An Evidential Logic for Multi-Relational Networks
Marko A. RodriguezT-5, Center for Nonlinear StudiesLos Alamos National Laboratory
http://markorodriguez.com
Joe GeldartComputer Science Department
University of Durhamhttp://www.dur.ac.uk/j.r.c.geldart
March 23, 2009
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
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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.
Reasoner
read/write
Knowledge Representation
f(x)
AAAI Symposium on Technosocial Predictive Analytics – Stanford University, California – March 24, 2009
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Outline
• Structure
? Network Representations? Resource Description Framework
• Process
? Description Logics? Evidential Logics
AAAI Symposium on Technosocial Predictive Analytics – Stanford University, California – March 24, 2009
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Outline
• Structure
? Network Representations? Resource Description Framework
• Process
? Description Logics? Evidential Logics
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Undirected Single-Relational Network
Human-B
Human-C
Human-D
Human-E
Human-F
Human-A
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Directed Single-Relational Network
Article-B
Article-C
Article-D
Article-E
Article-F
Article-A
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Directed Multi-Relational Network
Article-A
Journal-A
Publisher-A
Article-B
Human-B
Human-A
authored
authored
authoredcontainedIn
editorOf
publishedBy
AAAI Symposium on Technosocial Predictive Analytics – Stanford University, California – March 24, 2009
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The Resource Description Framework
• The Resource Description Framework (RDF) is the standard forrepresenting the relationship between URIs and literals (e.g. float, string,date time, etc.).
• Relationships are directed, labeled links between URIs. A subject URIpoints to an object URI or literal by means of a predicate URI.
lanl:marko lanl:jhwfoaf:knows
foaf:name
"Marko A. Rodriguez"^^xsd:string
foaf:name
"Jennifer H. Watkins"^^xsd:string
subject objectpredicate
AAAI Symposium on Technosocial Predictive Analytics – Stanford University, California – March 24, 2009
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lanl:marko lanl:jhwfoaf:knows
foaf:name
"Marko A. Rodriguez"^^xsd:string
foaf:name
"Jennifer H. Watkins"^^xsd:string
foaf:member
lanl:lanl
foaf:member
foaf:name
"Los Alamos National Laboratory"^^xsd:string
unm:unm
foaf:member
foaf:name
"University of New Mexico"^^xsd:string
urn:doi:10.1016/j.joi.2008.04.002
foaf:publicationsrdf:type
foaf:Person
rdf:type
foaf:Document
rdf:type
foaf:Organization
rdf:type rdf:type
AAAI Symposium on Technosocial Predictive Analytics – Stanford University, California – March 24, 2009
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Outline
• Structure
? Network Representations? Resource Description Framework
• Process
? Description Logics? Evidential Logics
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Description Logics - Introduction
• The purpose of description logics is to infer subsumption relationshipsin a knowledge structure.
• Given a set of individuals (i.e. real-world instances), determine whichconcept descriptions subsume the individuals. For example, is marko atype 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
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Description Logics - Reasoner
• Inference rules: a collection of pattern descriptions are used to assertnew 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
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Description Logics - Example
• Terminological Box (T-Box): a collection of descriptions. Also knownas an ontology.
? Human ≡ (= 2 numberOfLegs) u (= false hasFur) u ∃bestFriend.Canine? Canine ≡ (= 4 numberOfLegs) u (= true hasFur)? Human v Mammal? Canine v Mammal
• Assertion Box (A-Box): a collection of individuals and their relationshipsto 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
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Description Logics - Example
marko fluffybestFriend
2 false 4 true
numberOfLegs hasFur numberOfLegs hasFur
Mammal
Human Canine
type
subClassOf
type
subClassOf
T-Box
A-Box
* The T-Box includes other description information, but for diagram clarity, this was left out.
type type
inferred
Yes — marko is a type of Mammal.
AAAI Symposium on Technosocial Predictive Analytics – Stanford University, California – March 24, 2009
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Description Logics - Drawbacks
• With “nested” descriptions and complex quantifiers, you can run intoexponential running times.
• Requires that all assertions in the A-Box are “true”. For example, ifthe T-Box declares that a country can have only one president and youassert that barack is the president of the United States and that markois the president of the United States, then it is inferred that barack andmarko are the same person. And this can have rippling effects such astheir 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
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Evidential Logics - Introduction
Evidential logics are multi-valued logics founded on AIKIR (Assumption ofInsufficient Knowledge and Insufficient Resources) and are:
• non-bivalent: there is no absolute truth in a statement, only differingdegrees of support or negation.
• non-monotonic: the evaluation of the “truth” of a statement is notimmutable, but can change as new experiences occur. In other words, asnew evidence is accumulated.
Wang, P., “Cognitive Logic versus Mathematical Logic”, Proceedings of the Third
International Seminar on Logic and Cognition, May 2004.[3]
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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 fluffy1It is helpful to think of the copula as “inherits the properties of” instead of “is a”.2Exemplification is a much less used syllogism in evidential reasoning.
AAAI Symposium on Technosocial Predictive Analytics – Stanford University, California – March 24, 2009
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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
Human Canine
Mammal
<1,0> <1,0>
4-legs2-legs fur
<1,0><1,0> <0,1> <1,0>
3The 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
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Evidential Logics - Example
marko
Human Canine
fluffy
Mammal
<1,0> <1,0>
4-legs2-legs fur
<1,0><1,0> <0,1> <1,0>
<0,1> <1,0><1,0><1,0>
experienced
AAAI Symposium on Technosocial Predictive Analytics – Stanford University, California – March 24, 2009
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Evidential Logics - Example
marko
Human Canine
fluffy
Mammal
<1,0> <1,0>
4-legs2-legs fur
<1,0><1,0> <0,1> <1,0>
<0,1> <1,0><1,0>
<1,0> <2,0>
<1,0>
D D
A
deductioninductionabduction
DI
inferred
AAAI Symposium on Technosocial Predictive Analytics – Stanford University, California – March 24, 2009
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Evidential Logics - Example
marko
Human Canine
fluffy
Mammal
<1,0> <1,0>
4-legs2-legs fur
<1,0><1,0> <0,1> <1,0>
<0,1> <1,0><1,0>
<1,0> <2,0>
<1,0>I
<1,0>A
A
deductioninductionabduction
DI
inferred
<0,1>
AAAI Symposium on Technosocial Predictive Analytics – Stanford University, California – March 24, 2009
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Evidential Logics - Example
marko
Human Canine
fluffy
Mammal
<1,0> <1,0>
4-legs2-legs fur
<1,0><1,0> <0,1> <1,0>
<0,1> <1,0><1,0>
<1,0> <2,0>
<1,0>
<1,0>
<0,1> <1,0>
inferred<1,0>D
A
deductioninductionabduction
DI
Yes — currently, marko is believed to be a type of Mammal.
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Conclusion
The associated article demonstrates provides a framework for doingevidential logic on multi-relational networks (e.g. RDF graphs). Thereasoner is based on algebraic manipulations of an evidence-basedmulti-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]
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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