Harold Boley: RuleML/Grailog: The Rule Metalogic Visualized with Generalized Graphs
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RuleML/Grailog provides a graph-based visualization syntax for logic called Grailog. It uses generalized directed graphs to represent logical concepts like predicates, variables, constants, and formulas. The graphs extend basic directed labeled graphs to allow for hyperedges, recursive nesting, and labelnodes. Various graphical elements like shapes, lines, and hatching are used to represent logical elements and their properties in a visual format that is cognitively easier for humans to understand and work with compared to symbolic logical representations.
Harold Boley: RuleML/Grailog: The Rule Metalogic Visualized with Generalized Graphs
- 1. RuleML/Grailog: The Rule Metalogic Visualized with Generalized Graphs Harold Boley NRC-IIT Fredericton Faculty of Computer Science University of New Brunswick Canada PhiloWeb 2011 Thessaloniki, Greece, 5 October 2011
- 2. 1 Graph Visualization of Logic • Enrich logical knowledge specifications with a convenient 2-dimensional syntax for logic-as-graph visualization – Supports human in the loop in knowledge elicitation, validation, and processing • Complementary to the use of graphs for the efficient implementation of such specifications
- 3. 2 Rule MetaLogic Provides Family of Language Standards for Web Knowledge Interchange • Developed on the Web: http://ruleml.org/metalogic • Principal (family-uniform) and variant semantics • Family-uniform syntaxes for humans and machines
- 4. 3 Three RuleML Syntaxes (1) Syntax Visualization Symbolic Presentation Serialization RuleML/Grailog RuleML/POSL RuleML/XML
- 5. 4 Three RuleML Syntaxes (2) Serialization = RuleML/XML: Specified in XML Schema and recently in Relax NG: http://ruleml.org Presentation = RuleML/POSL: Integrates Prolog and F-logic, and translates to RuleML/XML: http://ojs.academypublisher.com/index.php/jetwi/article/view/0204343353 Visualization = RuleML/Grailog: Based on Directed Recursive Labelnode Hypergraphs (DRLHs): http://www.dfki.uni-kl.de/~boley/drlhops.abs.html
- 6. 5 Grailog Graph inscribed logic invokes imagery for logic Proposed cognitively adequate graph standard for visualized knowledge: Easy to learn and draw, read and remember, e.g. for eScience, eLearning, and eBusiness Permits logic-cognitive nets on the Semantic Web
- 7. 6 Generalized Graphs for the Representation and Mapping of Logic Languages • We have used generalized graphs for representing various logic languages, where basically: – Graph nodes (vertices) represent individuals, classes, etc. – Graph arcs (edges) represent relations • Next slides: What are the principles of this representation and what graph generalizations are required? • Later slides: How are these graphs (invertibly) mapped to logic?
- 8. 7 Grailog Principles • Graphs should make it easier for humans to read and write logic constructs by exploiting the 2-dimensional representation • Graphs should be natural extensions of Directed Labeled Graphs (DLGs), used to represent simple semantic nets, i.e. of atomic ground formulas in function-free binary predicate logic (cf. binary Datalog ground facts and RDF triples) • Graphs should allow stepwise refinements for all logic constructs, e.g. description logic TBoxes
- 9. 8 Searle’s Chinese Room Argument Classes with relations subsumes understand hasInstance Language understand negation English Chinese lang lang lang lang lang haveLanguage to with for rules texts questions replies apply use with for Searle Wang Searle-replyi Wang-replyi distinguishable Instances with relations
