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1 From Data to Knowledge through Grailog Visualization (Long version: Harold Boley Faculty of Computer Science University of New Brunswick Fredericton, NB, Canada Ontology, Rules, and Logic Programming for Reasoning and Applications (RulesReasoningLP) Ontolog Mini-Series, Session 2, 31 October 2013

2 1 Acknowledgements Thanks for feedback on various versions and parts of this presentation (the long version has all parts, hence gapless slide numbers):long version From Data to Knowledge through Grailog Visualization ISO and Semantic Technologies 2013 Conference, Sogndal, Norway, 5-6 September 2013 Grailog 1.0: Graph-Logic Visualization of Ontologies and Rules The 7th International Web Rule Symposium (RuleML 2013), University of Washington, Seattle WA, July 2013 The Grailog Systematics for Visual-Logic Knowledge Representation with Generalized Graphs Faculty of Computer Science Seminar Series, University of New Brunswick, Fredericton, Canada, 26 September 2012 High Performance Computing Center Stuttgart (HLRS), Stuttgart, Germany, 14 August 2012 Grailog: Mapping Generalized Graphs to Computational Logic Grailog: Mapping Generalized Graphs to Computational Logic Symposium on Natural/Unconventional Computing and its Philosophical Significance, AISB/IACAP World Congress - Alan Turing 2012, 2-6 July 2012, Birmingham, UK The Grailog User Interface for Knowledge Bases of Ontologies & Rules The Grailog User Interface for Knowledge Bases of Ontologies & Rules OMG Technical Meeting, Ontology PSIG, Cambridge, MA, 21 June 2012 Grailog: Knowledge Representation with Extended Graphs for Extended Logics SAP Enterprise Semantics Forum, 24 April 2012 Grailog: Towards a Knowledge Visualization Standard BMIR Research Colloquium, Stanford, CA, 4 April 2012 PARC Research Talk, Palo Alto, CA, 29 March 2012 RuleML/Grailog: The Rule Metalogic Visualized with Generalized Graphs RuleML/Grailog: The Rule Metalogic Visualized with Generalized Graphs PhiloWeb 2011, Thessaloniki, Greece, 5 October 2011 Grailog: Graph inscribed logic Course about Logical Foundations of Cognitive Science, TU Vienna, Austria, 20 October -10 December 2008

3 2 Abstract Directed labeled graphs (DLGs) provide a good starting point for visual data & knowledge representation but cannot straightforwardly represent nested structures, non-binary relationships, and relation descriptions. These advanced features require encoded constructs with auxiliary nodes and relationships, which also need to be kept separate from straightforward constructs. Therefore, various extensions of DLGs have been proposed for data & knowledge representation, including graph partitionings (possibly interfaced as complex nodes), n-ary relationships as directed labeled hyperarcs, and (hyper)arc labels used as nodes of other (hyper)arcs. Meanwhile, a lot of AI / Semantic Web research and development on ontologies & rules has gone into extended logics for knowledge representation such as object (frame) logics, description logics, general modal logics, and higher-order logics. The slides demonstrate how data & knowledge representation with graphs and logics can be reconciled. They proceed from simple to extended graphs for logics needed in AI and the Semantic Web. Along with its visual introduction, each graph construct is mapped to its corresponding symbolic logic construct. These graph- logic extensions constitute a systematics defined by orthogonal axes, which has led to the Grailog 1.0 language as part of the Web-rule industry standard RuleML 1.0 (http://ruleml.org/#Grailog). While Grailog's DLG sublanguage corresponds to binary-associative memories, its hypergraph sublanguage corresponds to n-ary content-addressable memories, and its complex-node modules offer various further opportunities for parallel processinghttp://ruleml.org/#Grailog

4 3 Visualization of Data Useful in many areas, needed for big data Gain knowledge insights from data analytics, ideally with the entire pipeline visualized Statistical visualization Logical visualization Sample data visualization ( ) : Word cloud for frequency of words from BMIR abstract of this talk

5 4 Visualization of Data & Knowledge: Graphs Remove Entry Barrier to Logic From 1-dimensional symbol-logic knowledge specification to 2-dimensional graph-logic visualization in a systematic 2D syntax –Supports human in the loop across knowledge elicitation, specification, validation, and reasoning Combinable with graph transformation, (associative) indexing & parallel processing for efficient implementation of specifications Move towards model-theoretic semantics –Unique names, as graph nodes, mapped directly/ injectively to elements of semantic interpretation

6 5 Rule MetaLogic Provides Family of Language Standards for Web Data & Knowledge Interchange Developed on the Web: Principal (family-uniform) and variant semantics Family-uniform syntaxes for humans and machines

7 6 Three RuleML Syntaxes (1) Syntax Visualization RuleML/Grailog Symbolic Presentation Serialization RuleML/POSLRuleML/XML

8 7 Three RuleML Syntaxes (2) Serialization RuleML/XML: Specified in XML Schema and recently in Relax NG: Presentation RuleML/POSL: Integrates Prolog and F-logic, and translates to RuleML/XML: Visualization RuleML/Grailog: Based on Directed Recursive Labelnode Hypergraphs (DRLHs):

9 8 Grailog Graph inscribed logic provides intuition for logic Advanced cognitively motivated systematic graph standard for visual-logic data & knowledge: Features orthogonal easy to learn, e.g. for (Business) Analytics Generalized-graph framework as one uniform 2D syntax for major (Semantic Web) logics:Semantic Web Pick subset for each targeted knowledge base, map to/fro RuleML sublanguage, and exchange & validate it, posing queries again in Grailog

10 9 Note on Grailog and API4KB Besides mapping Grailog to/fro RuleML, RDF and UML+OCL can be targeted, with uniform access to be provided by API4KBGrailog to/fro RuleMLAPI4KB Grailog and API4KB strive to cover main data & knowledge representation paradigms: –RDF (directed-labeled-graph) and Relational (Datalog-fact-like) data –Ontology (RDFS and description-logic) and Rule (Horn- and general-logic) knowledge An API can be (initially) designed and tested with a human in the loop much like a GUI

11 10 Generalized Graphs to Represent and Map Logic Languages According to Grailog 1.0 Systematics We have used generalized graphs for representing various logic languages, where basically: –Graph nodes (vertices) represent individuals, classes, etc. –Graph arcs (edges) represent relationships Next slides: What are the principles of this representation and what graph generalizations are required? Later slides: How are these graphs mapped (invertibly) to logic, thus specifying Grailog as a GUI for knowledge? Final slides: What is the systematics of Grailog features?

