Regular Expressions and Languages

In this article, we are introducing a notation known as “regular expressions.” Regular expressions are a form of language-defining notation, briefly mentioned in the below article. Think of them as a type of “programming language” used to express important applications, such as text-search operations or components in a compiler.

Regular expressions are closely tied to nondeterministic finite automata (NFA) and can be seen as a more user-friendly alternative to NFA notation for describing software components.

After defining regular expressions, we demonstrate that they have the capability to define precisely the class of languages known as “regular languages.” We explore how regular expressions are applied in various software systems. Additionally, we delve into the algebraic laws that govern regular expressions. These laws exhibit similarities to the algebraic laws of arithmetic, although there are noteworthy differences between the algebraic structures of regular expressions and arithmetic expressions.

Regular Expressions

In the current discussion, we’re shifting our focus from machine-oriented language descriptions, like deterministic and nondeterministic finite automata, to an algebraic approach known as “regular expressions.” What we’ll discover is that regular expressions have the ability to precisely define the same languages that various types of automata describe, specifically the regular languages.

Here’s where regular expressions stand out: they provide a declarative way to express the strings we want to accept. In other words, they allow us to state what we’re looking for in a more direct manner. This feature makes regular expressions particularly valuable as the input language for many systems that deal with string processing. To illustrate, some examples of such systems are:

Example 1: Imagine you’re using a search command like UNIX’s grep or a similar command in a web browser or text-formatting system. These commands are designed to find specific strings or patterns within a file. Here’s where regular expressions come into play. The user can describe patterns they want to find using a notation similar to regular expressions.

For instance, if you’re looking for all email addresses in a document, you might use a regular expression like ^[a-zA-Z0-9._%+-]+@[a-zA-Z0-9.-]+\.[a-zA-Z]{2,}$. This regular expression is designed to match common patterns for email addresses.

Now, what happens behind the scenes is interesting. Different search systems might convert this regular expression into either a Deterministic Finite Automaton (DFA) or a Nondeterministic Finite Automaton (NFA). These automata essentially represent a set of rules or steps to recognize the specified pattern.

Once the regular expression is converted, the search system applies this automaton to the file being searched. It simulates the automaton’s steps on the content of the file, checking for matches with the specified pattern. In our example, it would identify and extract all instances of email addresses in the file, helping the user efficiently locate the information they are looking for.

Example 2: Consider lexical-analyzer generators like Lex or Flex. In the context of compiler construction, a lexical analyzer is a crucial component that breaks down the source program into logical units known as tokens. Tokens are meaningful chunks of the source code, such as keywords (e.g., “while”), identifiers (e.g., any letter followed by zero or more letters and/or digits), and symbols like “+” or “<=”.

Now, these lexical-analyzer generators, such as Lex or Flex, simplify the process of creating a lexical analyzer. They operate by accepting descriptions of token forms, which are essentially regular expressions. For instance, if we want to define an identifier as any letter followed by zero or more letters and/or digits, the corresponding regular expression might be [a-zA-Z][a-zA-Z0-9]*.

The lexical-analyzer generator takes these regular expressions describing the forms of tokens and produces a Deterministic Finite Automaton (DFA). This DFA is a state machine that recognizes which token appears next in the input based on the provided regular expressions.

So, if we use the example regular expression for identifiers, the generated DFA would effectively identify and extract identifiers from the source code. This streamlined process helps in efficiently tokenizing the source program, a crucial step in the compilation process.

The Operators of Regular Expressions

Regular expressions are a way of representing languages. For example, the regular expression 01* + 10* represents a language containing strings that are either a single ‘0’ followed by any number of ‘1’s or a single ‘1’ followed by any number of ‘0’s. Even if you’re not familiar with interpreting regular expressions at this point, trust that our interpretation of the language described by this expression will become clear as we define the symbols used.

