Generalized Satisfiability for the Description Logic
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Hybrid Logic As Extension of Modal and Temporal Logics
Revista de Humanidades de Valparaíso Humanities Journal of Valparaiso Año 7, 2019, 1er semestre, No 13, págs. 34-67 No 13 (2019): 34-67 eISSN 0719-4242 Hybrid Logic as Extension of Modal and Temporal Logics La lógica híbrida como extensión de las lógicas modal y temporal Daniel Álvarez-Domínguez Universidad de Salamanca, España [email protected] DOI: https://doi.org/10.22370/rhv2019iss13pp34-67 Recibido: 16/07/2019. Aceptado: 14/08/2019 Abstract Developed by Arthur Prior, Temporal Logic allows to represent temporal information on a logical system using modal (temporal) operators such as P, F, H or G, whose intuitive meaning is “it was sometime in the Past...”, “it will be sometime in the Future...”, “it Has always been in the past...” and “it will always Going to be in the future...” respectively. Val- uation of formulae built from these operators are carried out on Kripke semantics, so Modal Logic and Temporal Logic are consequently related. In fact, Temporal Logic is an extension of Modal one. Even when both logics mechanisms are able to formalize modal-temporal information with some accuracy, they suffer from a lack of expressiveness which Hybrid Logic can solve. Indeed, one of the problems of Modal Logic consists in its incapacity of naming specific points inside a model. As Temporal Logic is based on it, it cannot make such a thing neither. But First-Order Logic does can by means of constants and equality relation. Hybrid Logic, which results from combining Modal Logic and First-Order Logic, may solve this shortcoming. The main aim of this paper is to explain how Hybrid Logic emanates from Modal and Temporal ones in order to show what it adds to both logics with regard to information representation, why it is more expressive than them and what relation it maintains with the First-Order Correspondence Language. -
Tense Or Temporal Logic Gives a Glimpse of a Number of Related F-Urther Topics
TI1is introduction provi'des some motivational and historical background. Basic fonnal issues of propositional tense logic are treated in Section 2. Section 3 12 Tense or Temporal Logic gives a glimpse of a number of related f-urther topics. 1.1 Motivation and Terminology Thomas Miiller Arthur Prior, who initiated the fonnal study of the logic of lime in the 1950s, called lus logic tense logic to reflect his underlying motivation: to advance the philosophy of time by establishing formal systems that treat the tenses of nat \lrallanguages like English as sentence-modifying operators. While many other Chapter Overview formal approaches to temporal issues have been proposed h1 the meantime, Pri orean tense logic is still the natural starting point. Prior introduced four tense operatm·s that are traditionally v.rritten as follows: 1. Introduction 324 1.1 Motiv<Jtion and Terminology 325 1.2 Tense Logic and the History of Formal Modal Logic 326 P it ·was the case that (Past) 2. Basic Formal Tense Logic 327 H it has always been the case that 2.1 Priorean (Past, Future) Tense Logic 328 F it will be the case that (Future) G it is always going to be the case that 2.1.1 The Minimal Tense Logic K1 330 2.1 .2 Frame Conditions 332 2.1.3 The Semantics of the Future Operator 337 Tims, a sentence in the past tense like 'Jolm was happy' is analysed as a temporal 2.1.4 The lndexicai'Now' 342 modification of the sentence 'John is happy' via the operator 'it was the case thai:', 2.1.5 Metric Operators and At1 343 and can be formally expressed as 'P HappyO'olm)'. -
Querying Description Logic Knowledge Bases
QUERYING DESCRIPTION LOGIC KNOWLEDGE BASES A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Engineering and Physical Sciences 2007 By Birte Glimm School of Computer Science Contents Abstract 7 Declaration 8 Copyright 9 Acknowledgements 10 1 Introduction 11 1.1 Description Logics .......................... 12 1.1.1 Description Logic Knowledge Bases ............. 13 1.1.2 Historical Background of Description Logics ........ 14 1.1.3 Application Areas of Description Logics ........... 15 1.1.4 Semantics of Description Logics ............... 17 1.2 Reasoning Services .......................... 17 1.2.1 Standard Reasoning Services ................. 18 1.2.2 Conjunctive Queries ..................... 18 1.2.3 Challenges of Query Answering ............... 23 1.3 Aims and Objectives ......................... 27 1.4 A Guide for Readers ......................... 27 2 Foundations of Description Logics 29 2.1 Syntax and Semantics ........................ 29 2.2 Standard Reasoning Tasks ...................... 33 2.3 Conjunctive Queries ......................... 36 2.3.1 Query Answering ....................... 40 2.3.2 Conjunctive Queries in Databases .............. 41 2.3.3 Why Conjunctive Queries .................. 41 2 2.4 Combined and Data Complexity ................... 42 3 Related Work and Alternative Approaches 44 3.1 Conjunctive Queries for Expressive Description Logics ....... 44 3.2 Conjunctive Queries for Tractable Description Logics ....... 45 3.3 Modal Correspondence Theory .................... 46 3.4 Query Containment .......................... 47 3.4.1 The Difficulty of Regular Expressions ............ 49 3.5 Rule Formalisms ........................... 50 3.5.1 The Carin System ....................... 52 3.5.2 Extensions of the Carin System ............... 57 3.6 Hybrid Logics ............................. 60 3.6.1 Hybrid Logic Binders for Query Answering ......... 60 3.7 First-Order Logic .......................... -
