Thomas Meyer

Nicholas Leisegang, Thomas Meyer, Ivan Varzniczak

Recent work in defeasible reasoning has seen notions of preferential semantics and entailment in the style of Kraus et al. applied to modal logics. However, work in this field has focussed primarily on satisfiability checking, and monotonic notions of entailment, which may be inferentially weak. One particular modal logic where this has been introduced is propositional standpoint logics, where modalities can express the views of different viewpoints. This has resulted in the formalisation of propositional defeasible standpoint logic (PDSL). In this paper, we propose a means of lifting the class of (non-monotonic) rational entailment relations from traditional KLM-style reasoning to a fragment of PDSL. In order to do so, we extend the expressivity of PDSL via situated standpoint conditionals, allowing us to talk about a defeasible conditional holding in the context of a given standpoint. This allows us to re-characterise the syntax of PDSL in terms of situated conditionals, and shows that a large fragment of PDSL is expressible as a set of situated conditionals. We then focus on characterising non-monotonic entailment in this fragment, defining a method to transport any ranking-based entailment relation from the propositional case into the PDSL case. This is first described in the general case and then considered in the specific cases of rational and lexicographic closures, providing a faithful translation of each inference into PDSL. We also show that entailment-checking in this fragment of PDSL can be done largely using algorithms from the propositional case, while preserving complexity bounds.

Racquel Dennison, Jesse Heyninck, Thomas Meyer

Defeasible entailment is concerned with drawing plausible conclusions from incomplete information. A foundational framework for modelling defeasible entailment is the KLM framework. Introduced by Kraus, Lehmann, and Magidor, the KLM framework outlines several key properties for defeasible entailment. One of the most prominent algorithms within this framework is Rational Closure (RC). This paper presents a declarative definition for computing RC using Answer Set Programming (ASP). Our approach enables the automatic construction of the minimal ranked model from a given knowledge base and supports entailment checking for specified queries. We formally prove the correctness of our ASP encoding and conduct empirical evaluations to compare the performance of our implementation with that of existing imperative implementations, specifically the InfOCF solver. The results demonstrate that our ASP-based approach adheres to RC's theoretical foundations and offers improved computational efficiency.

Nicholas Leisegang, Giovanni Casini, Thomas Meyer

Weighted-knowledge bases and cost-based semantics represent a recent formalism introduced by Bienvenu et al. for Ontology Mediated Data Querying in the case where a given knowledge base is inconsistent. This is done by adding a weight to each statement in the knowledge base (KB), and then giving each DL interpretation a cost based on how often it breaks rules in the KB. In this paper we compare this approach with c-representations, a form of non-monotonic reasoning originally introduced by Kern-Isberner. c-Representations describe a means to interpret defeasible concept inclusions in the first-order case. This is done by assigning a numerical ranking to each interpretations via penalties for each violated conditional. We compare these two approaches on a semantic level. In particular, we show that under certain conditions a weighted knowledge base and a set of defeasible conditionals can generate the same ordering on interpretations, and therefore an equivalence of semantic structures up to relative cost. Moreover, we compare entailment described in both cases, where certain notions are equivalently expressible in both formalisms. Our results have the potential to benefit further work on both cost-based semantics and c-representations

Nicholas Leisegang, Thomas Meyer, Ivan Varzinczak

In this paper, we introduce a new defeasible version of propositional standpoint logic by integrating Kraus et al.'s defeasible conditionals, Britz and Varzinczak's notions of defeasible necessity and distinct possibility, along with Leisegang et al.'s approach to defeasibility into the standpoint logics of Gómez Álvarez and Rudolph. The resulting logical framework allows for the expression of defeasibility on the level of implications, standpoint modal operators, and standpoint-sharpening statements. We provide a preferential semantics for this extended language and propose a tableaux calculus, which is shown to be sound and complete with respect to preferential entailment. We also establish the computational complexity of the tableaux procedure to be in PSpace.

Kai Sauerwald, Kenneth Skiba, Eduardo Fermé et al.

We study how linear orders can be employed to realise choice functions for which the set of potential choices is restricted, i.e., the possible choice is not possible among the full powerset of all alternatives. In such restricted settings, constructing a choice function via a relation on the alternatives is not always possible. However, we show that one can always construct a choice function via a linear order on sets of alternatives, even when a fallback value is encoded as the minimal element in the linear order. The axiomatics of such choice functions are presented for the general case and the case of union-closed input restrictions. Restricted choice structures have applications in knowledge representation and reasoning, and here we discuss their applications for theory change and abstract argumentation.

