The Probabilistic Description Logic

2020 ◽  
pp. 1-24
Author(s):  
LEONARD BOTHA ◽  
THOMAS MEYER ◽  
RAFAEL PEÑALOZA
Keyword(s):  
Knowledge Representation ◽  
Description Logic ◽  
Description Logics ◽  
Light Weight ◽  
Classical Form ◽  
First Order ◽  
Probabilistic Description

Abstract Description logics (DLs) are well-known knowledge representation formalisms focused on the representation of terminological knowledge. Due to their first-order semantics, these languages (in their classical form) are not suitable for representing and handling uncertainty. A probabilistic extension of a light-weight DL was recently proposed for dealing with certain knowledge occurring in uncertain contexts. In this paper, we continue that line of research by introducing the Bayesian extension of the propositionally closed DL . We present a tableau-based procedure for deciding consistency and adapt it to solve other probabilistic, contextual, and general inferences in this logic. We also show that all these problems remain ExpTime-complete, the same as reasoning in the underlying classical .

scholarly journals Unifying Class-Based Representation Formalisms

1999 ◽  
Vol 11 ◽  
pp. 199-240 ◽  
Author(s):  
D. Calvanese ◽  
M. Lenzerini ◽  
D. Nardi
Keyword(s):  
Knowledge Representation ◽  
Description Logic ◽  
Description Logics ◽  
Expressive Power ◽  
Finite Model ◽  
Functional Dependencies ◽  
Semantic Data ◽  
Representation Systems ◽  
Frame Systems ◽  
Essential Set

The notion of class is ubiquitous in computer science and is central in many formalisms for the representation of structured knowledge used both in knowledge representation and in databases. In this paper we study the basic issues underlying such representation formalisms and single out both their common characteristics and their distinguishing features. Such investigation leads us to propose a unifying framework in which we are able to capture the fundamental aspects of several representation languages used in different contexts. The proposed formalism is expressed in the style of description logics, which have been introduced in knowledge representation as a means to provide a semantically well-founded basis for the structural aspects of knowledge representation systems. The description logic considered in this paper is a subset of first order logic with nice computational characteristics. It is quite expressive and features a novel combination of constructs that has not been studied before. The distinguishing constructs are number restrictions, which generalize existence and functional dependencies, inverse roles, which allow one to refer to the inverse of a relationship, and possibly cyclic assertions, which are necessary for capturing real world domains. We are able to show that it is precisely such combination of constructs that makes our logic powerful enough to model the essential set of features for defining class structures that are common to frame systems, object-oriented database languages, and semantic data models. As a consequence of the established correspondences, several significant extensions of each of the above formalisms become available. The high expressiveness of the logic we propose and the need for capturing the reasoning in different contexts forces us to distinguish between unrestricted and finite model reasoning. A notable feature of our proposal is that reasoning in both cases is decidable. We argue that, by virtue of the high expressive power and of the associated reasoning capabilities on both unrestricted and finite models, our logic provides a common core for class-based representation formalisms.

Logics for the Semantic Web

2011 ◽  
pp. 24-43
Author(s):  
J. Bruijn
Keyword(s):  
Semantic Web ◽  
Logic Programming ◽  
Description Logic ◽  
Description Logics ◽  
Order Logic ◽  
First Order Logic ◽  
Horn Logic ◽  
First Order ◽  
Logical Language ◽  
Rule Languages

This chapter introduces a number of formal logical languages which form the backbone of the Semantic Web. They are used for the representation of both ontologies and rules. The basis for all languages presented in this chapter is the classical first-order logic. Description logics is a family of languages which represent subsets of first-order logic. Expressive description logic languages form the basis for popular ontology languages on the Semantic Web. Logic programming is based on a subset of first-order logic, namely Horn logic, but uses a slightly different semantics and can be extended with non-monotonic negation. Many Semantic Web reasoners are based on logic programming principles and rule languages for the Semantic Web based on logic programming are an ongoing discussion. Frame Logic allows object-oriented style (frame-based) modeling in a logical language. RuleML is an XML-based syntax consisting of different sublanguages for the exchange of specifications in different logical languages over the Web.