A Hybrid Intuitionistic Logic: Semantics and Decidability
A Hybrid Intuitionistic Logic: Semantics and Decidability Rohit Chadha1 Damiano Macedonio2 Vladimiro Sassone2 1 Center for Logic and Computation, IST Lisboa 2 University of Sussex Abstract An intuitionistic, hybrid modal logic suitable for reasoning about distribution of resources was introduced in [17, 18]. The modalities of the logic allow validation of properties in a particular place, in some place and in all places. We give a sound and complete Kripke semantics for the logic extended with disjunctive connectives. The extended logic can be seen as an instance of Hybrid IS5. We also give a sound and complete birelational semantics, and show that it enjoys the finite model property: if a judgement is not valid in the logic, then there is a finite birelational counter-model. Hence, we prove that the logic is decidable. Keywords: spatial distribution of resources, spatial modalities, Kripke and birelational semantics, soundness and completeness, finite-model property. 1 Introduction In current computing, paradigm distributed resources spread over and shared amongst different nodes of a computer system are very common. For example, printers may be shared in local area networks, or distributed data may store documents in parts at different locations. The traditional rea- soning methodologies are not easily scalable to these systems as they may lack implicitly trust-able objects such as a central control. This has resulted in the innovation of several reasoning techniques. A popular approach in the literature has been the use of algebraic systems such as process algebra [10, 21, 16]. These algebras have rich theories in terms of semantics [21], logics [9, 8, 15, 23], and types [16]. -
Schroder.Pdf (Pdf, 179
Symposium on Theoretical Aspects of Computer Science year (city), pp. numbers www.stacs-conf.org NAMED MODELS IN COALGEBRAIC HYBRID LOGIC LUTZ SCHRODER¨ 1 AND DIRK PATTINSON 2 1 DFKI Bremen and Department of Computer Science, Universitat¨ Bremen E-mail address: [email protected] 2 Department of Computing, Imperial College London E-mail address: [email protected] ABSTRACT. Hybrid logic extends modal logic with support for reasoning about individual states, designated by so-called nominals. We study hybrid logic in the broad context of coalgebraic seman- tics, where Kripke frames are replaced with coalgebras for a given functor, thus covering a wide range of reasoning principles including, e.g., probabilistic, graded, default, or coalitional operators. Specif- ically, we establish generic criteria for a given coalgebraic hybrid logic to admit named canonical models, with ensuing completeness proofs for pure extensions on the one hand, and for an extended hybrid language with local binding on the other. We instantiate our framework with a number of examples. Notably, we prove completeness of graded hybrid logic with local binding. Introduction Modal logics have traditionally played a central role in Computer Science, appearing, e.g., in the guise of temporal logics, program logics such as PDL, epistemic logics, and later as description logics. The development of modal logics has seen extensions along (at least) two axes: the enhance- ment of the expressive power of basic (relational) modal logic on the one hand, and the continual extension, beyond the purely relational realm, of the class of structures described using modal logics on the other hand. -
Model Theoretic Characterisations of Description Logics