12 11 Grailog Principles Graphs should make it easier for humans to read and write logic constructs via 2D state-of-the-art representation with shorthand & normal forms, from Controlled English to logic Graphs should be natural extensions (e.g. n-ary) of Directed Labeled Graphs (DLGs), often used to represent simple semantic nets, i.e. of atomic ground formulas in function-free dyadic predicate logic (cf. binary Datalog ground facts, RDF triples, the Open Graph, and the Knowledge Graph)DatalogRDFOpen GraphKnowledge Graph Graphs should allow stepwise refinements for all logic constructs: Description Logic constructors, F-logic frames, general PSOA RuleML terms, etc.Description Logic F-logicPSOA RuleML Extensions to boxes & links should be orthogonal

13 12 Informal Grailog Preview: Searles Chinese Room Argument John Searle (emphasis added):... whatever purely formal principles you put into the computer, they will not be sufficient for understanding, since a human will be able to follow the formal principles without understanding anything. (Minds, Brains and Programs, 1980)

14 13 English Chinese textquestion ruleset reply Searle Language Wang understand lang Searle-reply i Wang-reply i distinguishable SubClassOf HasInstance apply for with use Classes with relationships Instances with relationships with for to negation understand langhasLanguage Searles Chinese Room Scenario: Grailog for Visual Controlled English understand

15 14 Grailog Generalizations Directed hypergraphs: For n-ary relationships, directed relation-labeled (binary) arcs will be generalized to directed relation-labeled (n-ary) hyperarcs, e.g. representing relational-database tuples Recursive (hierarchical) graphs: For nested terms and formulas, modal logics, and modularization, flat graphs will be generalized to allow other graphs as complex nodes to any level of depth Labelnode graphs: For allowing higher-order logics describing both instances and relations (predicates), arc labels will also become usable as nodes

16 15 Graphical Elements: Names Written into boxes (nodes): Unique (canonical, distinct) names –Unique Name Assumption (UNA) refined to Unique Name Specification (UNS) Written onto boxes (node labels): Non-unique (alternate, aka) names –Non-unique Name Assumption (NNA) refined to Non-unique Name Specification (NNS) Grailog combines UNS and NNS: xNS, with x = U or N unique non-unique

17 16 Instances: Individual Constants with Unique Name Specifications unique Warren Buffett General:Graph (node)Logic US$ General Electric Examples:GraphLogic unique Warren Buffett US$ General Electric mapping

18 17 Instances: Individual Constants with Non-unique Name Specifications General:Graph (node)Logic (vertical bar for non-uniqueness) Examples:GraphLogic |non-unique |WB |US$ 3B |GE mapping WB GE US$ 3B non-unique

19 18 Graphical Elements: Hatching Patterns No hatching (boxes): Constant Hatching (elementary boxes): Variable

20 19 Parameters: Individual Variables General:Graph (hatched node)Logic (italics font, POSL uses ? prefix )POSL Examples:GraphLogic variable X A Y X Y A

21 20 Predicates: Binary Relations (1) inst 1 General:Graph (labeled arc) Logic Example:Graph Logic binrel(inst 1, inst 2 ) inst 2 binrel Warren BuffettGeneral Electric Trust Trust(Warren Buffett, General Electric )

22 21 Predicates: Binary Relations (2) General:Graph (labeled arc)Logic Example:GraphLogic binrel(var 1, var 2 ) binrel Trust Trust(X,Y) X Y var 1 var 2

23 22 Ground Equality: Identifying Pairs of Constants inst 1 General:Graph (unlabeled Logic (with equality) undirected arc) Example:Graph Logic inst 1 = inst 2 inst 2 General Electric | GE = General Electric Inspired by Charles Sanders Peirces line of identity, as a co-reference linkline of identity GE

24 23 Ground Equality: Defining Constants with Constants inst 1 Example:Graph Logic inst 1 := inst 2 inst 2 General Electric | GE := General Electric GE General:Graph (unlabeled Logic (with oriented undirected, equality) colon-tailed arc) : :

25 24 inst General:Graph (unlabeled Logic (with oriented undirected, equality, webized) colon-tailed arc) Example:Graph Logic inst := IRI IRI | GenElec := Ground Equality: Defining Symbolic Constants as IRIs Definitional equality can also be used for the prefix part of the CURIE notationCURIE notation : : GenElec

26 25 Negated Predicates: Binary Relations (Shorthand) inst 1 General:Graph (dashed arc)Logic Example:GraphLogic binrel(inst 1, inst 2 ) inst 2 binrel Joe Smallstock Trust( Joe Smallstock, General Electric ) General Electric Trust

27 26 Negated Predicates: Binary Relations (Long Form) inst 1 General:Graph (dashed box)Logic Example:GraphLogic (binrel(inst 1, inst 2 )) inst 2 binrel Joe Smallstock (Trust( Joe Smallstock, General Electric )) General Electric Trust

28 27 Ground Inequality: Pairwise Difference (Shorthand) inst 1 General:Graph (dashed Logic (with equality) unlabeled undirected arc) Example:Graph Logic (with equality) inst 1 inst 2 inst 2 Joe Smallstock Warren Buffett Joe Smallstock

29 28 Ground Inequality: Pairwise Difference (Long Form) inst 1 General:Graph (dashed box, Logic (with equality) unlabeled undirected arc) Example:Graph Logic (with equality) (inst 1 = inst 2 ) inst 2 (Joe Smallstock = Warren Buffett) Warren BuffettJoe Smallstock

30 29 Graphical Elements: Arrows (1) Labeled arrows (directed links) for arcs and hyperarcs (where hyperarcs cut through nodes intermediate between first and last)

31 30 Predicates: n-ary Relations (n>1) inst 1 General:Graph (hyperarc) Logic Example:Graph Logic (n=3) rel(inst 1, inst 2,..., inst n-1, inst n ) inst 2 rel Invest Invest(|WB, |GE, US$ 3·10 9 ) inst n inst n-1 US$ 3·10 9 WB GE

32 31 Negated Predicates: n-ary Relations (Shorthand) inst 1 General:Graph (dashed hyperarc ) Logic Example:Graph Logic (n=3) rel(inst 1, inst 2,..., inst n-1, inst n ) inst 2 rel inst n inst n-1 Invest US$ 4·10 9 WB GE Invest(|WB, |GE, US$ 4·10 9 )

