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The Boolean Model

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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

The Boolean Model

Ka

Kb

- 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

D1 = “computer information retrieval”

D2 = “computer retrieval”

D3 = “information”

D4 = “computer information”

Q1 = “information retrieval”

Q2 = “information ¬computer”

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}

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

- 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

Set Theoretic Models

Algebraic

Set Theoretic

Generalized Vector

Lat. Semantic Index

Neural Networks

Structured Models

Fuzzy

Extended Boolean

Non-Overlapping Lists

Proximal Nodes

Classic Models

Probabilistic

Boolean

Vector

Probabilistic

Inference Network

Belief Network

Browsing

Flat

Structure Guided

Hypertext

U

s

e

r

T

a

s

k

Retrieval:

Adhoc

Filtering

Browsing

- 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

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)

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

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

for query term

a class

- A fuzzy subsetA 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:

document collection

- 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}

- 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

- 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

- 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,j1.ki is a good fuzzy index
- When all index terms of dj are only loosely related to ki, i,j0.ki is not a good fuzzy index

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

cc3

Da

djDa 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

Query q=ka (kb kc)

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

- 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 3/3arrived = gold = shipment = truck = 0.176 = log 3/2damaged = delivery = fire = silver = 0.477 = log 3/1

- 8 Keywords (Dimensions) are selected
- arrived(1), damaged(2), delivery(3), fire(4), gold(5), silver(6), shipment(7), truck(8)

- Sim(q,d): Alternative 1
Sim(q,d3) > Sim(q,d2) > Sim(q,d1)

- Sim(q,d): Alternative 2
Sim(q,d3) > Sim(q,d2) > Sim(q,d1)

- 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

- 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.

- 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?

ky

ky

kx

kx

- 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.

- For query q=Kx and Ky, (1,1) is the most desirable point.
- We use
to rank the documents.

- The bigger the better.

- qor=k1 p k2 p … p Km
- qand=k1 p k2 p … p km

- 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)

- 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.