Regular Expressions—The Full Story

“I define UNIX as 30 definitions of regular expressions living under one roof.”

Turing Award winner Donald Knuth said that, and it doesn’t stop there. All mainstream programming languages host regular expressions as a domain-specific language (DSL). To finally learn regular expressions thoroughly will provide you with a handy tool for the rest of your programming career.

Don’t get left behind. You can have the skills of knowing when and how to write clean, effective, maintainable regular expressions. This is what’s at stake here and you can become The Regular Expresssions Hero.

PART I: The Automaton

As you awake one morning from uneasy dreams, you find yourself transformed in your bed into a gigantic email spammer. To send unsolicited electronic mails, you write a regular expressions-based spider …. No, wait! We’re going vastly too fast now. Regular expressions-based spiders are still a few sections ahead.

To truly understand regular expressions, we must start under the hood. We must climb down into the automaton and watch the oily gears interact with each other and hear the plops and clicks.

The simplest kind of regular expressions core is a finite automaton (FA), which is a part of a young science that took shape in the 20th century. Don’t get horrified at the thought of states, symbols, and transition tables. We’ll discover this in baby steps, and the return on investment will be generous.

Two-up is a traditional Australian casino game.

Take a look at the directed graph above, which contains the following elements:

  • States: The small circles. We’re jumping from state to state. The only thing that’s certain is that at a particular time, exactly one state is the current one. The state with a small arrow pointing at it is the start state. All states surrounded by a ring are accept states.
  • Alphabet: Only two symbols exist in this alphabet: Heads and Tails. Now, you might wonder why they’re drawn multiple times. It’s because the same symbol can be part of many transitions.
  • Transitions: The long arrows between states are called transitions. They’re armed with at least one symbol from the alphabet. Transitions are the rules for how we can travel, e.g., we may go from the start state to the accept state in two transitions if we first get a tails, then another tails.

I call this type of directed graph a transition diagram, which is handy for us humans. It’s easy to draw a transition diagram. It’s also easy to read them and get a quick idea of ​​what they mean.

This graph describes how to win the final game of a Two-up night. Two-up is a traditional Australian casino game in which you win if you get heads. Twice. In a row. We all know that it’s really difficult to win the final game at a casino. Every time we win, there’s a human desire to bet the money we just won — until our wallets are empty.

Every time you toss, you follow one of the transitions based on whether you had heads or tails, from your current state to the next state. If you end up in the accept state — the state surrounded by a ring — you’ve won the final game.

Four different alphabets: coin, English lowercase, binary digits, and two dots.

Think about heads and tails. We say that they’re two symbols, but together they comprise the complete set of possible outcomes when you toss a coin once. We call a finite set of symbols an alphabet. So, heads and tails are the coin alphabet. What other alphabets can you think of? Here are some examples of alphabets:

  • A coin alphabet comprising the symbols H (heads), T (tails).
  • The English lowercase alphabet comprises the symbols a, b, c, … z.
  • The binary digits alphabet comprises the symbols 𝟶 and 𝟷.
  • The ASCII alphabet comprises 128 symbols (characters), including some that are not printable.
  • A Five-Emojis alphabet comprising the symbols 😜, 🤔, 🙄, 🤭, 🙃.
  • The Unicode alphabet comprises more than 100,000 symbols (characters) from hundreds of scripts.

Note that even Unicode, with more than 100,000 characters, is still finite.

A string or a word is a finite sequence of symbols from an alphabet. Here are some strings from the alphabets above:

  • HTTHHT from the coin alphabet.
  • artichoke and grxsttt from the English lowercase alphabet.
  • 𝟶𝟷𝟷𝟶𝟷, 𝟷𝟶, and 𝟷 from the binary digits alphabet.
  • 4$, a@a, and BIG from the ASCII alphabet.
  • 😜🤔🤔, 🙄🤔, and 🙄🤭🙃 from the Five-Emojis alphabet.
  • fänkål, apple, and teşekkürler from the Unicode alphabet.

We may construct an infinite number of possible strings from any alphabet consisting of at least one symbol, even though the alphabet itself is finite. If you’re not convinced about this, then imagine a string created with an arbitrary alphabet, e.g., 𝟷𝟶𝟷 created with the binary alphabet. We may concatenate another symbol — say, 𝟶 — at the end of our string. Voila, we have another string: 𝟷𝟶𝟷𝟶. This procedure can go on forever; thus, the number of strings that can be constructed from a finite alphabet is infinite.

Four different glyphs.

In Unicode, and sometimes in this essay, a symbol is called a grapheme, which is the smallest semantically distinguishing unit in a written language. How a grapheme actually is drawn — i.e., what it looks like — is called a glyph. Note that 𝒶 and 𝕒 are two different glyphs that represent the same grapheme in the English lowercase alphabet. Having said this, I’ll now tell you that in most cases, symbols will be called characters in this essay. Thus, I interchangeably use the words symbol, grapheme, and character to mean the same thing.

Every symbol has a unique index number.

Not in automata theory, but in regular expressions, alphabets must be an ordered list of symbols. This means that every symbol has a unique index number. In the ASCII alphabet, 𝙰 has an index of 65, 𝙸 has an index of 73, and 𝙺 has an index of 75. The numbers aren’t important in any way other than that they designate and order the symbols. No two symbols have the same index. From indices, we can make ranges, e.g., the ASCII range 𝚊–𝚏, i.e., the set 𝚊, 𝚋, 𝚌, 𝚍, 𝚎, and 𝚏. In Unicode, these indices are termed code points.

A language is a subset of all the strings that we possibly can construct from a specific alphabet. E.g., all binary strings with a trailing zero is a language: 𝟶, 𝟶𝟶, 𝟷𝟶, 𝟶𝟶𝟶, 𝟶𝟷𝟶, 𝟷𝟶𝟶, 𝟷𝟷𝟶 etc. We’ll see later that a regular expression defines a language. As a matter of fact, a directed graph defines a language as well.

You can be in only one country at a time. Suppose you’re traveling by train on a continent comprising eight countries. You always start your trips from your home country. With the right ticket, you can travel from the country in which you’re currently located to a neighboring country. You may also take a trip within a country, i.e., the journey starts and ends in the same country.

Buy a ticket booklet, and you can make it a tour, i.e., you can take several trips, but you’re forced to use the tickets in the order in which they’re placed in the ticket booklet. If the next ticket in the booklet isn’t valid for any trip departing from the country you currently are in, you’ve failed. You’ve also failed when you use the entire ticket booklet and end up in a country for which you don’t have a permanent residence permit. However, if you start in your home country, take a tour so that you use the entire ticket booklet, and end up in a country in which you do have permanent residence status — then you’ve succeeded! However, as I noted above, you can be in only one country at a time.

Now imagine an automaton that describes the system above:

  • Tickets are symbols.
  • A trip with departure and arrival in the same country is a loop.
  • The countries are the states of the automaton.
  • The ticket booklet is a string.
  • Your home country is the start state.
  • The countries where you have permanent residence are accept states.
  • A successful tour is a string (of symbols from your alphabet) that the automaton accepts.
  • A failed tour is a string that the automaton doesn’t accept.
  • The trips are termed transitions, and a tour is termed a walk.

We can visualize the automaton as a transition diagram. The states are drawn as rings. A small arrow points at the special start state, and all the accept states have double rings. There might be many accept states, but there can be only one start state. The directed arcs between the rings demonstrate the possible transitions that we may do. Each transition is decorated with one or more allowed input symbols.

This automaton accepts all even binary numbers, i.e., the strings of zeroes and ones ending in one or more zeroes.

The directed graphs — i.e., transition diagrams — that we’ve seen so far are easy for humans to read, as we quickly develop an intuitive idea of ​​how this system works. The problem is that computers cannot really compete with us when it comes to reading this kind of graphical information because it sometimes looks ambiguous to them. Furthermore, if the computer can’t understand the system we want to describe, then it can’t help us answer questions about the system’s nature and condition. Thus, the $64,000 question is: Can we find a notation that the computer understands?

The transition diagram to the left below describes all strings that don’t end with two blue dots. The alphabet seems to comprise two symbols: white and blue dots. We have three states — 𝙰, 𝙱, and 𝙲. The small arrow that points at 𝙰 indicates that 𝙰 is the start state. Both 𝙰 and 𝙱 have double rings; thus, they’re accept states. If we end up in one of these when we’ve fed the automaton a whole string, then the string doesn’t end with two blue dots.

Finite automaton: all strings that don’t end with two blue dots.

But now we want a notation for computers, such as a table. Computers love tables, as they use them as directives on how they should behave. To the right in the picture above is a transition table describing exactly the same automaton as the graph to the left. In the table’s left column, we have the three states — 𝙰, 𝙱, and 𝙲. A small arrow to the left of 𝙰 indicates that 𝙰 is the start state. 𝙰 and 𝙱 are underlined, i.e., they’re valid accept states. The only two symbols in this alphabet, the blue and white dots, are in the top row.

At the intersection of the states to the left and the icons at the top, we can see what state we’ll transition to next. Assume that we’re in state 𝙱, and we read a white dot from our input string. We can see in the table that we’d then end up in state 𝙰.

The intersection of the state B and the white dot symbol says: “Go to state A.”

Humans love graphs, but computers love tables. So, can we find a format suitable for both? What’s the ultimate interface between humans and computers, a language through which both can understand and express themselves? It’s text, of course. As we shall see, Steven Kleene created a text-based language — an algebra — through which we can describe the same automata that we describe with transition tables, as well as with transition diagrams. This algebra is termed regular expressions.

So, we’ve seen that directed graphs — so-called transition graphs — are easier for us humans to understand, compared with transition tables. We also outshine computers when it comes to interpreting these graphs. However, the opposite is true for non-deterministic finite automata.

Deterministic finite automata (DFA) have exactly one transition — i.e., one rule — for each unique pair of a state and an upcoming input symbol. Non-deterministic finite automata (NFA) have a less-rigid definition. Instead of providing a unique alternative for each state and symbol, we allow for multiple possible alternatives. This sounds ambiguous, doesn’t it? Here’s the catch: We’ll let the computer try out all options and see if any of them one works — hence, the nondeterministic descriptor. Here are some consequences and details:

  1. If we’re in a specific state and have a specific upcoming input symbol, then we specify a set of possible transitions.
  2. We have spontaneous transitions — denoted as ε — shifting from one state to another, without consuming any input symbol!
  3. The sets in (1) can be empty. When we’re in a specific state and have a specific symbol, we sometimes lack any possible transition away from there.

Computers want clear deterministic rules, e.g., an NFA that accepts a string if at least one possible walk starts in the start state, consumes the string symbols in order, and ends up in one of the automaton’s accept states.

Enough theory for now. Here are two examples. In the first, we built an automaton — depicted as an NFA (left) and a DFA — that accepts all strings in which no white dot can ever be preceded by a blue dot. Notice how much easier the NFA to the left is to read compared with the DFA to the right.

NFA and DFA: all strings in which no white dot can ever be preceded by a blue dot

In the second example, an automaton accepts any string that ends with a sequence of first one white, then one blue dot. Here, as well, the NFA is much easier to read (and write) than our equivalent DFA.

NFA and DFA: any string that ends with a sequence of first one white, then one blue dot.

You’ve probably already figured out that all the DFAs are also NFAs, but that not all NFAs are DFAs. However, what you may not have figured out is that if we take an arbitrary NFA, we can use a general algorithm to transform it into a DFA that accepts exactly the same language, i.e., the same set of strings.

Venn diagram: All DFAs are NFAs. Some NFAs are DFAs.

Thus, DFAs and NFAs solve the same problems. A DFA is always faster, or at least as fast as an NFA. Why? An intuitive explanation is provided below.

Automaton for Two-up. Symbol H represents heads. Symbol T represents tails.

The NFA’s description left us with some question marks. If alternative transitions exist, which ones will the computer try first? How does it know that it has tried all possible walks?

The computer saves a marker each time more than one optional transition occurs from the current state. One marker is saved in the input string and another in the automaton, indicating in what automaton state it was and how much of the input string it had consumed. Then it continues, and the computer may backtrack to these markers later if it happens to reach a dead end, i.e., part of the consumed string may be re-consumed in a new walk.