33 32 Negated Predicates: n-ary Relations (Long Form) inst 1 General:Graph (dashed box) Logic Example:Graph Logic (n=3) (rel(inst 1, inst 2,..., inst n-1, inst n )) inst 2 rel inst n inst n-1 Invest US$ 4·10 9 WB GE (Invest(|WB, |GE, US$ 4·10 9 ))

34 33 Implicit Conjunction of Formula Graphs: Co-Occurrence on Graph Top-Level inst 1,1 rel 1 (inst 1,1, inst 1,2,..., inst 1,n 1 ) inst 1,2 inst 1,n 1 inst m,1 rel m (inst m,1, inst m,2,...,inst m,n m ) inst m,2 inst m,n m... rel 1 rel m General:Graph (m hyperarcs) Logic Example:Graph (2 hyperarcs) Logic... Invest(|WB, |GE, US$ 3·10 9 ) Invest(|JS, |VW, US$ 2·10 4 ) Invest US$ 3·10 9 Invest US$ 2·10 4 WB GE JSVW

35 34 Explicit Conjunction of Formula Graphs: Co-Occurrence in ( parallel-processing ) And Node inst 1,1 (rel 1 (inst 1,1, inst 1,2,..., inst 1,n 1 ) inst 1,2 inst 1,n 1 inst m,1 rel m (inst m,1, inst m,2,...,inst m,n m )) inst m,2 inst m,n m... rel 1 rel m General:Graph (solid+linear) Logic Example:Graph Logic... Invest (Invest(|WB, |GE, US$ 3·10 9 ) US$ 3·10 9 Invest Invest(|JS, |VW, US$ 2·10 4 )) US$ 2·10 4 WB GE JSVW

36 35 Not of And of Formula Graphs: Co-Occurrence in a Nots And Node inst 1,1 (rel 1 (inst 1,1, inst 1,2,..., inst 1,n 1 ) inst 1,2 inst 1,n 1 inst m,1 rel m (inst m,1, inst m,2,...,inst m,n m )) inst m,2 inst m,n m... rel 1 rel m General:Graph ( dashed/solid+linear ) Logic Example:Graph Logic... Invest (Invest(|WB, |GE, US$ 3·10 9 ) US$ 3·10 9 Invest Invest(|JS, |VW, US$ 2·10 4 )) US$ 2·10 4 WB GE JSVW

37 36 Not of And (Nand) of Formula Graphs: Co-Occurrence in Nand Node (Shorthand) inst 1,1 (rel 1 (inst 1,1, inst 1,2,..., inst 1,n 1 ) inst 1,2 inst 1,n 1 inst m,1 rel m (inst m,1, inst m,2,...,inst m,n m )) inst m,2 inst m,n m... rel 1 rel m General:Graph (dashed+linear) Logic Example:Graph Logic... Invest (Invest(|WB, |GE, US$ 3·10 9 ) US$ 3·10 9 Invest Invest(|JS, |VW, US$ 2·10 4 )) US$ 2·10 4 WB GE JSVW

38 37 Disjunction of Formula Graphs: Co-Occurrence in Or Node inst 1,1 (rel 1 (inst 1,1, inst 1,2,..., inst 1,n 1 ) inst 1,2 inst 1,n 1 inst m,1 rel m (inst m,1, inst m,2,...,inst m,n m ) ) inst m,2 inst m,n m... rel 1 rel m General: Graph (solid+wavy) Logic Example:Graph Logic... (Invest(|WB, |GE, US$ 3·10 9 ) Invest(|JS, |VW, US$ 2·10 4 )) Invest US$ 3·10 9 Invest US$ 2·10 4 WB GE JSVW

39 38 Not of Or of Formula Graphs: Co-Occurrence in a Nots Or Node inst 1,1 (rel 1 (inst 1,1, inst 1,2,..., inst 1,n 1 ) inst 1,2 inst 1,n 1 inst m,1 rel m (inst m,1, inst m,2,...,inst m,n m ) ) inst m,2 inst m,n m... rel 1 rel m General: Graph ( dashed/solid+linear/wavy ) Logic Example:Graph Logic... (Invest(|WB, |GE, US$ 3·10 9 ) Invest(|JS, |VW, US$ 2·10 4 )) Invest US$ 3·10 9 Invest US$ 2·10 4 WB GE JSVW

40 39 Not of Or (Nor) of Formula Graphs: Co-Occurrence in Nor Node (Orthogonal Shorthand) inst 1,1 (rel 1 (inst 1,1, inst 1,2,..., inst 1,n 1 ) inst 1,2 inst 1,n 1 inst m,1 rel m (inst m,1, inst m,2,...,inst m,n m ) ) inst m,2 inst m,n m... rel 1 rel m General: Graph (dashed+wavy) Logic Example:Graph Logic... (Invest(|WB, |GE, US$ 3·10 9 ) Invest(|JS, |VW, US$ 2·10 4 )) Invest US$ 3·10 9 Invest US$ 2·10 4 WB GE JSVW

41 40 John Latin Paul Mary Hypergraph (2 hyperarcs, crossing inside a node) John Latin Paul Mary Kate to From Hyperarc Crossings to Node Copies as a Normalization Sequence (1) Kate Teach Show DLG (4 arcs, do not specify to whom Latin is shown or taught) Symbolic Controlled English John shows Latin to Kate. Mary teaches Latin to Paul.

42 41 From Hyperarc Crossings to Node Copies as a Normalization Sequence (1*) John Latin Show Paul Mary Kate Hypergraph (2 hyperarcs, crossing outside nodes) John Latin Show Paul Mary Kate DLG (4 arcs, do not specify to whom Latin is shown or taught) to Teach

43 42 Hypergraph (2 hyperarcs, parallel-cutting a node) John Latin Kate Mary Teach Paul to John Latin Kate Mary Teach Paul From Hyperarc Crossings to Node Copies as a Normalization Sequence (1**) Show DLG (4 arcs, do not specify to whom Latin is shown or taught) The hyperarc for, e.g., ternary Show(John,Latin,Kate) can be seen as the path composition of 2 arcs for binary Show(John,Latin) and binary to(Latin,Kate)

44 43 Hypergraph (2 hyperarcs, parallel-cutting a node) John Latin Kate Mary Teach1 Paul John Latin Kate Mary Teach Paul From Hyperarc Crossings to Node Copies Insert on Correct Binary ReductionCorrect Binary Reduction Show Show1 DLG (8 arcs with 4 reified relation/ship nodes to point to arguments) arg1arg2arg3 arg1arg2arg3 Show Teach

