# Christoph Beierle's research while affiliated with FernUniversität in Hagen and other places

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## Publications (286)

As a semantics for conditional knowledge bases, ranking functions order possible worlds by mapping them to a degree of plausibility. c-Representations are special ranking functions that are obtained by assigning individual integer impacts to the conditionals in a knowledge base \(\mathcal R\) and by defining the rank of each possible world as the s...

Inductive inference operators have been introduced to formalize the process of completing a conditional belief base to a full inference relation. In this paper, we investigate the approximation of inductive inference operator system W with combinations of system Z (or equivalently rational closure) and c-inference, both of which are known to be ext...

This paper considers whether the sequential application of a combination of a merging operator and a ranking construction operator predicts human propositional reasoning. Our formally sound approach is benchmarked on data from a psychological experiment, demonstrating that the approach is cognitively more adequate than classical logic.

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 no...

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 mo...

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 o...

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 con...

The recently introduced notion of an inductive inference operator captures the process of completing a given conditional belief base to an inference relation. System W is such an inductive inference operator exhibiting some notable properties like extending rational closure and satisfying syntax splitting for inference from conditional belief bases...

In nonmonotonic reasoning, a conditional of the form ‘If A then usually B’ is typically accepted if a situation where both A and B hold is deemed to be more plausible, more probable, or less surprising, etc., than a situation where A holds, but B does not hold. In a propositional setting, this leads to a relation on the propositional interpretation...

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 r...

In this paper, we bring together marginalization and forgetting of signature elements in the framework of epistemic states. Marginalization of epistemic states is a specific approach to actively reduce signatures, also aiming at forgetting atoms which are very well known from probability theory. On the other hand, rooted in Boole’s variable elimina...

System W is an approach to reasoning from conditional beliefs that exhibits many properties desirable for nonmonotonic reasoning like extending rational closure, avoiding the drowning problem, and complying with syntax splitting. Multipreference closure, initially considered for reasoning in description logics with exceptions, has recently been rec...

System W is a recently introduced reasoning method for conditional belief bases. While system W exhibits various desirable properties for nonmonotonic reasoning like extending rational closure and fully complying with syntax splitting, an implementation of it has been missing so far. In this paper, we present a first implementation of system W. The...

System W is a recently introduced inference method for conditional belief bases with some notable properties like capturing system Z and thus rational closure and, in contrast to system Z, fully satisfying syntax splitting. This paper further investigates properties of system W. We show how system W behaves with respect to postulates put forward fo...

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 s...

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 f...

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.

Semantically based on Spohn’s ranking functions, c-representations are special ranking models obtainedby assigning individual integer impacts to the conditionals in a knowledge base R and by defining the rank of eachpossible world as the sum of the impacts of falsified conditionals. c-Inference is the inference relation taking allc-representations...

Many modern artificial intelligence (AI) systems like human-interacting smart devices or expert systems adapt to specific users' information processes but the underlying AI methods commonly lack a theory of mind. Thus, there is a need to better understand human thinking and to integrate the resulting cognitive models into AI methods. By taking the...

Normal forms of syntactic entities play an important role in many different areas in computer science. In this paper, weaddress the question of how to obtain normal forms and minimal normal forms of conditional belief bases in order to,e.g., ease reasoning with them or to simplify their comparison. We introduce notions of equivalence of belief base...

In the field of knowledge representation, the considered epistemic states are often based on propositional interpretations, also called worlds. E.g., epistemic states of agents can be modelled by ranking functions or total preorders on worlds. However, there are usually different ways of how to describe a real world situation in a propositional lan...

Conditionals are defeasible rules of the form If A then usually B , and they play a central role in many approaches to nonmonotonic reasoning. Normal forms of conditional knowledge bases consisting of a set of such conditionals are useful to create, process, and compare the knowledge represented by them. In this article, we propose several new norm...

Iterated Belief Change is the research area that investigates principles for the dynamics of beliefs over (possibly unlimited) many subsequent belief changes. In this paper, we demonstrate how iterated belief change is connected to computation. In particular, we show that iterative belief revision is Turing complete, even under the condition that b...