The picture above describes the automaton for Two-up. We flip a coin twice in each game. If the final two games give us heads and heads, then we’ve succeeded. Suppose we flip the following sequence: heads, tails, heads, and heads (H-T-H-H). We then end up in the accept state. Here’s how the computer works this:

Another example is H-H-H-T, in which we don’t reach the accept state:

The nondeterministic finite automaton contains decision points in which there are two options: Either let the current state consume one more symbol of the input string — i.e., loop — or else transition to another state. Thus, we have two mutually exclusive strategies that the computer may use, which I term greedy and reluctant. Some people say lazy rather than reluctant.

  • With the greedy strategy, we try — as long as possible — to be in a self-transitioning loop of the current state. In this strategy, each state is greedy and tries to consume as much as possible from the input string.
  • With the reluctant strategy, we try — as quickly as possible — to transition from the current state to another state, ultimately to arrive early in an accept state. Each state is reluctant to consume more of the input string and, if possible, tries to pass the initiative to the next state instead.

Below is an NFA that accepts all non-empty sequences of blue dots.

Suppose that our automaton is fed with a string comprising two blue dots (●-●). With the greedy strategy, it would be:

However, if consume the same string with the reluctant strategy it would instead be:

The loop may be longer, such as in this automaton, which accepts all strings with alternating blue and white dots, in which the first and last dots are blue.

Imagine that the input is a white dot surrounded by two blue dots (●-○-●). With the greedy strategy, the loop tries to consume as much as possible:

However, the reluctant algorithm would instead give us:

The conclusion is that backtracking is a waste of time for the computer. If our algorithm finds the best walk in the first shot, then our NFA will be as fast as a DFA. We’ll see later in this essay that we may influence exactly that.

The next section explains how an NFA can be translated into a DFA. You may skip this if you’re not utterly fascinated with the most advanced parts of regular expressions, but I think you are.

DFAs are faster than NFAs, so why bother with NFAs anyway? As we’ll see later in this essay, NFAs can be extended with features that are either impossible or at least very difficult to implement in DFAs. When we don’t use these features, DFAs are superior. The good news is that any NFA can be transformed into a DFA that accepts exactly the same set of strings. Here’s an example of how to do this.

Below, we have an NFA that accepts every integer, i.e., digit sequences. The integers may be preceded by a plus or minus sign. We have two constraints:

  1. Only the integer zero can start with the digit 𝟶.
  2. The integer zero can’t be preceded by a plus or minus sign.
Nondeterministic finite automaton.

Remember that the symbol ε is used to indicate that we can make a spontaneous transition, i.e., a transition without consuming any symbol from the input. To translate an NFA into a DFA, we need something called ε-closure. The ε-closure for a state is the set of states that can be reached with ε-transitions. Thus, the ε-closure of 𝙲 above is {𝙲, 𝙳}.

Considering that 𝙰 is our NFA’s start state, the ε-closure of 𝙰 — which is {𝙰, 𝙱} — becomes the start state of our new DFA that we’re creating. From the DFA state {𝙰, 𝙱}, we can get to DFA state {𝙳} if we read the symbol 𝟶, to DFA state {𝙱} if we read a plus or minus sign, and to DFA state {𝙲, 𝙳} (the ε-closure of 𝙲) with symbols in the range of 𝟷–𝟿. Thus, the first attempt in our DFA transition table is:

The question marks will soon be replaced by ε-closures that we can reach from {𝙱} and {𝙳}.

Let’s continue with {𝙱}, from which we can reach {𝙲, 𝙳} if we read a digit in the 1–9 range. That’s it. No other transitions from DFA state {𝙱} exist:

The pillow symbol, ¤, means the empty set { }, which is for sure a set, even though it doesn’t include any of our NFA-states. We use it e.g. at the intersection above between the DFA state {𝙱} and the symbols +- to visualize that reading plus or minus while in {𝙱} is a failure. If that happens then we’ll simply not accept the input string — it’s not an integer — and therefore transition to the sink state.

Perhaps you’re now wondering which of these new states are accept states. The answer is that any DFA state containing an accept state from the NFA is an accept state. So far, {𝙳} and {𝙲, 𝙳} are accept states because they contain 𝙳 which was the only accept state in our NFA.

We continue to fill the transition table the same way:

We now have four states, plus the sink state. The latter is where we end up when we know that this isn’t an integer. Below is a transition diagram for our DFA. Note that from each and every state, exactly one transition exists for each and every symbol in our alphabet {+, -, 𝟶, 𝟷, 𝟸, 𝟹, 𝟺, 𝟻, 𝟼, 𝟽, 𝟾, 𝟿}, i.e., this finite automaton is deterministic.

Deterministic finite automaton.

To summarize:

  • Transitioning in NFAs may include alternatives, but in DFAs the path is deterministic.
  • All DFAs are NFAs, but all NFAs are not DFAs.
  • We have an algorithm to transform any given NFA to a DFA.

A problem with finite automata is that they don’t have any memory. Once they’re in a state, they have no idea how they got there. Wise people invented the stack. When we add a stack to a finite automaton, it becomes very powerful, but also quite complicated. Thus, it’s not a finite automaton anymore, of course — it’s a pushdown automaton.

We have seen that the finite automaton reads an input string serially. Every new symbol read from the input string initiates a transition in the automaton from the current state to a new state (the new state may be the same as the current state, i.e. looping). If we’re in an accept state when we’ve read the whole input string, then we have a match. Bingo! However, in a pushdown automaton, we also add a stack, i.e., a data structure operating last in, first out (LIFO).

Supported by the stack, our pushdown automaton iterates in two steps. For each iteration, our automaton may read a symbol from the input string, pop a symbol from the top of the stack, or both:

  • First Step Option 1: Read a symbol from the input string.
  • First Step Option 2: Pop a symbol from the top of the stack.
  • First Step Option 3: Read a symbol AND pop a symbol.

After the first step, our pushdown automaton needs to make a decision based on the current state, the symbol read from the input string, and/or the symbol popped from the stack. The possible decisions are to initiate a state transition, push a symbol onto the top of the stack, or both:

  • Second Step Option 1: Initiate a state transition.
  • Second Step Option 2: Push a symbol onto the top of the stack.
  • Second Step Option 3: Initiate a state transition AND push a symbol.

If we end up in an accept state when the whole input string is consumed (read) and the stack is empty at that time, then we have a match. Pushdown Bingo!

Pushdown automaton.

The picture above depicts a pushdown automaton that matches any input with the same number of blue and white dots — something impossible to describe with a finite automaton. This automaton has two states: a start state and accept state. The four icons in the middle indicate that a dot is popped or pushed from the stack. Suppose that the input is blue-white-white-blue:

  1. Read the blue dot and make a transition from the start state, through the third stack icon from the top — i.e., a blue dot is pushed to the stack — and back to the start state.
  2. Read the white dot and make a transition from the start state, through the first stack icon — i.e., a blue dot is popped from the stack — and back to the start state. Now the stack is empty.
  3. Read the white dot and make a transition from the start state, through the second stack icon from the top — i.e., a white dot is pushed to the stack — and back to the start state.
  4. Read the blue dot and make a transition from the start state, through the fourth stack icon — i.e., a white dot is popped from the stack — and to the accept state. Now the stack is empty again, and all the input is read. It’s a match.

You might suspect by now that the pushdown automaton gives us a tool for all kinds of recursive implementations. Now that you understand how, I’ll go back to finite automata in all explanations wherever possible. It’s very important to truly understand how, for example, backtracking and greediness impact performance and also what input will be matched. You don’t need a complicated pushdown automaton to learn this. We’ll stick to finite automata in this essay and simultaneously remember that many operators in modern regular expressions automata rely on a stack, which is why modern regular expressions cores are implemented as pushdown automata.

By the way, did you notice the corner case bug in the pushdown automaton above? Doesn’t an empty input string have the same number of blue and white dots?

PART II: Two Operations and One Function

Where nothing else is said, examples are written in Ruby. You may install Ruby on your computer and then start IRB (Interactive Ruby Shell) in a terminal to try out the code. You may also run IRB online without any installation.

In Part 1 of this essay (The Automaton), we observed that the same problem could be expressed in a human-friendly way (a graph) and a computer-friendly way (a table). We, the humans, are creative. We formulate new problems that we want to solve. On the other hand, computers solve problems at a high speed, and they never make mistakes — at least not if we implement the correct algorithms. How do we find a notation that’s easy for people to express themselves with and also easy for computers to interpret?

Directed graphs, i.e., transition diagrams, give us a good overview, as long as they don’t contain too many details. We can focus on a piece of the graph, and we may also zoom out to see the big picture as an abstract description. When we walk around in the graph, we’re in an area. Our brains thrive like a fish in water with this way of presenting and digesting information.

The computer excels at having full control over every detail of a large amount of data, as it doesn’t need any abstraction to understand the big picture. It just wants deterministic instructions on how to behave in its current state. If we provide all possible transitions in tabular form, the computer will remain happy, but for us humans, it’s difficult to get an overview of a transition table. We will wonder what problem this table is solving.

When it comes to details, the ultimate interface between humans and computers has long proven to be text. Visual programming languages are launched recurrently in our industry, but the market never takes off. We still must teach the computer all the details about how it should behave. Too many details make the graphical programs as difficult to interpret for us humans as they are for computers. Programming languages such as C, Java, C#, and Ruby are completely text-based and still are uncrowned queens of expression for us programmers.

Even a text-based programming language needs an effective and simple notation. In this part of the essay, we’ll see that regular expressions (one regex, many regexes) is a language with only two operations and one function. Sounds powerful, doesn’t it? Two operations and one function are sufficient to describe all the transition diagrams and transition tables that can be constructed. It’s enough to describe every possible DFA and NFA in text form.

The regular expressions pioneers.

Regular expressions is a programming language with which we can specify a set of strings. Supported by only two operations and one function, we can be very concise. A non-concise alternative would be to list all the strings included in the set. Where does this regular expressions language come from, why is it called regular, and how does it differ from regex?

The story begins with a neuroscientist and a logician who together tried to understand how the human brain could produce complex patterns using simple cells that are bound together. In 1943, Warren McCulloch and Walter Pitts published “A logical calculus of the ideas immanent in nervous activity” in the Bulletin of Mathematical Biophysics 5:115–133. Although it was not the objective, it turned out that this paper greatly influenced computer science in our time. In 1956, mathematician Stephen Kleene took McCulloch and Pitts’ theories one step further. In the paper “Representation of events in nerve nets and finite automata” in the Annals of Mathematics Studies, Number 34, Kleene presented a simple algebra, and somewhere along the line, the terms regular sets and regular expressions were born. As mentioned above, Kleene’s algebra had only two operations and one function.

In 1968, Unix pioneer Ken Thompson published the article “Regular Expression Search Algorithm” in Communications of the ACM (CACM), Volume 11. With code and prose, he described a regular expression compiler that creates IBM 7094 object code. Thompson’s efforts did not end there. He also implemented Kleene’s notation in the editor QED. The value was that users could do advanced pattern matching in text files. The same feature appeared later in the editor ed.

To search for a regular expression in ed, the user wrote 𝚐/<𝚛𝚎𝚐𝚞𝚕𝚊𝚛 𝚎𝚡𝚙𝚛𝚎𝚜𝚜𝚒𝚘𝚗>/𝚙. The letter 𝚐 meant global search, and p meant print the result. The command — 𝚐/𝚛𝚎/𝚙 — resulted in the stand-alone program grep, released in the fourth edition of Unix in 1973. However, grep didn’t have a complete implementation of regular expressions, and it was not until 1979, in the seventh edition of Unix, when we were blessed with Alfred Aho’s egrep (extended grep). Now the circle was complete. The egrep program translated any regular expressions into a corresponding DFA.

