The Boolean model of information retrieval

Information retrieval

Information retrieval is the area of study concerned with searching for documents, for information within documents, and for metadata about documents, as well as that of searching structured storage, relational databases, and the World Wide Web...

 (BIR) is a classical information retrieval (IR) model and, at the same time, the first and most adopted one. It is used by virtually all commercial IR systems today.

Definitions

The BIR is based on Boolean Logic and classical Set theory

Set theory

Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...

 in that both the documents to be searched and the user's query are conceived as sets of terms. Retrieval is based on whether or not the documents contain the query terms. Given a finite set

T = {t1, t2, ..., tj, ..., tm}

of elements called index terms (e.g. words or expressions - which may be stemmed - describing or characterising documents such as keywords given for a journal article), a finite set

D = {D1, ..., Di, ..., Dn}, where Di is an element of the powerset of T

of elements called documents. Given a Boolean expression - in a normal form - Q called a query as follows:

Q = (Wi OR Wk OR ...) AND ... AND (Wj OR Ws OR ...),
with Wi=ti, Wk=tk, Wj=tj, Ws=ts, or Wi=NON ti, Wk=NON tk, Wj=NON tj, Ws=NONts

where ti means that the term ti is present in document Di, whereas NON ti means that it is not.

Equivalently, Q can be given in a disjunctive normal form, too. An operation called retrieval, consisting of two steps, is defined as follows:

1. The sets Sj of documents are obtained that contain or not term tj (depending on whether
Wj=tj or Wj=NON tj) :

Sj = {Di|Wj element of Di}

2. Those documents are retrieved in response to Q which are the result of the corresponding sets operations, i.e. the answer to Q is as follows:

UNION ( INTERSECTION Sj)

Example

Let the set of original (real) documents be, for example

O = {O1, O2, O3}

where

O1 = Bayes' Principle: The principle that, in estimating a parameter, one should initially assume that each possible value has equal probability (a uniform prior distribution).

O2 = Bayesian Decision Theory: A mathematical theory of decision-making which presumes utility and probability functions, and according to which the act to be chosen is the Bayes act, i.e. the one with highest Subjective Expected Utility. If one had unlimited time and calculating power with which to make every decision, this procedure would be the best way to make any decision.

O3 = Bayesian Epistemology: A philosophical theory which holds that the epistemic status of a proposition (i.e. how well proven or well established it is) is best measured by a probability and that the proper way to revise this probability is given by Bayesian conditionalisation or similar procedures. A Bayesian epistemologist would use probability to define, and explore the relationship between, concepts such as epistemic status, support or explanatory power.

Let the set T of terms be:

T = {t1 = Bayes' Principle, t2 = probability, t3 = decision-making, t4 =
Bayesian Epistemology}

Then, the set D of documents is as follows:

D = {D1, D2, D3}

where

D1 = {Bayes' Principle, probability}

D2 = {probability, decision-making}

D3 = {probability, Bayesian Epistemology}

Let the query Q be:

Q = probability AND decision-making

1. Firstly, the following sets S1 and S2 of documents Di are obtained (retrieved):

S1 = {D1, D2, D3}

S2 = {D2}

2. Finally, the following documents Di are retrieved in response to Q:
{D1, D2, D3} INTERSECTION {D2} = {D2}

This means that the original document O2 (corresponding to D2) is the answer to Q.

Obviously, if there are more than one documents with the same representation, every such document is retrieved. Such documents are, in the BIR, indistinguishable (or, in other words, equivalent).

Disadvantages

• Exact matching may retrieve too few or too many documents
• Difficult to rank output, some documents are more important than others
• Hard to translate a query into a Boolean expression
• All terms are equally weighted
• More like data retrieval than information retrieval

Data structures and algorithms

From a pure formal mathematical point of view, the BIR is straightforward. From a practical point of view, however, several further problems should be solved that relate to algorithms and data structures, such as, for example, the choice of terms (manual or automatic selection or both), stemming, hash tables, inverted file structure, and so on.

Hash Sets

Another possibility is to use Hash Sets. Each document is represented by a Hash-table which contains every single term of that document. Since Hash-table size increases and decreases in real time with the addition and removal of terms, each document will occupy much less space in memory. However, it will have a slowdown in performance because the operations are more complex than with bit vectors. On the worst case performance can degrade from O(n) to O(n²). On the average case, the performance slowdown will not be that much worse than bit vectors and the space usage is much more efficient.