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 f...

In this article, we consider iteration principles for contraction, with the goal of identifying properties for contractions that respect conditional beliefs. Therefore, we investigate and evaluate four groups of iteration principles for contraction which consider the dynamics of conditional beliefs. For all these principles, we provide semantic cha...

For nonmonotonic reasoning in the context of a knowledge base $\mathcal {R}$ R containing conditionals of the form If A then usually B , system P provides generally accepted axioms. Inference solely based on system P, however, is inherently skeptical because it coincides with reasoning that takes all ranking models of $\mathcal {R}$ R into account....

Probability kinematics is a leading paradigm in probabilistic belief change. It is based on the idea that conditional beliefs should be independent from changes of their antecedents’ probabilities. In this paper, we propose a re-interpretation of this paradigm for Spohn’s ranking functions which we call Generalized Ranking Kinematics as a new princ...

Activation-based conditional inference applies conditional reasoning to ACT-R, a cognitive architecture developed to formalize human reasoning. The idea of activation-based conditional inference is to determine a reasonable subset of a conditional belief base in order to draw inductive inferences in time. Central to activation-based conditional inf...

Delgrande's knowledge level account of forgetting provides a general approach to forgetting syntax elements from sets of formulas with links to many other forgetting operations, in particular, to Boole's variable elimination. On the other hand, marginalisation of epistemic states is a specific approach to actively reduce signatures in more complex...

The research on non-prioritized revision studies revision operators which do not accept all new beliefs. In this paper, we contribute to this line of research by introducing the concept of dynamic-limited revision, which are revisions expressible by a total preorder over a limited set of worlds. For a belief change operator, we consider the scope,...

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 fo...

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 infere...

An important concept for nonmonotonic reasoning in the context of a conditional belief base \(\mathcal R\) is syntax splitting, essentially stating that taking only the syntactically relevant part of \(\mathcal R\) into account should be sufficient. In this paper, for the semantics of ordinal conditional functions (OCF) of \(\mathcal R\), we introd...

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 contractio...

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 rev...

Conditional knowledge bases consisting of qualitativeconditionals play a predominant role in knowledge representationand reasoning. In this paper, we develop a full map of allconsistent conditional knowledge bases over a small signature indifferent normal forms. We introduce two new normal formsthat take the induced system P inference relation into...

Given a belief base ∆ consisting of a set of conditionals,there are many different ways an agent may inductivelycomplete the knowledge represented by ∆ to a completeepistemic state; two well-known approaches are given by systemP and system Z, and also each ranking model of ∆ induces afull inference relation. C-representations are special rankingmod...

This paper presents a fully automatic approach to the scansion of Classical Greek hexameter verse. In particular, the paper describes an algorithm that uses deterministic finite-state automata and local linguistic rules to implement a targeted search for valid spondeus patterns and, in addition, a weighted finite-state transducer to correct and com...

Qualitative conditionals of the form “If A, then usually B” are often used to model nonmonotonic inference relations. Evaluating conditionals as three valued logical objects, allows for a classification of all conditionals over a given propositional signature. These classes of conditionals and their properties in terms of nonmonotonic inference are...

A conditional knowledge base R is a set of conditionals of the form “If A then usually B”. Using structural information derived from the conditionals in R, we introduce the preferred structure relation on worlds. The preferred structure relation is the core ingredient of a new inference relation called system W inference that inductively completes...

Descriptor revision by Hansson is a framework for addressing the problem of belief change. In descriptor revision, different kinds of change processes are dealt with in a joint framework. Individual change requirements are qualified by specific success conditions expressed by a belief descriptor, and belief descriptors can be combined by logical co...

Different approaches have been investigated for the modelling of real-world situations, especially in the medical field, many of which are based on probabilities or other numerical parameters. In this paper, we show how real world situations from the biomedical domain can be conveniently modelled with qualitative conditionals by presenting three ca...

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 se...

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 alon...

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...

Descriptor revision by Hansson is a framework for addressing the problem of belief change. In descriptor revision, different kinds of change processes are dealt with in a joint framework. Individual change requirements are qualified by specific success conditions expressed by a belief descriptor, and belief descriptors can be combined by logical co...

