Reasoning with Incomplete Information in Generalized Galois Logics Without Distribution: The Case of Negation and Modal Operators

Hartonas, Chrysafis

doi:10.1007/978-3-319-29300-4_14

Cited by 12 publications

(4 citation statements)

References 33 publications

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“…We call this rule weak Johansson's constructive contraposition as it is a special case of Johansson's constructive contraposition named (LLJ) by Hartonas (2016).…”

Section: Ortholatticesmentioning

confidence: 99%

“…So a fine-grained treatment of the kind of uncertainty in our partial models must rely on intensional semantics and could proceed by considering partial models as done by Hartonas (2016) or by providing an epistemic operator □ (Donini, Lenzerini, Nardi, Nutt, & Schaerf, 1998, for example), i.e. a special modal logic operator where the accessibility relation expresses a kind of accessibility between epistemic states.…”

Section: Using Cone-based Embeddings For Learningmentioning

confidence: 99%

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Embedding Ontologies in the Description Logic ALC by Axis-Aligned Cones

Özcep,

Leemhuis,

Wolter

2023

jair

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This paper is concerned with knowledge graph embedding with background knowledge, taking the formal perspective of logics. In knowledge graph embedding, knowledge— expressed as a set of triples of the form (a R b) (“a is R-related to b”)—is embedded into a real-valued vector space. The embedding helps exploiting geometrical regularities of the space in order to tackle typical inductive tasks of machine learning such as link prediction. Recent embedding approaches also consider incorporating background knowledge, in which the intended meanings of the symbols a, R, b are further constrained via axioms of a theory. Of particular interest are theories expressed in a formal language with a neat semantics and a good balance between expressivity and feasibility. In that case, the knowledge graph together with the background can be considered to be an ontology. This paper develops a cone-based theory for embedding in order to advance the expressivity of the ontology: it works (at least) with ontologies expressed in the description logic ALC, which comprises restricted existential and universal quantifiers, as well as concept negation and concept disjunction. In order to align the classical Tarskian Style semantics for ALC with the sub-symbolic representation of triples, we use the notion of a geometric model of an ALC ontology and show, as one of our main results, that an ALC ontology is satisfiable in the classical sense iff it is satisfiable by a geometric model based on cones. The geometric model, if treated as a partial model, can even be chosen to be faithful, i.e., to reflect all and only the knowledge captured by the ontology. We introduce the class of axis-aligned cones and show that modulo simple geometric operations any distributive logic (such as ALC) interpreted over cones employs this class of cones. Cones are also attractive from a machine learning perspective on knowledge graph embeddings since they give rise to applying conic optimization techniques.

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“…We call this rule weak Johansson's constructive contraposition as it is a special case of Johansson's constructive contraposition named (LLJ) by Hartonas (2016).…”

Section: Ortholatticesmentioning

confidence: 99%

Section: Using Cone-based Embeddings For Learningmentioning

confidence: 99%

Embedding Ontologies in the Description Logic ALC by Axis-Aligned Cones

Özcep,

Leemhuis,

Wolter

2023

jair

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“…The logic L min pOM is a non-distributive propositional logic with an orthonegation. Non-distributive logics are investigated thoroughly in [14]. The semantical models considered there are of an abstract kind and not geometrically motivated.…”

Section: Related Workmentioning

confidence: 99%

“…A different example of KR-relevant structures consisting of geometric objects is that of closed subspaces in a Hilbert space. Closed subspaces can be used to model partial information [14] regarding the states and measurements of particles on the micro-level [2].…”

Section: Introductionmentioning

confidence: 99%

Orthologics for Cones

Leemhuis¹,

Özçep²,

Wolter³

2020

Preprint

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In applications that use knowledge representation (KR) techniques, in particular those that combine data-driven and logic methods, the domain of objects is not an abstract unstructured domain, but it exhibits a dedicated, deep structure of geometric objects. One example is the class of convex sets used to model natural concepts in conceptual spaces, which also links via convex optimisation techniques to machine learning. In this paper we study logics for such geometric structures. Using the machinery of lattice theory, we describe an extension of minimal orthologic with a partial modularity rule that holds for closed convex cones. This logic combines a feasible data structure (exploiting convexity/conicity) with sufficient expressivity, including full orthonegation (exploiting conicity).

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