Before delving into the regular-expression notation, it’s essential to understand three operations on languages that the operators in regular expressions represent:

  1. Union (): The union of two languages, denoted L ∪ M, is the set of strings that are in either L or M, or both. For example, if L = {001, 10, 111} and M = {ε, 001}, then L ∪ M = {ε, 10, 001, 111}.
  2. Concatenation ( or just placing two expressions together): The concatenation of languages L and M is the set of strings formed by taking any string in L and concatenating it with any string in M. For instance, if L = {001, 10, 111} and M = {ε, 001}, then L ⋅ M (or LM) is {001, 10, 111, 001001, 10001, 111001}. The last three strings in LM are formed by taking each string in L and concatenating it with the second string in M.
  3. Closure (L* or Kleene closure): The closure of a language L, denoted L*, represents the set of strings that can be formed by taking any number of strings from L, possibly with repetitions, and concatenating all of them. For example, if L = {0, 1}, then L* is all strings of ‘0’s and ‘1’s. If Z = {0, 11}, then L* consists of strings of ‘0’s and ‘1’s such that the ‘1’s come in pairs, like 011, 11110, and ε, but not 01011 or 101. Formally, L* is the infinite union of Li, where L0 = {ε}, L1 = L, and Li (for i > 1) is the concatenation of i copies of L.

This set of operations forms the basis for constructing regular expressions and understanding the languages they represent.

Building Regular Expressions

Think of algebras as systems that start with basic things, like numbers or letters. These basic things are called constants and variables. In algebra, we can use operators (like plus or times) to create more complex expressions from these basic elements. We also need a way to group things together, often using parentheses.

For example, in regular arithmetic, we start with numbers and build more complicated expressions using operators like + and ×.

Now, the algebra of regular expressions works in a similar way. Instead of dealing with regular numbers, we deal with languages. The basic elements are constants and variables that represent languages, and we have operators for three actions: union (combining), dot (joining), and star (repeating).

To explain regular expressions, we use a recursive approach. This means we describe them by breaking them down into smaller parts. For each regular expression (let’s call it E), we not only say what it looks like, but we also describe the language it represents. We use L(E) to talk about the language represented by the expression E.


  1. Constants: We have two special words, € and @. Each of them is like a secret code representing a language. When we see €, it means the language with nothing in it (just an empty language), and when we see @, it represents the language @ (whatever that language is).
  2. Symbols: If we have any regular letter or symbol (let’s call it a), that’s also a secret code. When we see ‘a’, it represents a language containing just that symbol (like a secret language for that symbol).
  3. Variables: Now, there are special placeholders, like Z. When we see Z (or any other variable), it’s like a blank space that can represent any language.

So, in simple terms, we have special words (€ and @), secret codes for letters (like ‘a’), and blank spaces (variables like Z) that can represent any language. These are the building blocks we use to create more complex expressions in this language system.


When we’re talking about induction, we’re figuring out how to make more interesting expressions in four different ways:

  1. Combining Expressions: Imagine we have two expressions, let’s call them E and F. If we put them together with a plus sign (E + F), it’s like saying we want a new expression that has all the words from both E and F. It’s like making a list that includes words from E and words from F. For example, if E is “sun” and F is “moon,” then (E + F) is like having a list of words with both “sun” and “moon.”
  2. Putting Words Together: Now, if we have two expressions, E and F, and we write them side by side (EF), it’s like saying we want a new expression that has words made by putting the words from E and F together. So, if E is “big” and F is “apple,” then (EF) is like having words like “bigapple.”
  3. Repeating Things: If we have an expression E, and we put a star (*) after it (E*), it’s like saying we can repeat the words from E as many times as we want. For example, if E is “hello,” then (E*) is like having words like “hellohellohello…” as many times as we like.
  4. Using Parentheses: Lastly, if we have an expression E, and we put parentheses around it ((E)), it’s like saying this expression represents the same words as E. It’s a way of grouping things together.

So, induction helps us understand how to create more expressions by combining, repeating, and grouping words using symbols like plus, concatenation, star, and parentheses.

