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 theare sameretrieved time,based on whether they satisfy the firstconditions andof mosta adoptedquery one.that Ituses is[[Boolean usedlogic]]. byAs manythe IRfirst systemsand tomost-adopted thisinformation day.retrieval model,<ref>{{citationCite web needed|title=Information Retrieval |url=https://mitpress.mit.edu/9780262528870/information-retrieval/ |access-date=November2023-12-09 2010|website=MIT Press |language=en-US}}</ref> it treats each document as a set of words, or ''terms''. A user's query uses logical operators like AND, OR, and NOT to create a rule for retrieval. The system then returns all documents that match the rule.

==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''' (or ''term'') is a [[keyword]] that characterizes the content of a document. Terms are the fundamental units of the model. Common, low-information words (called [[stop words]]) like "a", "the", and "is" are typically excluded from being used as index terms.

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. Retrieval is based on whether or not the documents contain the query terms. Given a finite set<math display="block">T = \{t_1, t_2,\ ...,\ t_m\}</math>of elements called index terms (e.g. words or expressions - which may be [[stemming|stemmed]] - describing or characterizing documents such as keywords given for a journal article), a finite set<math display="block">D = \{D_1,\ ...\ ,D_n\} \text{ where } D_i \in Powerset(T)</math>of elements called documents, and a Boolean expression - in a normal form - <math display="inline">Q</math><math display="block">Q = (W_1\ \or\ W_2\ \or\ \ldots) \and\ \ldots\ \and\ (W_i\ \or\ W_j\ \or\ \ldots)</math>called a query where <math display="inline">W_i</math> either implies that <math>t_i \in D_j</math> or that <math>t_i \notin D_j</math> (equivalently, <math display="inline">Q</math> can be given in a [[disjunctive normal form]] as well). Then an operation called retrieval, consisting of two steps, is defined as follows:

* 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===

: 1. Find the sets <math display="inline">S_j</math> of documents that satisfy <math display="inline">W_j</math> (that is, whether <math display="inline">t_j \in D_k</math> or <math>t_j \notin D_k</math> ) :<math display="block">S_j = \{D_i\ |\ W_j\}</math>2. Those documents are retrieved in response to Q which are the result of the corresponding sets operations, i.e. the answer to <math display="inline">Q</math> is as follows:<math display="block">(S_1 \cup S_2 \cup \ \ldots) \cap\ \ldots\ \cap\ (S_i \cup S_j \cup\ \ldots)</math>

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 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>.

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>.

==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' Principleprinciple: 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 Decisiondecision Theorytheory]]: 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 [[Epistemologyepistemology]]: 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:

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

<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 Epistemologyepistemology}\}

\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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* Easy to implement

* Intuitive concept

* If the resulting document set is either too small or too big, it is directly clear which operators will produce respectively a bigger or smaller set.

*It gives (expert) users a sense of control over the system. It is immediately clear why a document has been retrieved given a query.

== Disadvantages ==

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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''

* 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

* The model frequently returns either too few or too many documents in response to a user query

== 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 table]]s, [[Inverted index|inverted file]] structure, and so on.<ref name="wartik">{{cite book | last = Wartik | first = Steven | title = Information Retrieval Data Structures & Algorithms | chapter = Boolean operations | publisher = Prentice-Hall, Inc. | year = 1992 | isbn = 0-13-463837-9 | url = https://www.scribd.com/doc/13742235/Information-Retrieval-Data-Structures-Algorithms-William-B-Frakes | url-status = dead | archiveurl = https://web.archive.org/web/20130928060217/http://www.scribd.com/doc/13742235/Information-Retrieval-Data-Structures-Algorithms-William-B-Frakes | archivedate = 2013-09-28 }}</ref>

=== Hash sets ===

{{ 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-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" >

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"].

</ref><ref name="goodwin" >

Bob Goodwin; et al.

[https://safari.ethz.ch/architecture/fall2018/lib/exe/fetch.php?media=p605-goodwin.pdf "BitFunnel: Revisiting Signatures for Search"].

2017.

</ref><ref>

Richard Startin.

[https://richardstartin.github.io/posts/bit-sliced-signatures-and-bloom-filters "Bit-Sliced Signatures and Bloom Filters"].

</ref>

The signature file approached is used in [[BitFunnel]].

=== Inverted file ===

Another possibility is to use [[hash set]]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(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.

{{ main | inverted index }}

An inverted index file contains two parts:

a vocabulary containing all the terms used in the collection,

and for each distinct term an inverted index that lists every document that mentions that term.<ref name="zobel" /><ref name="goodwin" />

== References ==

{{reflist}}

* {{ citation | last1= Lashkari | first1=A.H. | last2=Mahdavi | first2= F.| last3=Ghomi | first3= V. | title=2009 International Conference on Information Management and Engineering | doi= 10.1109/ICIME.2009.101 | titlechapter= A Boolean Model in Information Retrieval for Search Engines | year=2009 | pages=385–389 | isbn=978-0-7695-3595-1 | s2cid=18147603 }}

[[Category:Operations research]]

[[Category:Mathematical modeling]]

[[Category:Information retrieval techniques]]