Larry Wall’s Perl programming language from the late 1980s helped regular expressions become mainstream. Perl integrated regular expressions seamlessly, even with regular expression literals. Perl also added new features to regular expressions. The language was extended with abstractions, syntactic sugar, and also some brand new features that may not even be possible to implement in finite automata. This raises the question of whether modern regular expressions can be called regular expressions. Perhaps we can override that discussion if we use the term regex instead of regular expressions when we refer to the not-so-regular regular expressions?

An alphabet is a finite, nonempty set of symbols, i.e., at least one symbol and a limited number. Here are some examples of alphabets:

  • The binary alphabet {0, 1} contains only two symbols: 1 and 0.
  • The American Standard Code for Information Interchange (ASCII) contains 128 symbols, only 95 of which are printable. The others, such as Backspace (ASCII 8), are also symbols in our definition.
  • The set of 95 printable symbols in ASCII is also an alphabet.
  • Unicode contains over 100,000 symbols that include Arabic, Chinese, Latin, and many other types. However, even if it’s a very large number, it’s still a finite number of symbols. Therefore, Unicode is an alphabet.
  • The set of the Cyrillic symbols in Unicode is an alphabet.
  • We can construct an alphabet comprising a blue dot and a white dot, i.e., an alphabet comprising two symbols, just like the binary alphabet.

Did you, by any chance, know that the word alphabet comes from alpha and beta, the first two letters of the 3,000-year-old Phoenician alphabet, and that alpha meant ox and beta meant house? However, a specific order of the symbols doesn’t make an alphabet unique. Thus, {a, b} is the same alphabet as {b, a}. The empty set — the set comprising zero symbols — is by definition not an alphabet.

A string is a finite ordered sequence of symbols chosen from an alphabet. Here are some examples of strings:

  • 0110 and 1001 are two strings from the binary alphabet.
  • The strings kadikoy and uskudar are constructed with symbols from the ASCII alphabet.
  • футбол is a string from the Cyrillic alphabet.
  • اللوز is a string from the Arabic alphabet.
  • From any alphabet, it’s possible to create an empty string, i.e., a string comprising zero symbols. By convention, we denote that string with the character ε.

The very same symbol can occur multiple times in a string, but the order is significant, e.g., gof isn’t the same string as fog.

A language is a countable set of strings from a fixed alphabet. The restriction is that an alphabet is finite. A language may well include an infinite number of strings. Here are some examples of languages:

  • All binary numbers ending in 0, e.g., 10, 100, 110, etc.
  • All palindromes — strings that are the same forward and backward — constructed from ASCII symbols.
  • All strings with an even number of symbols from Unicode.
  • A finite set {dog, cat, bird}.
  • An empty set, i.e., a language with no strings. Such an alphabet is by convention denoted as Ø.
  • The language {ε}, which comprises only one symbol: the empty string ε. (Note that {ε} is very different from Ø, for example in string cardinality.)

Our focus in this essay is on regular expressions. A regular expression is an algebraic description of an entire language. Not all languages can be described using regular expressions. For example, no regular expression — without non-regular extensions — can describe the language comprising all palindromes created with symbols from ASCII. However, I’ll try to convince you that regular expressions are ridiculously easy to learn and amazingly powerful to use.

Regular expressions starts with two basic rules:

(1) Empty String Language. The empty string ε is a regular expression that describes a language comprising only one string, which happens to be the empty string.

(2) Single Symbol Language. If a symbol, 𝚊, is a member of an alphabet, then a is a regular expression. It describes a language that has the string “𝚊” as its only member.

That was easy, wasn’t it? In addition to these two basic rules, we have two operations and one function that we’ll cover with another four rules below. Out of pure laziness, we have conventions for precedence between these operations and the function. More on this later.

First, you should try the two basic rules in the Interactive Ruby Shell (IRB). Either you install Ruby on your computer and start IRB in a shell, or you may use an online version of IRB.

With IRB, we write regular expressions between two dashes. For example, you may write the regular expression a as /a/. IRB can help us figure out whether a string is in a language described by a particular regular expression:

Only when IRB returns the entire string is it matched by the regular expression. The number sign # (hash tag) and everything to the right of it is ignored by the computer. We can put messages to humans there, for example what data we expect to match.

Now that we have two basic rules, we’d like to add four more. We then can build more advanced regular expressions recursively from very basic regular expressions. The first three rules (№3–5) describe regular expressions’ only two necessary binary operations and one and only necessary function. The fourth rule (№6) deals with parentheses:

(3) Alternation. If 𝚙 and 𝚚 are regular expressions, then 𝚙|𝚚 also will be a regular expression. The expression 𝚙|𝚚 matches the union of the strings matched by 𝚙 and 𝚚. Think of it as either 𝚙 or 𝚚.

(4) Concatenation. If 𝚙 and 𝚚 are two regular expressions, then 𝚙𝚚 also will be a regular expression. Note that the symbol for concatenation is invisible. Some literature uses × for concatenation, e.g., 𝚙×𝚚. The expression 𝚙𝚚 denotes a language comprising all strings with a prefix matched by 𝚙, directly followed by a suffix matched by 𝚚 — and nothing between the prefix and suffix.

(5) Closure. If 𝚙 is a regular expression, then 𝚙* also will be a regular expression. This is the closure of concatenation with the expression itself. The expression 𝚙* matches all strings that may be divided into zero or more substrings, each of which is matched by 𝚙.

(6) Parentheses. If 𝚙 is a regular expression, then (𝚙) will be a regular expression as well, i.e., we can enclose an expression in parentheses without altering its meaning.

In addition to these rules, we’ll add some convenience rules for operator precedence shortly. They’re not necessary, but allow us to write shorter and more readable regular expressions. Quite soon, you’ll also see real regular expression examples based on these two operations and one function. It might be difficult to imagine that these six rules are sufficient for writing every possible regular expression in the world.

Do you remember George Bernard Shaw’s quote “The golden rule is that there are no golden rules”? What about Mark Twain’s “It is a good idea to obey all the rules when you’re young just so you’ll have the strength to break them when you’re old”? This is exactly how we should think. For now, these rules are all we need, but modern regex automata contain powerful functions, e.g., a back reference and lookarounds. To implement these functions, we need more than these six rules, but until we’re there, it’ll be very useful to think of regular expressions as a system comprising only these six rules.

Thus, regular expression is a mathematical theory, and modern regex automata are based on a super set of this theory. With the help of the theory, we can prove the following:

  • For each regular expression, we may construct at least one DFA and at least one NFA, so that all three (regular expression, DFA, and NFA) solve the same problem.
  • For every finite automaton — deterministic (DFA) as well as non-deterministic (NFA) — we may write a regular expression, so that both (automaton and regular expression) solve the same problem.

Solving a problem here means determining whether a string is part of a language, i.e., a specific set of strings. The proofs mentioned above aren’t reproduced in this essay, but they’re easily found in every textbook on automata theory. The beauty of this analogy between regular expressions and finite automata is that I can explain several key features of regular expressions for you, with the help of graphs of finite automata. Furthermore, in almost every mainstream programming language, there is a compiler that translates our handwritten regular expressions into computer-friendly finite automata, or possibly more advanced pushdown automata.

Using Rule №2 (Single Symbol Language) and №4 (Concatenation), any possible sequence of symbols from our alphabet may be written as a regular expression. Rule №2 said that if the symbol 𝚊 is in the alphabet, then “𝚊” is a regular expression. Rule №4 said that if 𝚙 and 𝚚 are two regular expressions, then the concatenation 𝚙𝚚 is a regular expression as well. The concatenation symbol itself is invisible. Just write the two regular expressions right after each other:

We usually use some handy terms for substrings:

  • A prefix is the substring we leave if we remove zero or more symbols from the end of a string. The strings “m”, “mo”, “mod”, and “moda” are all prefixes of the string “moda”. Even the empty string ε is a prefix of “moda”.
  • A suffix is the substring that remains if we remove zero or more characters from the beginning of the string. The strings “moda”, “oda”, “da”, “a” and ε are all possible suffixes of the string “moda”.
  • A substring is what we have left if we remove a prefix and a suffix from a string. Note that even ε is a valid prefix and suffix. Symbols in the substrings must be consecutive in the original string. The strings “od” and “moda”, but not “mda”, are substrings of “moda”.

For any regular expression 𝚙, it’s true that ε𝚙 = 𝚙ε = 𝚙; thus, we say that the empty string ε is the identity operand under concatenation. Concatenation isn’t commutative because 𝚙𝚚 isn’t equal to 𝚚𝚙, but it’s associative because for any regular expressions, 𝚙 and 𝚚, it’s true that 𝚙(𝚚𝚛) = (𝚙𝚚)𝚛.

If we view concatenation as a multiplication product, then regular expressions also support exponentiation, i.e., 𝚡ⁿ. We write the exponent enclosed in brackets {𝚗} to the right of the regular expression:

This is obviously just syntactic sugar. We can unfold all regular expressions that we may write using the exponential operation. More shortcuts can be used for finite repeated concatenations:

We’ll soon see that concatenation of two regular expressions isn’t the same as concatenation of two literal strings. This is obvious when we recall that a regular expression corresponds to a set of strings, not a single literal string. For example, if 𝚙 = {𝚊, 𝚋} and 𝚚 = {𝚌, 𝚍}, then 𝚙𝚚 = {𝚊𝚌, 𝚊𝚍, 𝚋𝚌, 𝚋𝚍}

Venn diagram showing exclusive disjunction (XOR).

From Rule №2 (Single Symbol Language) and №3 (Alternation), we may define paradigms — a number of possible patterns, i.e., we add two or more languages by applying the union set operation to them. The union of the sets {𝚊, 𝚋} and {𝚌, 𝚍} is {𝚊, 𝚋, 𝚌, 𝚍}; thus, it’s all the elements that are either in one or more of the sets that we unite. In Boolean logic, we call this the inclusive or. In regular expressions, it’s called alternation and is written with a vertical bar: |. Here are some examples:

Note that when more than one alternative is correct, most regex dialects select the leftmost alternative, but exceptions to this rule exist. A regex automaton based on DFA or POSIX NFA selects the longest alternative. However, most other regex automata select the leftmost.

Can you write a regular expression that matches all binary strings of length one? The binary alphabet is {𝟶, 𝟷}. Considering that few binary strings are of length one, you may even list them: {𝟶, 𝟷}. The regular expression with alternation then becomes /𝟶|𝟷/:

Four binary strings of length two {𝟶𝟶, 𝟶𝟷, 𝟷𝟶, 𝟷𝟷} can be captured with /𝟶𝟶|𝟷𝟶|𝟶𝟷|𝟷𝟷/:

Did you notice that we used concatenation in the regular expression above? (Can you see the invisible concatenation symbol between the two binary digits in the regular expression? If not, maybe you should make an appointment with an optometrist — or maybe not. Not even an optometrist can help you see invisible symbols, which is what the concatenation symbol is.) Each of the binary strings of length two comprises two concatenated binary strings of length one. Considering that concatenation takes precedence over alternation, we didn’t need any parentheses.

Alternation is commutative, i.e., for two regular expressions, 𝚙 and 𝚚, it holds that 𝚙|𝚚 = 𝚚|𝚙. It’s also associative: 𝚙|(𝚚|𝚛) = (𝚙|𝚚)|𝚛. An interesting and very useful fact is that concatenation distributes over alternation, i.e., for all regular expressions 𝚙, 𝚚, and 𝚛, it’s true that 𝚙(𝚚|𝚛) = 𝚙𝚚|𝚙𝚛 and (𝚙|𝚚)𝚛 = 𝚙𝚚|𝚙𝚛. A consequence of this is that /(𝟶|𝟷)(𝟶|𝟷)/ = /(𝟶|𝟷)𝟶|(𝟶|𝟷)𝟷/ = /𝟶𝟶|𝟷𝟶|𝟶𝟷|𝟷𝟷/. So, another way to match any binary strings of length two is:

Of course, the brackets are necessary because concatenation takes precedence over alternation. We also may add the empty string ε as one of our alternatives:

Given all this, here’s an opportunity to convince you that all regular expressions have an infinite number of synonyms. A revealing example: (𝚊) = (𝚊|) = (𝚊||) etc.

