An Evidential Logic for Multi-Relational Networks

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

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.

Reasoner

read/write

Knowledge Representation

f(x)


3

Outline

• Structure

? Network Representations? Resource Description Framework

• Process

? Description Logics? Evidential Logics


4

Outline

• Structure


• Process



5

Undirected Single-Relational Network

Human-B

Human-C

Human-D

Human-E

Human-F

Human-A


6

Directed Single-Relational Network

Article-B

Article-C

Article-D

Article-E

Article-F

Article-A


7

Directed Multi-Relational Network

Article-A

Journal-A

Publisher-A

Article-B

Human-B

Human-A

authored

authored

authoredcontainedIn

editorOf

publishedBy


8

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


9

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


10

Outline

• Structure


• Process



11

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]


12

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)

? . . .


13

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).


14

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.


15

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.


16

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]


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 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.


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

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.


19


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


20


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


21


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


DI

inferred

<0,1>


22


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


DI

Yes — currently, marko is believed to be a type of Mammal.


23

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]


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.


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An Evidential Logic for Multi-Relational Networks

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