45 44 Hypergraph (2 hyperarcs, employing a node copy) Logic (2 relations, employing a symbol copy) John Latin Show(John, Latin, Kate) Teach(Mary, Latin, Paul) Kate Mary Latin Paul From Hyperarc Crossings to Node Copies as a Normalization Sequence (1***) Teach Show Both Latin occurrences remain one node even when copied for easier layout: Having a unique name, Latin copies can be merged again. This fully node copied normal form can help to learn the symbolic form, is implemented by Grailog KS Viz, and demoed in the Loan Processor test suiteGrailog KS Viztest suite

46 45 From Predicate Labels on Hyperarcs to Labelnodes Starting Hyperarcs inst 1 General:Graph (hyperarc with Logic rect4vex-shaped labelnode) Example:Graph Logic (n=3) rel(inst 1, inst 2,..., inst n-1, inst n ) inst 2 rel Invest(|WB, |GE, US$ 3·10 9 ) inst n inst n-1 rel inst 1 inst 2 inst n inst n-1 US$ 3·10 9 Invest US$ 3·10 9 WB GE WB GE Invest (Shorthand) (Normal Form)

47 46 inst 1 General:Graph (arc) Logic Example:GraphLogic binrel(inst 1, inst 2 ) inst 2 binrel Warren BuffettGeneral Electric Trust Trust(Warren Buffett, General Electric) From Predicate Labels on Arcs to Labelnodes Starting Binary Hyperarcs inst 1 inst 2 binrel Warren BuffettGeneral Electric Trust

48 47 General:GraphLogic Example:GraphLogic unaryrel(inst 1 ) Frugal(Warren Buffett) Arities Smaller than in Binary DLGs: Labelnodes Starting Unary Hyperarcs (cf. Slide on Classes, Concepts, Types) inst 1 unaryrel Warren Buffett Frugal

49 48 Example:Graph Logic Antonym(Frugal, Prodigal) Labelnodes Starting Hyperarcs Enable Second-Order Predicates (e.g. Binary): Hyperarcs with Labelnode Arguments General:Graph (2-adorned rect- Logic 4vex: 2 nd order) binrel2ndord(rel 1, rel 2 ) binrel2ndordrel 1 rel 2 Antonym Prodigal Frugal 2 2

50 49 General:GraphLogic Example:GraphLogic nullaryrel() Sunny() Arities Smaller than in Binary DLGs: Labelnodes for Nullary Hyperarcs (cf. Propositional Logic) nullaryrel Sunny

51 50 Functions Actively Applied to Arguments (Shorthand) General:Graph ( hyperarc with Logic rect4cave-shaped labelnode ) Example:Graph Logic (n=3) fun(inst 1, inst 2,..., inst n-1, inst n ) Profit( | WB, | YY, 2011) fun inst 1 inst 2 inst n inst n Profit WB YY

52 51 Function Apps Value-Returning: Active Call (Long Form) General:Graph ( round2cave-shaped Logic enclosing box ) Profit( | WB, | YY, 2011) inst 1 inst 2 inst n inst n fun Profit WB YY fun(inst 1, inst 2,..., inst n-1, inst n ) Example:Graph Logic (n=3)

53 52 Function Apps Value-Returning: Result for Definition of Next Slide General:Graph Logic Example:Graph Logic (n=3) val US$ 2·10 6 val

54 53 General:Graph (ground) Logic Profit( | WB, | YY, 2011) := US$ 2·10 6 inst 1 inst 2 inst n inst n Function Apps Value-Returning: Logic with Equality Definition (1) : : US$ 2·10 6 val fun Profit WB YY Example:Graph Logic (n=3) fun(inst 1, inst 2,..., inst n-1, inst n ):= val

55 54 General:Graph (inst/var terms) Logic Example:Graph Logic (n=1) term 1 term 2 term n Double Function Apps Value-Returning: Logic with Equality Definition (2) : : val Mult 2 fun X X ( X) Double(X) := Mult(X, 2) ( var i ) fun(term 1, term 2,..., term n ) := val

56 55 Double(3) Double Function Sample Call: Rewriting Trace (1) 3 Graph Logic Call/query of Double instantiates equality definition of previous slide (X=3) Double

57 56 Double Function Sample Call: Rewriting Trace (1) Graph Logic 3 Mult(3, 2) Call/query of Mult assumed to be computed by a built-in definition (3*2) Mult 2

58 57 Double Function Sample Call: Rewriting Trace (1) Graph Logic 6 6 More in slides about Functional-Logic Programming with (oriented) equationsFunctional-Logic Programming

59 58 Function Apps Value-Denoting: Passive Data Construction General:Graph ( rect2cave-shaped Logic (POSL) enclosing box ) Example:Graph Logic (n=3) fun[inst 1, inst 2,..., inst n-1, inst n ] Profit[ | WB, | YY, 2011] inst 1 inst 2 inst n inst n fun Profit WB YY

60 59 Predicates: Unary Relations (Classes, Concepts, Types) inst 1 Example:GraphLogic class(inst 1 ) class Warren Buffett Billionaire Billionaire( Warren Buffett) General:Graph (class appliedLogic to instance node) HasInstance

61 60 Negated Predicates: Unary Relations inst 1 General:Graph (class dash-appliedLogic to instance node) Example:GraphLogic class(inst 1 ) class Joe Smallstock Billionaire( Joe Smallstock) not HasInstance Billionaire

62 61 Graphical Elements: Arrows (2) Arrows for special arcs and hyperarcs –HasInstance: Connects class, as labelnode, with instance (hyperarc of length 1) As in DRLHs and shown earlier, labelnodes can also be used (instead of labels) for hyperarcs of length > 1DRLHs –SubClassOf: Connects subclass, unlabeled, with superclass (arc, i.e. of length 2) –Implies: Hyperarc from premise(s) to conclusion –Object-IDentified slots and shelves: Bulleted arcs and hyperarcs

63 62 Class Hierarchies (Taxonomies): Subclass Relation General:Graph (two nodes) (Description) Logic Example:Graph (Description) Logic class 1 class 2 class 2 Rich Billionaire Rich class 1 SubClassOf Billionaire

64 63 Class Hierarchies (Taxonomies): Negated Subclass Relation General:Graph (two nodes) (Description) Logic Example:Graph (Description) Logic class 1 class 2 class 2 Poor Billionaire Poor class 1 not SubClassOf Billionaire

65 64 General:Graph (solid+linear node, (Description) as for conjunction)Logic Example:Graph (Description) Logic class 1 class 2 class n... Billionaire Benefactor Environmentalist Intensional-Class Constructions (Ontologies): Class Intersection class 2 class 1 class n... BillionaireBenefactor Environmentalist

66 65 General:Graph (complex class (xNS-Description) applied to instance node) Logic Example:Graph (xNS-Description) Logic ( Billionaire Benefactor Environmentalist) (Warren Buffett) Intensional-Class Applications: Class Intersection Warren Buffett (class 1 class 2 class n ) (inst 1 )... inst 1 BillionaireBenefactor Environmentalist class 2 class 1 class n...

