7 IR models based on predicate logic

<strong>IR</strong> <strong>models</strong> <strong>based</strong> on predicate logic 1
7 <strong>IR</strong> <strong>models</strong> <strong>based</strong> on predicate
logic
7.1 General considerations
7.1.1 <strong>IR</strong> as inference
q - query
d – document
retrieval:
search for documents which imply the query:
d → q
example:
d = {t 1 , t 2 , t 3 }
q = {t 1 , t 3 }
logical view:
d = t 1 ∧ t 2 ∧ t 3
q = t 1 ∧ t 3
⇒: d → q
Norbert Fuhr

<strong>IR</strong> <strong>models</strong> <strong>based</strong> on predicate logic 2
advantage of inference-<strong>based</strong> approach:
step from term-<strong>based</strong> to knowledge-<strong>based</strong> retrieval
e.g. easy incorporation of additional knowledge
example:
d: ’squares’
q: ’rectangles’
thesaurus: ’squares’ → ’rectangles’
⇒: d → q
Norbert Fuhr

<strong>IR</strong> <strong>models</strong> <strong>based</strong> on predicate logic 3
7.1.2 <strong>IR</strong> as uncertain inference
d: ’quadrangles’
q: ’rectangles’
⇒ uncertain knowledge required
’quadrangles’ 0.3
→ ’rectangles’
[Rijsbergen 86]:
<strong>IR</strong> as uncertain inference
Retrieval ˆ=
estimate probability P (d → q) = P (q|d)
q
t 1
t 4
t 2
t 5
t 3
t 6
d
Norbert Fuhr

<strong>IR</strong> <strong>models</strong> <strong>based</strong> on predicate logic 4
7.1.3 Propositional vs. predicate logic
spatio-temporal relationships:
• document attributes
query: documents published after 1990?
?- pubyear(D,Y) & Y>1990
• multimedia retrieval
Norbert Fuhr
conventional indexing (<strong>based</strong> on propositional logic):
d = {tree, house}
query: Is there a tree on the left of the house?
⇒ query cannot be expressed in propositional logic
predicate logic:
d: tree(t1). house(h1). tree(t2).
left(t1, h1). left(h1,t2).
?- tree(X) & house(Y) & left(X,Y).

<strong>IR</strong> <strong>models</strong> <strong>based</strong> on predicate logic 5
Ontologies
Thesaurus
polygon
regular
polygon
triangle quadrangle ...
rectangle
regular
triangle
square
thesaurus knowledge:
can be expressed in propositional logic
square = quadrangle ∧ regular-polygon
description logics
• instances of concepts
• roles (relationships) between concepts/instances
Norbert Fuhr

<strong>IR</strong> <strong>models</strong> <strong>based</strong> on predicate logic 6
Advantages of predicate logic:
• modelling of spatial and temporal relationships
(e.g. for multimedia retrieval)
• instances of concepts
≈ combination of controlled vocabulary and free text search
• roles/relationships between concepts or instances
→ higher expressiveness for concept definition and description of document
content
Norbert Fuhr

<strong>IR</strong> <strong>models</strong> <strong>based</strong> on predicate logic 7
7.2 RDF
7.2.1 RDF: basic concepts
Resource object on the WWW, e.g. Web page, database
naming of resources: Uniform Resource Identifier (URI)
Literal special type of resource, with string value, no explicit URI
Property aspect / attribute / characteristics / relation
Statement resource + named property + value of property
(subject, predicate, object)
visits
Norbert Pisa
Norbert Fuhr

<strong>IR</strong> <strong>models</strong> <strong>based</strong> on predicate logic 8
RDF example
organized−by
ISSDL
M.Agosti
isPartOf
Name Email
"Maristella Agosti" "agosti@..."
<strong>IR</strong>−Course
title
"Introduction to <strong>IR</strong>"
teaches
N.Fuhr
Name
"Norbert Fuhr"
Email
"fuhr@cs.uni−..."
Norbert Fuhr

<strong>IR</strong> <strong>models</strong> <strong>based</strong> on predicate logic 9
RDF schemas
similar to semantic networks / description logics
describes relationships between types of resources and/or properties
• fundamental concepts
– rdfs:Resource
– rdf:Property
– rdfs:Class
• schema definition concepts
– rdf:type
– rdfs:subClassOf
– rdfs:subPropertyOf
– rdfs:seeAlso
– rdfs:isDefinedBy
Norbert Fuhr