Precedence of Regular-Expression Operators

The precedence of regular-expression operators determines the order in which operations are performed when constructing or evaluating a regular expression. Here’s a breakdown of the precedence of common regular-expression operators:

  1. Concatenation (AB): Concatenation has the highest precedence. If two symbols or expressions are written together without an intervening operator, they are implicitly concatenated. For example, AB is interpreted as (A)(B).
  2. Closure/Repetition (A*, A+, A?): The repetition operators have the next level of precedence. These include * (zero or more occurrences), + (one or more occurrences), and ? (zero or one occurrence). They apply to the symbol or expression immediately preceding them. For example, A* means (A)*.
  3. Alternation/Union (A | B or A + B): Alternation or union has the lowest precedence among the common operators. It is denoted by | or +. If two expressions are separated by | or +, the alternation applies to the expressions on both sides. For example, A | B means (A) | (B).

Explicitly using parentheses can alter the default precedence, just as in arithmetic expressions. Parentheses group operations, ensuring that they are evaluated together. For instance, (AB)* means repeating the concatenation of A and B, not repeating just A and then concatenating B.

Understanding the precedence of regular-expression operators is crucial for creating accurate and effective regular expressions.

Finite Automata and Regular Expressions

Imagine you have two ways of describing languages (sets of words): one using a method called regular expressions (kind of like fancy search patterns), and the other using a method called finite automata (like simple machines that read and process words).

Even though these two methods seem different, it turns out they describe exactly the same types of languages, which we call “regular languages.” This is like saying two different ways of talking about the same thing.

We’ve already shown that certain types of finite automata (machines) all describe the same set of languages. Now, to complete the picture, we need to show that regular expressions also describe the same set of languages.

Here’s how we do it:

  1. If a language can be accepted by one of these finite automata (let’s say a deterministic finite automaton or DFA), then we can also describe that language using a regular expression. It’s like saying both methods can tell us the same things.
  2. If a language can be described by a regular expression, then we can also find a type of finite automaton (specifically, a non-deterministic finite automaton or NFA with epsilon transitions) that describes the same language. Again, it’s showing that both methods are saying the same stuff.
Fig 1: Plan for showing the equivalence of four different notations for regular languages

The diagram in Fig 1 summarizes all these connections. Each arrow from one type to another means that the languages described by the first method are also described by the second method. Since we can trace a path between any two methods in the diagram, it means that all four methods are basically describing the same set of languages. It’s like having different ways to talk about the same thing, and they all end up meaning the same.

From DFA’s to Regular Expressions

Imagine you have a special kind of machine called a DFA (Deterministic Finite Automaton), and you want to describe all the words it can understand using something called a regular expression.

Describing these words turns out to be a bit tricky. Here’s how we do it:

  1. We start by making expressions that describe paths in the DFA’s picture (like a map of how the machine moves between states).
  2. But here’s the catch: these paths can only go through certain states, not all of them.
  3. We start with really simple expressions that describe paths not going through any states (just jumping from one point to another).
  4. Then, step by step, we add more complexity to our expressions, allowing paths to go through larger sets of states.
  5. Finally, we end up with expressions that let paths go through any state in the DFA.

In other words, we build these expressions gradually, starting from the simplest paths and working our way up to describing all possible paths the machine can take.

These ideas are used to prove a theorem, which basically says we can always create a regular expression to describe all the words a DFA can understand. It’s like creating a special language pattern that matches exactly what the machine can process.

Theorem : If (L = L(A)) for some DFA (Deterministic Finite Automaton) (A), then there is a regular expression (R) such that (L = L(R)).

Theorem 3.4: If L = L(A) for some DFA A, then there is a regular expression R such that L = L(R).