All finite languages can be described using regular expressions. We simply list the strings as an string1|string2|string3|… alternation. We may also use regular expressions to describe some languages that have an infinite number of strings. To achieve that, we use a function we call the kleene star, named after the aforementioned American mathematician Stephen Cole Kleene. If 𝚙 is a regular expression, then 𝚙* is the smallest superset of 𝚙 that contains ε (the empty string) and is closed under the concatenation operation. Huh? Read that definition again, then try this definition: It’s the set of finite length strings that can be created by concatenating strings that match the expression 𝚙. If 𝚙 can match any string other than ε, then 𝚙* will match an infinite number of possible strings.

The real name of the typographic symbol (glyph) that denotes the kleene star is an asterisk, a Greek (not geek) word that means, appropriately enough, little star. Normally, this glyph has five or six arms, but its original purpose was to describe birth dates in a family tree, and it had seven arms. This is a very popular symbol. In Unicode, loads of asterisk variants are adorned with interesting names, e.g., Heavy Teardrop-Spoked Pinwheel Asterisk and Balloon-Spoked Asterisk. Many fields have assigned their own meanings to the asterisk. In typography, it means a footnote. In musical notation, it may mean that the piano’s sustain pedal should be lifted. On our cell phones, we use the asterisk to navigate menus in touch-tone systems. Thus, no worldwide consensus interpretation of the asterisk * exists, but in this essay, it always means the function kleene star.

Do you want to see a simple example? Of course, you do! The concatenation closure of one single symbol — e.g., 𝚊 — is /𝚊*/ = { ε, 𝚊, 𝚊𝚊, 𝚊𝚊𝚊, … }. Want to see a more academic example? The concatenation closure of the set comprising solely the empty string ε is — ♫ ta–da♫ — well, the set comprising solely the empty string, i.e., ε* = ε. Want to see a more complicated example? /(𝟷|𝟶)*/ = { ε, 𝟶, 𝟷, 𝟶𝟷, 𝟷𝟶, 𝟶𝟶𝟷, 𝟶𝟷𝟶, 𝟶𝟷𝟷, … }. Yay, it’ll match every possible binary string, i.e., a language comprising infinitely many strings. Can you write a regular expression that matches all binary strings that contain at least one zero? Or all binary strings with an even number of ones?, I urge you to do some trial-and-error in IRB.

The kleene star function is pronounced KLAYnee star/ˈkleɪniː stɑr/. (Don’t think that the last thing in that sentence was a regular expression, even though it’s flanked by the oblique slanting line punctuation mark we usually call a slash. It happens to be a representation of speech sounds in written form. But wait, it’s actually a regular expression with symbols from the IPA alphabet, and it’ll match nothing but the exact string: ˈkleɪniː stɑr).

The kleene star function is unary, i.e., it takes only one operand. The operand is the regular expression to the left, which allows us to say that it’s a postfix operator. The operand comes first, and the operator (the asterisk) comes last. It takes precedence over concatenation and alternation, and it’s associative. The latter means that if two operators of the same precedence are competing, then the operator closest to the operand wins. Considering that 𝚙** = 𝚙*, we say that the kleene star is idempotent. I want to emphasize again that 𝚙* = (𝚙|)*, i.e., the empty string ε is always present in a closure. We’ll see later that a very common — and disastrous — mistake is to forget that important fact.

Here are some possible answers to the questions above:

The positive closure operation + and the at least n operation {n,} are abstractions for expressing infinite concatenation. We will examine them in more detail later in this essay.

You might be tempted to read the following regular expression as “(third or fifth) row”:

Unfortunately, as you can see, it’s more like either “third” (only) or “fifth row” because of something called order of operations or operator precedence. The invisible operator for concatenation takes precedence over the alternation operator |.

To oil these wheels, we now add parentheses to our three operators. In a regular expression, the subexpression enclosed in parentheses gets the highest priority:

Note that the parentheses are metacharacters, not literals. They won’t match anything in the subject string.

You’ve probably guessed by now that we also may nest parentheses:

We must understand three things to predict in what order and with what operands the regular expression automata will execute the operators:

  • Operator precedence is an ordered list of all operators. It tells us which operator must be executed before another operator in a regular expression. Several operators can have the same priority. In mathematics, the terms inside parentheses have the highest priority. Multiplication and division have a lower priority, and addition and subtraction have the lowest. This is why 6+6/(2+1) = 8.
  • Operator position indicates where the operands are located in relation to the operator. The position can be prefix, infix, or postfix. If the operator is prefix, then the operand resides to the right of the operator, as the mathematical unary minus sign in, e.g., -3. An infix operator has an operand on each side, as in addition, e.g., 1+2. Finally, a postfix operator stands to the right of its operand, as the exclamation point that represents the faculty operator, e.g., 5!.
  • Operator associativity tells us how to group two operators on the same precedence level. An infix operator can be right-associative, left-associative, or non-associative. In mathematics, the infix operations addition and subtraction have the same precedence. Considering that both are left-associative, the following equation holds: 1–2+3 = (1–2)+3 = 2. Prefix or postfix operators are either associative or non-associative. If they’re associative, we start with the operator closest to the operand. A non-associative operator can’t compete with operators of the same precedence.

Here’s a table of the regular expressions operators we’ve studied so far:

If you find that difficult to remember, then try to memorize the mnemonic SCA, which stands for Star-Concat-Alter, i.e., the order of precedence in regular expressions.

Currently, we have three operators and a small framework. After all this theory, you might wonder whether it’s possible for us to solve any problems. Of course we can, and here are some examples:

All binary strings with no more than one zero:

All binary strings with at least one pair of consecutive zeroes:

All binary strings that have no pair of consecutive zeroes:

All binary strings ending in 01:

Remember that concatenation takes precedence over alternation when we match all binary strings not ending in 01:

All binary strings that have every pair of consecutive zeroes before every pair of consecutive ones:

See if you can find even better regular expressions that solve these problems. Remember that there’s an infinite number of synonyms for each regular expression.

How to design finite automata for concatenation (top left), alternation (top right), kleene star (bottom left), and a combined pattern (bottom right).

For each regular expression — and I mean the two operations, the function, and the six recursive rules style — a finite automaton accepts exactly the same strings as the regular expression. Considering that this isn’t a university mathematics textbook, I’ll show you some inductive reasoning on this and not a formal proof.

Thus, the hypothesis is that for an arbitrary regular expression 𝚙, we can create a finite automaton that has exactly one start state, no transitions toward the start state, no transitions leaving the accept state, and that accepts the exact same strings matched by 𝚙.

We’ll need the following three automata:

  • The empty string ε is a regular expression corresponding to a finite automaton with a start state, a transition that accepts the empty string ε and leads from the start state to an accept state. We’ll call this an ε-transition.
  • The empty set Ø is equivalent to a regular expression that can’t match any single string — not even the empty string ε. It’s the same as a two-state automaton, with no single transition. One state is the start state, and the other is the accept state. Unfortunately, they’re not linked.
  • A regular expression that only matches the symbol 𝚋 corresponds to a finite automaton with two states: start and accept. There’s a transition from start to accept, and this transition only accepts the symbol 𝚋.

All three finite automata above have two states. One is the start state and the other is the accept state. The differences between these three automata are that the first one has an ε-transition from start to accept, the second has no transition, and the third has a b-transition.

Imagine that we have two regular expressions, 𝚙 and 𝚚, corresponding to two finite automata, 𝚜 and 𝚝, respectively:

  • Concatenation of 𝚙 and 𝚚 means that we first match a string with 𝚙, directly followed by a string matched with 𝚚. To create this finite automaton, we first add ε-transitions from every accept state in 𝚜 to the start state of 𝚝, then we remove the accept status from all accept states in s. We also withdraw the start status of the start state in 𝚝, i.e., 𝚜 cannot accept a string, and 𝚝 cannot start reading. However, 𝚜 and 𝚝 now are connected, and we may start in 𝚜 and accept in 𝚝. In the top left graph above, this is the method we use to transform the two regular expressions, /○/ and /●/, into /○●/.
  • Alternation of 𝚙 and 𝚚, i.e., 𝚙|𝚚, is like a finite automaton with a new start state that has ε-transitions to all start states of 𝚜 and 𝚝. The new finite automaton also has a new accept state reached with ε-transitions from all accept states of 𝚜 and 𝚝. Thus, the former start and accept states of 𝚜 and 𝚝 aren’t start and accept states in our new automaton. In this way, we transform the two regular expressions, /○/ and /●/, into /○|●/ in the top right graph.
  • Kleene star is the concatenation closure. Assume that 𝚙 = 𝚚*. Then 𝚜 is the finite automaton that we get when we take 𝚝 and add two states and four transitions as follows: One new state is the start state, and the other is an accept state. All accept states of 𝚜 lose their accept status and instead get an ε-transition to the new accept state. We add two ε-transitions from the new initial state — one to the old start state and one to the new accept state. Furthermore, we insert one ε-transition from each of the old accept states to the old start state. Using this method helps us transform the regular expression /●/ into /●*/ in the lower left graph.

Now look again at the first three graphs in the picture above. Take your time, then take a deep breath and ponder whether you can transform any arbitrary regular expression into a finite automaton. Finally, assess the last picture, in which the regular expression /(○|●●)*/ is depicted as a graph using the methods described above. Does it feel reasonable?

Now that we’ve created a small programming language, i.e., regex, and are comfortable with all the theory, we can examine what separates this language from other programming languages. Three special, but not unique, regex characteristics are that it’s declarative, it’s a domain-specific language, and it (usually) has no whitespace, code comments, or separators.

Declarative programming languages express logic in a calculation without specifying its control flow. In regex, we declare what the solution is, but not how the computer can get there. This can be compared with imperative languages such as C, Java, C#, and Ruby, in which we instruct the computer to first do this, then do that, and finally we inspect the results. Thus, in regex, we describe a pattern that the solution must meet, e.g., “the string begins with a one, which is followed by one or more zeroes.” By understanding how a regex automaton behaves, e.g., how backtracking works under the hood, we may write the regex that requires minimum energy and time from the computer.

Domain-specific languages (DSLs) are designed to solve a specific type of problem. SQL is a domain-specific language for querying relational databases, VHDL is a specialized language for describing hardware, and regex is a domain-specific language for describing sets of strings. The complement to DSL is general-purpose programming languages — e.g., C, Java, C#, and Ruby — which are designed to solve problems found in most applications. Regex isn’t suitable for solving every type of problem; it’s even beyond what it, by all means, can do. However, when it comes to describing a set of strings, it’s phenomenal.

Separators and space are present in most programming languages. We use separators, such as semicolons, to signal that a statement is complete. Most programming languages also allow the programmer to add extra white space in strategic places in the code. It’s not unusual for the programmer to write code comments in the middle of the code. Neither separators, whitespace, nor code comments are really meaningful to the computer. They don’t add, remove, or change any computer calculations, but they may make the code more readable for us humans. In regex, no redundant characters exist. Everything has a meaning. The code comprises literals, metacharacters, and operators. In some regex dialects, you can add an extra flag to get permission to insert code comments and whitespace, but in the normal case, this isn’t possible.

Finally, a few words about dialects. “The nice thing about standards is that there are so many of them to choose from” is a quote that has been attributed to many celebrities. Almost every regex dialect has its own syntax and semantics. They’re mostly the same, but certain details differ. The biggest difference among the dialects is which operators are actually available, i.e., it’s not enough to find a regex on the web. It must be compatible with the regex dialect in which you intend to use it.

The normal regex usage flow comprises three steps:

  1. Compile a regex pattern.
  2. Feed input into an application.
  3. Iterate the matched results.

Sometimes the three steps are combined in a macro, and sometimes you need to go through all three steps, one by one.

(1) In the first step, our carefully written regex pattern is compiled into an automaton. If we store a reference to our compiled automaton in a variable, we may reuse the automaton however many times we want. Here’s how to compile in IRB:

(2) Our shiny automaton, referred to by the variable 𝚏𝚊, now offers several services. It can validate, find, replace, parse, and filter text that we feed it, then we select a service and feed the automaton with input. In IRB, we use the method 𝚜𝚌𝚊𝚗 to find all instances of our pattern:

(3) Now we can iterate over our result set:

(1), (2), and (3) Usually, we don’t need to perform each step explicitly. Instead, we may compile them implicitly, like this:

Note that our regex pattern was compiled on the fly in this example.

