Boolean model of information retrieval: Difference between revisions

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{{short description|Classical information retrieval model}}

{{technical|date=June 2018}}The (standard) '''Boolean model of information retrieval''' ('''BIR''')<ref>{{citation | author1=Lancaster, F.W. | author2=Fayen, E.G. | title=Information Retrieval On-Line | publisher=Melville Publishing Co., Los Angeles, California | year=1973}}</ref> is a classical [[information retrieval]] (IR) model and,where atdocuments are retrieved based on whether they satisfy the sameconditions time,of a query that uses [[Boolean logic]]. As the first and most-adopted one.information retrieval model,<ref>{{Cite web |title=Information Retrieval |url=https://mitpress.mit.edu/9780262528870/information-retrieval/ |access-date=2023-12-09 |website=MIT Press |language=en-US}}</ref> Theit BIRtreats iseach baseddocument onas [[Boolean logic]] and classicala [[set theory]]of inwords, thator both''terms''. the documents to be searched and theA user's query areuses conceivedlogical asoperators setslike ofAND, termsOR, (aand [[bag-of-wordsNOT model]]).to Retrievalcreate isa basedrule onfor whetherretrieval. orThe notsystem thethen returns all documents containthat match the query termsrule.

{{technical|date=June 2018}}<!-- Needs a plain-English definition without mathematical symbols -->

The (standard) '''Boolean model of information retrieval''' ('''BIR''')<ref>{{citation | author1=Lancaster, F.W. | author2=Fayen, E.G. | title=Information Retrieval On-Line | publisher=Melville Publishing Co., Los Angeles, California | year=1973}}</ref> is a classical [[information retrieval]] (IR) model and, at the same time, the first and most-adopted one.<ref>{{Cite web |title=Information Retrieval |url=https://mitpress.mit.edu/9780262528870/information-retrieval/ |access-date=2023-12-09 |website=MIT Press |language=en-US}}</ref> The BIR is based on [[Boolean logic]] and classical [[set theory]] in that both the documents to be searched and the user's query are conceived as sets of terms (a [[bag-of-words model]]). Retrieval is based on whether or not the documents contain the query terms.

==Definitions==

In the Boolean model, documents and queries are represented using concepts from [[set theory]]. A document is seen as a simple collection (a set) of terms, and a query is a formal statement (a Boolean expression) that specifies which terms must or must not be present in a retrieved document.

* An '''index term''' is a word (or expression'',term'') whichis may bea [[stemming|stemmedkeyword]], describingthat orcharacterizes characterizingthe content of a document,. Terms suchare asthe afundamental keywordunits givenof forthe amodel. journalCommon, article.low-information Let<mathwords display="block">T(called =[[stop \{t_1,words]]) t_2like "a",\ \ldots"the",\ t_m\}</math>beand the"is" setare oftypically allexcluded suchfrom being used as index terms.

* A '''document''' is represented as a [[set]] of index terms. This is a [[bag-of-words model]], meaning the order and frequency of terms in the original document are ignored. For example, a document about Bayes' theorem might be represented simply as the set <math>\{\text{Bayes' theorem, probability, decision-making}\}</math>.

* A '''query''' is a formal expression of the user's information need, written using index terms and Boolean operators (AND, OR, NOT). The model retrieves every document that is considered a "match" for this logical expression.

===Formal representation===

A ''document'' is any subset of <math>T</math>. Let<math display="block">D = \{D_1,\ \ldots\ ,D_n\}</math>be the set of all documents.

The model can be defined formally as follows:

* Let <math>T = \{t_1, t_2, \ldots, t_k\}</math> be a set of all index terms.

A ''query'' is a Boolean expression <math display="inline">Q</math> in normal form:<math display="block">Q = (W_1\ \or\ W_2\ \or\ \cdots) \and\ \cdots\ \and\ (W_i\ \or\ W_{i+1}\ \or\ \cdots)</math>where <math display="inline">W_i</math> is true for <math>D_j</math> when <math>t_i \in D_j</math>. (Equivalently, <math display="inline">Q</math> could be expressed in [[disjunctive normal form]].)

* A document <math>D_j</math> is any subset of <math>T</math>.

* A query <math>Q</math> is a Boolean expression, typically in [[conjunctive normal form]]:<math display="block">Q = (t_a \lor t_b) \land (\lnot t_c \lor t_d) \land \dots</math>where <math>t_a, t_b, \dots \in T</math>.

We seek to find the set of documents that satisfy <math display="inline">Q</math>. This operation is called ''retrieval'' and consists of the following two steps:

Retrieval is the process of identifying the set of all documents <math>\{D_j\}</math> that satisfy the query <math>Q</math>. For example, for the simple query <math>Q = t_a \land t_b</math>, the system would retrieve all documents whose set of terms contains both <math>t_a</math> and <math>t_b</math>.