  1. Understanding the DFA: Let A be a DFA that recognizes the language L.
  2. Defining Regular Expressions: We want to show that there exists a regular expression R that describes the same language L. To do this, we’ll build R to mimic the paths that the DFA A can take.
  3. States of the DFA: Consider the states of the DFA A. Let Q be the set of states in A, and let n be the number of states in Q.
  4. Construction of Regular Expression: For each pair of states p, q in Q, we want to find a regular expression Rpq that represents all paths from state p to state q.
    1. If there is a direct transition from p to q with label a, then Rpq = a.
    2. If there is no direct transition from p to q, but there is a path that goes through an intermediate state r, then we use the expression Rpq = Rpr ⋅ Rrq, where ⋅ denotes concatenation.
  5. Base Case: Start with Rpp for paths that do not pass through any state (Rpp is the regular expression for paths that stay in the same state).
  6. Inductive Step: Build up Rpq for paths passing through progressively larger sets of states until allowing paths to pass through any state.
  7. Regular Expression for the Entire Language: The regular expression R for the entire language L is constructed as the union of all Rpq for p, q in Q, representing all paths from any state to any other state.

Converting DFA’s to Regular Expressions by Eliminating States

Converting DFAs to Regular Expressions by Eliminating States

Process Overview:

  1. Start with a DFA:
    • We begin with a DFA that recognizes a certain language.
  2. Eliminate States:
    • Identify a state to eliminate.
    • For each pair of states p and q, find a regular expression Rpq representing paths from p to q without going through the eliminated state.
  3. Update Transitions:
    • Modify DFA transitions by replacing those involving the eliminated state with corresponding regular expressions.
  4. Repeat:
    • Continue eliminating states until only two states remain.
  5. Combine Expressions:
    • The transitions between the remaining two states now represent the final regular expression for the entire language.
  6. Example:
    • Let’s say you have a DFA with states q0, q1, and q2.
    • Eliminate q1: Find R02 representing paths from q0 to q2 without q1, and update transitions.
    • Eliminate q0 or q2: Repeat until only two states remain.
    • Combine the expressions between the remaining two states to get the final regular expression.

Converting Regular Expressions to Automata

Converting Regular Expressions to Automata

Conversion Process:

  1. Start with a Regular Expression:
    • We begin with a regular expression that describes a language.
  2. Regular Expression Components:
    • Break down the regular expression into its basic components, including symbols, concatenation (.), union (|), and closure (*).
  3. Constructing NFA Fragments:
    • For each component, construct a corresponding NFA fragment.
      • For symbols, create a simple NFA with two states (initial and accepting) and a transition between them.
      • For concatenation, connect the NFAs of the concatenated components.
      • For union, create NFAs for each component and connect them to a new initial state.
      • For closure, create an NFA for the component and add epsilon transitions for closure.
  4. Combining NFA Fragments:
    • Combine the NFA fragments to form the complete NFA for the entire regular expression.
      • Connect the accepting states of one fragment to the initial state of the next fragment based on the operation.
  5. Determinizing (Optional):
    • If needed, convert the NFA to a DFA using the subset construction algorithm.
  6. Final Automaton:
    • The resulting automaton now recognizes the language described by the original regular expression.

Applications of Regular Expressions

Regular expressions, often abbreviated as regex or regexp, find applications in various fields due to their powerful and flexible pattern-matching capabilities. Here are some common applications of regular expressions:

  1. Text Search and Manipulation:
  • Find and Replace: Text editors and word processors use regular expressions for searching and replacing text patterns.
  • String Operations: Programming languages and scripting tools use regular expressions for string manipulation.
  1. Data Validation and Verification:
  • Form Validation: Regular expressions are commonly used to validate user input in web forms, ensuring that data follows a specific format (e.g., email addresses, phone numbers).
  • Password Policies: Regular expressions can enforce password strength requirements.
  1. Data Extraction and Parsing:
  • Log File Analysis: Regular expressions help extract relevant information from log files.
  • Data Extraction: They are used to parse and extract data from structured or semi-structured text formats.
  1. Programming and Scripting:
  • Pattern Matching: Regular expressions are integral to pattern matching in programming languages, aiding in the identification of specific patterns within strings.
  • Data Processing: Regular expressions are used for complex text processing tasks in scripts and programs.
  1. Database Operations:
  • Pattern Matching Queries: Regular expressions can be used in database queries for pattern matching and filtering.
  • Data Cleansing: Regular expressions assist in cleaning and normalizing data stored in databases.
  1. Web Development:
  • URL Routing: Regular expressions are employed in defining patterns for URL routing in web applications.
  • Input Validation: They help validate and sanitize user input in web applications.
  1. Network Security:
  • Firewall Rules: Regular expressions are used to define rules for filtering and inspecting network traffic in firewalls.
  • Intrusion Detection Systems (IDS): Regular expressions aid in pattern matching for identifying potential security threats.
  1. Natural Language Processing (NLP):
  • Text Processing: Regular expressions play a role in tokenization, stemming, and other text processing tasks in NLP applications.
  • Pattern Recognition: They are used to identify specific linguistic patterns.
  1. Data Science and Analytics:
  • Data Cleaning: Regular expressions help clean and preprocess textual data in data science workflows.
  • Pattern Matching in Analytics: They are used for extracting specific patterns from datasets.
  1. File and Directory Operations:
    • File Renaming: Regular expressions assist in batch renaming files based on patterns.
    • Directory Searches: They are used for searching and filtering files within directories.