Verify (e.g. a card number), find, replace, filter, and parse are five regex applications.

In IRB, it’s good enough to write a regex between two slashes to get the regex compiled implicitly into an automaton. That puts us in a position in which we may invoke many different functions on our automaton object.

(1) We may verify that the input matches a specific pattern:

(2) We may find a pattern in the input:

(3) We may replace text in the input that matches a pattern:

(4) We may filter out a pattern from the input:

(5) And we also may parse a pattern from the input:

PART III: Syntactic Sugar, Abstractions, and Extensions

Where nothing else is said, examples are written in Ruby. You may install Ruby on your computer and then start IRB (Interactive Ruby Shell) in a terminal to try out the code. You may also run IRB online without any installation.

In Part II (Two Operations and One Function), we learned how powerful concatenation, alternation, and the kleene star work together. However, you’re probably aware that in modern regex dialects, many other operators — e.g., quantifiers, groups, lookarounds, etc. — exist, most of which are abstractions. They help us write more generic regexes without necessarily adding new regex functionality. Others are just syntactic sugar, making us write easier to read regexes. Finally, some extensions cannot be implemented with a finite automaton. We refer to them as path-dependent operations. While consuming the input string, we must know how we arrived in the current state; thus, we need a memory. Under the hood, this memory is implemented as a stack.

Sometimes, we want to repeat an expression to make it match more than one instance in the same input string. If the number of repetitions is known upfront, i.e., it’s a fixed number, we may simply repeat the whole expression. For example, the expression /𝙻𝚊𝙻𝚊𝙻𝚊/ is a repetition of the expression /𝙻𝚊/ three times. Any kind of fixed or variable number of repetitions is possible with the two original operations (concatenation and alternation) and the function (kleene star). However, this might be difficult to read. For example, /(𝙻𝚊|𝙻𝚊𝙻𝚊|𝙻𝚊𝙻𝚊𝙻𝚊|𝙻𝚊𝙻𝚊𝙻𝚊𝙻𝚊)/ expresses between one and four instances of 𝙻𝚊 in an annoyingly verbose way. This is why quantifier functions exist. These functions don’t add any new functionality to regular expressions, but instead support us with crispness.

The two most popular repetition requirements are to match either at least one instance, or at most one instance.

At least one means one, two, three, or any other positive integer. What will happen if we replace the first instance of /𝚊*/ with 𝚎 in the input string 𝚌𝚊𝚊𝚕𝚎𝚛𝚢?

The answer is 𝚎𝚌𝚊𝚊𝚕𝚎𝚛𝚢 because 𝚌𝚊𝚊𝚕𝚎𝚛𝚢 starts with (i.e. before the 𝚌) zero instances of 𝚊 and, /𝚊*/ says that we must match zero to many instances. As I told you before, most regex automata return the leftmost match. What can be more leftmost than before the initial character of a string? However, our intention was to replace repetitions of 𝚊 with an 𝚎. The handy positive closure function, written as a plus sign, +, comes to the rescue. The expression /𝚊+/ should be read as: match at least one 𝚊:

Thus, replacing /𝚊+/ with 𝚎 in 𝚌𝚊𝚊𝚕𝚎𝚛𝚢 gives us 𝚌𝚎𝚕𝚎𝚛𝚢 — a delicious vegetable.

The optional quantifier function — written as a question mark — matches at most one instance of an expression, i.e., either zero or one instance. We want to match the word chickpeas in singular and plural forms, which is easy. The expression /𝚌𝚑𝚒𝚌𝚔𝚙𝚎𝚊𝚜?/ matches 𝚌𝚑𝚒𝚌𝚔𝚙𝚎𝚊, as well as 𝚌𝚑𝚒𝚌𝚔𝚙𝚎𝚊𝚜:

The optional quantifier function binds to the 𝚜, i.e., it makes the 𝚜 optional. Why didn’t this function bind to the whole 𝚌𝚑𝚒𝚌𝚔𝚙𝚎𝚊𝚜 subpattern? You guessed it! It’s because the optional quantifier function takes precedence over concatenation, the invisible glue between the literal characters in 𝚌𝚑𝚒𝚌𝚔𝚙𝚎𝚊𝚜.

There’s also a generic quantifier function. With braces {}, we can express any kind of repetition. The normal form has a lowest and a highest acceptable number of matches, and these two numbers are separated by a comma. The expression /(𝙻𝚊){𝟷, 𝟺}/ matches at least one and at most four 𝙻𝚊. Let’s see what we get when we replace matchings with 𝚘𝚑 in the input 𝙻𝚊𝙻𝚊𝙻𝚊𝙻𝚊𝙻𝚊𝙻𝚊.

The default value for the first argument is zero; thus, the expression /(𝙻𝚊){, 𝟺}/ matches at most four and at least zero 𝙻𝚊 uses:

What was that? Another oh was added, but no 𝙻𝚊 was removed because most regex automata prefer to return the match that’s leftmost in the input string. We explicitly said “at least zero instances,” didn’t we? Thus, we matched zero instances right at the start of the string and replaced that empty string with 𝚘𝚑. Perhaps it’s more clear in this example:

No 𝙻𝚊 actually was needed to achieve zero matches. I also said before that quantifiers are greedy. Why not match four 𝙻𝚊’s rather than none? Is preferring leftmost and being greedy a contradiction? Not at all. However, our leftmost strategy exceeds our greedy strategy (or reluctant when this is our flavor). We found a match farest left, let’s be greedy right there.

If we keep the first generic quantifier argument, then we can omit the second, in which case, there’s no upper limit. The expression /(𝙻𝚊){𝟷,}/ matches at least one La and is equivalent to /𝙻𝚊+/:

The final version of the generic quantifier function has no comma and contains only one number. The expression /(𝙻𝚊){𝟸}/ matches exactly two instances of 𝙻𝚊 — nothing more and nothing less:

Quantifiers are unary, left-associative, and take precedence over alternation and concatenation. Consider why the following are true:

  • Concatenation: /𝚊𝚋{𝟷, 𝟸}/ equals /𝚊(𝚋{𝟷, 𝟸})/
  • Concatenation: /𝚊{𝟷, 𝟸}𝚋/ equals / (𝚊{𝟷, 𝟸})𝚋/
  • Alternation: /𝚊|𝚋{𝟷, 𝟸}/ equals /𝚊|(𝚋{𝟷, 𝟸})/
  • Alternation: /𝚊{𝟷, 𝟸}|𝚋/ equals /(𝚊{𝟷, 𝟸})|𝚋/

The horizontal axis in the picture above states how many instances a quantifier matches. It starts with zero, which matches the empty string, and ends in infinity, i.e., there’s no limit to how many times an instance can be repeated and still match our expression.

The first column (red dots) indicates that kleene star function *, optional quantifier function ?, and generic quantifier function with first argument zero {𝟶, 𝚖} or omitted {, 𝚖} may match zero instances.

The second column (blue dots) indicates that kleene star function *, positive closure function +, and generic quantifier function with second argument omitted {𝚗,} have no upper limit when it comes to the number of possible matches.

The notation of regular expressions, as Stephen Kleene defined it, has only two operations — concatenation and alternation — and a function: kleene star. This is enough to define shorthand functions, like some quantifiers. The shorthand provides us with a syntax that makes our expressions more readable and maintainable, though they don’t add any new functionalities. Everything they can do also can be done with the original two operations (concatenation and alternation) and the function (kleene star). However, the latter would be more verbose.

To convince you that a quantifier really is just a shorthand, I’ll give you what I call regex equations. An equation is a mathematical statement that says two expressions are equal.

Let’s start with the optional quantifier function, written as a question mark. Like all quantifiers, it binds the expression to the left. In the figure above, you can see that saying that the blue dot is optional is the equivalent of using alternation to say either we match nothing or else we match one blue dot.

The positive closure function is written as a plus sign. Positive closure means we must match at least one instance of the expression to the left. In the second equation in the figure above, I claim that a blue dot’s positive closure is equivalent to one mandatory blue point concatenated with the closure we get by applying the kleene star, written as an asterisk, to another blue dot. One blue dot is mandatory, then follows zero-to-many blue dots. This wasn’t a very provocative proposition, was it?

Do you ever say curly brackets when referring to “{” and “}”? Programmers in the US often call them braces, and in the UK, they sometimes are called squiggly brackets. India has the more poetic name flower brackets, and in Sweden, we say gull wings. However, in regex, we use them as another shorthand quantifier. They hold a pair of arguments that define the minimum and maximum number of repetitions of the expression to the left. The third equation in the figure above states that it’s equivalent to a) repeat the blue dot at least two times and at most four times, and b) match two blue dots concatenated with the alternation zero, one, or two blue dots.

Here are some more equations. Inspect them to ensure that you agree with me that they’re true:

  • 𝚊{𝟹} equals 𝚊{𝟹, 𝟹}
  • 𝚊* equals 𝚊{𝟶, }
  • 𝚊+ equals 𝚊{𝟷, }
  • 𝚊? equals 𝚊{𝟶, 𝟷}
  • (𝚊*)* equals 𝚊*
  • (𝚊+)+ equals 𝚊+
  • (𝚊?)? equals 𝚊?
  • (𝚊*)+ equals (𝚊+)* equals 𝚊*
  • (𝚊*)? equals (𝚊?)* equals 𝚊*
  • (𝚊+)? equals (𝚊?)+ equals 𝚊*
  • 𝚊* equals (𝚊+|)
  • 𝚊? equals (𝚊|)? equals (𝚊|)
  • (𝚊|)* equals 𝚊*
Greedy (left) and reluctant (right) kleene star.

Because of something called backtracking, we sometimes might match more than we hoped for. The task below is to catch all the div tags in an HTML document and put them in a vector. Our naïve solution provides the wrong answer:

What’s backtracking? Regex quantifiers are naturally greedy, i.e., they consume as much as they can. The period symbol in the idiom /.*/ matches anything except newline (more on this later in the essay), and the kleene star * means that this anything is repeated as many times as possible. In its first attempt, /.*/ matches 𝚊</𝚍𝚒𝚟><𝚜𝚙𝚊𝚗>𝚌</𝚜𝚙𝚊𝚗><𝚍𝚒𝚟>𝚋</𝚍𝚒𝚟>. Unfortunately, this means that the last part of the regex /<\/𝚍𝚒𝚟>/ is starving, which makes /.*/ very sad. The latter then backtracks — i.e., un-consumes — the last part of the input string, character by character, until the whole expression matches. Considering that the string ends with a </𝚍𝚒𝚟>, the substring 𝚊</𝚍𝚒𝚟><𝚜𝚙𝚊𝚗>𝚌</𝚜𝚙𝚊𝚗><𝚍𝚒𝚟>𝚋 will be consumed by /.*/. Problems may arise from greed.

Many stories tell tales of greedy people who claim more than they need. As Louis Blanc wrote in 1840 in The Organization of Work: “From each according to his abilities, to each according to his needs.” We don’t think that the kleene star * and the period symbol in the above example need to consume more than the substring <𝚍𝚒𝚟>𝚊</𝚍𝚒𝚟>. Alexander Pushkin describes in The Tale of the Fisherman and the Fish how a magic fish promises to fulfill whatever the fisherman wishes. The fisherman’s wife eventually starts asking the fish for bigger and better things — and she gets them — until she eventually wants to become Ruler of the Sea. The magic fish then takes back everything he gave the fisherman’s wife.

Quantifiers in regex are naturally greedy. They attempt to consume as much of the input string as possible. The good news is that regex provides an alternative: the reluctant (sometimes called lazy) quantifier modifier. You guessed it: The reluctant modifier makes the quantifiers attempt to consume as little as possible of the input string. The verb attempt is important here. After the attempt to consume as much (greedy) or little (reluctant) as possible, there might be subexpressions further to the right that can’t match the remaining input string, i.e., the entire expression can’t be matched. If this happens, the automaton will backtrack, and our quantifier will make a new attempt. This time, it will consume one less (greedy) or more (reluctant) character compared with its first uncompromising attempt. The method is repeated until we either can match everything or confirm that there’s no possible way to create a match.