scholarly journals A Brief Roadmap into Uncertain Knowledge Representation via Probabilistic Description Logics

Algorithms ◽  
2021 ◽  
Vol 14 (10) ◽  
pp. 280
Author(s):  
Rafael Peñaloza
Keyword(s):  
Artificial Intelligence ◽  
Knowledge Representation ◽  
Classical Logic ◽  
Description Logics ◽  
Building Blocks ◽  
Main Building ◽  
Probabilistic Description ◽  
The Family ◽  
Uncertain Knowledge ◽  
Classical Knowledge

Logic-based knowledge representation is one of the main building blocks of (logic-based) artificial intelligence. While most successful knowledge representation languages are based on classical logic, realistic intelligent applications need to handle uncertainty in an adequate manner. Over the years, many different languages for representing uncertain knowledge—often extensions of classical knowledge representation languages—have been proposed. We briefly present some of the defining properties of these languages as they pertain to the family of probabilistic description logics. This limited view is intended to help pave the way for the interested researcher to find the most adequate language for their needs, and potentially identify the remaining gaps.

scholarly journals Probabilistic Description Logics for Subjective Uncertainty

2017 ◽  
Vol 58 ◽  
pp. 1-66 ◽  
Author(s):  
Victor Gutierrez-Basulto ◽  
Jean Christoph Jung ◽  
Carsten Lutz ◽  
Lutz Schröder
Keyword(s):  
Description Logics ◽  
Order Logic ◽  
First Order Logic ◽  
Two Dimensional ◽  
Subjective Probabilities ◽  
Subjective Uncertainty ◽  
First Order ◽  
New Family ◽  
Probabilistic Description

We propose a family of probabilistic description logics (DLs) that are derived in a principled way from Halpern's probabilistic first-order logic. The resulting probabilistic DLs have a two-dimensional semantics similar to temporal DLs and are well-suited for representing subjective probabilities. We carry out a detailed study of reasoning in the new family of logics, concentrating on probabilistic extensions of the DLs ALC and EL, and showing that the complexity ranges from PTime via ExpTime and 2ExpTime to undecidable.

scholarly journals On Free Description Logics with Definite Descriptions

2021 ◽  
Author(s):  
Alessandro Artale ◽  
Andrea Mazzullo ◽  
Ana Ozaki ◽  
Frank Wolter
Keyword(s):  
Description Logic ◽  
Description Logics ◽  
Expressive Power ◽  
Definite Descriptions ◽  
First Order ◽  
Domain Semantics ◽  
Automated Support ◽  
Suitable Notion ◽  
Dual Domain ◽  
Polynomial Time Reduction

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.

scholarly journals Semantic Matchmaking as Non-Monotonic Reasoning: A Description Logic Approach

2007 ◽  
Vol 29 ◽  
pp. 269-307 ◽  
Author(s):  
T. Di Noia ◽  
E. Di Sciascio ◽  
F. M. Donini
Keyword(s):  
Semantic Web ◽  
Knowledge Representation ◽  
Web Service ◽  
Description Logic ◽  
Supply And Demand ◽  
Description Logics ◽  
Semantic Web Services ◽  
Logical Framework ◽  
Electronic Marketplace ◽  
Logic Approach

Matchmaking arises when supply and demand meet in an electronic marketplace, or when agents search for a web service to perform some task, or even when recruiting agencies match curricula and job profiles. In such open environments, the objective of a matchmaking process is to discover best available offers to a given request. We address the problem of matchmaking from a knowledge representation perspective, with a formalization based on Description Logics. We devise Concept Abduction and Concept Contraction as non-monotonic inferences in Description Logics suitable for modeling matchmaking in a logical framework, and prove some related complexity results. We also present reasonable algorithms for semantic matchmaking based on the devised inferences, and prove that they obey to some commonsense properties. Finally, we report on the implementation of the proposed matchmaking framework, which has been used both as a mediator in e-marketplaces and for semantic web services discovery.

scholarly journals Consequence-Based Reasoning for Description Logics with Disjunctions and Number Restrictions

