An Evidential Logic for Multi-Relational Networks

                    Marko A. Rodriguez
             T-5, Center for Non...
1

                                      Background

• Collective Decision Making Systems
    Decision markets, voting sys...
2

          Knowledge Representation and Reasoning

• Knowledge representation: a model of a domain of discourse – struct...
3

                                             Outline

• Structure
    Network Representations
    Resource Description ...
4

                                             Outline

• Structure
    Network Representations
    Resource Description ...
5

   Undirected Single-Relational Network

                                                  Human-D
       Human-B




 ...
6

       Directed Single-Relational Network

                                                     Article-D
            A...
7

      Directed Multi-Relational Network

                                                 Publisher-A
       Article-A
...
8

            The Resource Description Framework

• The Resource Description Framework (RDF) is the standard for
  repres...
9

                                               foaf:Organization                "University of New
      "Los Alamos Na...
10

                                             Outline

• Structure
    Network Representations
    Resource Description...
11

                  Description Logics - Introduction

• The purpose of description logics is to infer subsumption relat...
12

                     Description Logics - Reasoner

• Inference rules: a collection of pattern descriptions are used t...
13

                    Description Logics - Example

• Terminological Box (T-Box): a collection of descriptions. Also kno...
14

                       Description Logics - Example
          inferred                              Mammal

          ...
15

                  Description Logics - Drawbacks

• With “nested” descriptions and complex quantifiers, you can run int...
16

                    Evidential Logics - Introduction

Evidential logics are multi-valued logics founded on AIKIR (Assu...
17

                           Evidential Logics - The Process
Evidential reasoning is done using various syllogisms:1

• ...
18

                               Evidential Logics - Example

Assume that the past experience of the evidential system h...
19

                   Evidential Logics - Example

experienced                              Mammal

                     ...
20

                   Evidential Logics - Example

inferred                                Mammal

                      ...
21

                   Evidential Logics - Example

inferred                                    Mammal

                  ...
22

                         Evidential Logics - Example

                          <1,0>                     Mammal
     ...
23

                                               Conclusion

The associated article demonstrates provides a framework fo...
24




References

[1] Franz Baader, Diego Calvanese, Deborah L. Mcguinness, Daniele Nardi, and Peter F.
    Patel-Schneid...
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An Evidential Logic for Multi-Relational Networks