A conditional knowledge base R is a set of conditionals of the form "If A, the usually B". Using structural information derived from the conditionals in R, we introduce the preferred structure relation on worlds. The preferred structure relation is the core ingredient of a new inference relation called system W inference that inductively completes...

Normal forms of conditional knowledge bases are useful to create, process and compare the knowledge represented by them. In this paper, we propose the reduced antecedent normal form (RANF) for conditional knowledge bases. Compared to the antecedent normal form, it represents conditional knowledge with significantly fewer conditionals. A set of tran...

According to Boutillier, Darwiche, 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. This is especially because for iterated belief change the connection between revision and contraction via the Levi and Harper identit...

Several different semantics have been proposed for conditional knowledge bases \(\mathcal {R}\) containing qualitative conditionals of the form “If A, then usually B”, leading to different nonmonotonic inference relations induced by \(\mathcal {R}\). For the notion of c-representations which are a subclass of all ranking functions accepting \(\math...

Desirable properties of a normal form for conditional knowledge are, for instance, simplicity, minimality, uniqueness, and the respecting of adequate equivalences. In this paper, we propose the notion of antecedentwise equivalence of knowledge bases. It identifies more knowledge bases as being equivalent and allows for a simpler and more compact no...

While research on iterated revision is predominant in the field of iterated belief change, the class of iterated contraction operators received more attention in recent years. In this article, we examine a non-prioritized generalisation of iterated contraction. In particular, the class of weak Open image in new window operators is introduced, which...

Conditionals of the form “If A, then usually B” are often used to define nonmonotonic inference relations. Many ways have been proposed to inductively complete a knowledge base consisting of a finite set of conditionals to a complete inference relation. Implementations of these semantics are usually used to answer specific queries on demand. Howeve...

Computations with real numbers are decisive for all scientific and technical applications, in particular cyber-physical systems, and precision in the results is essential for quality and safety. The type-2 theory of effectivity (TTE) is a well established theory of computability on infinite strings, which can be used to represent real numbers by ra...

Ein Nachteil des üblichen probabilistischen Ansatzes ist die Erfordernis präziser Wahrscheinlichkeitswerte. Zum einen ist oft die Spezifikation solcher exakten Werte problematisch, zum anderen lässt sich probabilistische Unsicherheit nicht von der durch fehlendes Wissen bedingten Unsicherheit unterscheiden. Werfen wir eine Münze, so schätzen wir di...

In Kapitel 2 haben wir bereits ein regelbasiertes System kennengelernt: den Geldautomaten. Wir führten einige Regeln an, die die Bewilligung einer Auszahlung gestatten oder verweigern. Damit der Geldautomat korrekt arbeitet, muss ein Regelinterpreter die Anwendung der Regeln steuern.

Wie das Ziehen von Schlussfolgerungen und das Lernen ist das zielgerichtete Planen etwas, in dem sich intelligentes Verhalten in besonderer Weise manifestiert. Während es aber beim Schließen darum geht festzustellen, ob ein bestimmter Sachverhalt vorliegt oder nicht, ist das Ziel des Planens ein anderes. Gegeben sei ein vorliegender Zustand und die...

Schon der Name “Nichtmonotone Logik(en)” mag Unbehagen einflößen. Man ist froh, endlich die formalen Hürden klassischer Logik genommen zu haben, hat ihre Formalismen verinnerlicht und weiß vielleicht sogar ihre Klarheit und verlässliche Stärke zu schätzen – wozu sich also nun mit einem Thema beschäftigen, das wie eine abstruse und höchst artifiziel...

Der Bereich der Argumentation hat eine lange Tradition. Historisch gesehen ist Argumentation eine Teildisziplin der Philosophie, die sich mit der Analyse von Diskursen und mit Rhetorik beschäftigt. Dabei wird Argumentation zur dialektischen Erörterung benutzt, in der ein Sachverhalt unter unterschiedlichen Gesichtspunkten beleuchtet wird.