- 10. 9 Grailog Generalizations • Directed hypergraphs: For n-ary relations, directed (binary) arcs should be generalized to directed (n-ary) hyperarcs, e.g. representing relational tuples • Recursive (hierarchical) graphs: For nested terms and formulas, modal logics, and modularization, ‘flat’ graphs should be generalized to allow other graphs as complex nodes to any level of ‘depth’ • Labelnode graphs: For allowing hybrid logics describing both instances and predicates, arc labels should also become usable as nodes
- 11. 10 Graphical Elements: Basic Shapes (1) • Boxes for atomic and complex nodes – Oval: Classes, as labelnodes, for unary relations – Rectangle: Atomic for instances. Complex for (‘passive’) instance-denoting constructor-function applications – Roundangle (Rounded rectangle): (‘active’) function, predicate, and connective applications – Octagon: Embedded propositions and modules • Labeled arrows (directed links) for arcs and hyperarcs (where hyperarcs ‘cut through’ nodes intermediate between first and last)
- 12. 11 Graphical Elements: Basic Shapes (2) • Arrows for special arcs and hyperarcs – subsumes: Connects superclass, unlabeled, with subclass (arc, i.e. of length 2) – hasInstance: Connects class, as labelnode, with instance (hyperarc of length 1) • As in DRLHs, labelnodes can also be used (instead of labels) for hyperarcs of length > 1 – Implies: Hyperarc from premise(s) to conclusion
- 13. 12 Graphical Elements: Line Styles • Solid lines (boxes & links): Positive • Dashed lines (boxes & links): Negative • Dotted lines (boxes): Disjunctive • Heavy single lines (unlabeled arrows): subsumes • Light single lines (unlabeled arrows): hasInstance • Heavy double lines (unlabeled arrows): Implies • Light double lines (unlabeled undirected links): Equal
- 14. 13 Graphical Elements: Hatching Patterns • No hatching (boxes): Constant • Hatching (atomic boxes): Variable
- 15. 14 Instances: Individual Constants mapping General: Graph (node) Logic (and POSL) instance instance Examples: Graph Logic Warren Buffett Warren Buffett General Electric General Electric US$ 3 000 000 000 US$ 3 000 000 000
- 16. 15 Unknowns: Individual Variables General: Graph (hatched node) Logic (POSL uses “?” prefix) variable variable Examples: Graph Logic X X Y Y A A
- 17. 16 Predicates: Binary Relations (1) General: Graph (labeled arc) Logic binrel inst1 inst2 binrel(inst1, inst2) Example: Graph Logic Trust Warren Buffett General Electric Trust(Warren Buffett, General Electric )
- 18. 17 Predicates: Binary Relations (2) General: Graph (labeled arc) Logic binrel var1 var2 binrel(var1, var2) Example: Graph Logic Trust X Y Trust(X,Y)
- 19. 18 Equality Predicate: Distinguished General: Graph (unlabeled Logic (with equality) undirected double arc) inst1 inst2 inst1 = inst2 Example: Graph Logic (with equality) GE General Electric GE = General Electric
- 20. 19 Negated Predicates: Binary Relations General: Graph (dashed arc) Logic binrel inst1 inst2 ¬ binrel(inst1, inst2) Example: Graph Logic Trust Joe Smallstock General Electric ¬ Trust( Joe Smallstock, General Electric )
- 21. 20 Inequality Predicate: Distinguished General: Graph (dashed Logic (with equality) unlabeled undirected double arc) inst1 inst2 inst1 ≠ inst2 Example: Graph Logic (with equality) Joe Smallstock Warren Buffett Joe Smallstock ≠ Warren Buffett
- 22. 21 Predicates: n-ary Relations (n>1) General: Graph (hyperarc) Logic rel inst1 inst2 instn-1 instn rel(inst1, inst2, ..., instn-1, instn) Example: Graph Logic (n=3) Invest WB GE US$ 3 ·109 Invest(WB, GE, US$ 3·109)
- 23. 22 Negated Predicates: n-ary Relations General: Graph (dashed: not) Logic rel inst1 inst2 instn-1 instn ¬ rel(inst1, inst2, ..., instn-1, instn) Example: Graph Logic (n=3) Invest WB GE US$ 4 ·109 ¬ Invest(WB, GE, US$ 4·109)