  1. 1. An Evidential Logic for Multi-Relational Networks Marko A. Rodriguez T-5, Center for Nonlinear Studies Los Alamos National Laboratory http://markorodriguez.com Joe Geldart Computer Science Department University of Durham http://www.dur.ac.uk/j.r.c.geldart March 23, 2009
  2. 2. 1 Background • Collective Decision Making Systems Decision markets, voting systems, recommender systems http://cdms.lanl.gov • Multi-Relational Graph Analysis Novel/practical reasoning mechanisms Graph metrics on multi-relational/semantic networks Designing programming languages that exploit such structures AAAI Symposium on Technosocial Predictive Analytics – Stanford University, California – March 24, 2009
  3. 3. 2 Knowledge Representation and Reasoning • Knowledge representation: a model of a domain of discourse – structure. • Reasoning: an algorithm by which implicit knowledge is made explicit – process. f (x) Reasoner read/write Knowledge Representation AAAI Symposium on Technosocial Predictive Analytics – Stanford University, California – March 24, 2009
  4. 4. 3 Outline • Structure Network Representations Resource Description Framework • Process Description Logics Evidential Logics AAAI Symposium on Technosocial Predictive Analytics – Stanford University, California – March 24, 2009
  5. 5. 4 Outline • Structure Network Representations Resource Description Framework • Process Description Logics Evidential Logics AAAI Symposium on Technosocial Predictive Analytics – Stanford University, California – March 24, 2009
  6. 6. 5 Undirected Single-Relational Network Human-D Human-B Human-F Human-C Human-A Human-E AAAI Symposium on Technosocial Predictive Analytics – Stanford University, California – March 24, 2009
  7. 7. 6 Directed Single-Relational Network Article-D Article-B Article-F Article-C Article-A Article-E AAAI Symposium on Technosocial Predictive Analytics – Stanford University, California – March 24, 2009
  8. 8. 7 Directed Multi-Relational Network Publisher-A Article-A publishedBy authored editorOf Human-B Journal-A containedIn authored Human-A authored Article-B AAAI Symposium on Technosocial Predictive Analytics – Stanford University, California – March 24, 2009
  9. 9. 8 The Resource Description Framework • The Resource Description Framework (RDF) is the standard for representing the relationship between URIs and literals (e.g. float, string, date time, etc.). • Relationships are directed, labeled links between URIs. A subject URI points to an object URI or literal by means of a predicate URI. subject predicate object lanl:marko foaf:knows lanl:jhw foaf:name foaf:name "Marko A. Rodriguez"^^xsd:string "Jennifer H. Watkins"^^xsd:string AAAI Symposium on Technosocial Predictive Analytics – Stanford University, California – March 24, 2009
  10. 10. 9 foaf:Organization "University of New "Los Alamos National Laboratory"^^xsd:string Mexico"^^xsd:string rdf:type rdf:type foaf:name foaf:name foaf:Document lanl:lanl unm:unm rdf:type foaf:member foaf:member urn:doi:10.1016/j.joi.2008.04.002 foaf:member foaf:Person foaf:publications rdf:type rdf:type lanl:marko foaf:knows lanl:jhw foaf:name foaf:name "Marko A. Rodriguez"^^xsd:string "Jennifer H. Watkins"^^xsd:string AAAI Symposium on Technosocial Predictive Analytics – Stanford University, California – March 24, 2009
  11. 11. 10 Outline • Structure Network Representations Resource Description Framework • Process Description Logics Evidential Logics AAAI Symposium on Technosocial Predictive Analytics – Stanford University, California – March 24, 2009
  12. 12. 11 Description Logics - Introduction • The purpose of description logics is to infer subsumption relationships in a knowledge structure. • Given a set of individuals (i.e. real-world instances), determine which concept descriptions subsume the individuals. For example, is marko a type of Mammal? F. Baader, D. Calvanese, D. L. McGuinness, D. Nardi, P. F. Patel-Schneider: The Description Logic Handbook: Theory, Implementation, Applications. Cambridge University Press, Cambridge, UK, 2003.[1] AAAI Symposium on Technosocial Predictive Analytics – Stanford University, California – March 24, 2009
  13. 13. 12 Description Logics - Reasoner • Inference rules: a collection of pattern descriptions are used to assert new statements: (?x, subClassOf, ?y) ∧ (?y, subClassOf, ?z) ⇒ (?x, subClassOf, ?z) (?x, subClassOf, ?y) ∧ (?y, subClassOf, ?x) ⇒ (?x, equivalentClass, ?y) (?x, subPropertyOf, ?y) ∧ (?y, subPropertyOf, ?z) ⇒ (?x, subPropertyOf, ?z) (?x, type, ?y) ∧ (?y, subClassOf, ?z) ⇒ (?x, type, ?z) (?x, onProperty, ?y) ∧ (?x, hasValue, ?z) ∧ (?a, subClassOf, ?x) ⇒ (?a, ?y, ?z) (?x, onProperty, ?y) ∧ (?x, hasValue, ?z) ∧ (?a, ?y, ?z) ⇒ (?a, type, ?x) . . . AAAI Symposium on Technosocial Predictive Analytics – Stanford University, California – March 24, 2009
  14. 14. 13 Description Logics - Example • Terminological Box (T-Box): a collection of descriptions. Also known as an ontology. Human ≡ (= 2 numberOfLegs) (= false hasFur) ∃bestFriend.Canine Canine ≡ (= 4 numberOfLegs) (= true hasFur) Human Mammal Canine Mammal • Assertion Box (A-Box): a collection of individuals and their relationships to one another. numberOfLegs(marko, 2), hasFur(marko, false), bestFriend(marko, fluffy), numberOfLegs(fluffy, 4), hasFur(fluffy, true). AAAI Symposium on Technosocial Predictive Analytics – Stanford University, California – March 24, 2009
  15. 15. 14 Description Logics - Example inferred Mammal subClassOf subClassOf Human Canine type type T-Box A-Box type type marko bestFriend fluffy numberOfLegs hasFur numberOfLegs hasFur 2 false 4 true * The T-Box includes other description information, but for diagram clarity, this was left out. Yes — marko is a type of Mammal. AAAI Symposium on Technosocial Predictive Analytics – Stanford University, California – March 24, 2009
