ignorations, and Revisions on Ranking Functions Using Selection Strategies

• Jonas Haldimann
• Christoph Beierle
• Gabriele Kern-Isberner

Abstract For characterizing belief sets consisting of independent parts, Parikh introduced the notion of syntax splitting. Corresponding postulates have been developed for the reasoning from and for the revision of belief bases with respect to syntax splitting. Kern-Isberner and Brewka introduced syntax splitting for epistemic states and iterated belief revision. Only recently, syntax splitting has also been studied for contractions and iterated contractions of epistemic states; however, all of the evaluated contractions proposed in the literature failed to fulfil the full syntax splitting postulates. In this paper, we study syntax splitting for iteratively contracting and revising epistemic states, represented by ranking functions, not only with respect to a set of formulas, but with respect to a set of conditionals. Using a framework of belief change governed by the principle of conditional preservation, we employ the concept of selection strategies. We develop axioms for selection strategies ensuring that the induced contractions and revisions fully obey the desired syntax splitting properties. Furthermore, we transfer our approach to ignorations and prove a theorem showing how selection strategies satisfying the axioms can effectively be constructed.

ECAI Conference 2020 Conference Paper

A Conditional Perspective for Iterated Belief Contraction

• Kai Sauerwald
• Gabriele Kern-Isberner
• Christoph Beierle

According to Boutillier, Darwiche and Pearl and others, principles for iterated revision can be characterised in terms of changing beliefs about conditionals. For iterated contraction, a similar formulation is not known. In particular, the characterisation for revision does not immediately yield a characterisation for contraction, because in the setting of iterated belief change, revision and contraction are not easily interdefinable. In this article, we develop two axiomatisations of iterated contraction, the first one in terms of changing conditional beliefs, and the second one by employing a new notion of equivalence. We prove that each of these new sets of postulates conforms semantically to the class of operators like the ones given by Konieczny and Pino Pérez for iterated contraction.

ECAI Conference 2020 Conference Paper

Cognitive Logics - Features, Formalisms, and Challenges

• Marco Ragni
• Gabriele Kern-Isberner
• Christoph Beierle
• Kai Sauerwald

Logic is responsible for scientific progress in many disciplines. In particular, computer science and AI would be impossible without it. Classical logics have long been accepted as a normative framework for human reasoning to capture correct reasoning. However, many psychological findings such as the Wason Selection Task have demonstrated that classical logic cannot serve as a possible descriptive language for the human inference process, which is the aim of what we call a cognitive logic. Recently, some nonmonotonic logics have been employed to explain human inferences for some well-known examples. In this paper, we discuss possible features of cognitive logics, present first results, and highlight new challenges.

KR Conference 2020 Conference Paper

Syntax Splitting = Relevance + Independence: New Postulates for Nonmonotonic Reasoning From Conditional Belief Bases

• Gabriele Kern-Isberner
• Christoph Beierle
• Gerhard Brewka

Syntax splitting, first introduced by Parikh in 1999, is a natural and desirable property of KR systems. Syntax splitting combines two aspects: it requires that the outcome of a certain epistemic operation should only depend on relevant parts of the underlying knowledge base, where relevance is given a syntactic interpretation (relevance). It also requires that strengthening antecedents by irrelevant information should have no influence on the obtained conclusions (independence). In the context of belief revision the study of syntax splitting already proved useful and led to numerous new insights. In this paper we analyse syntax splitting in a different setting, namely nonmonotonic reasoning based on conditional knowledge bases. More precisely, we analyse inductive inference operators which, like system P, system Z, or the more recent c-inference, generate an inference relation from a conditional knowledge base. We axiomatize the two aforementioned aspects of syntax splitting, relevance and independence, as properties of such inductive inference operators. Our main results show that system P and system Z, whilst satisfying relevance, fail to satisfy independence. C-inference, in contrast, turns out to satisfy both relevance and independence and thus fully complies with syntax splitting.

KR Conference 2020 Conference Paper

Syntax Splitting for Iterated Contractions

• Jonas Philipp Haldimann
• Gabriele Kern-Isberner
• Christoph Beierle

Parikh developed the notion of syntax splitting to describe belief sets with independent parts. He also formulated a postulate demanding that belief revisions respect syntax splittings in belief sets. The concept of syntax splitting was later transferred to epistemic states with total preorders and ranking functions by Kern-Isberner and Brewka along with corresponding postulates for belief revisions. Besides revision, contraction is also a central operation in the field of general belief change. In this paper, we analyse belief contractions with respect to syntax splitting. Based on the work on syntax splitting for revision, we develop syntax splitting postulates for contractions on ranking functions, on epistemic states with total preorder, and on belief sets. Finally, we evaluate different contractions from the literature, namely moderate contraction, natural contraction, lexicographic contraction, and c-contractions with respect to the newly developed contraction postulates.