Does it matter whether we use the greedy or reluctant approach? Well, the strings that can be matched with greedy are also matched with reluctant, and vice versa. However, when multiple matches are possible, greedy tracking sometimes will choose a different match than reluctant tracking. These two approaches also differ in performance. In some cases, reluctant is faster, while in other cases, greedy is. It’s a question of how many backtrackings we must make.

No special symbol for reluctant quantifiers exists. Instead, we have a modifier symbol — written as a question mark — that may be added to the right of any quantifier. While * says “repeat as many times as possible,” *? says “repeat as few times as possible.” What a great duo. Similarly, we may modify any other quantifier:

  • Positive closure function with reluctant modifier: at least one, as few as possible: +?
  • Optional quantifier function with reluctant modifier: zero or one, preferably zero: ??
  • Generic quantifier function with reluctant modifier: between three and five and as few as possible: {𝟹, 𝟻}?

Note that the question mark that modifies quantifiers isn’t the same question mark that’s used as the optional quantifier function. They may even be used in conjunction with each other, as ?? indicates above. It’s context dependent whether a question mark represents the reluctant modifier or the optional quantifier function. Of course, in some contexts, the question mark also can be a literal, i.e., we want to match a question mark in the input string.

Here are some quantifier examples with and without the reluctant modifier. The first one is a solution to the div tag problem:

Greedy and reluctant (lazy) quantifiers find the same matches. Considering that you always ask the quantifier for just one match, they choose the first one they find. The difference is the search method they use and, therefore, the order in which they find matches, i.e., which match they find first.

Possessive quantifiers will only find a subset of the matches found by reluctant and greedy; therefore, possessive may ignore some perfectly correct matches. If things go badly, a possessive quantifier sometimes returns empty-handed despite the fact that possible matches existed.

So, what value do possessive quantifiers bring? They sometimes come up with a result in less time. The quantifiers we’ve seen so far use brute force. When they notice that they’re on the wrong track, they backtrack, then try the next possible way. It’s all these failures — the backtrackings — that waste our time! A possessive quantifier never, ever backtracks. It refuses to backtrack. Thus, it’ll quickly give you an answer: yes or no.

If possessive quantifiers deliver faster due to the strategy of not backtracking, then they must choose the shortest search path (to maximize consumption of the input string) up front, mustn’t they? Is it even possible to know the shortest path up front? The badly kept secret is that possessive quantifiers don’t know. They just possess (that word again!) an unusually high dose of self-esteem. Possessive quantifiers use the greedy algorithm to consume as much as possible. They won’t notice whether a subexpression further to the right in your regex fails simply because Mr. Possessive Operator speculatively consumed too many characters from the input string. You see, this is a hungry rascal who doesn’t care about its fellow subexpressions.

Enough is enough — show me some code! The possessive quantifier syntax is a modifier written as a plus, +, to the right of the quantifier. All quantifiers can have the possessive modifier: possessive optional ?+; possessive kleene star *+; possessive positive closure ++; and possessive generic quantifier {𝟸, 𝟻}+.

Whenever there’s a match below, the match is replaced with the character pillow symbol ¤. Note that the period symbol matches any character except line breaks.

Possessive optional:

Possessive kleene star:

Possessive positive closure:

As of this essay’s writing, Ruby/IRB doesn’t support the possessive quantifier in default mode. Instead, it treats {𝚖, 𝚗}+ as two consecutive quantifiers {𝚖, 𝚗} and +:

In Scala, all four types of quantifiers — ?, *, +, and {𝚖, 𝚗} — support the possessive modifier:

To sum up, the possessive quantifier modifier is written as a trailing plus, +; is greedy; and can improve performance by refusing to re-track. But be careful: It might ignore matches that you’re expecting to receive.

The twelve literals who doesn‘t match literally.

Most characters in a regex match literally, i.e., you simply search for a text the same way as when you’re searching in your word processor:

A dozen characters normally carry special meaning in a regex; thus, they don’t match literally.

  • Caret ^ and dollar $ assert a position at the beginning or the end.
  • Left and right parantheses ( and ) start and end a capturing group.
  • Asterisk *, plus +, questionmark ?, and left brace { repeat the preceding subexpression.
  • Period . matches any character except line breaks.
  • Left square bracket [ starts a character class.
  • Backslash \ lets a metacharacter be matched literally.
  • Vertical line | is an alternation.

I’ll call these not-literally-matching characters metacharacters. In specific contexts, these characters might have other meanings, yet they don’t match literally. Note that the closing square bracket ], hyphen -, and closing curly bracket } normally do match literally in regex. If you still want to match one of the twelve characters above literally, you must escape them with a backslash \:

Suppose you get a string from the user. You’d like to search for that string in the 𝚒𝚍 attribute values in some HTML, then show the user the matching results, if any:

Unfortunately, a user realizes that he can make a regex injection attack:

Luckily, most programming languages ​​that support regex have a function that neutralizes all metacharacters. In Ruby, it’s called 𝚁𝚎𝚐𝚎𝚡𝚙.𝚎𝚜𝚌𝚊𝚙𝚎():

In some regex dialects, it’s possible to escape with a \𝚀 to \𝙴 block. It works inside, as well as outside, character classes:

“Unicode provides a unique number for every character, no matter what the platform, no matter what the program, no matter what the language.”

That’s what the Unicode Consortium writes on its website. The Unicode standard covers the characters from most writing systems of the world — modern and ancient. It also includes technical symbols, punctuation, and many other characters used in writing text. As of this essay’s writing, Unicode supports more than 100 scripts and 100,000 characters.

In the finite automata theory sections above, I explained alphabets. A regex automaton is equipped with an alphabet, i.e., an ordered list of all possible input characters it understands. To understand what a character is, we first must sort out some other concepts:

  • Grapheme: The smallest units of meaning in written language, e.g., letters or numbers, are called graphemes. A single grapheme can be visualized in different ways, i.e., the grapheme for the number seven can be crossed or not crossed. Thus, grapheme is something semantic, but not anything visual.
  • Glyph: The different ways we can visualize a grapheme on paper or on screen are called glyphs. A glyph is a shape associated with an abstract character, e.g., the grapheme for the Ohm sign may have a glyph that looks like this: Ω. Several different glyphs may represent the same grapheme.
  • Code Point: A code point normally is assigned to an abstract character, which is a level above encodings of bits and bytes. Think of it as an index in an array of characters. In Unicode, the Ohm sign code point is U+2126, which is hexadecimal. Thus, the decimal index is 8486.
  • Character encoding: A pairing of codes, e.g., bitmaps or natural numbers and characters is called character encoding. The most popular encodings are UTF-8, UTF-16, and UCS-2. Unicode can be implemented with different encodings. The ohm sign is encoded as 𝟶𝚡𝙴𝟸 𝟶𝚡𝟾𝟺 𝟶𝚡𝙰𝟼 in UTF-8 and as 𝟶𝚡𝟸𝟷𝟸𝟼 in UTF-16.

Most, but not all, regex dialects support Unicode. The easiest way to match a character is by writing it literally:

Of course, you also can match a Unicode code point:

After \𝚞 comes four hexadecimal digits that add up to a code point number, but the syntax varies. Some regex dialects limit the number of digits in the hexadecimal number to one, two, four, or eight, while other dialects allow unlimited large numbers. Some have \𝚡, while others have \𝚞, like the one above. In some dialects, you actually write \𝚡{𝟸𝟷𝟸𝟼} to match code point U+2126. With the \𝚡{} syntax, you can choose how many digits you want. Here are some examples:

  • Python: \𝚡𝙳𝙵
  • Perl: \𝚡𝙳, \𝚡𝙳𝙵, and \𝚡{𝟸𝟷𝟸𝟼}
  • Java and C#: \𝚡𝙳𝙵 and \𝚞𝟸𝟷𝟸𝟼

It’s also possible to point out a code point with octal numbers. This syntax varies as well between regex dialects. It may be one, two, three, and even four digits. The common denominator is that in all cases, it starts with a backslash \, directly followed by numbers. Some regex dialects (not Ruby) require the first digit to be zero:

The \𝚌 syntax allows you to address any of the first 26 characters in the ASCII character set. First, there’s a \𝚌, which is followed by a character from 𝙰 to 𝚉 in the English alphabet. For example, considering that newline is the tenth character in the character set in my Linux computer terminal, I can reference newline with 𝙹, which is the tenth letter of the English alphabet. The code then will be \𝚌𝙹:

Depending on the regex dialects that you’re using and also what character encoding the input text uses, the pairing of characters and code points may vary. The first 128 code points are almost always the same, and if you use Unicode, everything, of course, is standard. It’s not like herding cats, but you must be careful.

The easiest way to match a character is, of course, literally. However, in some cases, it’s not possible. For example, how do you match the escape sign \, the one that usually has code point 27? And is there a way to match compound characters, e.g., the German ß, also called double-s?

Seven invisible (i.e., they’re not associated with any glyph) characters have their own aliases, each of which pairs a backslash \ with an intuitive letter:

  • \𝚊 — Sound alert, ASCII index 0x07
  • \𝚎 — Escape-character, ASCII index 0x1B (Visible as \.)
  • \𝚏 — Form feed, ASCII index 0x0C
  • \𝚗 — Newline, ASCII index 0x0A on Windows/Linux and 0x0D on OSX
  • \𝚛 — Carriage return, ASCII index 0x0D on Windows/Linux and 0x0A on OSX
  • \𝚁 — Any line break and also CRLF as a combo
  • \𝚝 — Horizontal tab, ASCII index 0x09
  • \𝚟 — Vertical tab, ASCII index 0x0B

Newline and carriage return are, of course, not the same:

Matching line breaks may be tricky because conventions for how they are coded differs between operative systems. Sometimes \𝚁 comes to the rescue:

An odd feature is POSIX’s collating sequences. If a locale defines composite characters, e.g., the German ß or Spanish ll, as a character, with its own place in the local alphabet, then you can match the combination with the syntax [[.𝚜𝚙𝚊𝚗-𝚕𝚕.]]. In the Spanish alphabet, ll lies between l and m, and in the German alphabet, ß lies between s and t. Collating sequences also may be used in ranges; see the Generic Character Classes section of this essay.

Speaking of odd POSIX features, I mustn’t forget to mention character equivalents. In some locales, the two characters with and without diacritics are viewed as equivalent. If you want to match the Spanish ñ, as well as the ordinary n, then the character equivalents feature provides the syntax [[=𝚗=]]. Another example is [[=𝚊=]], which (in some locales) matches a, à, and á.

When I say that collating sequences and character equivalents are odd features, I don’t mean they’re bad or useless — just that they’re rare. Collating sequences and character equivalents can be used only inside generic character classes. The outer pair of square brackets above are actually the generic character class boundaries.

Some composite glyphs can be coded both as single code points and as pairs of diacritical marks and main characters. For example, à can be encoded either as U+00E0 or as the sequence U+0061 U+0300. Some regex dialects have a flag that makes both of these representations equivalent, i.e., canonically equivalent.

The period symbol matches any character except line breaks.

Do you remember Barbapapa from Annette Tison and Talus Taylor’s children’s books and films from the 1970s? The hero was a pink, pear-shaped guy with the ability to take on almost any shape whatsoever. The equivalent in regex is the period symbol, i.e., the punctuation mark sometimes called dot or full stop.

The period symbol is a character class — a generic character. Rather than writing a literal character, like 2, a, or #, we may use the period symbol to specify that we accept almost any character:

However, two human problems surface when we use the period symbol, and they should be noted:

  1. The character class period . and the kleene star function * are together and separately the most abused features of regex. If you use them frequently without thinking, you’ll often end up with overly general regexes — sometimes they are even incorrect. Every time you intend to write period ., asterisk *, or even the combo .*, you first may want to consider whether you really mean something more specific.
  2. Most regex books, including the most popular ones, are unclear or even entirely incorrect, as they claim that “period symbol matches any character.” In most cases, the period symbol matches “any character except line breaks,” which is a very, very important difference.