- 24. 23 Implicit Conjunction of Formula Graphs: Co-Occurrence on Top-Level General: Graph (m hyperarcs) Logic rel1(inst1,1, inst1,2, inst1,1 rel1 inst1,2 inst1,n1 ..., inst1,n1) ∧ ... ... ∧ instm,1 relm instm,2 instm,nm relm(instm,1, instm,2, ...,instm,nm) Example: Graph (2 hyperarcs) Logic Invest(WB, GE, WB Invest GE US$ 3 ·109 US$ 3·109) ∧ Invest(JS, VW, JS Invest VW US$ 2 ·104 US$ 2·104)
- 25. 24 Explicit Conjunction of Formula Graphs: Co-Occurrence in Complex Node General: Graph (m hyperarcs) Logic (rel1(inst1,1, inst1,2, inst1,1 rel1 inst1,2 inst1,n1 ..., inst1,n1) ∧ ... ... ∧ instm,1 relm instm,2 instm,nm relm(instm,1, instm,2, ...,instm,nm)) Example: Graph (2 hyperarcs) Logic (Invest(WB, GE, WB Invest GE US$ 3 ·109 US$ 3·109) ∧ JS VW US$ 2 ·104 Invest(JS, VW, Invest US$ 2·104))
- 26. 25 Disjunction of Formula Graphs: Co-Occurrence in Disjunctive Node General: Dotted Graph Logic (rel1(inst1,1, inst1,2, inst1,1 rel1 inst1,2 inst1,n1 ..., inst1,n1) ∨ ... ... ∨ instm,1 relm instm,2 instm,nm relm(instm,1, instm,2, ...,instm,nm) ) Example: Dotted Graph Logic (Invest(WB, GE, WB Invest GE US$ 3 ·109 US$ 3·109) ∨ JS VW US$ 2 ·104 Invest(JS, VW, Invest US$ 2·104))
- 27. 26 Predicates: Unary Relations (Classes, Concepts, Types) General: Graph (class applied Logic to instance node) class hasInstance class(inst1) inst1 Example: Graph Logic Billionaire Billionaire( Warren Buffett) Warren Buffett
- 28. 27 Negated Predicates: Unary Relations General: Graph (class dash-applied Logic to instance node) class not hasInstance ¬ class(inst1) inst1 Example: Graph Logic Billionaire ¬ Billionaire( Joe Smallstock) Joe Smallstock
- 29. 28 Class Hierarchies (Taxonomies): Subclass Relation General: Graph (two nodes) (Description) class2 Logic subsumes class1 class2 class1 Example: Graph (Description) Rich Logic Billionaire Rich Billionaire
- 30. 29 Class Hierarchies (Taxonomies): Negated Subclass Relation General: Graph (two nodes) (Description) class2 Logic not subsumes class1 class2 class1 Example: Graph (Description) Poor Logic Billionaire Poor Billionaire
- 31. 30 Class Hierarchies (Taxonomy Trees): Class Union General: Graph (blank node over n) (Description) Logic subsumes ... class1 class2 ... class1 class2 ... classn classn Example: Graph (blank node over 3) (Description) Logic Billionaire Benefactor Billionaire Benefactor Environmentalist Environmentalist
- 32. 31 Class Hierarchies (Taxonomy DAGs): Class Intersection General: Graph (blank node under n) (Description) Logic class1 class2 ... classn ... class1 class2 subsumes ... classn Example: Graph (blank node under 3) (Description) Logic Billionaire Benefactor Environmentalist Billionaire Benefactor Environmentalist
- 33. 32 Class Hierarchies (Taxonomies): Class Complement General: Graph (Description) (dashed node Logic contains node to be complemented) Arbitrary class Atomic class (abbreviation) class class ¬ class Example: Graph (Description) Logic Billionaire Billionaire ¬ Billionaire
- 34. 33 Intensional Class Constructions (Ontologies): Class-Property-Restricting TBox (Existential) General: Graph (Description) Logic ∃binrel class ∃binrel . class Example: Graph (Description) Logic ∃ Substance Physical ∃Substance . Physical
- 35. 34 Instance Assertions (Populated Ontologies): ABox for Restriction TBox (Existential) General: Graph (Description) Logic ∃binrel class ∃binrel.class(inst1) class(inst2) binrel inst1 inst2 binrel(inst1, inst2) Example: Graph (Description) Logic ∃ Substance Physical ∃Substance.Physical (Socrates) Substance Physical(P1) Socrates P1 Substance(Socrates,P1)
- 36. 35 Intensional Class Constructions (Ontologies): Class-Property-Restricting TBox (Universal) General: Graph (Description) Logic ∀binrel class ∀binrel . class Example: Graph (Description) Logic ∀ Substance Physical ∀Substance . Physical
