1. The Boolean Model Simple model based on set theory
Queries specified as boolean expressions
precise semantics
neat formalism
q = ka  (kb  kc)
Terms are either present or absent. Thus, wij  {0,1}
Consider
q = ka  (kb  kc)
vec(qdnf) = (1,1,1)  (1,1,0)  (1,0,0)
vec(qcc) = (1,1,0) is a conjunctive component
Each query can be transformed in DNF form

2. The Boolean Model q = ka  (kb  kc)
sim(q,dj) = 1, if document satisfies the boolean query
0 otherwise
- no in-between, only 0 or 1
(1,1,0)
(1,0,0)
(1,1,1) Kc

3. Exercise D1 = “computer information retrieval”
D2 = “computer retrieval”
D3 = “information”
D4 = “computer information”
Q1 = “information  retrieval”
Q2 = “information  ¬computer”

4. Exercise กำหนด Index term ของแต่ละเอกสาร
D1 = {love, need, person, possess, understand}
D2 = {heart, listen, love, practice, suffer}
D3 = {compassion, love, mind, person, practice}
D4 = {death, health, languor, life, suffer}
D5 = {energy, love, nourish, practice, teach}
Q = {love ^ suffer}

5. Drawbacks of the Boolean Model
Retrieval based on binary decision criteria with no notion of partial matching
No ranking of the documents is provided (absence of a grading scale)
Information need has to be translated into a Boolean expression which most users find awkward
The Boolean queries formulated by the users are most often too simplistic
As a consequence, the Boolean model frequently returns either too few or too many documents in response to a user query

6. Drawbacks of the Boolean Model
The Boolean model imposes a binary criterion for deciding relevance
The question of how to extend the Boolean model to accomodate partial matching and a ranking has attracted considerable attention in the past
Two extensions of boolean model:
Fuzzy Set Model
Extended Boolean Model

7. Set Theoretic Models

8. Non-Overlapping Lists
IR Models
Set Theoretic
Fuzzy
Extended Boolean
Classic Models
Boolean
Vector
Probabilistic
Algebraic
Generalized Vector
Lat. Semantic Index
Neural Networks U s e r T a k
Retrieval:
Adhoc
Filtering
Non-Overlapping Lists
Proximal Nodes
Structured Models
Probabilistic
Inference Network
Belief Network
Browsing
Browsing
Flat
Structure Guided
Hypertext

9. Set Theoretic Models
The Boolean model imposes a binary criterion for deciding relevance
The question of how to extend the Boolean model to accomodate partial matching and a ranking has attracted considerable attention in the past
Two set theoretic models for this:
Fuzzy Set Model
Extended Boolean Model

10. Fuzzy Set Model
Queries and docs represented by sets of index terms: matching is approximate from the start
This vagueness can be modeled using a fuzzy framework, as follows:
with each term is associated a fuzzy set
each doc has a degree of membership in this fuzzy set
This interpretation provides the foundation for many models for IR based on fuzzy theory
In here, we discuss the model proposed by Ogawa, Morita, and Kobayashi (1991)

11. Fuzzy Set Theory
Framework for representing classes whose boundaries are not well defined
Key idea is to introduce the notion of a degree of membership associated with the elements of a set
This degree of membership varies from 0 to 1 and allows modeling the notion of marginal membership
Thus, membership is now a gradual notion, contrary to the crispy notion enforced by classic Boolean logic

12. Fuzzy Set Theory Model A query term: a fuzzy set
A document: degree of membership in this test
Membership function
Associate membership function with the elements of the class
0: no membership in the test
1: full membership
0 ~1: marginal elements of the test
documents

13. Fuzzy Set Theory
a class
for query term
document collection
A fuzzy subset A of a universe of discourse U is characterized by a membership function µA: U[0,1] which associates with each element u of U a number µA(u) in the interval [0,1]
complement:
union:
intersection:

14. Examples Assume U={d1, d2, d3, d4, d5, d6}
Let A and B be {d1, d2, d3} and {d2, d3, d4}, respectively.
Assume A={d1:0.8, d2:0.7, d3:0.6, d4:0, d5:0, d6:0} and B={d1:0, d2:0.6, d3:0.8, d4:0.9, d5:0, d6:0}
= {d1:0.2, d2:0.3, d3:0.4, d4:1, d5:1, d6:1} =
{d1:0.8, d2:0.7, d3:0.8, d4:0.9, d5:0, d6:0}
= {d1:0, d2:0.6, d3:0.6, d4:0, d5:0, d6:0}