Model Theoretic Characterisations of Description Logics esis submied in accordance with the requirements of the University of Liverpool for the degree of Doctor in Philosophy by Robert Edgar Felix Piro From Celle, Germany August 2012 Degree of Doctor in Philosophy P S Prof. Dr. Frank Wolter Department of Computer Science University of Liverpool S S Dr. Boris Konev Department of Computer Science University of Liverpool A Dr. Clare Dixon Department of Computer Science University of Liverpool I E Dr. Clare Dixon Department of Computer Science University of Liverpool E E Prof. Dr. Ulrike Saler School of Computer Science University of Manchester Abstract e growing need for computer aided processing of knowledge has led to an in- creasing interest in description logics (DLs), which are applied to encode know- ledge in order to make it explicit and accessible to logical reasoning. DLs and in particular the family around the DL have therefore been thor- oughly investigated w.r.t. their complexity theory and proof theory. e question arises which expressiveness these logics actually have. e expressiveness of a logic can be inferred by a model theoretic characterisa- tion. On concept level, these DLs are akin to modal logics whose model theoretic properties have been investigated. Yet the model theoretic investigation of the DLs with their TBoxes, which are an original part of DLs usually not considered in context of modal logics, have remained unstudied. is thesis studies the model theoretic properties of , , , as well as , , and . It presents model theoretic proper- ties, which characterise these logics as fragments of the first order logic (FO). -
A Fitch-Style Derivation System for Hybrid Logic by Jared Smith
A Fitch-Style Derivation System for Hybrid Logic by Jared Smith (Under the Direction of Charles Cross) ABSTRACT Standard modal logic is typically parsed as concerning possible worlds. However there is no way for normal modal logic to refer to those possible worlds directly, which hampers its expressiveness. Hybrid logic develops tools to talk about particular possible worlds and so avoids the curtailed expressive power modal logic possesses. There are several proof systems for hybrid logic. Here I develop one in Fitch-style. This type of proof system naturally fits the goals of hybrid logic because of its clarity and resemblance to our natural process of reasoning. I prove this newly developed system is equivalent to established systems for building proofs in hybrid logic. Finally I examine what traits we should look for in a proof system. INDEX WORDS: Hybrid Logic, Modal Logic, Fitch-style Natural Deduction A Fitch-Style Derivation System for Hybrid Logic by JARED SMITH A.B., University of Georgia, 2011 B.B.A., University of Georgia, 2011 A Thesis Submitted to the Graduate Faculty of The University of Georgia in Partial Fulfillment of the Requirements for the Degree MASTER OF ARTS ATHENS,GEORGIA 2014 c 2014 Jared Smith A Fitch-Style Derivation System for Hybrid Logic by Jared Smith Approved: Major Professor: Charles Cross Committee: Charles Cross René Jagnow Yuri Balashov Electronic Version Approved: Maureen Grasso Dean of the Graduate School The University of Georgia April 2014 Acknowledgments The easiest way to double the length of this project is to simply express my gratitude to everyone who has aided its preparation. -
Introduction to Hybrid Logic
Introduction to Hybrid Logic Patrick Blackburn Department of Philosophy, Roskilde University, Denmark NASSLLI 2012, University of Austin, Texas Goals of the course This mini-course (or, more accurately, extended lecture) introduces and explores hybrid logic, a form of modal logic in which it is possible to name worlds (or times, or computational states, or situations, or nodes in parse trees, or people | indeed, whatever it is that the elements of Kripke Models are taken to represent). The course has two main goals. The first is to convey, as clearly as possible, the ideas and intuitions that have guided the development of hybrid logic. The second is to gently hint at some technical themes, such as the role of bisimulation and why hybrid logic is so useful proof theoretically. Course Outline • Lecture 1: From Modal to Hybrid Logic • Lecture 2: Hybrid deduction • Lecture 3: X marks the spot, or Living Locally with Downarrow • Lecture 4: Going First-Order • Lecture 5: And Prior to that. • Lecture 6: Now for Yesterday, Today and Tomorrow, or Actually Now Kamp Rules! or I Can, You Can, We Can, Kaplan Today: From Modal to Hybrid Logic In today's lecture we discuss: • Orthodox modal logic | from an Amsterdam perspective. • A problem with orthodox modal logic. • Fixing this problem with basic hybrid logic. • Technical theme: bisimulations. • Conceptual theme: What hybrid logic is, and why it is genuinely modal. The slides will uploaded to the NASSLLI website. What is modal logic? Slogan 1: Modal languages are simple yet expressive languages for talking about relational structures. Slogan 2: Modal languages provide an internal, local perspective on relational structures. -
Description Logics for Ontologies