- 37. 36 Instance Assertions (Populated Ontologies): ABox for Restriction TBox (Universal) General: Graph (Description) ∀binrel Logic class ∀binrel.class(inst1) class(inst2) ... binrel ... class(instn) inst2 inst1 ... binrel(inst1, inst2) instn ... binrel binrel(inst1, instn) Example: Graph (Description) Logic ∀ Substance Physical ∀Substance.Physical (Socrates) Substance P1 Physical(P1) Socrates Physical(P2) Substance P2 Substance(Socrates, P1) Substance(Socrates, P2)
- 38. 37 Modally Embedded Propositions General: Graph (Modal) Logic (complex octagon node used to ‘quarantine’ what another agent believes, wants, etc.) believe agent graph believeagent(graph) Example: Graph (Modal) Logic believe Invest GE WB GE US$ 4 ·109 believeGE(Invest(WB, GE, US$ 4 ·109))
- 39. 38 Rules: Relations Imply Relations (1) General: Graph (ground) Logic rel1 rel1(inst1,1, inst1,2, inst1,1 inst1,2 inst1,n1 ..., inst1,n1) ⇒ inst2,1 rel2 inst2,2 inst2,n2 rel2(inst2,1, inst2,2, ...,inst2,n2) Example: Graph Logic Invest Invest(WB, GE, WB GE US$ 3 ·109 US$ 3·109) ⇒ JS GE US$ 5 ·103 Invest(JS, GE, Invest US$ 5·103)
- 40. 39 Rules: Relations Imply Relations (2) General: Graph (non-ground) Logic rel1 (∀vari,j) var1,1 var1,2 var1,n1 rel1(var1,1, var1,2, ..., var1,n1) ⇒ var2,1 rel2 var2,2 var2,n2 rel2(var2,1, var2,2, ...,var2,n2) Example: Graph Logic Invest (∀ X, Y, A, U, V, B) X Y A Invest(X, Y, A) ⇒ U Invest V B Invest(U, V, B)
- 41. 40 Rules: Relations Imply Relations (3) General: Graph (inst/var terms) Logic rel1 (∀vari,j) term1,1 term1,2 term1,n1 rel1(term1,1, term1,2, ..., term1,n1) ⇒ term2,1 rel2 term2,2 term2,n2 rel2(term2,1, term2,2, ..., term2,n2) Example: Graph Logic Invest (∀ Y, A) WB Y A Invest(WB,Y,A) ⇒ JS Y US$ 5 ·103 Invest(JS, Y, Invest US$ 5·103)
- 42. 41 Rules: Conjunctions Imply Relations General: Graph (inst/var terms) Logic rel1 (∀vari,j) term1,1 term1,2 term1,n1 rel1(term1,1, term1,2, rel2 ..., term1,n1) ∧ term2,1 rel term2,2 term2,n2 rel2(term2,1, term2,2, 3 ..., term2,n2) ⇒ term3,1 term3,2 term3,n3 rel3(term3,1, term3,2, ..., term3,n3) Example: Graph Logic Invest (∀ Y, A) WB Trust Y A Invest(WB,Y,A) ∧ JS Y Trust(JS, Y) ⇒ Invest Invest(JS, Y, JS Y US$ 5 ·103 US$ 5·103)
- 43. 42 Beliefs and Desires as Propositional Attitudes (1) Propositional attitude: a mental state relating a person to a proposition “If George desires action A and believes (the proposition) that originator O will cause A, then George desires O.” desire Grailog: George believe A cause O desire
- 44. 43 Beliefs and Desires as Propositional Attitudes (2) Example: “If John fears (state of affairs) X, then John wants that not X.” fear Grailog: John X want 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
- 45. 44 Conclusions (1) • Refining/extending Grailog for the Rule Metalogic • Comparing it with other graph formalisms – Conceptual Graphs: http://conceptualstructures.org – Unified Modeling Language: http://www.uml.org • Use cases from philosophy to technology to business – E.g. “Logical Foundations of Cognitive Science”: http://www.ict.tuwien.ac.at/lva/Boley_LFCS/index.html • Implementing tools – Mapping between graphs, logic (as shown) & RuleML/XML – Graph indexing & querying (cf. http://www.hypergraphdb.org) – Graph-to-graph transformations (normal forms, merges, ...) – Advanced graph-theoretical operations (e.g., path tracing) • Submitting to standards body
- 46. 45 Conclusions (2) • Proceeding from the 2-dimensional (planar) Grailog to a 3-dimensional (spatial) one – Exploiting advantages of crossing-free layout, spatial shortcuts, and analogical representation of 3D worlds – Mitigating disadvantages of occlusion and of harder spatial orientation and navigation • Considering the 4th (temporal) dimension of animations to visualize logical inferences, graph processing, etc. • See also: http://ruleml.org/#Grailog