JELIA Conference 2019 Conference Paper

Systematic Generation of Conditional Knowledge Bases up to Renaming and Equivalence

• Christoph Beierle
• Steven Kutsch

Abstract A conditional of the form “If A then usually B” establishes a plausible connection between A and B, while still allowing for exceptions. A conditional knowledge base consists of a finite set of conditionals, inducing various nonmonotonic inference relations. Sets of knowledge bases are of interest for, e. g. , experimenting with systems implementing conditional reasoning and for empirically evaluating them. In this paper, we present an approach for systematically generating knowledge bases over a given signature. The approach is minimal in the sense that no two knowledge bases are generated that can be transformed into each other by a syntactic renaming or that are elementwise equivalent. Furthermore, the approach is complete in the sense that, taking renamings and equivalences into account, every consistent knowledge base is generated.

FLAP Journal 2017 Journal Article

A Framework for Versatile Knowledge and Belief Management Operations in a Probabilistic Conditional Logic.

• Christoph Beierle
• Marc Finthammer
• Nico Potyka
• Julian Varghese
• Gabriele Kern-Isberner

Intelligent agents equipped with epistemic capabilities are expected to carry out quite different knowledge and belief management tasks like answering queries, performing diagnosis and hypothetical reasoning, or revising and updating their own state of belief in the light of new information. In any realistic setting, such agents must also take vagueness and uncertainty into account. In this paper, we report on an approach that uses probabilistic conditional logic for modelling such an intelligent agent. We propose a simple, yet powerful agent model supporting versatile knowledge and belief management operations. The semantics of a knowledge base consisting of a set of probabilistic conditionals is obtained by employing the principle of maximum entropy. We give an overview of the MEcore system providing the core functionalities needed for realizing the agent model. In order to illustrate the use of MEcore’s functionalities, we present a case-study of applying probabilistic logic to the analysis of clinical patient data in neurosurgery. Probabilistic conditionals are used to build a knowledge base for modelling and representing both clinical brain tumor data and expert knowledge of physicians working in this area.

ECAI Conference 2016 Conference Paper

Skeptical, Weakly Skeptical, and Credulous Inference Based on Preferred Ranking Functions

• Christoph Beierle
• Christian Eichhorn 0001
• Gabriele Kern-Isberner
• Steven Kutsch

While the axiomatic system P is an important standard for plausible nonmonotonic reasoning, inference relations obtained from system Z or from c-representations have been designed which go beyond system P. In this paper, we propose the new concept of weakly skeptical inference that properly extends the recently introduced skeptical c-inference, but avoids disadvantages of a too liberal credulous inference. We extend the concepts of skeptical, weakly skeptical, and credulous c-inference by taking preferred models obtained from different minimality criteria into account. We illustrate the usefulness of the obtained inference relations, show that they fulfill various desirable properties, and elaborate on their interrelationships.

YNIMG Journal 2014 Journal Article

A novel meta-analytic approach: Mining frequent co-activation patterns in neuroimaging databases

• Julian Caspers
• Karl Zilles
• Christoph Beierle
• Claudia Rottschy
• Simon B. Eickhoff

In recent years, coordinate-based meta-analyses have become a powerful and widely used tool to study co-activity across neuroimaging experiments, a development that was supported by the emergence of large-scale neuroimaging databases like BrainMap. However, the evaluation of co-activation patterns is constrained by the fact that previous coordinate-based meta-analysis techniques like Activation Likelihood Estimation (ALE) and Multilevel Kernel Density Analysis (MKDA) reveal all brain regions that show convergent activity within a dataset without taking into account actual within-experiment co-occurrence patterns. To overcome this issue we here propose a novel meta-analytic approach named PaMiNI that utilizes a combination of two well-established data-mining techniques, Gaussian mixture modeling and the Apriori algorithm. By this, PaMiNI enables a data-driven detection of frequent co-activation patterns within neuroimaging datasets. The feasibility of the method is demonstrated by means of several analyses on simulated data as well as a real application. The analyses of the simulated data show that PaMiNI identifies the brain regions underlying the simulated activation foci and perfectly separates the co-activation patterns of the experiments in the simulations. Furthermore, PaMiNI still yields good results when activation foci of distinct brain regions become closer together or if they are non-Gaussian distributed. For the further evaluation, a real dataset on working memory experiments is used, which was previously examined in an ALE meta-analysis and hence allows a cross-validation of both methods. In this latter analysis, PaMiNI revealed a fronto-parietal “core” network of working memory and furthermore indicates a left-lateralization in this network. Finally, to encourage a widespread usage of this new method, the PaMiNI approach was implemented into a publicly available software system.