67 66 General:Graph (solid+wavy node, (Description) as for disjunction) Logic Example:Graph (Description) Logic... Billionaire Benefactor Environmentalist Intensional-Class Constructions (Ontologies): Class Union class 1 class 2 class n class 2 class 1 class n... BillionaireBenefactor Environmentalist

68 67 General:Graph (complex class (xNS-Description) applied to instance node) Logic Example:Graph (xNS-Description) Logic Intensional-Class Applications: Class Union Warren Buffett inst 1 (class 1 class 2 class n ) (inst 1 )... ( Billionaire Benefactor Environmentalist) (Warren Buffett) BillionaireBenefactor Environmentalist class 2 class 1 class n...

69 68 General:Graph (Description) (dashed+linear node,Logic as for negation contains node to be complemented) Example:Graph (Description) Logic class Billionaire class Arbitrary class Atomic class (shorthand) Intensional Class Constructions (Ontologies): Class Complement Billionaire class Billionaire

70 69 Class Hierarchies (Taxonomy DAGs): Top and Bottom General: Top (special node) (Description) Logic General: Bottom (special node) (Description) Logic (owl:Thing) (owl:Nothing)

71 70 Intensional Class Constructions (Ontologies): Class-Property Restriction Existential (1) General:Graph (shorthand) (Description) Logic Example:Graph (Description) Logic binrel. class binrel Substance. Physical Substance Physical A kind of schema, where Top class is specialized to have (multi-valued) attribute/property, Substance, with at least one value typed by class Physical class

72 71 Intensional Class Constructions (Ontologies): Class-Property Restriction Existential (1*) General:Graph (normal) (Description) Logic Example:Graph (Description) Logic binrel. class Substance. Physical binrel class Substance Physical A kind of schema, where Top class is specialized to have (multi-valued) attribute/property, Substance, with at least one value typed by class Physical

73 72 Instance Assertions (Populated Ontologies): Using Restriction for ABox Existential (1) General:Graph (shorthand) (xNS-Description) Logic Example:Graph (xNS-Description) Logic binrel.class(inst 0 ) binrel Substance.Physical (Socrates) Physical(P1) Substance(Socrates, P1) SocratesP1 Substance inst 0 binrel binrel(inst 0, inst 1 ) class(inst 1 ) inst 1 Substance class Physical

74 73 Instance Assertions (Populated Ontologies): Using Restriction for ABox Existential (1*) General:Graph (normal) (xNS-Description) Logic Example:Graph (xNS-Description) Logic binrel.class(inst 0 ) binrel Substance.Physical (Socrates) Physical(P1) Substance(Socrates, P1) SocratesP1 Substance inst 0 binrel binrel(inst 0, inst 1 ) class(inst 1 ) inst 1 Substance class Physical

75 74 Intensional Class Constructions (Ontologies): Class-Property Restriction Universal (1) General:Graph (shorthand) (Description) Logic Example:Graph (Description) Logic binrel. class binrel Substance. Physical Substance A kind of schema, where Top class is specialized to have (multi-valued) attribute/property, Substance, with each value typed by class Physical class Physical

76 75 Intensional Class Constructions (Ontologies): Class-Property Restriction Universal (1*) General:Graph (normal) (Description) Logic Example:Graph (Description) Logic binrel. class binrel Substance. Physical Substance class Physical A kind of schema, where Top class is specialized to have (multi-valued) attribute/property, Substance, with each value typed by class Physical

77 76 Instance Assertions (Populated Ontologies): Using Restriction for ABox Universal (1) General:Graph (shorthand) (xNS-Description) Logic Example:Graph (xNS-Description) Logic binrel.class(inst 0 ) Substance.Physical (Socrates) Physical(P1) Physical(P2) Substance(Socrates, P1) Substance(Socrates, P2) P1 P2 Substance inst 0 inst 1 inst n binrel... binrel(inst 0, inst 1 ) binrel(inst 0, inst n ) class(inst 1 ) class(inst n )... Substance Socrates binrel class Substance Physical

78 77 Instance Assertions (Populated Ontologies): Using Restriction for ABox Universal (1*) General:Graph (normal) (xNS-Description) Logic Example:Graph (xNS-Description) Logic binrel.class(inst 0 ) Substance.Physical (Socrates) Physical(P1) Physical(P2) Substance(Socrates, P1) Substance(Socrates, P2) P1 Substance inst 0 inst 1 inst n binrel... binrel(inst 0, inst 1 ) binrel(inst 0, inst n ) class(inst 1 ) class(inst n )... Substance Socrates binrel class P2 Substance Physical

79 78 Existential vs. Universal Restriction (Physical/Mental Assumed Disjoint: Can Be Explicated via Bottom Intersection) Example:Graph (xNS-Description) Logic Substance.Physical (Socrates) Physical(P1) Mental(P3) Substance(Socrates, P1) Substance(Socrates, P3) Substance P1 P3 Substance Socrates Example:Graph (xNS-Description) Logic Substance.Physical (Socrates) Physical(P1) Mental(P3) Substance(Socrates, P1) Substance(Socrates, P3) Substance P1 P3 Substance Socrates ConsistentConsistent InconsistentInconsistent Physical Mental Physical Mental

80 79 LuckyParent Example (1) Poor Doctor Child Spouse LuckyParent EquivalentClasses Person LuckyParent Person Spouse.Person Child.( Poor Child.Doctor)

81 80 LuckyParent Example (1*) Child Spouse LuckyParent LuckyParent Person Spouse.Person Child.( Poor Child.Doctor) Spouse : Poor Doctor Person

82 81 LuckyParent Example (1**) Child LuckyParent LuckyParent Person Spouse.Person Child.( Poor Child.Doctor) : Child Spouse Person Poor Doctor

