Extended Boolean model

The Extended Boolean model was described in a Communications of the ACM article appearing in 1983, by Gerard Salton, Edward A. Fox, and Harry Wu. The goal of the Extended Boolean model is to overcome the drawbacks of the Boolean model that has been used in

Thus, the extended Boolean model can be considered as a generalization of both the Boolean and vector space models; those two are special cases if suitable settings and definitions are employed. Further, research has shown effectiveness improves relative to that for Boolean query processing. Other research has shown that relevance feedback and query expansion can be integrated with extended Boolean query processing.

Definitions

In the Extended Boolean model, a document is represented as a vector (similarly to in the vector model). Each i dimension corresponds to a separate term associated with the document.

The weight of term K_x associated with document d_j is measured by its normalized

and can be defined as:

${\displaystyle w_{x,j}=f_{x,j}*{\frac {Idf_{x}}{max_{i}Idf_{i}}}}$

where Idf_x is

and f_x,j the term frequency for term x in document j.

The weight vector associated with document d_j can be represented as:

${\displaystyle \mathbf {v} _{d_{j}}=[w_{1,j},w_{2,j},\ldots ,w_{i,j}]}$

The 2 Dimensions Example

Considering the space composed of two terms K_x and K_y only, the corresponding term weights are w₁ and w₂.^[2] Thus, for query q_or = (K_x ∨ K_y), we can calculate the similarity with the following formula:

${\displaystyle sim(q_{or},d)={\sqrt {\frac {w_{1}^{2}+w_{2}^{2}}{2}}}}$

For query q_and = (K_x ∧ K_y), we can use:

${\displaystyle sim(q_{and},d)=1-{\sqrt {\frac {(1-w_{1})^{2}+(1-w_{2})^{2}}{2}}}}$

Generalizing the idea and P-norms

We can generalize the previous 2D extended Boolean model example to higher t-dimensional space using Euclidean distances.

This can be done using

• A generalized conjunctive query is given by:

${\displaystyle q_{or}=k_{1}\lor ^{p}k_{2}\lor ^{p}....\lor ^{p}k_{t}}$

• The similarity of ${\displaystyle q_{or}}$ and ${\displaystyle d_{j}}$ can be defined as:

:${\displaystyle sim(q_{or},d_{j})={\sqrt[{p}]{\frac {w_{1}^{p}+w_{2}^{p}+....+w_{t}^{p}}{t}}}}$

• A generalized disjunctive query is given by:

${\displaystyle q_{and}=k_{1}\land ^{p}k_{2}\land ^{p}....\land ^{p}k_{t}}$

• The similarity of ${\displaystyle q_{and}}$ and ${\displaystyle d_{j}}$ can be defined as:

${\displaystyle sim(q_{and},d_{j})=1-{\sqrt[{p}]{\frac {(1-w_{1})^{p}+(1-w_{2})^{p}+....+(1-w_{t})^{p}}{t}}}}$

Examples

Consider the query q = (K₁ ∧ K₂) ∨ K₃. The similarity between query q and document d can be computed using the formula:

${\displaystyle sim(q,d)={\sqrt[{p}]{\frac {(1-{\sqrt[{p}]{({\frac {(1-w_{1})^{p}+(1-w_{2})^{p}}{2}}}}))^{p}+w_{3}^{p}}{2}}}}$

Improvements over the Standard Boolean Model

Lee and Fox[4] compared the Standard and Extended Boolean models with three test collections, CISI, CACM and INSPEC. Using P-norms they obtained an average precision improvement of 79%, 106% and 210% over the Standard model, for the CISI, CACM and INSPEC collections, respectively.
The P-norm model is computationally expensive because of the number of exponentiation operations that it requires but it achieves much better results than the Standard model and even

is still the most efficient.

Further reading

• Adaptive Feedback Methods in an Extended Boolean Model by Dr.Jongpill Choi
• Interpolation of the extended Boolean retrieval model
• Fox, E.; Betrabet, S.; Koushik, M.; Lee, W. (1992), Information Retrieval: Algorithms and Data structures; Extended Boolean model, Prentice-Hall, Inc., archived from the original on 2013-09-28, retrieved 2017-09-09
• Skorkovská, Lucie; Ircing, Pavel (2009), "Experiments with Automatic Query Formulation in the Extended Boolean Model", Text, Speech and Dialogue, Lecture Notes in Computer Science, vol. 5729, Springer Berlin / Heidelberg, pp. 371–378,
  ISBN 978-3-642-04207-2

See also

• Information retrieval

References