15. Fuzzy Information Retrieval
basic idea
Expand the set of index terms in the query with related terms (from the thesaurus) such that additional relevant documents can be retrieved
A thesaurus can be constructed by defining a term-term correlation matrix c whose rows and columns are associated to the index terms in the document collection
keyword connection matrix

16. Fuzzy Information Retrieval (Continued)
normalized correlation factor ci,l between two terms ki and kl (0~1)
In the fuzzy set associated to each index term ki, a document dj has a degree of membership µi,j
ni is # of documents containing term ki
where
nl is # of documents containing term kl
ni,l is # of documents containing ki and kl

17. Fuzzy Information Retrieval (Continued)
physical meaning
A document dj belongs to the fuzzy set associated to the term ki if its own terms are related to ki, i.e., i,j=1.
If there is at least one index term kl of dj which is strongly related to the index ki, then i,j1. ki is a good fuzzy index
When all index terms of dj are only loosely related to ki, i,j0. ki is not a good fuzzy index

18. Example
q = (ka  (kb  kc)) = (ka  kb  kc)  (ka  kb   kc) (ka  kb  kc) = cc1+cc2+cc3
Da: the fuzzy set of documents
associated to the index ka
cc2 Da
cc3
djDa has a degree of membership
a,j > a predefined threshold K
cc1 Db
Da: the fuzzy set of documents
associated to the index ka
(the negation of index term ka) Dc

19. Example Query q=ka  (kb   kc) Recall
disjunctive normal form qdnf=(1,1,1)  (1,1,0)  (1,0,0)
(1) the degree of membership in a disjunctive fuzzy set is computed
using an algebraic sum (instead of max function) more smoothly
(2) the degree of membership in a conjunctive fuzzy set is computed
using an algebraic product (instead of min function)
Recall

20. Fuzzy Set Model
Q: “gold silver truck” D1: “Shipment of gold damaged in a fire” D2: “Delivery of silver arrived in a silver truck” D3: “Shipment of gold arrived in a truck”
IDF (Select Keywords)
a = in = of = 0 = log 3/3 arrived = gold = shipment = truck = = log 3/2 damaged = delivery = fire = silver = = log 3/1
8 Keywords (Dimensions) are selected
arrived(1), damaged(2), delivery(3), fire(4), gold(5), silver(6), shipment(7), truck(8)

21. Fuzzy Set Model

22. Fuzzy Set Model

23. Fuzzy Set Model Sim(q,d): Alternative 1 Sim(q,d): Alternative 2
Sim(q,d3) > Sim(q,d2) > Sim(q,d1)
Sim(q,d): Alternative 2

24. Extended Boolean Model
Disadvantages of “Boolean Model” :
No term weight is used
Counterexample: query q=Kx AND Ky.
Documents containing just one term, e,g, Kx is considered as irrelevant as another document containing none of these terms.
The size of the output might be too large or too small

25. Extended Boolean Model
The Extended Boolean model was introduced in 1983 by Salton, Fox, and Wu
The idea is to make use of term weight as vector space model.
Strategy: Combine Boolean query with vector space model.
Why not just use Vector Space Model?
Advantages: It is easy for user to provide query.

26. Extended Boolean Model
Each document is represented by a vector (similar to vector space model.)
Remember the formula.
Query is in terms of Boolean formula.
How to rank the documents?

27. Fig. Extended Boolean logic considering the space composed of two terms kx and ky only.

28. Extended Boolean Model
For query q=Kx or Ky, (0,0) is the point we try to avoid. Thus, we can use
to rank the documents
The bigger the better.

29. Extended Boolean Model
For query q=Kx and Ky, (1,1) is the most desirable point.
We use
to rank the documents.
The bigger the better.

30. Extend the idea to m terms
qor=k1 p k2 p … p Km
qand=k1 p k2 p … p km

31. Properties
The p norm as defined above enjoys a couple of interesting properties as follows. First, when p=1 it can be verified that
Second, when p= it can be verified that
Sim(qor,dj)=max(xi)
Sim(qand,dj)=min(xi)

32. Example
For instance, consider the query q=(k1 k2)  k3. The similarity sim(q,dj) between a document dj and this query is then computed as
Any boolean can be expressed as a numeral formula.