  16. 16. 15 Description Logics - Drawbacks • With “nested” descriptions and complex quantifiers, you can run into exponential running times. • Requires that all assertions in the A-Box are “true”. For example, if the T-Box declares that a country can have only one president and you assert that barack is the president of the United States and that marko is the president of the United States, then it is inferred that barack and marko are the same person. And this can have rippling effects such as their mothers and fathers must be the same people, etc. • Not very “organic” as concepts descriptions are driven, not by the system, but by a human designer. AAAI Symposium on Technosocial Predictive Analytics – Stanford University, California – March 24, 2009
  17. 17. 16 Evidential Logics - Introduction Evidential logics are multi-valued logics founded on AIKIR (Assumption of Insufficient Knowledge and Insufficient Resources) and are: • non-bivalent: there is no absolute truth in a statement, only differing degrees of support or negation. • non-monotonic: the evaluation of the “truth” of a statement is not immutable, but can change as new experiences occur. In other words, as new evidence is accumulated. Wang, P., “Cognitive Logic versus Mathematical Logic”, Proceedings of the Third International Seminar on Logic and Cognition, May 2004.[3] AAAI Symposium on Technosocial Predictive Analytics – Stanford University, California – March 24, 2009
  18. 18. 17 Evidential Logics - The Process Evidential reasoning is done using various syllogisms:1 • deduction: (?x, ?y) ∧ (?y, ?z) ⇒ (?x, ?z) fluffy is a canine, canine is a mammal ⇒ fluffy is a mammal • induction: (?x, ?y) ∧ (?z, ?y) ⇒ (?x, ?z) fluffy is a canine, fifi is a canine ⇒ fluffy is a fifi • abduction: (?x, ?y) ∧ (?x, ?z) ⇒ (?y, ?z) fluffy is a canine, fluffy is a dog ⇒ canine is a dog • exemplification: (?x, ?y) ∧ (?y, ?z) ⇒ (?z, ?x)2 fluffy is a canine, canine is a mammal ⇒ mammal is a fluffy 1 It is helpful to think of the copula as “inherits the properties of” instead of “is a”. 2 Exemplification is a much less used syllogism in evidential reasoning. AAAI Symposium on Technosocial Predictive Analytics – Stanford University, California – March 24, 2009
  19. 19. 18 Evidential Logics - Example Assume that the past experience of the evidential system has provided these w+, w− evidential tuples for the following relationships, where w+ is positive evidence and w− is negative evidence.3 Mammal <1,0> <1,0> Human Canine <1,0> <0,1> <1,0> <1,0> 2-legs fur 4-legs 3 The example to follow is not completely faithful to NAL-* (Non-Axiomatic Logic). Please refer to Pei, W., “Rigid Flexibility”, Springer, 2006.[4] for more expressive NAL constructs. AAAI Symposium on Technosocial Predictive Analytics – Stanford University, California – March 24, 2009
  20. 20. 19 Evidential Logics - Example experienced Mammal <1,0> <1,0> Human Canine <1,0> <0,1> <1,0> <1,0> 2-legs fur 4-legs <1,0> <0,1> <1,0> <1,0> marko fluffy AAAI Symposium on Technosocial Predictive Analytics – Stanford University, California – March 24, 2009
  21. 21. 20 Evidential Logics - Example inferred Mammal <1,0> <1,0> Human Canine <1,0> <0,1> <1,0> <1,0> <1,0> D <2,0> D 2-legs fur 4-legs <1,0> <0,1> <1,0> <1,0> D deduction marko I induction fluffy A abduction AAAI Symposium on Technosocial Predictive Analytics – Stanford University, California – March 24, 2009
  22. 22. 21 Evidential Logics - Example inferred Mammal <1,0> <1,0> Human Canine <1,0> <0,1> <1,0> <1,0> <1,0> <2,0> 2-legs <0,1> fur <1,0> 4-legs I A <1,0> <0,1> <1,0> <1,0> D deduction marko I induction fluffy A abduction AAAI Symposium on Technosocial Predictive Analytics – Stanford University, California – March 24, 2009
  23. 23. 22 Evidential Logics - Example <1,0> Mammal inferred D <1,0> <1,0> Human <1,0> Canine <1,0> <0,1> <1,0> <1,0> <1,0> <2,0> 2-legs <0,1> fur <1,0> 4-legs <1,0> <0,1> <1,0> <1,0> D deduction marko I induction fluffy A abduction Yes — currently, marko is believed to be a type of Mammal. AAAI Symposium on Technosocial Predictive Analytics – Stanford University, California – March 24, 2009
  24. 24. 23 Conclusion The associated article demonstrates provides a framework for doing evidential logic on multi-relational networks (e.g. RDF graphs). The reasoner is based on algebraic manipulations of an evidence-based multi-relational structure. Rodriguez, M.A., Geldart, J., “An Evidential Path Logic for Multi-Relational Networks”, Association for the Advancement of Artificial Intelligence (AAAI): Technosocial Predictive Analytics Symposium, AAAI Press, LA-UR-08-06397, Stanford University, March 2009.[2] AAAI Symposium on Technosocial Predictive Analytics – Stanford University, California – March 24, 2009
  25. 25. 24 References [1] Franz Baader, Diego Calvanese, Deborah L. Mcguinness, Daniele Nardi, and Peter F. Patel-Schneider, editors. The Description Logic Handbook: Theory, Implementation and Applications. Cambridge University Press, January 2003. [2] Marko A. Rodriguez and Joe Geldart. An evidential logic for multi-relational networks. In Proceedings of the Association for the Advancement of Artificial Intelligence. Association for the Advancement of Artificial Intelligence, May 2009. [3] Pei Wang. Cognitive logic versus mathematical logic. In Proceedings of the Third International Seminar on Logic and Cognition, May 2004. [4] Pei Wang. Rigid Flexibility. Springer, 2006. AAAI Symposium on Technosocial Predictive Analytics – Stanford University, California – March 24, 2009
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