Extending the Description Logic ALC with More Expressive Cardinality Constraints on Concepts

Baader, Franz; Ecke, Andreas

doi:10.29007/f3hh

Cited by 6 publications

(25 citation statements)

References 12 publications

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“…The reason why the coding of numbers is irrelevant in the presence of ECBoxes is that one can use iterated multiplication to create large numbers from small ones (see [8] for a more detailed argument).…”

Section: Reasoning Wrt Cboxes and Ecboxesmentioning

confidence: 99%

“…These conditions can be Boolean combinations of atomic properties required for the individual (expressed by concept names) or properties that refer to relationships with other individuals and their properties (expressed as role restrictions). Adapting an example from [8,3] to a different domain, the concept of a Computer Science author can be formalized by the concept description which uses the concept names Person, Paper and CS, and the role names published and topic, as well as the concept constructors conjunction ( ), existential restriction (∃r.C), and value restriction (∀r.C). It describes the set of all persons that have published a paper all of whose topics lie in the area of Computer Science.…”

Section: Introductionmentioning

confidence: 99%

“…Just like classical number restrictions, CRs can only relate the cardinality of a concept to a fixed number. In [8], we have introduced and investigated more general constraints on the cardinalities of concepts, which we called extended cardinality constraints. The main idea was again to use QFBAPA to formulate and combine these constraints.…”

Section: Introductionmentioning

confidence: 99%

“…An example of a constraint expressible this way, but not expressible using CRs is 2 • |Paper ∀topic.DL| |Paper ∀topic.AR| which states that papers with topic Automated Reasoning outnumber papers with topic Description Logic by a factor of at least two. In [8] it is shown that, in the DL ALC, the complexity of reasoning w.r.t. extended cardinality constraints (NExpTime for binary coding of numbers) is the same as for reasoning w.r.t.…”

Section: Introductionmentioning

confidence: 99%

“…In [2,3], we combined the work in [1] and [8] by considering extended cardinality constraints in ALCSCC. This turned out to be non-trivial since the local cardinality constraints of ALCSCC may interact with the global ones in the extended cardinality constraints.…”

Section: Introductionmentioning

confidence: 99%

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Description Logics That Count, and What They Can and Cannot Count

Baader¹,

Bortoli²

EPiC Series in Computing

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Simple counting quantifiers that can be used to compare the number of role successors of an individual or the cardinality of a concept with a fixed natural number have been employed in Description Logics (DLs) for more than two decades under the respective names of number restrictions and cardinality restriction on concepts. Recently, we have considerably extended the expressivity of such quantifiers by allowing to impose set and cardinality constraints formulated in the quantifier-free fragment of Boolean Algebra with Presburger Arithmetic (QFBAPA) on sets of role successors and concepts, respectively. We were able to prove that this extension does not increase the complexity of reasoning.In the present paper, we investigate the expressive power of the DLs obtained this way, using appropriate bisimulation characterizations and 0--1 laws as tools for distinguishing the expressiveness of different logics. In particular, we show that, in contrast to most classical DLs, these logics are no longer expressible in first-order predicate logic (FOL), and we characterize their first-order fragments. In most of our previous work on DLs with QFBAPA-based set and cardinality constraints we have employed finiteness restrictions on interpretations to ensure that the obtained sets are finite. Here we dispense with these restrictions to make the comparison with classical DLs, where one usually considers arbitrary models rather than finite ones, easier. It turns out that doing so does not change the complexity of reasoning.

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Section: Reasoning Wrt Cboxes and Ecboxesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Description Logics That Count, and What They Can and Cannot Count

Baader¹,

Bortoli²

EPiC Series in Computing

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On Free Description Logics with Definite Descriptions

Artale

Mazzullo

Ozaki

et al. 2021

Proceedings of the Eighteenth International Conference on Principles of Knowledge Representation and Reasoning

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Definite descriptions are phrases of the form ‘the x such that φ’, used to refer to single entities in a context. They are often more meaningful to users than individual names alone, in particular when modelling or querying data over ontologies. We investigate free description logics with both individual names and definite descriptions as terms of the language, while also accounting for their possible lack of denotation. We focus on the extensions of ALC and, respectively, EL with nominals, the universal role, and definite descriptions. We show that standard reasoning in these extensions is not harder than in the original languages, and we characterise the expressive power of concepts relative to first-order formulas using a suitable notion of bisimulation. Moreover, we lay the foundations for automated support for definite descriptions generation by studying the complexity of deciding the existence of definite descriptions for an individual under an ontology. Finally, we provide a polynomial-time reduction of reasoning in other free description logic languages based on dual-domain semantics to the case of partial interpretations.

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Succinctness and Complexity of ALC with Counting Perceptrons

Galliani

Kutz

Troquard

2023

Proceedings of the Twentieth International Conference on Principles of Knowledge Representation and Reasoning

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Perceptron operators have been introduced to knowledge representation languages such as description logics in order to define concepts by listing features with associated weights and by giving a threshold. Semantically, an individual then belongs to such a concept if the weighted sum of the listed features it belongs to reaches that threshold. Such operators have been subsequently applied to cognitively-motivated modelling scenarios and to building bridges between learning and reasoning. However, they suffer from the basic limitation that they cannot consider the weight or number of role fillers. This paper introduces an extension of the basic perceptron operator language to address this shortcoming, defining the language ALCP and answering some basic questions regarding the succinctness and complexity of the new language. Namely, we show firstly that in ALCP+, when weights are positive, the language is expressively equivalent to ALCQ, whilst it is strictly more expressive in the general case allowing also negative weights. Secondly, ALCP+ is shown to be strictly more succinct than ALCQ. Thirdly, capitalising on results concerning the logic ALCSCC, we show that despite the added expressivity, reasoning in ALCP remains EXPTIME-complete.

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Extending the Description Logic ALC with More Expressive Cardinality Constraints on Concepts

Cited by 6 publications

References 12 publications

Description Logics That Count, and What They Can and Cannot Count

Description Logics That Count, and What They Can and Cannot Count

On Free Description Logics with Definite Descriptions

Succinctness and Complexity of ALC with Counting Perceptrons

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