JELIA Conference 2012 Conference Paper

How to Exploit Parametric Uniformity for Maximum Entropy Reasoning in a Relational Probabilistic Logic

• Marc Finthammer
• Christoph Beierle

Abstract The relational probabilistic conditional logic FO-PCL employs the principle of maximum entropy (ME). We show that parametric uniformity of an FO-PCL knowledge base \(\mathcal R\) can be exploited for solving the optimization problem required for ME reasoning more efficiently. The original ME optimization problem containing a large number of linear constraints, one for each ground instance of a conditional, can be replaced by an optimization problem containing just one linear constraint for each conditional. We show that both optimization problems have the same ME distribution as solution. An implementation employing Generalized Iterative Scaling illustrates the benefits of our approach

LPAR Conference 2003 Conference Paper

A Logical Study on Qualitative Default Reasoning with Probabilities

• Christoph Beierle
• Gabriele Kern-Isberner

Only very special subclasses of probability distributions can be used for qualitative reasoning that meets basic logical demands. Snow’s atomic bound systems ( big-stepped probabilities ) provide one positive example for such a subclass. This paper presents a thorough investigation of the formal logical relationships between qualitative and probabilistic default reasoning. We start with formalizing qualitative conditional logic, as well as both standard and big-stepped probabilistic logic as abstract logical systems, using the notion of institutions. The institution of big-stepped probabilities turns out to be a proper combination of the other two. Moreover, the framework of institutions offers the possibility to elaborate exactly the properties that make probability distributions suitable for qualitative reasoning.

JELIA Conference 2002 Conference Paper

Using Institutions for the Study of Qualitative and Quantitative Conditional Logics

• Christoph Beierle
• Gabriele Kern-Isberner

Abstract It is well known that conditionals need a non-classical environment to be evaluated. In this paper, we present a formalization of conditional logic in the framework of institutions. In regarding both qualitative and probabilistic conditional logic as abstract logical systems, we investigate how they can be related to one another, on the one hand, and to the institution of propositional logic, on the other hand. In spite of substantial differences between these three logics, we find surprisingly clear formal relationships between them.

CSL Conference 1992 Conference Paper

Correctness Proof For the WAM with Types

• Christoph Beierle
• Egon Börger

Abstract We provide a mathematical specification of an extension of Warren's Abstract Machine for executing Prolog to type-constraint logic programming and prove its correctness. In this paper, we keep the notion of types and dynamic type constraints rather abstract to allow applications to different constraint formalisms like Prolog III or CLP(R). This generality permits us to introduce modular extensions of Börger's and Rosenzweig's formal derivation of the WAM. Starting from type-constraint Prolog algebras that are derived from Börger's standard Prolog algebras, the specification of the type-constraint WAM extension is given by a sequence of evolving algebras, each representing a refinement level. For each refinement step a correctness proof is given. Thus, we obtain the theorem that for every such abstract type-constraint logic programming system L and for every compiler satisfying the specified conditions, the WAM extension with an abstract notion of types is correct w. r. t. L. This is a first step towards our aim to provide a full specification and correctness proof of a concrete system, the PROTOS Abstract Machine (PAM), an extension of the WAM by polymorphic order-sorted unification as required by the logic programming language PROTOS-L.

CSL Conference 1989 Conference Paper

The Knowledge Representation Language L LILOG

• Christoph Beierle
• Jochen Dörre
• Udo Pletat
• Claus-Rainer Rollinger
• Peter H. Schmitt
• Rudi Studer

Abstract L LILOG is the knowledge representation language that is used in the LILOG project, a research project in the area of natural language understanding. L LILOG is based on full first order predicate logic and incorporates a powerful sort mechanism. Besides having a set of sorts together with an order relation on them sorts may also be defined by attribute-value pairs. Further concepts which have been included in L LILOG are reference objects for representing real world objects as well as knowledge packets for structuring knowledge bases. This paper describes the evolution of L LILOG and its basic design decisions, gives a complete abstract syntax, and provides a model theoretic semantics for the logical kernel of the language. A definition of the concrete syntax which is used in the first LILOG prototype is given in Appendix 1.

ECAI Conference 1982 Conference Paper

Synthesizing Minimal Programs from Traces of Observable Behaviour

• Christoph Beierle