83 82 LuckyParent Example (1**) Child LuckyParent LuckyParent Person Spouse.Person Child.( Poor Child.Doctor) : Child Spouse Doctor Poor Person

84 83 Object-Centered Logic: Grouping Binary Relations Around Instance General:Graph (Object-Centered) (inst 0 -centered) Logic Example:Graph (Object-Centered) (Socrates-centered) Logic Philosopher(Socrates) Substance(Socrates, P1) Teaching(Socrates, T1) P1 T1 Substance inst 0 inst 1 inst n binrel 1 binrel n... binrel 1 (inst 0, inst 1 ) binrel n (inst 0, inst n ) class(inst 0 )... Teaching Philosopher Socrates class

85 84 RDF-Triple (Subject-Centered) Logic: Grouping Properties Around Instance {(Socrates, rdf:type, Philosopher ), (Socrates, Substance, P1), (Socrates, Teaching, T1)} P1 T1 Substance inst 0 inst n property 1 property n... (inst 0, property 1, inst 1 ), (inst 0, property n, inst n )} {(inst 0, rdf:type, class),... Teaching Socrates General:Graph (Subject-Centered) (inst 0 -centered) Logic Example:Graph (Subject-Centered) (Socrates-centered) Logic inst 1 class Philosopher

86 85 Logic of Frames ( Records ): Associating Slots with OID-Distinguished Instance General:Graph (PSOA Frame) (bulleted arcs) Logic Example:Graph (PSOA Frame) Logic Socrates#Philosopher( Substance->P1 ; Teaching->T1) P1 T1 Substance inst 0 inst n slot 1 slot n... slot 1 ->inst 1 ; slot n ->inst n ) inst 0 #class(... Teaching Socrates inst 0 class, slot 1 = inst 1,... slot n = inst n inst 1 class Philosopher

87 86 Logic of Shelves ( Arrays ): Associating Tuple(s) with OID-Distinguished Instance General:Graph (PSOA Shelf) (bulleted hyperarc) Logic Example:Graph (PSOA Shelf) Logic Socrates#Philosopher( c. 469 BC, 399 BC ) 399 BC inst 0 inst 1, …, inst m ) inst 0 #class( Socrates inst 1 inst m... c. 469 BC class Philosopher

88 87 Positional-Slotted-Term Logic: Associating Tuple(s)+Slots with OID-Distinged Instance General:Graph (PSOA Positional- Slotted-Term) Logic Example:Graph (PSOA Positional- Slotted-Term) Logic Socrates#Philosopher( c. 469 BC, 399 BC; Substance->P1 ; Teaching->T1) 399 BC Substance inst 0 inst n slot 1 slot n... inst 1, …, inst m ; slot 1 ->inst 1 ; slot n ->inst n ) inst 0 #class(... Teaching Socrates inst 1 inst m... c. 469 BC P1 T1 inst 1 class Philosopher

89 88 Term and Formula Description: Associating Slots with Complex Node Complex Node (e.g. Roundangle) having Outward Slots Slots visible in outer context. Example: Like for elementary node inst n slot 1 slot n... inst 1... inst n slot 1 slot n... inst 1... Slots visible in inner context. Example: Later slide with closure slot Complex Node (e.g. Roundangle) having Inward Slots

90 89 inst 1,1 rel 1 (inst 1,1, inst 1,2,..., inst 1,n 1 ) inst 1,2 inst 1,n 1 inst 2,1 rel 2 (inst 2,1, inst 2,2,...,inst 2,n 2 ) inst 2,2 inst 2,n 2 rel 1 rel 2 General:Graph (ground, Logic shorthand) Example:Graph Logic Invest Invest(|WB, |GE, US$ 3·10 9 ) US$ 3·10 9 Invest Invest(|JS, |VW, US$ 5·10 3 ) US$ 5·10 3 Rules: Relations Imply Relations (1) WB GE JSVW

91 90 Rules: Relations Imply Relations (1*) inst 1,1 rel 1 (inst 1,1, inst 1,2,..., inst 1,n 1 ) inst 1,2 inst 1,n 1 inst 2,1 rel 2 (inst 2,1, inst 2,2,...,inst 2,n 2 ) inst 2,2 inst 2,n 2 rel 1 rel 2 General:Graph (ground, normal) Logic Example:Graph Logic Invest Invest(|WB, |GE, US$ 3·10 9 ) US$ 3·10 9 Invest Invest(|JS, |VW, US$ 5·10 3 ) US$ 5·10 3 WB GE JS VW

92 91 Rules: Relations Imply Relations (2) ( var i, j ) rel 1 (var 1,1, var 1,2,..., var 1,n 1 ) rel 2 (var 2,1, var 2,2,...,var 2,n 2 ) rel 1 rel 2 General:Graph (non-ground, Logic where Implies arrow creates universal closure) Example:Graph Logic Invest ( X, Y, A, U, V, B) Invest(X, Y, A) Invest Invest(U, V, B) XYA UVB var 1,1 var 1,2 var 1,n 1 var 2,1 var 2,2 var 2,n 2

93 92 Rules: Relations Imply Relations (3) General:Graph (inst/var terms) Logic Example:Graph Logic ( Y, A) Invest(|WB, Y, A) term 1,1 term 1,2 term 1,n 1 term 2,1 rel 2 ( term 2,1, term 2,2,..., t erm 2,n 2 ) term 2,2 term 2,n 2 rel 2 rel 1 ( var i, j ) rel 1 ( term 1,1, term 1,2,..., term 1,n 1 ) Invest US$ 5·10 3 Invest(|JS, Y, US$ 5·10 3 ) WB JS YA Y

94 93 Rules: Conjuncts Imply Relations (1) General:Graph (shorthand) Logic Example:Graph Logic ( Y, A) Invest(|WB, Y, A) Trust(|JS, Y) term 1,1 term 1,2 term 1,n 1 term 2,1 rel 2 ( term 2,1, term 2,2,..., t erm 2,n 2 ) term 2,2 term 2,n 2 rel 2 rel 1 ( var i, j ) rel 1 ( term 1,1, term 1,2,..., term 1,n 1 ) Invest US$ 5·10 3 Invest(|JS, Y, US$ 5·10 3 ) term 3,1 term 3,2 term 3,n 3 rel 3 rel 3 ( term 3,1, term 3,2,..., t erm 3,n 3 ) Trust WB JS YA Y Y