Während bei Truth Maintenance-Systemen die nichtmonotone Ableitung im Vordergrund steht und nichtmonotone Regeln hier – eher unscheinbar – als ein Mittel zum Zweck eingesetzt werden, rücken diese unsicheren Regeln (defeasible rules) in der Default-Logik in den Mittelpunkt des Geschehens. Sie bekommen einen eigenen Namen, Default-Regel oder einfach...

Ähnlich wie es grundlegende Schwierigkeiten gibt, den Begriff der künstlichen Intelligenz exakt zu definieren, gilt dies auch für den Begriff des maschinellen Lernens. Beide Begriffe stehen nämlich in einem ähnlichen Verhältnis zueinander, wie dies auch die Begriffe der Intelligenz und des Lernens tun. Intelligentes Verhalten wird oft eng mit der F...

Mit den regelbasierten Systemen haben wir eine wichtige Grundform wissensbasierter Systeme kennengelernt. Ihre Bedeutung verdanken sie nicht zuletzt dem Umstand, dass Regeln in besonderem Maße als Repräsentanten von Wissen akzeptiert und geschätzt werden. Regeln drücken generisches Wissen aus, also allgemeines, vom speziellen Kontext abstrahierende...

In diesem Kapitel werden wir auf die Grundlagen logikbasierter Ansätze zur Wissensrepräsentation und -inferenz eingehen. Neben einem allgemeinen Überblick werden wir dabei insbesondere die charakteristischen Eigenschaften klassisch-logischer Systeme herausarbeiten, die zum einen den Kern vieler Repräsentationssprachen bilden und zum anderen als Ref...

Neben den symbolischenMethoden zur Repräsentation unsicheren Wissens verfolgte man von Anfang an auch quantitative Ansätze zur Repräsentation und Verarbeitung von Wissen. Ein wegweisendes Beispiel hierfür war MYCIN, eines der ersten namhaften Expertensysteme (siehe Kapitel 4.7). Im Allgemeinen werden dabei den Aussagen bzw. Formeln numerische Größe...

Kein anderes Gebiet im gesamten Bereich der Deduktions- und Inferenzsysteme ist so erfolgreich in praktische Anwendungen vorgedrungen wie das logische Programmieren. Beim klassischen logischen Programmieren handelt es sich um einen normalen Resolutionskalkül mit einer syntaktisch sehr einfach zu charakterisierenden Restriktion: Es werden nur Hornkl...

Nach ersten Beispielen für wissensbasierte Systeme gehen wir auf die Unterscheidung zwischen Expertensystemen und wissensbasierten Systemen ein. Angaben zu der Geschichte des Gebietes werden ergänzt durch die Vorstellung des für die Geschichte so wichtigen medizinischen Expertensystems MYCIN. Danach beschreiben wir den generellen Aufbau eines wisse...

Kaum ein anderes Paradigma hat die Entwicklung der Künstlichen Intelligenz in den letzten Jahren so beeinflusst und vorangetrieben wie das Konzept des Agenten. Ein Agent ist letztendlich das Zielobjekt, in dem alle Forschungsrichtungen der KI zusammenlaufen und zu einem integrierten, in seine Umgebung eingebetteten und mit ihr kommunizierenden Syst...

Intentional forgetting means to deliberately give up information and is a crucial part of change or consolidation processes, or to make knowledge more compact. Two well-known forgetting operations are contraction in the AGM theory of belief change, and various types of variable elimination in logic programming. While previous work dealt with postul...

While research on iterated revision is predominant in the field of iterated belief change, the class of iterated contraction operators received more attention in recent years. In this article, we examine a non-prioritized generalisation of iterated contraction. In particular, the class of weak decrement operators is introduced, which are operators...

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...

Conditionals of the form "If A, then usually B" play an important role in formal approaches to knowledge representation and reasoning. Sets of such conditionals, called a knowledge base, may represent the explicitly given knowledge of an intelligent agent. For many operations involving kowledge bases, it is advantageous to have a compact and standa...

Skeptical inference of an intelligent agent in the context of a knowledge base \(\mathcal {R}\) containing conditionals of the form If A then usually B can be defined with respect to a set of models of \(\mathcal {R}\). For the semantics of ranking functions that assign a degree of surprise to each possible world, we develop a method for comparing...