Why doesn’t the period symbol normally match line breaks? The original implementations of regex operated line by line. Programs like grep digest one line at a time, and trailing line breaks are filtered out before processing. Thus, no line breaks are left. NASA engineer Larry Wall created Perl In the 1980s — the programming language that evangelized regex more than anything else. The original purpose was to make report processing easier. Thus, what could be more natural than continuing on the path of line-oriented work? Another argument sometimes heard is the following: If the period symbol would match line breaks, then the meaning of the idiomatic combo .* would change. Perl set the agenda, and now, a few decades later, we only can accept that the period symbol typically doesn’t match line breaks, no matter what you and I believe is natural and consistent:

Can we force the period symbol to match line breaks? Yes, by setting a flag. Unfortunately, this flag has different names in different regex dialects.

In Perl, it’s called single-line mode. Imagine what happens if the period symbol matches all characters, including line breaks. Input data become a long line in which the line break is a character like any other — hence, the name. Single-line mode should not be confused with what in Perl is called multi-line mode. Multi-line mode affects anchors $ and ^, and it’s orthogonal with single-line mode.

To add more confusion, Ruby uses the term multi-line to mean what in Perl is called single-line mode. And what Perl calls multi-line mode is mandatory in Ruby — with no flag available. The best approach to this mess is to call this flag period match all no matter how it’s written syntactically in a specific regex dialect. By the way, in Ruby, we add 𝚖 next to the regex literal when we want the period symbol to match any character:

In some regex dialects, most notably JavaScript’s, no flag exists for period match all. A workaround is to replace the period symbol with the idiom [\𝚜\𝚂]. This idiom matches exactly one character — either white space or anything that isn’t whitespace. These two classes are, of course, 100% of all the characters — including line breaks:

I argued above that the period symbol often is abused. What does this mean? Imagine that we want to find every time string in a text, and we have the following requirements:

  • Time should always include hours and minutes, sometimes even seconds.
  • Hours, minutes, and seconds should always be written with two digits.
  • We don’t have to ignore impossible numbers, such as minute 61.
  • In between hours, minutes, and seconds, there should be one of the separators period . or colon :.

The result of the simple regex /\𝚍\𝚍.\𝚍\𝚍(.\𝚍\𝚍)?/ might surprise you:

That’s not what we wanted. The period symbol matches space! If we replace the item with the more specific character class [.:], we aim more closely at our target. You mustn’t forget that the period symbol inside a character class means that we literally want to match the period character:

Four character classes are so frequent that they have their own separate shorthand, i.e., their own aliases. One of these four is the period symbol, which means match any single character that isn’t a line break. But you knew that already. The other three — \𝚍, \𝚜, and \𝚠 — also have aliases for their complementary character classes, i.e., the complement class matches everything not included in this particular character class. Thus, seven character classes exist overall:

  • Period symbol . matches any character except line breaks.
  • \𝚜 matches any whitespace character, such as tabs, line breaks, and spaces.
  • \𝚂 matches any character that’s not a whitespace; it’s the complement of \𝚜.
  • \𝚍 matches any digit, e.g., 3 or 9.
  • \𝙳 matches any character that isn’t a digit; it’s the complement of \𝚍.
  • \𝚠 means word character and is normally equivalent to the character class [𝚊-𝚣𝙰-𝚉𝟶-𝟿_], but some regex dialects also include letters and digits from other scripts, e.g., å or ñ.
  • \𝚆 matches any character that’s not matched by \𝚠.

The character classes \𝚠 and \𝚆 rarely are what you really want and should be used carefully. They must be around for backward compatibility reasons. If your regex dialect supports Unicode, then the letter class \𝚙{𝙻}, digit class \𝚙{𝙽}, not-a-letter class \𝙿{𝙻}, and not-a-digit class \𝙿{𝙽} are preferred. More about this later in the Unicode categories section of this essay.

It’s usually pretty straightforward to use a shorthand character classes:

Here are some other examples:

And, as noted many times in this essay, the period symbol doesn’t normally match line breaks:

We can divide Unicode code points in three ways. Each code point is a member of one category (also called property). They also belong to a block, and if they aren’t unassigned, they belong to a script.

Let’s start with categories. Unicode is divided into seven categories:

  • Other: \𝚙{𝙲}
  • Letter: \𝚙{𝙻}
  • Mark: \𝚙{𝙼}
  • Number: \𝚙{𝙽}
  • Punctuation: \𝚙{𝙿}
  • Symbol: \𝚙{𝚂}
  • Separator: \𝚙{𝚉}

The seven categories are subdivided further into more than 30 subcategories. The syntax calls for adding an extra letter to the subcategory, e.g., 𝚕 for lowercase. Here are some examples of subcategories:

  • Letter, Lowercase \𝚙{𝙻𝚕}.
  • Punctuation, Dash \𝚙{𝙿𝚍}.
  • Symbol, Currency \𝚙{𝚂𝚌}.

Another decomposition, completely orthogonal to the categories, is scripts. When I wrote this essay, Unicode contained roughly 70 ancient and 90 modern scripts, and more scripts are in the melting pot for encoding. All unassigned code points belong to exactly one script — no more, no less:

The third way to divide Unicode is in blocks. A block is a range of code points. Over 300 blocks already exist:

  • U+0600…U+06FF == \𝚙{𝙸𝚗𝙰𝚛𝚊𝚋𝚒𝚌}
  • U+0000…U+007F == \𝚙{𝙸𝚗𝙱𝚊𝚜𝚒𝚌_𝙻𝚊𝚝𝚒𝚗}

In some regex dialects, the syntax is \𝚙{𝙸𝚜…} instead of \𝚙{𝙸𝚗 …}. Note that even unassigned code points can be part of a block.

By typing a capital 𝙿, we negate the properties, scripts, and blocks.

You may recall that I warned you about the word character class \w, which doesn’t always match our naïve expectations on what a word character could be. If our regex dialect supports Unicode, it’s more likely that we’re looking for, for example, all numbers and letters — even the letters not in the sequence a–z:

By now, you most certainly know that the period symbol normally matches anything except line breaks. A Unicode counterpart can be found in some regex dialects. It’s written \𝚇, and it even matches line breaks:

Alternation (top left), character class (bottom left), character class range (top right), and negated character class (bottom right).

Imagine that we must find all the vowels in a text. A no-nonsense approach is to simply list them in a long alternation:

What if we have a few (or many) characters and want to ensure that exactly one of these characters — no more, no less — is matched? In regex, we may use a more concise way to write exactly that: the generic character class. A character class is a designated subset of the alphabet (do you remember what an alphabet is in regex?). We describe our subset inside a pair of square brackets []. The description can be written ​​in several ways. The plain syntax is simply to list the characters we want to include in our subset:

This was more crisp than when we made a verbose alternation above, but alternation is exactly what it means, i.e., the exclusive disjunction or XOR in programmer lingo.

Here’s another example, in which we find all letter characters in a hexadecimal number:

Now, a to f comprise a range, an unbroken chain. We describe a range by writing the start character of the range, the end character, and in between them, a hyphen (minus):

What’s a range then? In regex, the alphabet — all possible characters that may occur in a text — is ordered. As you know, each individual character is assigned a number that we call a code point, which also may be called an index. Each code point is unique, i.e., multiple characters cannot have the same code point. You can think of the complete set of characters as an array in which each entry has an array index. Note that not only the letters are characters, e.g., numbers, punctuation, and white spaces are characters as well. A range is a sequence of characters in our alphabet array. If this alphabet array is Unicode or ASCII, then the English lower letters a, b, c, etc. comprise a range. The digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 comprise another range.

Consider whether you understand the following:

In this case, we used the Unicode character database. The matched letters have these code points in Unicode:

  • ‘7’ has code point 55 in Unicode.
  • ’@’ has code point 64 in Unicode.
  • ‘d’ has code point 100 in Unicode.

Why did the @ symbol match, but not the period symbol? Because the period symbol has code point 46. When we enter the range 7 to d, we really mean all characters with a code point greater than or equal to 55 and less than or equal to 100.

We may mix ranges and individual characters:

If the hyphen comes first or last in the character class, it’s matched literally, of course:

Though sometimes forgotten, it’s worth noting that a character class matches exactly one character, i.e., not only max one, but also at least one. When we want to find all names that end with a digit, this isn’t the correct regex:

This is highly disappointing, considering that we actually wanted to get the whole 𝚐𝚘𝚘𝚍𝟺𝚢𝚘𝚞𝟷 in a single match. However, it’s easily fixed, of course:

Sometimes it’s easier to say what you don’t want. It’s been decades since I ate meat, and I have no allergies or food aversions. In response to the question of what I eat, it’s easier to say “anything except meat” than to say “pasta, fruit, olives, nuts, bread, beer, baklava…” etc. Regex character classes support exactly that functionality: negated character classes.

Say you want to find all 𝚌𝚕𝚊𝚜𝚜 attributes and their values ​​in HTML code. Our first attempt looks good:

Great! Unfortunately, there’s another HTML snippet in which the attributes 𝚌𝚕𝚊𝚜𝚜 and 𝚛𝚎𝚕 were swapped:

We can solve this problem with negated character classes. A negated character class begins with a caret, ^. We want to match exactly one character — anything except those we enumerate in our character class:

Of course, there may be several characters in a negated character class, and none will match:

Using [^"]*, we express that we want to find zero or more — as many as possible — of anything except quotes.

Note that the caret needs to start the character class. Otherwise, the caret is matched literally, and no classes will be negated:

Some regex dialects have three more types of character classes:

  • A character class union is written as a nested character class. For example, [[𝚊-𝚏][𝟶-𝟿]] will match any lower hexadecimal digit.
  • A character class intersection adds the meta-sequence double-ampersand, &&, to find what’s common in the two sets: [\𝚠&&[\𝚍𝚊-𝚣]].
  • Character class subtraction has at least two different syntaxes. In some regex dialects, we type a hyphen, -, before the set we want to subtract. The expression [\da-z[g-z]] will match any lower hexadecimal digit. In other regex dialects, an intersection is written as the main set and the complement of what we want to remove (do you remember De Morgan’s laws?): [\𝚍𝚊-𝚣&&[^𝚐-𝚣]]

The set operations union, intersection, and subtraction are a bit obscure. They only exist in a few regex dialects, and it might be even more difficult to find applications for them. However, applications do exist even though the examples above are more esoteric.

The five literals which does not always match literally inside a character class.

Look at these two expressions. Why are the results so different?

In the first case, the character class includes a range. In the second case, it’s just a list of three different characters — a hyphen -, a, and z. All characters except five can be matched literally at any position inside a character class, i.e., we must be careful with the following five inside a character class:

  • A slash, \, indicates that the next character is a metacharacter, e.g., newline \n or the digit character class shorthand \d.
  • A caret, ^, creates a negative character class — the complement of the characters listed — if the caret is positioned first in the character class.
  • A hyphen, -, is put in between two characters to describe a range, which entails the hyphen is matched literally if positioned first or last in the character class.
  • A right square bracket, ], marks the end of a character class.
  • A left square bracket, [, marks the beginning of a character class union, subtraction, or intersection — if our regex dialect supports that functionality.