95 94 Rules: Conjuncts Imply Relations (1*) General:Graph (prenormal) Logic Example:Graph Logic ( Y, A) Invest(|WB, Y, A) Trust(|JS, Y) term 1,1 term 1,2 term 1,n 1 term 2,1 rel 2 ( term 2,1, term 2,2,..., t erm 2,n 2 ) term 2,2 term 2,n 2 rel 2 rel 1 ( var i, j ) rel 1 ( term 1,1, term 1,2,..., term 1,n 1 ) Invest US$ 5·10 3 Invest(|JS, Y, US$ 5·10 3 ) term 3,1 term 3,2 term 3,n 3 rel 3 rel 3 ( term 3,1, term 3,2,..., t erm 3,n 3 ) Trust WB JS YA Y Y

96 95 Rules: Conjuncts Imply Relations (1**) General:Graph (normal) Logic Example:Graph Logic ( Y, A) (Invest(|WB, Y, A) Trust(|JS, Y) term 1,1 term 1,2 term 1,n 1 term 2,1 rel 2 ( term 2,1, term 2,2,..., t erm 2,n 2 ) term 2,2 term 2,n 2 rel 2 rel 1 ( var i, j ) ( rel 1 ( term 1,1, term 1,2,..., term 1,n 1 ) Invest(|JS, Y, US$ 5·10 3 )) term 3,1 term 3,2 term 3,n 3 rel 3 rel 3 ( term 3,1, term 3,2,..., t erm 3,n 3 )) closure universal closure universal Invest US$ 5·10 3 Trust WB JS YA Y Y

97 96 Invest WB Y A Trust JS Y Invest JS Y US$ 5·10 3 Example: RuleML/XMLLogic Rules: Conjuncts Imply Relations (2) ( Y, A) (Invest(|WB, Y, A) Trust(|JS, Y) Invest(|JS, Y, US$ 5·10 3 )) Proposing an attribute unique with value "no" for NNS, and "yes" for UNS as the default

98 97 Implication-Defined Predicate Odd: RuleML/XML Serialization Greater X 2 Prime X Odd X Datalog RuleML/XMLDatalog RuleML/XML Logic ( X) Greater(X, 2) Prime(X) Odd (X) Odd Greater 2 Graph (prenormal) Graph arrow normalizes to RuleML-like closure="universal" Prime X X X

99 98 Relations Equivalent to Relations General:Graph (inst/var terms) Logic Example:Graph Logic ( X, Y, A) Transfer(X, Y, A) term 1,1 term 1,2 term 1,n 1 term 2,1 rel 2 ( term 2,1, term 2,2,..., t erm 2,n 2 ) term 2,2 term 2,n 2 rel 2 rel 1 ( var i, j ) rel 1 ( term 1,1, term 1,2,..., term 1,n 1 ) Transfer Receive Receive(Y, X, A) X XY Y A A

100 99 Equivalence-Defined Predicate Even: RuleML/XML Serialization Even X Divisible X 2 FOL RuleML/XMLFOL RuleML/XML Logic ( X) Even(X) : Divisible(X, 2) Even : Divisible 2 Graph (prenormal) Graph : arrow normalizes to RuleML-like closure="universal" X X

101 100 Equality-Defined Function Double: RuleML/XML Serialization Double X Mult X 2 Functional RuleML/XMLFunctional RuleML/XML Logic ( X) Double(X) := Mult(X, 2) : Graph := arrow normalizes to RuleML-like closure="universal" Double Mult 2 X X Graph (prenormal)

102 Positional-Slotted-Term Logic: Rule-defined Anonymous Family Frame (Visualized from IJCAI-2011 Presentation)IJCAI-2011 Presentation Example:Graph (PSOA Positional- Slotted-Term) Logic Group ( Forall ?Hu ?Wi ?Ch ( ?1#family(husb->?Hu wife->?Wi child->?Ch) :- And(married(?Hu ?Wi) Or(kid(?Hu ?Ch) kid(?Wi ?Ch))) ) married(Joe Sue) kid(Sue Pete) ) husb wife family ?1?1 ?Hu ?Wi child ?Ch married ?Hu ?Wi ?Hu?Ch ?Wi Joe Sue kid SuePete married 101 kid

103 Positional-Slotted-Term Logic: Ground Facts, incl. Deduced Frame, Model Family Semantics Example:Graph (PSOA Positional- Slotted-Term) Logic Group ( o#family(husb->Joe wife->Sue child->Pete) married(Joe Sue) kid(Sue Pete) ) husb family child Joe Sue kid SuePete married Joe Sue Pete wife o Previous slides existential variable ?1 in rule head becomes new OID constant o in frame fact, deduced from relational facts For reference implementation of PSOA querying see PSOATransRunPSOATransRun 102

104 Positional-Slotted-Term Logic: Conversely, Given Facts, Rule Can Be Inductively Learned Example:Graph husb family child M1 W1 kid M1C1 married M1 W1 C1 wife o1 kid W1C1 husb family child Mn Wn Cz wife on... Mn Wn married... kid MnCx kid WnCy... Abstracting OID constants o1,..., on to regain existential variable ?1 of previous rule, now induced from matching relational and frame facts 103

105 104 Modally Embedded Propositions General:Graph(Modal) Logic ( complex snip2vex node, used to encapsulate what another agent believes, wants, etc. ) Example:Graph(Modal) Logic believe agent ( graph ) agent believe graph believe |GE (Invest(|WB, |GE, US$ 4·10 9 )) Invest US$ 4·10 9 GE believe WB Can be serialized in the evolving Modal RuleML/XML

106 105 George A desire Grailog: Beliefs and Desires as Propositional Attitudes (1) Propositional attitude: a mental state relating a person to a proposition (which can involve other persons) If George desires action A and believes (the proposition) that originator O will cause A, then George supports O. O believecause support

107 106 Example: If John desires the negation of (state of affairs) X, then he does not desire X. John Beliefs and Desires as Propositional Attitudes (2) Grailog: While variables A and O of the earlier example are bound to an action and originator individual, variable X here is bound to an entire proposition or an arbitrarily complex set of propositions desire X

108 107 Graphical Elements: Line Styles Solid lines (boxes & links): Positive Dashed lines (boxes & links): Negative Wavy lines (boxes): Disjunctive Light lines (unlabeled arrows): HasInstance Light lines ( unlabeled undirected links ): SameIndividual Heavy lines (unlabeled arrows): SubClassOf Heavy lines ( unlabeled undirected links ): EquivalentClasses Double lines (unlabeled arrows): Implies Double lines ( unlabeled double-headed arrows ): Equivalence Double lines (unlabeled undirected links): Equality Colon tails (unlabeled links): TailDefinedByHead