2018 ◽  
Vol 63 ◽  
pp. 625-690 ◽  
Author(s):  
Andrew Bate ◽  
Boris Motik ◽  
Bernardo Cuenca Grau ◽  
David Tena Cucala ◽  
František Simančík ◽  
...  
Keyword(s):  
Description Logic ◽  
Practical Reasoning ◽  
Description Logics ◽  
Practical Implementation ◽  
First Order ◽  
Implementation Techniques ◽  
Important Addition ◽  
Logic Reasoning ◽  
Extensive Performance

Classification of description logic (DL) ontologies is a key computational problem in modern data management applications, so considerable effort has been devoted to the development and optimisation of practical reasoning calculi. Consequence-based calculi combine ideas from hypertableau and resolution in a way that has proved very effective in practice. However, existing consequence-based calculi can handle either Horn DLs (which do not support disjunction) or DLs without number restrictions. In this paper, we overcome this important limitation and present the first consequence-based calculus for deciding concept subsumption in the DL ALCHIQ+. Our calculus runs in exponential time assuming unary coding of numbers, and on ELH ontologies it runs in polynomial time. The extension to disjunctions and number restrictions is technically involved: we capture the relevant consequences using first-order clauses, and our inference rules adapt paramodulation techniques from first-order theorem proving. By using a well-known preprocessing step, the calculus can also decide concept subsumptions in SRIQ---a rich DL that covers all features of OWL 2 DL apart from nominals and datatypes. We have implemented our calculus in a new reasoner called Sequoia. We present the architecture of our reasoner and discuss several novel and important implementation techniques such as clause indexing and redundancy elimination. Finally, we present the results of an extensive performance evaluation, which revealed Sequoia to be competitive with existing reasoners. Thus, the calculus and the techniques we present in this paper provide an important addition to the repertoire of practical implementation techniques for description logic reasoning.

scholarly journals A Semantics and Complete Algorithm for Subsumption in the CLASSIC Description Logic

1994 ◽  
Vol 1 ◽  
pp. 277-308 ◽  
Author(s):  
A. Borgida ◽  
P. F. Patel-Schneider
Keyword(s):  
Standard Model ◽  
Knowledge Representation ◽  
Description Logic ◽  
Description Logics ◽  
Representation System ◽  
Practical Applications ◽  
Current Implementation ◽  
The Standard Model ◽  
Soundness And Completeness ◽  
Knowledge Representation System

This paper analyzes the correctness of the subsumption algorithm used in CLASSIC, a description logic-based knowledge representation system that is being used in practical applications. In order to deal efficiently with individuals in CLASSIC descriptions, the developers have had to use an algorithm that is incomplete with respect to the standard, model-theoretic semantics for description logics. We provide a variant semantics for descriptions with respect to which the current implementation is complete, and which can be independently motivated. The soundness and completeness of the polynomial-time subsumption algorithm is established using description graphs, which are an abstracted version of the implementation structures used in CLASSIC, and are of independent interest.

scholarly journals A Characterization Theorem for a Modal Description Logic

2017 ◽  
Author(s):  
Paul Wild ◽  
Lutz Schröder
Keyword(s):  
Description Logic ◽  
Description Logics ◽  
Characterization Theorem ◽  
Expressive Power ◽  
Order Logic ◽  
First Order Logic ◽  
Standard Description ◽  
First Order ◽  
Information State ◽  
Modal Description

Modal description logics feature modalities that capture dependence of knowledge on parameters such as time, place, or the information state of agents. E.g., the logic S5-ALC combines the standard description logic ALC with an S5-modality that can be understood as an epistemic operator or as representing (undirected) change. This logic embeds into a corresponding modal first-order logic S5-FOL. We prove a modal characterization theorem for this embedding, in analogy to results by van Benthem and Rosen relating ALC to standard first-order logic: We show that S5-ALC with only local roles is, both over finite and over unrestricted models, precisely the bisimulation-invariant fragment of S5-FOL, thus giving an exact description of the expressive power of S5-ALC with only local roles.

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