Boolean model of information retrieval

The (standard) Boolean model of information retrieval (BIR) is a classical information retrieval (IR) model and, at the same time, the first and most-adopted one. It is used by many IR systems to this day. 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.
An "index term" is a word or expression"," which may be stemmed, describing or characterizing a document, such as a keyword given for a journal article. Letformula_1be the set of all such index terms. 
A "document" is any subset of formula_2. Letformula_3be the set of all documents. 
A "query" is a Boolean expression formula_4 in normal form:formula_5where formula_6 is true for formula_7 when formula_8. (Equivalently, formula_4 could be expressed in disjunctive normal form.)
We seek to find the set of documents that satisfy formula_4. This operation is called "retrieval" and consists of the following two steps:
Example.
Let the set of original (real) documents be, for example
where
formula_18 = "Bayes' principle: The principle that, in estimating a parameter, one should initially assume that each possible value has equal probability (a uniform prior distribution)."
formula_19 = "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."
formula_20 = "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 formula_21 of terms be:
formula_22
Then, the set formula_23 of documents is as follows:
where formula_25
Let the query formula_4 be:
formula_27Then to retrieve the relevant documents:
This means that the original document formula_19 (corresponding to formula_36) is the answer to formula_4.
Obviously, if there is more than one document with the same representation, every such document is retrieved. Such documents are indistinguishable in the BIR (in other words, equivalent).
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("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.
The signature file approached is used in BitFunnel.
Inverted file.
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.