All other characters are matched literally inside character classes, even the period symbol:

When we want to match \, ^, - or ], we may prefix them with a backslash \. In most regex dialects, we also may write those in a position in our character class in which they cannot possibly have a functional meaning:

Shorthand character classes, except the period symbol, work inside character classes as well:

As you’ve seen, some characters are matched literally in regex, and some are interpreted as metacharacters, which is why we must prefix (escape) the latter with a backslash \ when we want these characters to be matched literally. However, you might have noticed something peculiar: The escape rules inside and outside the character class differ. We discussed earlier how only five characters are metacharacters inside a character class. A dozen characters are interpreted as metacharacters outside a character class. Here’s a revealing example:

The POSIX counterpart is older than the Unicode character class constants. It’s only about a dozen classes. I say about because it varies with different locales and dialects:

  • [:𝚊𝚕𝚗𝚞𝚖:] English alphanumeric characters, equivalent to [𝚊-𝚣𝙰-𝚉𝟶-𝟿] 𝚘𝚛 [\𝚍\𝚙{𝙻𝚕}]
  • [:𝚊𝚕𝚙𝚑𝚊:] English alphabetic characters, equivalent to [𝚊-𝚣𝙰-𝚉] 𝚘𝚛 \𝚙{𝙻}
  • [:𝚊𝚜𝚌𝚒𝚒:] The seven-bit ASCII characters, equivalent to [\𝚡𝟶𝟶-\𝚡𝟽𝙵]
  • [:𝚋𝚕𝚊𝚗𝚔:] Spaces and tabs, but not line breaks, equivalent to [ \𝚝]
  • [:𝚌𝚗𝚝𝚛𝚕:] Control characters, equivalent to [\𝚡𝟶𝟶-\𝚡𝟷𝙵\𝚡𝟽𝙵]
  • [:𝚍𝚒𝚐𝚒𝚝:] Digits, equivalent to \𝚍
  • [:𝚐𝚛𝚊𝚙𝚑:] Visible characters, i.e., not spaces, control characters
  • [:𝚕𝚘𝚠𝚎𝚛:] Lowercase alphabetic characters, equivalent to [𝚊-𝚣] or \𝚙{𝙻𝚕}
  • [:𝚙𝚛𝚒𝚗𝚝:] Same as [:𝚐𝚛𝚊𝚙𝚑:], but including the space characters
  • [:𝚙𝚞𝚗𝚌𝚝:] Punctuation characters
  • [:𝚜𝚙𝚊𝚌𝚎:] Whitespace characters, equivalent to \𝚜
  • [:𝚞𝚙𝚙𝚎𝚛:] Uppercase alphabetic characters, equivalent to [𝙰-𝚉] or \𝚙{𝙻𝚞}
  • [:𝚠𝚘𝚛𝚍:] Word characters, equivalent to \𝚠
  • [:𝚡𝚍𝚒𝚐𝚒𝚝:] Hexadecimal digits, equivalent to [𝙰-𝙵𝚊-𝚏𝟶-𝟿]

POSIX character classes can be used only inside generic character classes:

Some dialects redefine POSIX character classes to also include letters in Unicode other than a–z as alphabetic characters:

Java supports its own mix of Unicode and POSIX character class syntax. Classes have POSIX names, but the Unicode syntax looks like this: \𝚙{𝙶𝚛𝚊𝚙𝚑}. Of course, these may be used outside generic character classes, as well as inside them.

Parentheses are metacharacters in regex, i.e., they’re not matched as literals:

When we want them to match literally, we may escape the parentheses with a backslash \:

Above all, we may use parentheses to override the order of operations. Like mathematics, parentheses have the authority to cancel the natural operator precedence in regex. Positive Closure + takes precedence over concatenation. Parentheses override that rule in the second example below:

However, in regex, we also may use parentheses to give a subexpression an identity. We count the left parentheses from left to right in our expression and give them indices 1, 2, 3, etc. Here’s an analysis of the expression /𝚊((𝚋𝚌)((𝚍)𝚎))/:

  • Index 1: (𝚋𝚌)((𝚍)𝚎)
  • Index 2: 𝚋𝚌
  • Index 3: (𝚍)𝚎
  • Index 4: 𝚍

These subexpressions — 1, 2, 3, and 4 — are known as groups. Note that a group may contain many characters, a single character, or even be empty.

The groups constructed from the parentheses aren’t just esoteric; we may benefit from them. The groups are captured, stored, and can be referenced later:

The reference number is based on the order of the group’s left parenthesis. Note how they’re ordered in the example above where nested groups are present.

We even may refer to a group inside our regex. Imagine that you want to find all occasions in which an arbitrary character is repeated:

The generic regex /(.)\𝟷/ isn’t as cryptic as it may look. The period symbol matches any character that isn’t a line break. The parentheses around the period symbol capture whatever was matched by the period symbol in Group №1. With \𝟷, we refer back to the thing captured in Group №1. So, we wanted to match precisely the character that the period symbol matched with \𝟷 no matter what it was.

In the same way, we may match words or parts of words that repeat themselves:

Regex dialects based on deterministic finite automata (DFA) normally cannot capture groups at all, but most regex dialects are based on non-deterministic finite automata (NFA).

It’s possible to use parentheses, (), and refuse to capture a group. It sometimes may provide a marginally better performance and keep the indices lower. The price we pay is a cluttered regex. The syntax for a non-capturing group is a question mark, ?, followed by a colon, :, immediately after the left parenthesis:

You might have noticed the recurring syntax with question marks and some additional characters directly after the left parenthesis. Here are some examples:

  • Atomic grouping: (?>
  • Comments: (?#
  • Conditional: (?(
  • Mode modifiers: (?i:, (?s:, and (?m:
  • Named capture: (?<, (?P<, or (?’
  • Negative Lookahead: (?!
  • Positive Lookahead: (?=
  • Positive Lookbehind: (?<=
  • Negative Lookbehind: (?<!

Captured groups can be referenced in the replacement text of a search-and-replace. For example, you can format a date from the U.S. to the more logical ISO 8601:

The regex lies between the exclamation marks, and ‘\3-\1-\2’ is what matches are replaced with. As in the repeated character example above, we refer to the three groups with \1, \2, and \3.

A few regex dialects support forward references, but most treat them as errors. Some people find it challenging to understand the following:

Even more obscure are relative backreferences, in which the index is negative:

Even the capture can be named. Eventually, it adds more clarity when we refer back to that more explanatory name later:

The named capture syntax varies. Sometimes we name a capture with (?𝙿<𝚗𝚊𝚖𝚎>…) or (?'𝚗𝚊𝚖𝚎'…), instead of (?<𝚗𝚊𝚖𝚎>…), as we do above. The backreference part need not be \𝚔<𝚗𝚊𝚖𝚎> either. In some regex dialects, it’s (?𝙿=𝚗𝚊𝚖𝚎), \𝚔'𝚗𝚊𝚖𝚎', or even ${𝚗𝚊𝚖𝚎} etc. Look at this example, which matches vowels preceded by v and consonants preceded by c:

Be careful not to mix numeric and named backreferences. Some dialects ignore named groups whenever groups are numbered. Others count all groups. Note the difference here between Ruby and C#:

Although it’s not trivial to understand how atomic groupings work, it’s easier than understanding why and when to use them. The short answer to why is that the regex automaton fails faster. Atomic groups refuse to backtrack, and this sometimes may improve performance.

Below are two variants of a little tangled regex that searches for strings of at least two 𝚊’s followed by one 𝚋. The first and third command contain the atomic grouping (?>…), while the second and fourth don’t:

The most peculiar thing about the above examples isn’t that the last two fail, but that the last one on my computer took 25,000 times longer to execute than the first three. Why? It turns out that no 𝚋 is present at the end of the input, then the regex automaton starts to backtrack. With atomic grouping, we explicitly state that we don’t want backtracking. So, the question why, is answered with performance. Do previous discussions about possessive quantifiers ring a bell?

Here’s a simple example:

We’re looking for some digits followed by the degree symbol, °. With 𝟷𝟸𝟹° and 𝟽𝟾𝟿°, it’s a perfect match, but what happens with 𝟺𝟻𝟼 in the middle? The regex automaton first matches 456 with the subexpression \𝚍+, then discovers that there’s a space instead of the degree symbol directly after 𝟺𝟻𝟼. That makes the regex automaton instead only try to match 𝟺𝟻 with \𝚍+. Considering that 𝟼 isn’t a degree symbol either, the automaton now will test to match \𝚍+ with only 𝟺. Nor is 𝟻 a degree symbol, so now the automaton finally realizes that this doesn’t work.

If we instead encapsulate \𝚍+ in an atomic group, then the automaton gives up right away when it understands that 𝟺𝟻𝟼 isn’t followed by a degree symbol, i.e., atomic groups refuse to backtrack:

Atomic groups have strong kinship with possessive quantifiers. I said earlier that greedy and reluctant quantifiers match the same text, but in a different order. However, atomic groups match only a subset. It’s more efficient when we don’t find anything anyway. However, if you’re not careful, you might miss something that happens to be correct, like this:

Here’s the big thing with anchors: they don’t match any characters in the input string. However, they’re still useful because they match a position, i.e., their match is of zero length:

  • Caret ^ matches the position before the first character in the input string and usually also directly after a line break.
  • Dollar $ matches the position after the last character in the input string and usually also directly before a line break.
  • \𝙰 matches only at the start of the input string and ignores line breaks.
  • \𝚉 matches only at the end of the input string and ignores line breaks.
  • \𝚋 matches word boundaries, mostly based on what \𝚠 matches

Some regex dialects differ slightly from the following. Check this before applying anchors.

Let’s start with caret ^ and dollar $. Note how an extra newline \𝚗 affects these patterns:

Now let’s see what happens if we use the same regular expressions, only changing from caret ^ and dollar $ to \𝙰 and \𝚉:

The \𝚋 anchor matches word breaks. In most regex dialects, the definition of a word break \𝚋 is based on what’s matched by the word character class \𝚠:

“This clock-shaped machine solves the slash-slash problem. The square window at the top of the machine reads a tape — from left to right — consisting of the input string, i.e., the C code. We have a state machine of type Nondeterministic Finite Automaton (NFA). To visualize and represent the automaton, we use a transition graph, in which the vertices represent states and the edges represent transitions. The initial state is identified by an incoming unlabeled arrow not originating at any vertex. The acceptance state is surrounded by a circle. This is the graph that you see printed on the clock dial. Now, whenever a new symbol is read from the tape, the clock dial rotates so that the drooping peak, just below the reading window, points at the current state. Ah, and this particular machine also has a special lookahead feature: it’s a long arm with an eye in the end and a light bulb. This eye can look ahead and tell if there’s any double slash. If there is, the bulb will glow and the machine will understand that it doesn’t matter if the input ends in a pair of parentheses — it’s in a comment anyway.” Source: De Morgan to the Rescue.

Last, but certainly not least, are lookarounds. They’re actually often very useful and sometimes irreplaceable. Like the anchors we just saw, lookarounds don’t consume anything from the input string. How they differ from the other anchors is that lookarounds generically can match any pattern, asserting that the pattern exists or doesn’t exist. The response to that existential doubt is, of course, of boolean type. Either the pattern exists or it doesn’t. Nothing from the input string is consumed.

Four lookaround operators exist:

  • Positive lookahead (?= ) — checks whether a pattern is present to the right
  • Negative lookahead (?! ) — checks whether a pattern isn’t present to the right
  • Positive lookbehind (?<= ) — checks whether a pattern is present to the left
  • Negative lookbehind (?<! ) — checks whether a pattern isn’t present to the left

Note that lookbehinds look a little bit like left arrows.

Here’s a challenge for the lookahead function: Which words are followed by the word 𝚃𝚑𝚘𝚛?

Remember that \𝚠 matches any word letter, and \𝚋 is the anchor for word breaks.

Which fragments starting with 𝚌 immediately follow a whitespace \𝚜?

Ouch! We accidentally matched the whitespace preceding our words, which is why we rather use lookbehind:

Afterword

“Some people, when confronted with a problem, think, ‘I know, I’ll use regular expressions.’ Now they have two problems.”

Jamie Zawinski wrote that in August 1997. So, what’s the second problem? It’s clear to me that regex is only usable if you learn regex. Now that you’ve finished this essay, you have this proficiency. You’re the regex hero!

And never forget that regex isn’t a silver bullet. Some problems can be solved with awesome regex solutions, while others can’t be solved with regexes on any level.

And that brings us to the end of this essay.

Thanks for reading.

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Staffan Nöteberg

Staffan Nöteberg

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🌱 Twenty Years of Agile Coaching and Leadership • Monotasking and Pomodoro books (700.000 copies sold)