Regular expressions are a versatile tool that contributes to the efficiency and effectiveness of tasks across a wide range of domains. Their concise syntax for pattern matching makes them an essential part of many software development and data processing workflows.

Regular Expressions in UNIX

In UNIX-like operating systems, regular expressions (regex) are widely used for pattern matching and text manipulation in various command-line utilities. Here are some common ways regular expressions are used in UNIX:

1. grep Command:

The grep command searches for patterns in files or input streams.

  • Example: Search for lines containing the word “error” in a file.
grep 'error' filename

2. sed Command:

The sed (stream editor) command is used for text stream processing, supporting regex for substitution and other operations.

  • Example: Replace “apple” with “orange” in a file.
sed 's/apple/orange/g' filename

3. awk Command:

The awk command is a versatile text processing tool that uses regular expressions to match and manipulate data.

  • Example: Print lines where the second column contains the word “success.”
awk '$2 ~ /success/' filename

4. File Globbing:

File globbing is used in shell commands to match file names using wildcards.

  • Example: List all text files in the current directory.
ls *.txt

5. egrep and fgrep Commands:

egrep (or grep -E) and fgrep (or grep -F) provide extended and fixed-string matching, respectively.

  • Example: Search for lines with “apple” or “orange” using egrep.
egrep 'apple|orange' filename

6. find Command:

The find command searches for files and directories, and regex can be used for pattern matching.

  • Example: Find all .log files in the current directory and its subdirectories.
find . -name '*.log'

7. expr Command:

The expr command evaluates expressions, including basic pattern matching.

  • Example: Extract the first two characters from a string.
expr "$string" : '\(.\{2\}\)'

8. Shell Parameter Expansion:

Regular expressions can be used in shell parameter expansion for string manipulation.

  • Example: Remove the prefix “prefix_” from a variable.
echo $result

Regular expressions in UNIX provide a powerful means for searching, matching, and manipulating text data directly from the command line. They are essential for tasks ranging from simple text searches to complex text processing operations.

Algebraic Laws for Regular Expressions

Algebraic Laws for Regular Expressions

Closure (Kleene Star) Laws:

Idempotent Law: (R*)* = R*
Zero or More Concatenation: εR = Rε = R
Zero or More Union: R* + R* = R*

Concatenation Laws:

Associative Law: (RS)T = R(ST)
Identity Element: Rε = εR = R

Union Laws:

Associative Law: (R + S) + T = R + (S + T)
Commutative Law: R + S = S + R
Identity Element: R + ∅ = ∅ + R = R

Distribution Laws:

Left Distribution (Concatenation over Union): R(S + T) = RS + RT
Right Distribution (Union over Concatenation): (R + S)T = RT + ST

Complement Laws:

Double Complement Law: ¬(¬R) = R

Absorption Laws:

Union Absorption: R + RS = R
Concatenation Absorption: R(R + S) = R

De Morgan’s Laws:

Union De Morgan: ¬(R + S) = ¬R · ¬S
Concatenation De Morgan: ¬(RS) = ¬R + ¬S

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