Lucas Carr, Nicholas Leisegang, Thomas Meyer et al.

Defeasible conditionals are a form of non-monotonic inference which enable the expression of statements like "if $φ$ then normally $ψ$". The KLM framework defines a semantics for the propositional case of defeasible conditionals by construction of a preference ordering over possible worlds. The pattern of reasoning induced by these semantics is characterised by consequence relations satisfying certain desirable properties of non-monotonic reasoning. In FCA, implications are used to describe dependencies between attributes. However, these implications are unsuitable to reason with erroneous data or data prone to exceptions. Until recently, the topic of non-monotonic inference in FCA has remained largely uninvestigated. In this paper, we provide a construction of the KLM framework for defeasible reasoning in FCA and show that this construction remains faithful to the principle of non-monotonic inference described in the original framework. We present an additional argument that, while remaining consistent with the original ideas around non-monotonic reasoning, the defeasible reasoning we propose in FCA offers a more contextual view on inference, providing the ability for more relevant conclusions to be drawn when compared to the propositional case.

James P. Delgrande, Birte Glimm, Thomas Meyer et al.

Knowledge Representation and Reasoning is a central, longstanding, and active area of Artificial Intelligence. Over the years it has evolved significantly; more recently it has been challenged and complemented by research in areas such as machine learning and reasoning under uncertainty. In July 2022 a Dagstuhl Perspectives workshop was held on Knowledge Representation and Reasoning. The goal of the workshop was to describe the state of the art in the field, including its relation with other areas, its shortcomings and strengths, together with recommendations for future progress. We developed this manifesto based on the presentations, panels, working groups, and discussions that took place at the Dagstuhl Workshop. It is a declaration of our views on Knowledge Representation: its origins, goals, milestones, and current foci; its relation to other disciplines, especially to Artificial Intelligence; and on its challenges, along with key priorities for the next decade.

Giovanni Casini, Thomas Meyer, Ivan Varzinczak

Conditionals are useful for modelling, but are not always sufficiently expressive for capturing information accurately. In this paper we make the case for a form of conditional that is situation-based. These conditionals are more expressive than classical conditionals, are general enough to be used in several application domains, and are able to distinguish, for example, between expectations and counterfactuals. Formally, they are shown to generalise the conditional setting in the style of Kraus, Lehmann, and Magidor. We show that situation-based conditionals can be described in terms of a set of rationality postulates. We then propose an intuitive semantics for these conditionals, and present a representation result which shows that our semantic construction corresponds exactly to the description in terms of postulates. With the semantics in place, we proceed to define a form of entailment for situated conditional knowledge bases, which we refer to as minimal closure. It is reminiscent of and, indeed, inspired by, the version of entailment for propositional conditional knowledge bases known as rational closure. Finally, we proceed to show that it is possible to reduce the computation of minimal closure to a series of propositional entailment and satisfiability checks. While this is also the case for rational closure, it is somewhat surprising that the result carries over to minimal closure.

Katarina Britz, Giovanni Casini, Thomas Meyer et al.

We extend description logics (DLs) with non-monotonic reasoning features. We start by investigating a notion of defeasible subsumption in the spirit of defeasible conditionals as studied by Kraus, Lehmann and Magidor in the propositional case. In particular, we consider a natural and intuitive semantics for defeasible subsumption, and investigate KLM-style syntactic properties for both preferential and rational subsumption. Our contribution includes two representation results linking our semantic constructions to the set of preferential and rational properties considered. Besides showing that our semantics is appropriate, these results pave the way for more effective decision procedures for defeasible reasoning in DLs. Indeed, we also analyse the problem of non-monotonic reasoning in DLs at the level of entailment and present an algorithm for the computation of rational closure of a defeasible ontology. Importantly, our algorithm relies completely on classical entailment and shows that the computational complexity of reasoning over defeasible ontologies is no worse than that of reasoning in the underlying classical DL ALC.

Richard Booth, Giovanni Casini, Thomas Meyer et al.

Propositional Typicality Logic (PTL) is a recently proposed logic, obtained by enriching classical propositional logic with a typicality operator capturing the most typical (alias normal or conventional) situations in which a given sentence holds. The semantics of PTL is in terms of ranked models as studied in the well-known KLM approach to preferential reasoning and therefore KLM-style rational consequence relations can be embedded in PTL. In spite of the non-monotonic features introduced by the semantics adopted for the typicality operator, the obvious Tarskian definition of entailment for PTL remains monotonic and is therefore not appropriate in many contexts. Our first important result is an impossibility theorem showing that a set of proposed postulates that at first all seem appropriate for a notion of entailment with regard to typicality cannot be satisfied simultaneously. Closer inspection reveals that this result is best interpreted as an argument for advocating the development of more than one type of PTL entailment. In the spirit of this interpretation, we investigate three different (semantic) versions of entailment for PTL, each one based on the definition of rational closure as introduced by Lehmann and Magidor for KLM-style conditionals, and constructed using different notions of minimality.

