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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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