Description Logics for Ontologies Ulrike Sattler TU Dresden 1 Introduction to Description Logics Description logics (DLs) [6, 8, 21] are a family of logic-based knowledge repre- sentation formalisms designed to represent and reason about the knowledge of an application domain in a structured and well-understood way. The basic notions in DLs are concepts (unary predicates) and roles (binary relations), and a specific DL is mainly characterised by the constructors it pro- vides to form complex concepts and roles from atomic ones. Intuitively, the following concept describes \A cooler connected to a reactor which, in turn, has a part that is a stirrer and whose functionality is to stir or to cool (or both)": Cooler u 9connectedTo:(Reactor u (9hasPart:Stirrer u (1) 8functionality:(Cooling t Stirring))) In addition to such a set of constructors, DLs are usually equipped with a ter- minological component, often called a TBox. In its simplest form, a TBox can be used to introduce names (abbreviations) for complex concepts. For exam- ple, we could introduce the abbreviation CooledStirringReactor for the concept in Concept 1 from above. More expressive TBox formalisms allow the statement of general concepts inclusion axioms (GCIs) such as 9hasPart:Stirrer v_ Reactor u 9functionality:Stirring; which says that only stirring reactors can have stirrers. Description logic systems provide their users with various reasoning capabil- ities that deduce implicit knowledge from the one explicitly stated in the TBox. The subsumption algorithm determines subconcept-superconcept relationships: a concept C is subsumed by a concept D w.r.t. a TBox if, in each model of the TBox, each instance of C is also an instance of D. -
Reasoning in Description Logics Using Resolution and Deductive Databases
Zur Erlangung des akademischen Grades eines Doktors der Wirtschaftswissenschaften (Dr. rer. pol.) von der Fakult¨atf¨urWirtschaftswissenschaften der Universit¨atFridericiana zu Karlsruhe genehmigte Dissertation. Reasoning in Description Logics using Resolution and Deductive Databases M.Sc. Boris Motik Tag der m¨undlichen Pr¨ufung: 09. Januar 2006 Referent: Prof. Dr. Rudi Studer, Universit¨atKarlsruhe (TH) Korreferentin 1: Dr. habil. Ulrike Sattler, University of Manchester Korreferent 2: Prof. Dr. Karl-Heinz Waldmann, Universit¨atKarlsruhe (TH) 2006 Karlsruhe ii Abstract Description logics (DLs) are knowledge representation formalisms with well-understood model-theoretic semantics and computational properties. The DL SHIQ(D) provides the logical underpinning for the family of Semantic Web ontology languages. Metadata management applications, such as the Semantic Web, often require reasoning with large quantities of assertional information; however, existing algorithms for DL reasoning have been optimized mainly for efficient terminological reasoning. Although techniques for terminological reasoning can be used for assertional reasoning as well, they often do not exhibit good performance in practice. Deductive databases are knowledge representation systems specifically designed to efficiently reason with large data sets. To see if deductive database techniques can be used to optimize assertional reasoning in DLs, we studied the relationship between the two formalisms. Our main result is a novel algorithm that reduces a SHIQ(D) knowledge base KB to a disjunctive datalog program DD(KB) such that KB and DD(KB) entail the same set of ground facts. In this way, we allow DL reasoning to be performed in DD(KB) using known, optimized algorithms. The reduction algorithm is based on several novel results. -
FULLTEXT01.Pdf
http://www.diva-portal.org This is the published version of a paper published in . Citation for the original published paper (version of record): Goranko, V. (2019) Hybrid Deduction–Refutation Systems Axioms, 8(4): 118 https://doi.org/10.3390/axioms8040118 Access to the published version may require subscription. N.B. When citing this work, cite the original published paper. Permanent link to this version: http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-177315 axioms Article Hybrid Deduction–Refutation Systems Valentin Goranko 1,2 1 Department of Philosophy, Stockholm University, SE-10691 Stockholm, Sweden; [email protected] 2 Visiting professorship at Department of Mathematics, University of Johannesburg, Johannesburg 2006, South Africa Received: 27 August 2019; Accepted: 10 October 2019; Published: 21 October 2019 Abstract: Hybrid deduction–refutation systems are deductive systems intended to derive both valid and non-valid, i.e., semantically refutable, formulae of a given logical system, by employing together separate derivability operators for each of these and combining ‘hybrid derivation rules’ that involve both deduction and refutation. The goal of this paper is to develop a basic theory and ‘meta-proof’ theory of hybrid deduction–refutation systems. I then illustrate the concept on a hybrid derivation system of natural deduction for classical propositional logic, for which I show soundness and completeness for both deductions and refutations. Keywords: deductive refutability; refutation systems; hybrid deduction–refutation rules, derivative hybrid rules, soundness, completeness, natural deduction, meta-proof theory 1. Introduction 1.1. Semantic vs. Deductive Refutability Consider a generic logical system L, comprising a formal logical language with a given semantics.