<strong>IR</strong> <strong>models</strong> <strong>based</strong> on predicate logic 10
RDFS example: resource hierarchy
rdfs:Resource
rdfs:Class
xyz:MotorVehicle
xyz:Van
xyz:Truck
xyz:PassengerVehicle
xyz:MiniVan
rdfs:subClassOf
rdf:Type
Norbert Fuhr

<strong>IR</strong> <strong>models</strong> <strong>based</strong> on predicate logic 11
RDFS example: resource and property hierarchies
rdf:Property
rdfs:Class
rdf:type
rdf:type
rdf:type
visits
Person
visits
Place
rdfs:subPropertyOf
rdfs:subClassOf
rdfs:subClassOf
tourist−visit business−visit ISSDL−Tutor
business−visit
Conf.−Loc.
rdf:type rdf:type
business−visit
N. Fuhr Pisa
Norbert Fuhr

<strong>IR</strong> <strong>models</strong> <strong>based</strong> on predicate logic 12
Norbert Fuhr

<strong>IR</strong> <strong>models</strong> <strong>based</strong> on predicate logic 13
RDF example: image description
picture
contains
rdf:type
parc
above
artifact
rdfs:subClassOf
sculpture
right−of
man woman
swan cherub
socle
Norbert Fuhr

<strong>IR</strong> <strong>models</strong> <strong>based</strong> on predicate logic 14
Retrieval with RDF
?X
artifact
right−of
?Y ?Z
man woman
Norbert Fuhr

<strong>IR</strong> <strong>models</strong> <strong>based</strong> on predicate logic 15
Norbert Fuhr

<strong>IR</strong> <strong>models</strong> <strong>based</strong> on predicate logic 16
Subsumption
retrieval as inference (implication) in description logic: subsumption
find implicit subclasses of query concept
Subsumption in RDF:
resource r2 has property rdfs:subClassOf r1 if
1. r2 is subclass of all superclasses of r1
2. each property of r1 subsumes the corresponding property of r2
a property p2 is subsumed by a property p1 if
1. the properties are equal, or the statement r2 rdfs:subPropertyOf r1 holds.
2. the range of p2 is subsumed by the range of p1.
Norbert Fuhr

<strong>IR</strong> <strong>models</strong> <strong>based</strong> on predicate logic 17
7.3 Modelling <strong>IR</strong> in Datalog
7.3.1 Introduction
Datalog:
• horn predicate logic
(most <strong>IR</strong> <strong>models</strong> <strong>based</strong> on propositional logic)
• no functions
• restricted forms of negation allowed
• sound and complete evaluation algorithms
Norbert Fuhr

<strong>IR</strong> <strong>models</strong> <strong>based</strong> on predicate logic 18
ground facts:
docTerm(d1,ir).
docTerm(d1,db).
docTerm(d2,ir).
docTerm(d2,oop).
rules:
irdoc(D) :- docTerm(D,ir).
iranddb(D) :- docTerm(D,ir) & docTerm(D,db).
irnotdb(D) :- docTerm(D,ir) & not(docTerm(D,db)).
recursive rules:
link(d1,d2). link(d2,d3). link(d3,d1).
linked(X,Y) :- link(X,Y).
linked(X,Y) :- linked(X,Z) & link(Z,Y).
queries:
?- docTerm(D,ir).
?- docTerm(D,ir) & docTerm(D,db).
?- docTerm(D,ir) \& not(docTerm(D,db)).
Norbert Fuhr

<strong>IR</strong> <strong>models</strong> <strong>based</strong> on predicate logic 19
7.3.2 Hypertext structure
docTerm(d1,ir). docTerm(d1,db).
link(d1,d2). link(d2,d3). link(d3,d1).
about(D,T) :- docTerm(D,T).
about(D,T) :- link(D,D1) & about(D1,T).
d3
docterm
d1
d2
link
ir
db
?- about(D,ir)
Norbert Fuhr

<strong>IR</strong> <strong>models</strong> <strong>based</strong> on predicate logic 20
7.3.3 Aggregation
book
chapter
section
part(D,P) :- chapter(D,P).
part(D,P) :- section(D,P).
retrieve node if at least one part is about the search
term:
about(D,T) :- part(D,P) & about(P,T).
retrieve node if all its parts are about the search term:
about(D,T) :- part(D,X) & about(X,T) &
not(anypart(D,T)).
anypart(D,T):- part(D,P)& not(about(P,T)).
Norbert Fuhr

<strong>IR</strong> <strong>models</strong> <strong>based</strong> on predicate logic 21
7.4 Probabilistic Datalog
7.4.1 Motivation
powerful retrieval logic
• expressiveness of Datalog
– predicate logic
(spatio-temporal relationships, instances of
concepts)
– recursion
(structured documents, hypertext links, terminological
structures)
• uncertain inference:
probabilistic inference
Norbert Fuhr