109 108 Orthogonal Graphical Features Axes of Grailog Systematics Box axes: Corners: pointed vs. snipped vs. rounded To quote/copy vs. reify/instantiate vs. evaluate contents (cf. Lisp, Prolog, Relfun, Hilog, RIF, CL, and IKL) LispPrologRelfunHilogRIFCLIKL Shapes (rectangle-derived): composed from sides that are straight vs. concave vs. convex For neutral vs. function vs. relation contents Contents: elementary vs. complex nodes Arrow axes: Shafts: single vs. double Heads: triangular vs. diamond Tails: plain vs. bulleted vs. colonized Box & Arrow (line-style) axes: solid vs. dashed, linear vs. (box only) wavy

110 109 Graphical Elements: Box Systematics Axes of Corners and Shapes Per … Copy … Instantiation … Value Rect- Snip- Round- Neutral -angle Individual (Function Application) -2cave Function -4cave Proposition (Relation Application) -2vex Relation (incl. Class) -4vex Shape: Corner:

111 110 Graphical Elements: Boxes Function/Relation-Neutral Shape of Angles Varied w.r.t. Corner Dimension –Rectangle: Neutral per copy nodes quote their contents –Snipangle (octagon): Neutral per instantiation nodes dereference contained variables to values from context –Roundangle (rounded angles): Neutral per value nodes evaluate their contents through instantiation of variables and activation of function/relation applications 2 X X=3 : 2 X X 6 Assuming Mult built-in function Mult 2 X

112 111 Graphical Elements: Boxes Concave –Rect2cave (rectangle with 2 concave - top/bottom - sides): Elementary nodes for individuals (instances). Complex nodes for quoted instance-denoting terms (constructor-function applications) –Snip2cave (snipped): Elementary nodes for variables. Complex nodes for instantiated (reified) function applications –Round2cave (rounded): Complex nodes for evaluated built-in or equation-defined function applications –Rect4cave (4 concave sides): Elementary nodes for fcts. Complex nodes for quoted functional ( function-denoting ) terms –Snip4cave: Complex nodes for instantiated functl terms –Round4cave: Complex nodes for evaluated functional applications (active, function-returning applications)

113 112 Graphical Elements: Boxes Convex –Rect2vex (rectangle with 2 convex - top/bottom - sides): Elementary nodes for truth constants ( true, false, unknown ). Complex nodes for quoted truth-denoting propositions (embedded relation applications) –Snip2vex: Complex nodes for instantiated (reified) relation applications –Round2vex: Complex nodes for evaluated relation applications (e.g. as atomic formulas) and for connective uses –Rect4vex: Elementary nodes for relations, e.g. unary ones (classes). Complex nodes for quoted relational ( relation-denoting ) terms –Snip4vex: Complex nodes for instantiated relatl terms –Round4vex (oval): Complex nodes for evaluated relatl applications (active, relation-returning applications)

114 113 Conclusions (1) Grailog 1.0 incorporates feedback on earlier versions Graphical elements for novel box & arrow systematics using orthogonal graphical features –Leaving color (except for IRIs) for other purposes, e.g. highlighting subgraphs (for retrieval and inference) Introducing Unique vs. Non-unique Name Specification Focus on mapping to a family of logics as in RuleML Use cases from cognition to technology to business –E.g. Logical Foundations of Cognitive Science: Processing of earlier Grailog-like DRLHs studied in Lisp, FIT, and Relfun For Grailog, aligned with Web-rule standard RuleML: (test suite) suite

115 114 Conclusions (2) Symbolic-to-visual translators started as Semantic Web Techniques Fall 2012 Projects: Semantic Web TechniquesFall 2012 Projects –Team 1 A Grailog Visualizer for Datalog RuleML via XSLT 2.0 Translation to SVG by Sven Schmidt and Martin Koch: An Int'l Rule Challenge 2013 paper & demo introduced Grailog KS VizTeam 1A Grailog Visualizer for Datalog RuleML via XSLT 2.0 Translation to SVGInt'l Rule Challenge 2013Grailog KS Viz –Team 8 Visualizing SWRLs Unary/Binary Datalog RuleML in Grailog by Bo Yan, Junyan Zhang, and Ismail Akbari: A Canadian Semantic Web Symposium 2013 paper gave an overviewTeam 8Visualizing SWRLs Unary/Binary Datalog RuleML in GrailogCanadian Semantic Web Symposium 2013overview Grailog invites feature choice or combination –E.g. n-ary hyperarcs or n-slot frames or both Grailog Initiative on open standardization calls for further feedback for future 1.x versions

116 115 Future Work (1) Refine/extend Grailog, e.g. along with API4KB effortAPI4KB –Compare with other graph formalisms, e.g. Conceptual Graphs (http://conceptualstructures.org) and CoGui toolhttp://conceptualstructures.orgCoGui –Define mappings to/fro UML structure diagrams + OCL, adopting UML behavior diagrams (http://www.uml.org)http://www.uml.org Implement further tools, e.g. as use case for (Functional) RuleML (http://ruleml.org/fun) engineshttp://ruleml.org/fun –More mappings between graphs, logic, and RuleML/XML: Grailog generators:Further symbolic-to-visual mappings Grailog parsers:Initial visual-to-symbolic mappings –Graph indexing & querying (cf. ) –Graph transformations (normal form, typing homomorphism, merge,...)typing homomorphism –Advanced graph-theoretical operations (e.g., path tracing) –Exploit Grailog parallelism in implementation

117 116 Future Work (2) Develop a Grailog structure editor, e.g. supporting: –Auto-specialize of neutral application boxes (angles) to function apps (2caves) or relation apps (2vexes), depending on contents –Auto-specialize of neutral operator boxes (angles) to functions (4caves) or relations (4vexes), depending on context Benefit from, and contribute to, Protégé visualization plug-ins such as Jambalaya/OntoGraf and OWLViz for OWL ontologies and Axiomé for SWRL rulesJambalayaOntoGrafOWLVizAxiomé Proceed from the 2-dimensional (planar) Grailog to a 3-dimensional (spatial) one –Utilize advantages of crossing-free layout, spatial shortcuts, and analogical representation of 3D worlds –Mitigate disadvantages of occlusion and of harder spatial orientation and navigation Consider the 4 th (temporal) dimension of animations to visualize logical inferences, graph processing, etc.


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