Die Darstellung und Verarbeitung jeglicher Art von Wissen – ob sicher oder unsicher, vage, unvollständig oder sogar widersprüchlich – ist in der Künstlichen Intelligenz die zentrale Aufgabe intelligenter, wissensbasierter Systeme. Den Autoren ist es gelungen, die unterschiedlichen Methoden anschaulich zu präsentieren, so dass dieses Werk zum Selbst...

While humans have developed extremely effective ways of forgetting outdated or currently irrelevant information, freeing them to process ever-increasing amounts of information, there seems to be a gap between the technically defined notions of forgetting in knowledge representation and the common-sense understanding of forgetting. In order to bring...

Current trends, like digital transformation and ubiquitous computing, yield in massive increase in available data and information. In artificial intelligence (AI) systems, capacity of knowledge bases is limited due to computational complexity of many inference algorithms. Consequently, continuously sampling information and unfiltered storing in kno...

While the axiomatic system P is an important standard for plausible, nonmonotonic inferences from conditional knowledge bases, it is known to be too weak to solve benchmark problems like Irrelevance or Subclass Inheritance. Ordinal conditional functions provide a semantic base for system P and have often been used to design stronger inference relat...

Probabilistic reasoning under the principle of maximum entropy (so-called MaxEnt principle) is a viable and convenient alternative to graph-based methodologies such as Bayesian networks that realises an idea of information economy, i.e., of being as unbiased as possible. For relational conditional knowledge, the aggregating semantics provides a sem...

First-order typed model counting extends first-order model counting by the ability to distinguish between different types of models. In this paper, we exploit this benefit in order to calculate weighted conditional impacts (WCIs) which play a central role in nonmonotonic reasoning based on conditionals. More precisely, WCIs store information about...

A propositional conditional of the form , representing the default rule “If A, then usually B”, goes beyond the limits of classical logic, and the semantics of a knowledge base consisting of such conditionals must take into account the three-valued nature of conditionals. Ordinal conditional functions (OCF), also called ranking functions, assign a...

Conditional knowledge bases consisting of sets of conditionals are used in inductive nonmonotonic reasoning and can represent the defeasible background knowledge of a reasoning agent. For the comparison of the knowledge of different agents, as well as of different approaches to nonmonotonic reasoning, it is beneficial if these knowledge bases are a...

Skeptical inference in the context of a conditional knowledge base \(\mathcal R\) can be defined with respect to a set of models of \(\mathcal R\). For the semantics of ranking functions that assign a degree of surprise to each possible world, we develop a method for comparing the inference relations induced by different sets of ranking functions....

Background knowledge is often represented by sets of conditionals of the form “if A then usually B”. Such knowledge bases should not be circuitous, but compact and easy to compare in order to allow for efficient processing in approaches dealing with and inferring from background knowledge, such as nonmonotonic reasoning. In this paper we present tr...

Skeptical c-inference based on a set of conditionals of the form If A then usually B is defined by taking the set of c-representations into account. C-representations are ranking functions induced by impact vectors encoding the conditional impact on each possible world. By setting a bound for the maximal impact value, c-inference can be approximate...

When reasoning qualitatively from a conditional knowledge base, two established approaches are system Z and p-entailment. The latter infers skeptically over all ranking models of the knowledge base, while system Z uses the unique pareto-minimal ranking model for the inference relations. Between these two extremes of using all or just one ranking mo...

The number of parameters leading to a defined medical cancer therapy is growing rapidly. A clinical decision support system intended for better managing the resulting complexity must be able to reason about the respective active ingredients and their interrelationships. In this paper, we present a corresponding ontology and illustrate its use for a...

The axiomatic system P is an important standard for plausible, nonmonotonic inferences that is, however, known to be too weak to solve benchmark problems like irrelevance, or subclass inheritance (so-called Drowning Problem). Spohn’s ranking functions which provide a semantic base for system P have often been used to design stronger inference relat...