<strong>IR</strong> <strong>models</strong> <strong>based</strong> on predicate logic 22
Syntax
ground facts with probabilistic weights
0.9 docTerm(d1,ir).
0.5 docTerm(d1,db).
0.8 docTerm(d2,ir).
0.3 docTerm(d2,oop).
?- docTerm(D,ir).
gives
d1 0.9
d2 0.8
?- docTerm(D,ir) & docTerm(D,db).
gives
d1 0.45
Norbert Fuhr

<strong>IR</strong> <strong>models</strong> <strong>based</strong> on predicate logic 23
Example: Image retrieval
<strong>IR</strong>IS (Univ. Bremen): automatic indexing of images with semantic concepts
imgobj(O,I,N,X1,X2,Y1,Y2)
O image object
I image
N name of semantic concept
L,R,B,T bounding rectangle of image object
query: images with water in front of stones
?- imgobj(OA,I,water,L1,R1,B1,T1) ,
imgobj(OB,I,stone,L2,R2,B2,T2),
B1

<strong>IR</strong> <strong>models</strong> <strong>based</strong> on predicate logic 24
Norbert Fuhr

<strong>IR</strong> <strong>models</strong> <strong>based</strong> on predicate logic 25
7.4.2 Semantics of probabilistic Datalog
Extensional vs. intensional semantics
0.9 docTerm(d1,ir).
0.5 docTerm(d1,db).
0.7 link(d2,d1).
about(D,T) :- docTerm(D,T).
about(D,T) :- link(D,D1) & about(D1,T)
q(D) :- about(D,ir) & about(D,db).
extensional semantics:
weight of derived fact as function of weights of subgoals
P (q(d2)) = P (about(d2,ir)) · P (about(d2,db)) =
(0.7 · 0.9) · (0.7 · 0.5)
Problem:
“improper treatment of correlated sources of evidence”
[Pearl]
→ extensional semantics only correct for tree-like
inference structures
intensional semantics:
weight of IDB fact as function of weights of underlying
ground facts
Norbert Fuhr

<strong>IR</strong> <strong>models</strong> <strong>based</strong> on predicate logic 26
Implementation of intensional semantics
event keys and event expressions
0.9 docTerm(d1,ir). [dT(d1,ir)]
0.5 docTerm(d1,db). [dT(d1,db)]
0.8 docTerm(d2,ir). [dT(d2,ir)]
0.3 docTerm(d2,oop). [dT(d2,oop)]
0.7 link(d2,d1). [l(d2,d1)]
?- docTerm(D,ir) & docTerm(D,db).
gives
d1 [dT(d1,ir) & dT(d1,db)] 0.9 · 0.5 = 0.45
about(D,T) :- docTerm(D,T).
about(D,T) :- link(D,D1) & about(D1,T)
?- about(D,ir) & about(D,db).
gives
d2
[l(d2,d1) & dT(d1,ir) & l(d2,d1) &
dT(d1,db)]
0.7 · 0.9 · 0.5 = 0.315
d1 [dT(d1,ir) & dT(d1,ir)] 0.7 · 0.5 = 0.35
Norbert Fuhr

<strong>IR</strong> <strong>models</strong> <strong>based</strong> on predicate logic 27
about(D,T) :- docTerm(D,T).
about(D,T) :- link(D,D1) & about(D1,T).
d3
0.8
0.4
docterm
d1
0.5
d2
link
0.9
0.5
ir
db
?- about(D,ir)
d1 [dT(d1,ir) | l(d1,d2) & l(d2,d3) &
l(d3,d1) &
dT(d1,ir) | ...] 0.900
d3 [l(d3,d1) & dT(d1,ir)] 0.720
d2 [l(d2,d3) & l(d3,d1) & dT(d1,ir)] 0.288
?- about(D,ir) & about(D,db)
d1 [dT(d1,ir) & dT(d1,db)] 0.450
d3 [l(d3,d1) & dT(d1,ir) & l(d3,d1) &
dT(d1,db)] 0.360
d2 [l(d2,d3) & l(d3,d1) & dT(d1,ir) &
dT(d1,db)] 0.144
Norbert Fuhr

<strong>IR</strong> <strong>models</strong> <strong>based</strong> on predicate logic 28
computation of probabilities for event expressions
1. transformation of expression into disjunctive normal
form
2. application of sieve formula:
c i – conjunct of event keys
P (c 1 ∨ . . . ∨ c n ) =
n∑
(−1) i−1
i=1
∑
1≤j 1

