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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
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
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
3



    Three RuleML Syntaxes (1)
                   Syntax

   Visualization             Symbolic

              Presentation
                              Serialization


RuleML/Grailog RuleML/POSL RuleML/XML
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
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
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?
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
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
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
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)
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
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
13




Graphical Elements: Hatching Patterns

 • No hatching (boxes): Constant
 • Hatching (atomic boxes): Variable
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
15



  Unknowns: Individual Variables
General:   Graph (hatched node) Logic (POSL uses “?” prefix)
           variable                  variable

Examples: Graph                      Logic

           X                         X

           Y                         Y

           A                         A
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
                                                  )
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)
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
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
                                                  )
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
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)
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)
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)
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))
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))
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
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
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
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
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
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
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
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
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)
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
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)
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))
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)
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)
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)
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)
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
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
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
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

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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