Default rules like “If A, then normally B” or probabilistic rules like “If A, then B with probability x” are powerful constructs for knowledge representation. Such rules can be formalized as conditionals, denoted by \((B|A)\) or \((B|A)[x]\), and a conditional knowledge base consists of a set of conditionals. Different semantical models have been p...

## Citations

... Recently there has been a growing interest in using conditionals in the field of AI and of knowledge representation. See [10,21,26,28,35,44,46,54]. ...

... We now show that two inductive inference operators, lexicographic inference and system W, which both satisfy syntax splitting (Heyninck, Kern-Isberner, and Meyer 2022;Haldimann and Beierle 2022a) also satisfy conditional syntax splitting, thus providing a proof of concept for the notion of conditional syntax splitting. Theorem 1. C lex satisfies CRel and CInd (and thus CSyn-Split). ...

... Conditional independence for OCFs has been studied in (Spohn 1988), for belief revision in (Lynn, Delgrande, and Peppas 2022), and for conditional belief revision in (Kern-Isberner, Heyninck, and Beierle 2022). To the best of our knowledge, it has not been considered for inductive inference operators. ...

... Consistent belief bases usually have infinitely many probabilistic models. For reasoning tasks such as drawing inferences, it is useful to select a unique model among them as calling for inferences that hold in all models often leads to only little and uninformative new information (Wilhelm et al. 2022). From a commonsense perspective, the maximum entropy (MaxEnt) model ME R of a consistent belief base R is the preferable model (Paris 1998). ...

... Inductive inference operators generate non-monotonic inference relations |∼ ∆ on the basis of a set of conditionals ∆ of the form (ψ|φ), read as "if φ holds, then typically ψ holds". Examples of inductive inference operators include rational closure (also known as system Z) (Goldszmidt and Pearl 1996), lexicographic inference (Lehmann 1995), system W (Komo and Beierle 2022) and c-representations (Kern-Isberner 2002). For these systems, known complexity results point to a high computational complexity (Eiter and Lukasiewicz 2000). ...

... Among all belief bases that are observationally equivalent with respect to an inductive inference operator and a set of queries we want to select a unique representative. For this, we employ an ordering on the set of belief bases over NFC (Σ) as it is developed in (Beierle and Haldimann 2020a). This ordering uses signature renamings, where a function ρ : Σ → Σ is a renaming if ρ is a bijection. ...

... Up to now, InfOCF (Beierle, Eichhorn, and Kutsch 2017;Kutsch and Beierle 2021) provides the only implementation of c-inference which takes all c-representations of R into account (Beierle et al. 2018). In this paper, we present an Copyright © 2022by the authors. ...

Reference: Realization of c-Inference as a SAT Problem

... Furthermore, we will study the normalizing systems Θ ρ , Θ ra , Θ ρra , and the generating algorithm KB ρra with respect to efficiency and complexity. Using implementations of these methods and the InfOCF system [25,26], we will empirically evaluate properties of given and of systematically generated conditional knowledge bases; some first results of this emperical evaluation are given in [27]. Another open question to be investigated is how the normal forms introduced in this article can be applied or extended to defeasible logics where conditionals occur in the form of defeasible concept subsumptions as in, e.g., [28]. ...

... It was followed by Paxos, which came up with a solution to fault tolerance. Castro et al. (Haldimann et al, 2021) proposed the practical BFT (PBFT) by extending Paxos to crash failures. It secures normal operations in a partial synchronization mode but with very high message complexity, it has been followed by many proposals to optimize its execution, such as Zyzzyva (Sohrabi et al, 2020). ...

... Two well known inference methods for conditional belief bases consisting of such conditionals are p-entailment that is characterized by the axioms of System P (Adams 1965;Kraus, Lehmann, and Magidor 1990) and system Z (Pearl 1990;Goldszmidt and Pearl 1996). Newer approaches include inference with c-representations (Kern-Isberner 2001;Kern-Isberner 2004), skeptical c-inference taking all c-representations into account (Beierle et al. 2018;Beierle et al. 2021), and the recently introduced system W (Komo and Beierle 2020; Komo and Beierle 2022). ...