<strong>IR</strong> <strong>models</strong> <strong>based</strong> on predicate logic 29
Interpretation of probabilistic weights
possible worlds semantics
0.9 docTerm(d1,ir).
P (W 1 ) = 0.9: {docTerm(d1,ir)}
P (W 2 ) = 0.1: {}
Norbert Fuhr

<strong>IR</strong> <strong>models</strong> <strong>based</strong> on predicate logic 30
0.9 docTerm(d1,ir).
0.5 docTerm(d1,db).
possible interpretations:
I 1 :
P (W 1 ) = 0.45: {docTerm(d1,ir)}
P (W 2 ) = 0.45: {docTerm(d1,ir),
docTerm(d1,db)}
P (W 3 ) = 0.05: {docTerm(d1,db)}
P (W 3 ) = 0.05: {}
I 2 :
P (W 1 ) = 0.5: {docTerm(d1,ir)}
P (W 2 ) = 0.4: {docTerm(d1,ir), docTerm(d1,db)}
P (W 3 ) = 0.1: {docTerm(d1,db)}
I 3 :
P (W 1 ) = 0.4: {docTerm(d1,ir)}
P (W 2 ) = 0.5: {docTerm(d1,ir), docTerm(d1,db)}
P (W 3 ) = 0.1: {}
probabilistic logic:
0.4 ≤ P (docTerm(d1, ir)&docTerm(d1, db)) ≤ 0.5
probabilistic Datalog with independence assumptions:
P (docTerm(d1, ir)&docTerm(d1, db)) = 0.45
Norbert Fuhr

<strong>IR</strong> <strong>models</strong> <strong>based</strong> on predicate logic 31
Disjoint events
example: imprecise attribute values
# py(dk,av).
0.2 py(d3,89).
0.7 py(d3,90).
0.1 py(d3,91).
interpretation:
P (W 1 ) = 0.2: {py(d3,89)}
P (W 2 ) = 0.7: {py(d3,90)}
P (W 3 ) = 0.1: {py(d3,91)}
?- py(X,Y) & Y > 89.
d3 [p(d3,90) | p(d3,91)] 0.7 + 0.1 = 0.8
Norbert Fuhr

<strong>IR</strong> <strong>models</strong> <strong>based</strong> on predicate logic 32
Probabilistic search term weighting
via disjoint events
0.8 docTerm(d1,db). 0.7 docTerm(d1,ir).
# qtw(av).
0.4 qtw(db). 0.6 qtw(ir).
s(D) :- qtw(X) & docTerm(D,X).
?- s(D).
0.4 qtw(db) 0.6 qtw(ir)
0.7 docTerm(d1,ir)
0.8 docTerm(d1,db)
d1 [q(db) & dT(d1,db) | q(ir) & dT(d1,ir)]
0.4 · 0.8 + 0.6 · 0.7 = 0.74
Norbert Fuhr

<strong>IR</strong> <strong>models</strong> <strong>based</strong> on predicate logic 33
Probabilistic rules
rules for deterministic facts:
0.7 likes-sports(X) :- man(X).
0.4 likes-sports(X) :- woman(X).
man(peter).
interpretation:
P (W 1 ) = 0.7: {man(peter),
likes-sports(peter)}
P (W 2 ) = 0.3: {man(peter)}
rules for uncertain facts:
# sex(dk,av).
0.7 l-s(X) :- sex(X,male).
0.4 l-s(X) :- sex(X,female).
0.5 sex(X,male) :- human(X).
0.5 sex(X,female) :- human(X).
human(peter).
interpretation:
P (W 1 ) = 0.35: {sex(peter,male), l-s(peter)}
P (W 2 ) = 0.15: {sex(peter,male)}
P (W 3 ) = 0.20: {sex(peter,female), l-s(peter)}
P (W 4 ) = 0.30: {sex(peter,female)}
Norbert Fuhr

<strong>IR</strong> <strong>models</strong> <strong>based</strong> on predicate logic 34
Vague predicates
pc(m1,486/dx50,8,540,900).
pc(m2,pe60,16,250,1000).
pc(m3,pe90,16,540,1100).
?- pc(MOD, CPU, MEM, DISK, PRICE), PRICE < 1000
vague predicate ˆ< (builtin)
1.00 ˆ

<strong>IR</strong> <strong>models</strong> <strong>based</strong> on predicate logic 35
applications of vague predicates:
• vague fact conditions
• proper name search (string similarity)
(also OCRed text)
• multimedia <strong>IR</strong>
(e.g. audio retrieval, image retrieval)
Norbert Fuhr