: 1. For each <math display="inline">W_j</math> in <math display="inline">Q</math>, find the set <math display="inline">S_j</math> of documents that satisfy <math display="inline">W_j</math>:<math display="block">S_j = \{D_i\mid W_j\}</math>2. Then the set of documents that satisfy Q is given by:<math display="block">(S_1 \cup S_2 \cup \cdots) \cap \cdots \cap (S_i \cup S_{i+1} \cup \cdots)</math>

==Example==

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

: <math>OD = \{O_1D_1,\ O_2D_2,\ O_3D_3\}</math>

where

<math display="inline">O_1D_1</math> = "Bayes' principle: The principle that, in estimating a parameter, one should initially assume that each possible value has equal probability (a uniform prior distribution)."

<math display="inline">O_2D_2</math> = "[[Bayes' theorem|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."

<math display="inline">O_3D_3</math> = "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 <math display="inline">T</math> of terms be:<math display="block">T = \{t_1=\text{Bayes' principle}, t_2=\text{probability}, t_3=\text{decision-making}, t_4=\text{Bayesian epistemology}\}</math>Then, the set <math display="inline">D</math> of documents is as follows:<math display="block">D = \{D_1,\ D_2,\ D_3\}</math>where <math display="block">\begin{align}

Let the set <math display="inline">T</math> of terms be:

<math display="block">T = \{t_1=\text{Bayes' principle}, t_2=\text{probability}, t_3=\text{decision-making}, t_4=\text{Bayesian epistemology}\}</math>

Then, the set <math display="inline">D</math> of documents is as follows:

: <math display="block">D = \{D_1,\ D_2,\ D_3\}</math>

where <math display="block">\begin{align}

D_1 &= \{\text{probability},\ \text{Bayes' principle}\} \\

D_2 &= \{\text{probability},\ \text{decision-making}\} \\

D_3 &= \{\text{probability},\ \text{Bayesian epistemology}\}

\end{align}</math>Let the query <math display="inline">Q</math> be ("probability" AND "decision-making"):<math display="block">Q = \text{probability} \and \text{decision-making}</math>Then to retrieve the relevant documents:

\end{align}</math>

Let the query <math display="inline">Q</math> be:

<math display="block">Q = \text{probability} \and \text{decision-making}</math>Then to retrieve the relevant documents:

# Firstly, the following sets <math display="inline">S_1</math> and <math display="inline">S_2</math> of documents <math display="inline">D_i</math> are obtained (retrieved):<math display="block">\begin{align}

S_1 &= \{D_1,\ D_2,\ D_3\} \\

S_2 &= \{D_2\}

\end{align}</math>Where <math>S_1</math> corresponds to the documents which contain the term "probability" and <math>S_2</math> contain the term "decision-making".

\end{align}</math>

# Finally, the following documents <math display="inline">D_i</math> are retrieved in response to <math display="inline">Q</math>: <math display="block">Q: \{D_1,\ D_2,\ D_3\}\ \cap\ \{D_2\}\ =\ \{D_2\}</math>Where the query looks for documents that are contained in both sets <math>S</math> using the intersection operator.

This means that the original document <math display="inline">O_2</math> (corresponding to <math display="inline">D_2</math>) is the answer to <math display="inline">Q</math>.

Obviously, ifIf there is more than one document with the same representation (the same subset of index terms <math>t_n</math>), every such document is retrieved. Such documents are indistinguishable in the BIR (in other words, equivalent).

== Advantages ==

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* [[String search algorithm|Exact matching]] may retrieve too few or too many documents

* Hard to translate a query into a Boolean expression

* Ineffective for Search-Resistant Concepts<ref>{{cite journal |last1=Shokraneh |first1=Farhad |title=Stop searching and you will find it: Search-Resistant Concepts in systematic review searches |journal=BMJ Evidence-Based Medicine |date=6 August 2024 |pages=bmjebm–2023–112798 |doi=10.1136/bmjebm-2023-112798}}</ref>

* All terms are equally weighted

* More like ''[[data retrieval]]'' than ''information retrieval''

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{{ main | feature hashing }}

Another possibility is to use [[Set (abstract data type)|hash setsets]]s. 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 vector]]s. On the worst-case performance can degrade from O(''n'') to O(''n''²). On the average case, the performance slowdown will not be that much worse than bit vectors and the space usage is much more efficient.

=== Signature file ===

Each document can be summarized by [[Bloom filter]] representing the set of words in that document, stored in a fixed-length bitstring, called a signature. The signature file contains one such [[superimposed code]] bitstring for every document in the collection. Each query can also be summarized by a [[Bloom filter]] representing the set of words in the query, stored in a bitstring of the same fixed length. The query bitstring is tested against each signature.<ref name="zobel" >

The signature file contains one such [[superimposed code]] bitstring for every document in the collection.

Each query can also be summarized by a [[Bloom filter]] representing the set of words in the query, stored in a bitstring of the same fixed length.

The query bitstring is tested against each signature.<ref name="zobel" >

Justin Zobel; Alistair Moffat; and Kotagiri Ramamohanarao.

[https://people.eng.unimelb.edu.au/jzobel/fulltext/acmtods98.pdf "Inverted Files Versus Signature Files for Text Indexing"].