Gavin Rens, Abhaya Nayak, Thomas Meyer

Many multi-agent systems (MASs) are situated in stochastic environments. Some such systems that are based on the partially observable Markov decision process (POMDP) do not take the benevolence of other agents for granted. We propose a new POMDP-based framework which is general enough for the specification of a variety of stochastic MAS domains involving the impact of agents on each other's reputations. A unique feature of this framework is that actions are specified as either undirected (regular) or directed (towards a particular agent), and a new directed transition function is provided for modeling the effects of reputation in interactions. Assuming that an agent must maintain a good enough reputation to survive in the network, a planning algorithm is developed for an agent to select optimal actions in stochastic MASs. Preliminary evaluation is provided via an example specification and by determining the algorithm's complexity.

Giovanni Casini, Umberto Straccia, Thomas Meyer

Description Logics (DLs) under Rational Closure (RC) is a well-known framework for non-monotonic reasoning in DLs. In this paper, we address the concept subsumption decision problem under RC for nominal safe $\mathcal{ELO}_\bot$, a notable and practically important DL representative of the OWL 2 profile OWL 2 EL. Our contribution here is to define a polynomial time subsumption procedure for nominal safe $\mathcal{ELO}_\bot$ under RC that relies entirely on a series of classical, monotonic $\mathcal{EL}_\bot$ subsumption tests. Therefore, any existing classical monotonic $\mathcal{EL}_\bot$ reasoner can be used as a black box to implement our method. We then also adapt the method to one of the known extensions of RC for DLs, namely Defeasible Inheritance-based DLs without losing the computational tractability.

Gavin Rens, Thomas Meyer

Imaging is a form of probabilistic belief change which could be employed for both revision and update. In this paper, we propose a new framework for probabilistic belief change based on imaging, called Expected Distance Imaging (EDI). EDI is sufficiently general to define Bayesian conditioning and other forms of imaging previously defined in the literature. We argue that, and investigate how, EDI can be used for both revision and update. EDI's definition depends crucially on a weight function whose properties are studied and whose effect on belief change operations is analysed. Finally, four EDI instantiations are proposed, two for revision and two for update, and probabilistic rationality postulates are suggested for their analysis.

Gavin Rens, Thomas Meyer, Giovanni Casini

We propose a method for an agent to revise its incomplete probabilistic beliefs when a new piece of propositional information is observed. In this work, an agent's beliefs are represented by a set of probabilistic formulae -- a belief base. The method involves determining a representative set of 'boundary' probability distributions consistent with the current belief base, revising each of these probability distributions and then translating the revised information into a new belief base. We use a version of Lewis Imaging as the revision operation. The correctness of the approach is proved. The expressivity of the belief bases under consideration are rather restricted, but has some applications. We also discuss methods of belief base revision employing the notion of optimum entropy, and point out some of the benefits and difficulties in those methods. Both the boundary distribution method and the optimum entropy method are reasonable, yet yield different results.

Richard Booth, Thomas Meyer, Ivan Varzinczak et al.

Standard belief change assumes an underlying logic containing full classical propositional logic. However, there are good reasons for considering belief change in less expressive logics as well. In this paper we build on recent investigations by Delgrande on contraction for Horn logic. We show that the standard basic form of contraction, partial meet, is too strong in the Horn case. This result stands in contrast to Delgrande's conjecture that orderly maxichoice is the appropriate form of contraction for Horn logic. We then define a more appropriate notion of basic contraction for the Horn case, influenced by the convexity property holding for full propositional logic and which we refer to as infra contraction. The main contribution of this work is a result which shows that the construction method for Horn contraction for belief sets based on our infra remainder sets corresponds exactly to Hansson's classical kernel contraction for belief sets, when restricted to Horn logic. This result is obtained via a detour through contraction for belief bases. We prove that kernel contraction for belief bases produces precisely the same results as the belief base version of infra contraction. The use of belief bases to obtain this result provides evidence for the conjecture that Horn belief change is best viewed as a hybrid version of belief set change and belief base change. One of the consequences of the link with base contraction is the provision of a representation result for Horn contraction for belief sets in which a version of the Core-retainment postulate features.